Gadgets › the places

The places

A fault-tolerant quantum computer on trapped ions is a town. Its places are where many ions go to get one job done: a logical block is born at a preparation place, rests in a logical zone, goes for a check-up at syndrome extraction, meets another block at a lattice-surgery bridge or a transversal-gate workshop, and ends at a readout; magic states come from a factory through an injection place. Ions travel between places on roads, the way people travel between a school, a hotel and a hospital.

Every place below is verified twice. Its hardware program replays with no hardware rule failing and ends where it promised. And the quantum circuit that program executes — read off its own gate, measurement and reset instructions — is checked, for every input state and every measurement outcome at once, to compute exactly what the place promises, and with fault distance 3 wherever the place promises that too.

The town the algorithms run on: every verified place, two rows between three streets. Logical Algorithm →

14places

103/107ops pass the hardware rules

100ops verified as logic

12/12algorithms signed off end to end

The places

Each category has its own colour and its own outline — on this page, in the tool, and in every drawing of a design.

Logical Preparation Syndrome Extraction Logical Zone Logical Memory Logical Operation Lattice Surgery Pauli Product Measurement Logical Readout Magic State Factory Rotation Source Magic State Injection Decoder Classical Memory Ion Depot Roads and Junctions Classical Data Bus

Logical Preparation

the school

Fresh ions come in; a logical |0̄⟩ or |+̄⟩ block goes out: data reset, then a syndrome round establishes the code state.

Logical preparation (d=3) · `prep_d3`

fresh ions are reset at the docks, then one round of the resident ancillas makes them a code block

op	hardware	logic	fault distance	time
`prep.Z` a fresh block in \|0̄⟩	✓ 25 rules	✓ 9/9 flows, rank 9/9	✓ d ≥ 3 (409 faults)	30.0 ms 24 two-qubit	watch · studio
`prep.X` a fresh block in \|+̄⟩	✓ 25 rules	✓ 9/9 flows, rank 9/9	✓ d ≥ 3 (436 faults)	30.1 ms 24 two-qubit	watch · studio

what was checked for prep.Z

The circuit read off the hardware program (17 R, 8 H, 24 CX, 8 M) was run once on a Choi state — every ion entangled with a reference — with every measurement outcome kept symbolic. 9 stabilizer flows hold, and their rank 9 is the 9 a complete specification needs, so no other channel satisfies them.

✓ Z̄ = +1 (exact)
✓ X0 holds (frame ⊕ 1 record)
✓ X1 holds (frame ⊕ 1 record)
✓ X2 holds (frame ⊕ 1 record)
✓ X3 holds (frame ⊕ 1 record)
✓ Z0 holds (frame)
✓ Z1 holds (frame)
✓ Z2 holds (frame)
✓ Z3 holds (frame)

Fault distance: the op between ideal encoders and decoders, 409 fault locations, 12 detectors, 1 observables — no single fault and no pair of faults flips a logical observable without a detector firing.

data reset at the docks, then one syndrome round: the Z checks come out deterministic, the others random (their values are the frame)

what was checked for prep.X

The circuit read off the hardware program (17 R, 17 H, 24 CX, 8 M) was run once on a Choi state — every ion entangled with a reference — with every measurement outcome kept symbolic. 9 stabilizer flows hold, and their rank 9 is the 9 a complete specification needs, so no other channel satisfies them.

✓ X̄ = +1 (exact)
✓ X0 holds (frame)
✓ X1 holds (frame)
✓ X2 holds (frame)
✓ X3 holds (frame)
✓ Z0 holds (frame ⊕ 1 record)
✓ Z1 holds (frame ⊕ 1 record)
✓ Z2 holds (frame ⊕ 1 record)
✓ Z3 holds (frame ⊕ 1 record)

Fault distance: the op between ideal encoders and decoders, 436 fault locations, 12 detectors, 1 observables — no single fault and no pair of faults flips a logical observable without a detector firing.

data reset and H at the docks, then one syndrome round: the X checks come out deterministic, the others random (their values are the frame)

Syndrome Extraction

the clinic

A block comes in for a check-up: resident ancillas measure every stabilizer, rounds at a time, and the block leaves with its syndrome taken.

Syndrome extraction (d=3) · `se_d3`

8 resident ancillas in dock traps on a 24-slot ring; a visiting block rides the ring past them

op	hardware	logic	fault distance	time
`se.1` 1 syndrome-extraction round on a visiting block	✓ 25 rules	✓ 18/18 flows, rank 18/18	✓ d ≥ 3 (400 faults)	33.2 ms 24 two-qubit	watch · studio
`se.3` 3 syndrome-extraction rounds on a visiting block	✓ 25 rules	✓ 18/18 flows, rank 18/18	✓ d ≥ 3 (1,200 faults)	59.0 ms 72 two-qubit	watch · studio

what was checked for se.1

The circuit read off the hardware program (8 R, 8 H, 24 CX, 8 M) was run once on a Choi state — every ion entangled with a reference — with every measurement outcome kept symbolic. 18 stabilizer flows hold, and their rank 18 is the 18 a complete specification needs, so no other channel satisfies them.

✓ X0 measured (measure ⊕ 1 record)
✓ X0 holds (frame ⊕ 1 record)
✓ X1 measured (measure ⊕ 1 record)
✓ X1 holds (frame ⊕ 1 record)
✓ X2 measured (measure ⊕ 1 record)
✓ X2 holds (frame ⊕ 1 record)
✓ X3 measured (measure ⊕ 1 record)
✓ X3 holds (frame ⊕ 1 record)
✓ Z0 measured (measure ⊕ 1 record)
✓ Z0 holds (frame ⊕ 1 record)
✓ Z1 measured (measure ⊕ 1 record)
✓ Z1 holds (frame ⊕ 1 record)
✓ Z2 measured (measure ⊕ 1 record)
✓ Z2 holds (frame ⊕ 1 record)
✓ Z3 measured (measure ⊕ 1 record)
✓ Z3 holds (frame ⊕ 1 record)
✓ X̄ kept (exact)
✓ Z̄ kept (exact)

Fault distance: the op between ideal encoders and decoders, 400 fault locations, 16 detectors, 2 observables — no single fault and no pair of faults flips a logical observable without a detector firing.

ancillas live in the dock traps; the ring turns and each data ion docks at its check's ancilla when the schedule allows (Tomita-Svore order, as a partial order)

what was checked for se.3

The circuit read off the hardware program (24 R, 24 H, 72 CX, 24 M) was run once on a Choi state — every ion entangled with a reference — with every measurement outcome kept symbolic. 18 stabilizer flows hold, and their rank 18 is the 18 a complete specification needs, so no other channel satisfies them.

✓ X0 measured (measure ⊕ 1 record)
✓ X0 holds (frame ⊕ 1 record)
✓ X1 measured (measure ⊕ 1 record)
✓ X1 holds (frame ⊕ 1 record)
✓ X2 measured (measure ⊕ 1 record)
✓ X2 holds (frame ⊕ 1 record)
✓ X3 measured (measure ⊕ 1 record)
✓ X3 holds (frame ⊕ 1 record)
✓ Z0 measured (measure ⊕ 1 record)
✓ Z0 holds (frame ⊕ 1 record)
✓ Z1 measured (measure ⊕ 1 record)
✓ Z1 holds (frame ⊕ 1 record)
✓ Z2 measured (measure ⊕ 1 record)
✓ Z2 holds (frame ⊕ 1 record)
✓ Z3 measured (measure ⊕ 1 record)
✓ Z3 holds (frame ⊕ 1 record)
✓ X̄ kept (exact)
✓ Z̄ kept (exact)

Fault distance: the op between ideal encoders and decoders, 1,200 fault locations, 32 detectors, 2 observables — no single fault and no pair of faults flips a logical observable without a detector firing.

ancillas live in the dock traps; the ring turns and each data ion docks at its check's ancilla when the schedule allows (Tomita-Svore order, as a partial order)

Logical Zone

the hotel

Holds logical blocks between operations. No lasers: blocks check in, rest, and check out in the order they arrived.

Logical zone (1 block) · `zone_d3`

a 12-slot storage ring with no lasers: one block rests here

op	hardware	logic	fault distance	time
`check_in` a block checks in and rests on the ring	✓ 25 rules	✓ 18/18 flows, rank 18/18	—	3.7 ms 0 two-qubit	watch · studio
`check_out` the block checks out, in the order it arrived	✓ 25 rules	✓ 18/18 flows, rank 18/18	—	3.5 ms 0 two-qubit	watch · studio

what was checked for check_in

The circuit read off the hardware program () was run once on a Choi state — every ion entangled with a reference — with every measurement outcome kept symbolic. 18 stabilizer flows hold, and their rank 18 is the 18 a complete specification needs, so no other channel satisfies them.

✓ X(d0) (exact)
✓ Z(d0) (exact)
✓ X(d1) (exact)
✓ Z(d1) (exact)
✓ X(d2) (exact)
✓ Z(d2) (exact)
✓ X(d3) (exact)
✓ Z(d3) (exact)
✓ X(d4) (exact)
✓ Z(d4) (exact)
✓ X(d5) (exact)
✓ Z(d5) (exact)
✓ X(d6) (exact)
✓ Z(d6) (exact)
✓ X(d7) (exact)
✓ Z(d7) (exact)
✓ X(d8) (exact)
✓ Z(d8) (exact)

what was checked for check_out

✓ X(d0) (exact)
✓ Z(d0) (exact)
✓ X(d1) (exact)
✓ Z(d1) (exact)
✓ X(d2) (exact)
✓ Z(d2) (exact)
✓ X(d3) (exact)
✓ Z(d3) (exact)
✓ X(d4) (exact)
✓ Z(d4) (exact)
✓ X(d5) (exact)
✓ Z(d5) (exact)
✓ X(d6) (exact)
✓ Z(d6) (exact)
✓ X(d7) (exact)
✓ Z(d7) (exact)
✓ X(d8) (exact)
✓ Z(d8) (exact)

Logical Memory

the ward

Holds a logical block for as long as the program needs it, and corrects it while it rests: resident ancillas run a syndrome round whenever the block's coherence budget asks for one.

Logical memory (d=3) · `mem_d3`

the clinic ring and its 8 resident ancillas, with one block kept on the ring and corrected while it rests

op	hardware	logic	fault distance	time
`store` a block checks in, takes one syndrome round, and rests on the ring	✓ 25 rules	✓ 18/18 flows, rank 18/18	✓ d ≥ 3 (400 faults)	24.1 ms 24 two-qubit	watch · studio
`refresh` one syndrome round on the resting block, which stays	✓ 25 rules	✓ 18/18 flows, rank 18/18	✓ d ≥ 3 (400 faults)	13.2 ms 24 two-qubit	watch · studio
`release` the block leaves, in the order it arrived	✓ 25 rules	✓ 18/18 flows, rank 18/18	—	9.3 ms 0 two-qubit	watch · studio

what was checked for store

✓ X0 measured (measure ⊕ 1 record)
✓ X0 holds (frame ⊕ 1 record)
✓ X1 measured (measure ⊕ 1 record)
✓ X1 holds (frame ⊕ 1 record)
✓ X2 measured (measure ⊕ 1 record)
✓ X2 holds (frame ⊕ 1 record)
✓ X3 measured (measure ⊕ 1 record)
✓ X3 holds (frame ⊕ 1 record)
✓ Z0 measured (measure ⊕ 1 record)
✓ Z0 holds (frame ⊕ 1 record)
✓ Z1 measured (measure ⊕ 1 record)
✓ Z1 holds (frame ⊕ 1 record)
✓ Z2 measured (measure ⊕ 1 record)
✓ Z2 holds (frame ⊕ 1 record)
✓ Z3 measured (measure ⊕ 1 record)
✓ Z3 holds (frame ⊕ 1 record)
✓ X̄ kept (exact)
✓ Z̄ kept (exact)

ancillas live in the dock traps; the ring turns and each data ion docks at the ancilla of its check when the schedule allows (Tomita-Svore order, as a partial order); the block stays on the ring between ops

what was checked for refresh

✓ X0 measured (measure ⊕ 1 record)
✓ X0 holds (frame ⊕ 1 record)
✓ X1 measured (measure ⊕ 1 record)
✓ X1 holds (frame ⊕ 1 record)
✓ X2 measured (measure ⊕ 1 record)
✓ X2 holds (frame ⊕ 1 record)
✓ X3 measured (measure ⊕ 1 record)
✓ X3 holds (frame ⊕ 1 record)
✓ Z0 measured (measure ⊕ 1 record)
✓ Z0 holds (frame ⊕ 1 record)
✓ Z1 measured (measure ⊕ 1 record)
✓ Z1 holds (frame ⊕ 1 record)
✓ Z2 measured (measure ⊕ 1 record)
✓ Z2 holds (frame ⊕ 1 record)
✓ Z3 measured (measure ⊕ 1 record)
✓ Z3 holds (frame ⊕ 1 record)
✓ X̄ kept (exact)
✓ Z̄ kept (exact)

ancillas live in the dock traps; the ring turns and each data ion docks at the ancilla of its check when the schedule allows (Tomita-Svore order, as a partial order); the block stays on the ring between ops

what was checked for release

✓ X(d0) (exact)
✓ Z(d0) (exact)
✓ X(d1) (exact)
✓ Z(d1) (exact)
✓ X(d2) (exact)
✓ Z(d2) (exact)
✓ X(d3) (exact)
✓ Z(d3) (exact)
✓ X(d4) (exact)
✓ Z(d4) (exact)
✓ X(d5) (exact)
✓ Z(d5) (exact)
✓ X(d6) (exact)
✓ Z(d6) (exact)
✓ X(d7) (exact)
✓ Z(d7) (exact)
✓ X(d8) (exact)
✓ Z(d8) (exact)

Logical Operation

the workshop

Two blocks meet and take a transversal gate: ion i of one block with ion i of the other, so no fault spreads within a block.

Transversal CNOT (d=3) · `tcx_d3`

two conveyor rails over nine gate traps; both blocks flow through west to east and meet ion by ion

op	hardware	logic	fault distance	time
`cx` transversal CNOT: block on rail A controls block on rail B	✓ 25 rules	✓ 40/40 flows, rank 36/36	✓ d ≥ 3 (135 faults)	6.9 ms 9 two-qubit	watch · studio

what was checked for cx

The circuit read off the hardware program (9 CX) was run once on a Choi state — every ion entangled with a reference — with every measurement outcome kept symbolic. 40 stabilizer flows hold, and their rank 36 is the 36 a complete specification needs, so no other channel satisfies them.

✓ X(d0) (exact)
✓ Z(d0) (exact)
✓ X(e0) (exact)
✓ Z(e0) (exact)
✓ X(d1) (exact)
✓ Z(d1) (exact)
✓ X(e1) (exact)
✓ Z(e1) (exact)
✓ X(d2) (exact)
✓ Z(d2) (exact)
✓ X(e2) (exact)
✓ Z(e2) (exact)
✓ X(d3) (exact)
✓ Z(d3) (exact)
✓ X(e3) (exact)
✓ Z(e3) (exact)
✓ X(d4) (exact)
✓ Z(d4) (exact)
✓ X(e4) (exact)
✓ Z(e4) (exact)
✓ X(d5) (exact)
✓ Z(d5) (exact)
✓ X(e5) (exact)
✓ Z(e5) (exact)
✓ X(d6) (exact)
✓ Z(d6) (exact)
✓ X(e6) (exact)
✓ Z(e6) (exact)
✓ X(d7) (exact)
✓ Z(d7) (exact)
✓ X(e7) (exact)
✓ Z(e7) (exact)
✓ X(d8) (exact)
✓ Z(d8) (exact)
✓ X(e8) (exact)
✓ Z(e8) (exact)
✓ X̄_A → X̄_A X̄_B (exact)
✓ Z̄_A kept (exact)
✓ X̄_B kept (exact)
✓ Z̄_B → Z̄_A Z̄_B (exact)

Fault distance: the op between ideal encoders and decoders, 135 fault locations, 16 detectors, 4 observables — no single fault and no pair of faults flips a logical observable without a detector firing.

the i-th ion of one block meets only the i-th ion of the other: a fault spreads to at most one ion per block

Lattice Surgery

the bridge

Two blocks are merged through a seam of resident ions for d rounds and split again: the product Z̄Z̄ or X̄X̄ is measured without touching either logical alone.

Lattice surgery (d=3) · `ls_d3`

20 resident ancillas and a 3-ion seam on a 60-slot ring; two blocks enter from the left, merge for d rounds, split, and leave right

op	hardware	logic	fault distance	time
`zz` measure Z̄_A Z̄_B by merging for 3 rounds	✓ 25 rules	✓ 36/36 flows, rank 36/36	✓ d ≥ 3 (3,168 faults)	206.6 ms 192 two-qubit	watch · studio
`xx` measure X̄_A X̄_B by merging for 3 rounds	✓ 25 rules	✓ 36/36 flows, rank 36/36	✓ d ≥ 3 (3,222 faults)	200.6 ms 192 two-qubit	watch · studio

what was checked for zz

The circuit read off the hardware program (63 R, 54 H, 192 CX, 63 M) was run once on a Choi state — every ion entangled with a reference — with every measurement outcome kept symbolic. 36 stabilizer flows hold, and their rank 36 is the 36 a complete specification needs, so no other channel satisfies them.

✓ A.X0 measured (measure ⊕ 1 record)
✓ A.X0 holds (frame ⊕ 1 record)
✓ A.X1 measured (measure ⊕ 1 record)
✓ A.X1 holds (frame ⊕ 1 record)
✓ A.X2 measured (measure ⊕ 1 record)
✓ A.X2 holds (frame ⊕ 1 record)
✓ A.X3 measured (measure ⊕ 1 record)
✓ A.X3 holds (frame ⊕ 3 records)
✓ A.Z0 measured (measure ⊕ 1 record)
✓ A.Z0 holds (frame ⊕ 1 record)
✓ A.Z1 measured (measure ⊕ 1 record)
✓ A.Z1 holds (frame ⊕ 1 record)
✓ A.Z2 measured (measure ⊕ 1 record)
✓ A.Z2 holds (frame ⊕ 1 record)
✓ A.Z3 measured (measure ⊕ 1 record)
✓ A.Z3 holds (frame ⊕ 1 record)
✓ B.X0 measured (measure ⊕ 1 record)
✓ B.X0 holds (frame ⊕ 3 records)
✓ B.X1 measured (measure ⊕ 1 record)
✓ B.X1 holds (frame ⊕ 1 record)
✓ B.X2 measured (measure ⊕ 1 record)
✓ B.X2 holds (frame ⊕ 1 record)
✓ B.X3 measured (measure ⊕ 1 record)
✓ B.X3 holds (frame ⊕ 1 record)
✓ B.Z0 measured (measure ⊕ 1 record)
✓ B.Z0 holds (frame ⊕ 1 record)
✓ B.Z1 measured (measure ⊕ 1 record)
✓ B.Z1 holds (frame ⊕ 1 record)
✓ B.Z2 measured (measure ⊕ 1 record)
✓ B.Z2 holds (frame ⊕ 1 record)
✓ B.Z3 measured (measure ⊕ 1 record)
✓ B.Z3 holds (frame ⊕ 1 record)
✓ Z̄_A kept (frame)
✓ Z̄_B kept (frame)
✓ X̄_AX̄_B kept (frame ⊕ 1 record)
✓ Z̄_AZ̄_B measured (measure ⊕ 8 records)

Fault distance: the op between ideal encoders and decoders, 3,168 fault locations, 72 detectors, 4 observables — no single fault and no pair of faults flips a logical observable without a detector firing.

the seam (3 resident ions) starts in |+⟩ and is measured in X at the split; the merged 7×3 patch has 20 checks
the outcome is the product of the 4 seam Z checks of any merged round

what was checked for xx

The circuit read off the hardware program (63 R, 72 H, 192 CX, 63 M) was run once on a Choi state — every ion entangled with a reference — with every measurement outcome kept symbolic. 36 stabilizer flows hold, and their rank 36 is the 36 a complete specification needs, so no other channel satisfies them.

✓ A.X0 measured (measure ⊕ 1 record)
✓ A.X0 holds (frame ⊕ 1 record)
✓ A.X1 measured (measure ⊕ 1 record)
✓ A.X1 holds (frame ⊕ 1 record)
✓ A.X2 measured (measure ⊕ 1 record)
✓ A.X2 holds (frame ⊕ 1 record)
✓ A.X3 measured (measure ⊕ 1 record)
✓ A.X3 holds (frame ⊕ 1 record)
✓ A.Z0 measured (measure ⊕ 1 record)
✓ A.Z0 holds (frame ⊕ 1 record)
✓ A.Z1 measured (measure ⊕ 1 record)
✓ A.Z1 holds (frame ⊕ 3 records)
✓ A.Z2 measured (measure ⊕ 1 record)
✓ A.Z2 holds (frame ⊕ 1 record)
✓ A.Z3 measured (measure ⊕ 1 record)
✓ A.Z3 holds (frame ⊕ 1 record)
✓ B.X0 measured (measure ⊕ 1 record)
✓ B.X0 holds (frame ⊕ 1 record)
✓ B.X1 measured (measure ⊕ 1 record)
✓ B.X1 holds (frame ⊕ 1 record)
✓ B.X2 measured (measure ⊕ 1 record)
✓ B.X2 holds (frame ⊕ 1 record)
✓ B.X3 measured (measure ⊕ 1 record)
✓ B.X3 holds (frame ⊕ 1 record)
✓ B.Z0 measured (measure ⊕ 1 record)
✓ B.Z0 holds (frame ⊕ 1 record)
✓ B.Z1 measured (measure ⊕ 1 record)
✓ B.Z1 holds (frame ⊕ 1 record)
✓ B.Z2 measured (measure ⊕ 1 record)
✓ B.Z2 holds (frame ⊕ 3 records)
✓ B.Z3 measured (measure ⊕ 1 record)
✓ B.Z3 holds (frame ⊕ 1 record)
✓ X̄_A kept (frame)
✓ X̄_B kept (frame)
✓ Z̄_AZ̄_B kept (frame ⊕ 1 record)
✓ X̄_AX̄_B measured (measure ⊕ 8 records)

Fault distance: the op between ideal encoders and decoders, 3,222 fault locations, 72 detectors, 4 observables — no single fault and no pair of faults flips a logical observable without a detector firing.

the seam (3 resident ions) starts in |0⟩ and is measured in Z at the split; the merged 3×7 patch has 20 checks
the outcome is the product of the 4 seam X checks of any merged round

Pauli Product Measurement

the observatory

Up to four blocks come in; a resident probe ion measures the product of one X̄ or Z̄ from each -- Z̄_A X̄_B Z̄_C, say -- without reading any of them alone: the operation a program written in Pauli-product measurements is made of.

Pauli-product bench (d=3) · `ppm_d3`

one probe ion in a dock on a 60-slot ring; up to four blocks enter on the left, the probe measures the product of one logical X̄ or Z̄ from each, and they leave right

op	hardware	logic	fault distance	time
`ppm.X` measure X̄_A with a probe ion, 3 rounds	✓ 25 rules	✓ 19/19 flows, rank 18/18	—	21.8 ms 9 two-qubit
`ppm.Z` measure Z̄_A with a probe ion, 3 rounds	✓ 25 rules	✓ 19/19 flows, rank 18/18	—	21.8 ms 9 two-qubit	studio
`ppm.XX` measure X̄_AX̄_B with a probe ion, 3 rounds	✓ 25 rules	✓ 39/39 flows, rank 36/36	—	42.7 ms 18 two-qubit
`ppm.XZ` measure X̄_AZ̄_B with a probe ion, 3 rounds	✓ 25 rules	✓ 39/39 flows, rank 36/36	—	42.7 ms 18 two-qubit	studio
`ppm.ZX` measure Z̄_AX̄_B with a probe ion, 3 rounds	✓ 25 rules	✓ 39/39 flows, rank 36/36	—	42.7 ms 18 two-qubit
`ppm.ZZ` measure Z̄_AZ̄_B with a probe ion, 3 rounds	✓ 25 rules	✓ 39/39 flows, rank 36/36	—	42.7 ms 18 two-qubit	studio
`ppm.XXX` measure X̄_AX̄_BX̄_C with a probe ion, 3 rounds	✓ 25 rules	✓ 59/59 flows, rank 54/54	—	64.0 ms 27 two-qubit
`ppm.XXZ` measure X̄_AX̄_BZ̄_C with a probe ion, 3 rounds	✓ 25 rules	✓ 59/59 flows, rank 54/54	—	61.6 ms 27 two-qubit
`ppm.XZX` measure X̄_AZ̄_BX̄_C with a probe ion, 3 rounds	✓ 25 rules	✓ 59/59 flows, rank 54/54	—	64.0 ms 27 two-qubit
`ppm.XZZ` measure X̄_AZ̄_BZ̄_C with a probe ion, 3 rounds	✓ 25 rules	✓ 59/59 flows, rank 54/54	—	61.6 ms 27 two-qubit
`ppm.ZXX` measure Z̄_AX̄_BX̄_C with a probe ion, 3 rounds	✓ 25 rules	✓ 59/59 flows, rank 54/54	—	64.0 ms 27 two-qubit
`ppm.ZXZ` measure Z̄_AX̄_BZ̄_C with a probe ion, 3 rounds	✓ 25 rules	✓ 59/59 flows, rank 54/54	—	61.6 ms 27 two-qubit	studio
`ppm.ZZX` measure Z̄_AZ̄_BX̄_C with a probe ion, 3 rounds	✓ 25 rules	✓ 59/59 flows, rank 54/54	—	64.0 ms 27 two-qubit
`ppm.ZZZ` measure Z̄_AZ̄_BZ̄_C with a probe ion, 3 rounds	✓ 25 rules	✓ 59/59 flows, rank 54/54	—	61.6 ms 27 two-qubit	studio
`ppm.XXXX` measure X̄_AX̄_BX̄_CX̄_D with a probe ion, 3 rounds	✓ 25 rules	✓ 79/79 flows, rank 72/72	—	85.4 ms 36 two-qubit
`ppm.XXXZ` measure X̄_AX̄_BX̄_CZ̄_D with a probe ion, 3 rounds	✓ 25 rules	✓ 79/79 flows, rank 72/72	—	83.0 ms 36 two-qubit
`ppm.XXZX` measure X̄_AX̄_BZ̄_CX̄_D with a probe ion, 3 rounds	✓ 25 rules	✓ 79/79 flows, rank 72/72	—	85.4 ms 36 two-qubit
`ppm.XXZZ` measure X̄_AX̄_BZ̄_CZ̄_D with a probe ion, 3 rounds	✓ 25 rules	✓ 79/79 flows, rank 72/72	—	83.0 ms 36 two-qubit
`ppm.XZXX` measure X̄_AZ̄_BX̄_CX̄_D with a probe ion, 3 rounds	✓ 25 rules	✓ 79/79 flows, rank 72/72	—	85.4 ms 36 two-qubit
`ppm.XZXZ` measure X̄_AZ̄_BX̄_CZ̄_D with a probe ion, 3 rounds	✓ 25 rules	✓ 79/79 flows, rank 72/72	—	83.0 ms 36 two-qubit
`ppm.XZZX` measure X̄_AZ̄_BZ̄_CX̄_D with a probe ion, 3 rounds	✓ 25 rules	✓ 79/79 flows, rank 72/72	—	85.4 ms 36 two-qubit
`ppm.XZZZ` measure X̄_AZ̄_BZ̄_CZ̄_D with a probe ion, 3 rounds	✓ 25 rules	✓ 79/79 flows, rank 72/72	—	83.0 ms 36 two-qubit
`ppm.ZXXX` measure Z̄_AX̄_BX̄_CX̄_D with a probe ion, 3 rounds	✓ 25 rules	✓ 79/79 flows, rank 72/72	—	85.4 ms 36 two-qubit
`ppm.ZXXZ` measure Z̄_AX̄_BX̄_CZ̄_D with a probe ion, 3 rounds	✓ 25 rules	✓ 79/79 flows, rank 72/72	—	83.0 ms 36 two-qubit
`ppm.ZXZX` measure Z̄_AX̄_BZ̄_CX̄_D with a probe ion, 3 rounds	✓ 25 rules	✓ 79/79 flows, rank 72/72	—	85.4 ms 36 two-qubit
`ppm.ZXZZ` measure Z̄_AX̄_BZ̄_CZ̄_D with a probe ion, 3 rounds	✓ 25 rules	✓ 79/79 flows, rank 72/72	—	83.0 ms 36 two-qubit
`ppm.ZZXX` measure Z̄_AZ̄_BX̄_CX̄_D with a probe ion, 3 rounds	✓ 25 rules	✓ 79/79 flows, rank 72/72	—	85.4 ms 36 two-qubit
`ppm.ZZXZ` measure Z̄_AZ̄_BX̄_CZ̄_D with a probe ion, 3 rounds	✓ 25 rules	✓ 79/79 flows, rank 72/72	—	83.0 ms 36 two-qubit
`ppm.ZZZX` measure Z̄_AZ̄_BZ̄_CX̄_D with a probe ion, 3 rounds	✓ 25 rules	✓ 79/79 flows, rank 72/72	—	85.4 ms 36 two-qubit
`ppm.ZZZZ` measure Z̄_AZ̄_BZ̄_CZ̄_D with a probe ion, 3 rounds	✓ 25 rules	✓ 79/79 flows, rank 72/72	—	83.0 ms 36 two-qubit	studio

what was checked for ppm.X

The circuit read off the hardware program (3 R, 6 H, 9 CX, 3 M) was run once on a Choi state — every ion entangled with a reference — with every measurement outcome kept symbolic. 19 stabilizer flows hold, and their rank 18 is the 18 a complete specification needs, so no other channel satisfies them.

✓ X(d0) kept (exact)
✓ X(d1) kept (exact)
✓ Z(d1) kept (exact)
✓ X(d2) kept (exact)
✓ Z(d2) kept (exact)
✓ X(d3) kept (exact)
✓ Z(d3)·Z(d0) kept (exact)
✓ X(d4) kept (exact)
✓ Z(d4) kept (exact)
✓ X(d5) kept (exact)
✓ Z(d5) kept (exact)
✓ X(d6) kept (exact)
✓ Z(d6)·Z(d0) kept (exact)
✓ X(d7) kept (exact)
✓ Z(d7) kept (exact)
✓ X(d8) kept (exact)
✓ Z(d8) kept (exact)
✓ X̄_A measured (measure ⊕ 1 record)
✓ X̄_A kept (exact)

a resident probe ion in |+⟩ takes a CX with each X̄-support ion and a CZ with each Z̄-support ion as the ring brings it past its dock, then is read in X: a controlled-P̄ measurement of the product, repeated 3 times
not fault-tolerant at circuit level (the user does not ask for circuit distance)

what was checked for ppm.Z

The circuit read off the hardware program (3 R, 6 H, 9 CZ, 3 M) was run once on a Choi state — every ion entangled with a reference — with every measurement outcome kept symbolic. 19 stabilizer flows hold, and their rank 18 is the 18 a complete specification needs, so no other channel satisfies them.

✓ Z(d0) kept (exact)
✓ X(d1)·X(d0) kept (exact)
✓ Z(d1) kept (exact)
✓ X(d2)·X(d0) kept (exact)
✓ Z(d2) kept (exact)
✓ X(d3) kept (exact)
✓ Z(d3) kept (exact)
✓ X(d4) kept (exact)
✓ Z(d4) kept (exact)
✓ X(d5) kept (exact)
✓ Z(d5) kept (exact)
✓ X(d6) kept (exact)
✓ Z(d6) kept (exact)
✓ X(d7) kept (exact)
✓ Z(d7) kept (exact)
✓ X(d8) kept (exact)
✓ Z(d8) kept (exact)
✓ Z̄_A measured (measure ⊕ 1 record)
✓ Z̄_A kept (exact)

a resident probe ion in |+⟩ takes a CX with each X̄-support ion and a CZ with each Z̄-support ion as the ring brings it past its dock, then is read in X: a controlled-P̄ measurement of the product, repeated 3 times
not fault-tolerant at circuit level (the user does not ask for circuit distance)

what was checked for ppm.XX

The circuit read off the hardware program (3 R, 6 H, 18 CX, 3 M) was run once on a Choi state — every ion entangled with a reference — with every measurement outcome kept symbolic. 39 stabilizer flows hold, and their rank 36 is the 36 a complete specification needs, so no other channel satisfies them.

✓ X(d0) kept (exact)
✓ X(d1) kept (exact)
✓ Z(d1) kept (exact)
✓ X(d2) kept (exact)
✓ Z(d2) kept (exact)
✓ X(d3) kept (exact)
✓ Z(d3)·Z(d0) kept (exact)
✓ X(d4) kept (exact)
✓ Z(d4) kept (exact)
✓ X(d5) kept (exact)
✓ Z(d5) kept (exact)
✓ X(d6) kept (exact)
✓ Z(d6)·Z(d0) kept (exact)
✓ X(d7) kept (exact)
✓ Z(d7) kept (exact)
✓ X(d8) kept (exact)
✓ Z(d8) kept (exact)
✓ X(e0) kept (exact)
✓ Z(e0)·Z(d0) kept (exact)
✓ X(e1) kept (exact)
✓ Z(e1) kept (exact)
✓ X(e2) kept (exact)
✓ Z(e2) kept (exact)
✓ X(e3) kept (exact)
✓ Z(e3)·Z(d0) kept (exact)
✓ X(e4) kept (exact)
✓ Z(e4) kept (exact)
✓ X(e5) kept (exact)
✓ Z(e5) kept (exact)
✓ X(e6) kept (exact)
✓ Z(e6)·Z(d0) kept (exact)
✓ X(e7) kept (exact)
✓ Z(e7) kept (exact)
✓ X(e8) kept (exact)
✓ Z(e8) kept (exact)
✓ X̄_AX̄_B measured (measure ⊕ 1 record)
✓ X̄_A kept (exact)
✓ X̄_B kept (exact)
✓ Z̄_AZ̄_B kept (exact)

a resident probe ion in |+⟩ takes a CX with each X̄-support ion and a CZ with each Z̄-support ion as the ring brings it past its dock, then is read in X: a controlled-P̄ measurement of the product, repeated 3 times
not fault-tolerant at circuit level (the user does not ask for circuit distance)

what was checked for ppm.XZ

The circuit read off the hardware program (3 R, 6 H, 9 CZ, 9 CX, 3 M) was run once on a Choi state — every ion entangled with a reference — with every measurement outcome kept symbolic. 39 stabilizer flows hold, and their rank 36 is the 36 a complete specification needs, so no other channel satisfies them.

✓ X(d0) kept (exact)
✓ X(d1) kept (exact)
✓ Z(d1) kept (exact)
✓ X(d2) kept (exact)
✓ Z(d2) kept (exact)
✓ X(d3) kept (exact)
✓ Z(d3)·Z(d0) kept (exact)
✓ X(d4) kept (exact)
✓ Z(d4) kept (exact)
✓ X(d5) kept (exact)
✓ Z(d5) kept (exact)
✓ X(d6) kept (exact)
✓ Z(d6)·Z(d0) kept (exact)
✓ X(d7) kept (exact)
✓ Z(d7) kept (exact)
✓ X(d8) kept (exact)
✓ Z(d8) kept (exact)
✓ X(e0)·Z(d0) kept (exact)
✓ Z(e0) kept (exact)
✓ X(e1)·Z(d0) kept (exact)
✓ Z(e1) kept (exact)
✓ X(e2)·Z(d0) kept (exact)
✓ Z(e2) kept (exact)
✓ X(e3) kept (exact)
✓ Z(e3) kept (exact)
✓ X(e4) kept (exact)
✓ Z(e4) kept (exact)
✓ X(e5) kept (exact)
✓ Z(e5) kept (exact)
✓ X(e6) kept (exact)
✓ Z(e6) kept (exact)
✓ X(e7) kept (exact)
✓ Z(e7) kept (exact)
✓ X(e8) kept (exact)
✓ Z(e8) kept (exact)
✓ X̄_AZ̄_B measured (measure ⊕ 1 record)
✓ X̄_A kept (exact)
✓ Z̄_B kept (exact)
✓ Z̄_AX̄_B kept (exact)

a resident probe ion in |+⟩ takes a CX with each X̄-support ion and a CZ with each Z̄-support ion as the ring brings it past its dock, then is read in X: a controlled-P̄ measurement of the product, repeated 3 times
not fault-tolerant at circuit level (the user does not ask for circuit distance)

what was checked for ppm.ZX

The circuit read off the hardware program (3 R, 6 H, 9 CX, 9 CZ, 3 M) was run once on a Choi state — every ion entangled with a reference — with every measurement outcome kept symbolic. 39 stabilizer flows hold, and their rank 36 is the 36 a complete specification needs, so no other channel satisfies them.

✓ Z(d0) kept (exact)
✓ X(d1)·X(d0) kept (exact)
✓ Z(d1) kept (exact)
✓ X(d2)·X(d0) kept (exact)
✓ Z(d2) kept (exact)
✓ X(d3) kept (exact)
✓ Z(d3) kept (exact)
✓ X(d4) kept (exact)
✓ Z(d4) kept (exact)
✓ X(d5) kept (exact)
✓ Z(d5) kept (exact)
✓ X(d6) kept (exact)
✓ Z(d6) kept (exact)
✓ X(d7) kept (exact)
✓ Z(d7) kept (exact)
✓ X(d8) kept (exact)
✓ Z(d8) kept (exact)
✓ X(e0) kept (exact)
✓ Z(e0)·X(d0) kept (exact)
✓ X(e1) kept (exact)
✓ Z(e1) kept (exact)
✓ X(e2) kept (exact)
✓ Z(e2) kept (exact)
✓ X(e3) kept (exact)
✓ Z(e3)·X(d0) kept (exact)
✓ X(e4) kept (exact)
✓ Z(e4) kept (exact)
✓ X(e5) kept (exact)
✓ Z(e5) kept (exact)
✓ X(e6) kept (exact)
✓ Z(e6)·X(d0) kept (exact)
✓ X(e7) kept (exact)
✓ Z(e7) kept (exact)
✓ X(e8) kept (exact)
✓ Z(e8) kept (exact)
✓ Z̄_AX̄_B measured (measure ⊕ 1 record)
✓ Z̄_A kept (exact)
✓ X̄_B kept (exact)
✓ X̄_AZ̄_B kept (exact)

a resident probe ion in |+⟩ takes a CX with each X̄-support ion and a CZ with each Z̄-support ion as the ring brings it past its dock, then is read in X: a controlled-P̄ measurement of the product, repeated 3 times
not fault-tolerant at circuit level (the user does not ask for circuit distance)

what was checked for ppm.ZZ

The circuit read off the hardware program (3 R, 6 H, 18 CZ, 3 M) was run once on a Choi state — every ion entangled with a reference — with every measurement outcome kept symbolic. 39 stabilizer flows hold, and their rank 36 is the 36 a complete specification needs, so no other channel satisfies them.

✓ Z(d0) kept (exact)
✓ X(d1)·X(d0) kept (exact)
✓ Z(d1) kept (exact)
✓ X(d2)·X(d0) kept (exact)
✓ Z(d2) kept (exact)
✓ X(d3) kept (exact)
✓ Z(d3) kept (exact)
✓ X(d4) kept (exact)
✓ Z(d4) kept (exact)
✓ X(d5) kept (exact)
✓ Z(d5) kept (exact)
✓ X(d6) kept (exact)
✓ Z(d6) kept (exact)
✓ X(d7) kept (exact)
✓ Z(d7) kept (exact)
✓ X(d8) kept (exact)
✓ Z(d8) kept (exact)
✓ X(e0)·X(d0) kept (exact)
✓ Z(e0) kept (exact)
✓ X(e1)·X(d0) kept (exact)
✓ Z(e1) kept (exact)
✓ X(e2)·X(d0) kept (exact)
✓ Z(e2) kept (exact)
✓ X(e3) kept (exact)
✓ Z(e3) kept (exact)
✓ X(e4) kept (exact)
✓ Z(e4) kept (exact)
✓ X(e5) kept (exact)
✓ Z(e5) kept (exact)
✓ X(e6) kept (exact)
✓ Z(e6) kept (exact)
✓ X(e7) kept (exact)
✓ Z(e7) kept (exact)
✓ X(e8) kept (exact)
✓ Z(e8) kept (exact)
✓ Z̄_AZ̄_B measured (measure ⊕ 1 record)
✓ Z̄_A kept (exact)
✓ Z̄_B kept (exact)
✓ X̄_AX̄_B kept (exact)

a resident probe ion in |+⟩ takes a CX with each X̄-support ion and a CZ with each Z̄-support ion as the ring brings it past its dock, then is read in X: a controlled-P̄ measurement of the product, repeated 3 times
not fault-tolerant at circuit level (the user does not ask for circuit distance)

what was checked for ppm.XXX

The circuit read off the hardware program (3 R, 6 H, 27 CX, 3 M) was run once on a Choi state — every ion entangled with a reference — with every measurement outcome kept symbolic. 59 stabilizer flows hold, and their rank 54 is the 54 a complete specification needs, so no other channel satisfies them.

✓ X(d0) kept (exact)
✓ X(d1) kept (exact)
✓ Z(d1) kept (exact)
✓ X(d2) kept (exact)
✓ Z(d2) kept (exact)
✓ X(d3) kept (exact)
✓ Z(d3)·Z(d0) kept (exact)
✓ X(d4) kept (exact)
✓ Z(d4) kept (exact)
✓ X(d5) kept (exact)
✓ Z(d5) kept (exact)
✓ X(d6) kept (exact)
✓ Z(d6)·Z(d0) kept (exact)
✓ X(d7) kept (exact)
✓ Z(d7) kept (exact)
✓ X(d8) kept (exact)
✓ Z(d8) kept (exact)
✓ X(e0) kept (exact)
✓ Z(e0)·Z(d0) kept (exact)
✓ X(e1) kept (exact)
✓ Z(e1) kept (exact)
✓ X(e2) kept (exact)
✓ Z(e2) kept (exact)
✓ X(e3) kept (exact)
✓ Z(e3)·Z(d0) kept (exact)
✓ X(e4) kept (exact)
✓ Z(e4) kept (exact)
✓ X(e5) kept (exact)
✓ Z(e5) kept (exact)
✓ X(e6) kept (exact)
✓ Z(e6)·Z(d0) kept (exact)
✓ X(e7) kept (exact)
✓ Z(e7) kept (exact)
✓ X(e8) kept (exact)
✓ Z(e8) kept (exact)
✓ X(f0) kept (exact)
✓ Z(f0)·Z(d0) kept (exact)
✓ X(f1) kept (exact)
✓ Z(f1) kept (exact)
✓ X(f2) kept (exact)
✓ Z(f2) kept (exact)
✓ X(f3) kept (exact)
✓ Z(f3)·Z(d0) kept (exact)
✓ X(f4) kept (exact)
✓ Z(f4) kept (exact)
✓ X(f5) kept (exact)
✓ Z(f5) kept (exact)
✓ X(f6) kept (exact)
✓ Z(f6)·Z(d0) kept (exact)
✓ X(f7) kept (exact)
✓ Z(f7) kept (exact)
✓ X(f8) kept (exact)
✓ Z(f8) kept (exact)
✓ X̄_AX̄_BX̄_C measured (measure ⊕ 1 record)
✓ X̄_A kept (exact)
✓ X̄_B kept (exact)
✓ X̄_C kept (exact)
✓ Z̄_AZ̄_B kept (exact)
✓ Z̄_BZ̄_C kept (exact)

a resident probe ion in |+⟩ takes a CX with each X̄-support ion and a CZ with each Z̄-support ion as the ring brings it past its dock, then is read in X: a controlled-P̄ measurement of the product, repeated 3 times
not fault-tolerant at circuit level (the user does not ask for circuit distance)

what was checked for ppm.XXZ

The circuit read off the hardware program (3 R, 6 H, 18 CX, 9 CZ, 3 M) was run once on a Choi state — every ion entangled with a reference — with every measurement outcome kept symbolic. 59 stabilizer flows hold, and their rank 54 is the 54 a complete specification needs, so no other channel satisfies them.

✓ X(d0) kept (exact)
✓ X(d1) kept (exact)
✓ Z(d1) kept (exact)
✓ X(d2) kept (exact)
✓ Z(d2) kept (exact)
✓ X(d3) kept (exact)
✓ Z(d3)·Z(d0) kept (exact)
✓ X(d4) kept (exact)
✓ Z(d4) kept (exact)
✓ X(d5) kept (exact)
✓ Z(d5) kept (exact)
✓ X(d6) kept (exact)
✓ Z(d6)·Z(d0) kept (exact)
✓ X(d7) kept (exact)
✓ Z(d7) kept (exact)
✓ X(d8) kept (exact)
✓ Z(d8) kept (exact)
✓ X(e0) kept (exact)
✓ Z(e0)·Z(d0) kept (exact)
✓ X(e1) kept (exact)
✓ Z(e1) kept (exact)
✓ X(e2) kept (exact)
✓ Z(e2) kept (exact)
✓ X(e3) kept (exact)
✓ Z(e3)·Z(d0) kept (exact)
✓ X(e4) kept (exact)
✓ Z(e4) kept (exact)
✓ X(e5) kept (exact)
✓ Z(e5) kept (exact)
✓ X(e6) kept (exact)
✓ Z(e6)·Z(d0) kept (exact)
✓ X(e7) kept (exact)
✓ Z(e7) kept (exact)
✓ X(e8) kept (exact)
✓ Z(e8) kept (exact)
✓ X(f0)·Z(d0) kept (exact)
✓ Z(f0) kept (exact)
✓ X(f1)·Z(d0) kept (exact)
✓ Z(f1) kept (exact)
✓ X(f2)·Z(d0) kept (exact)
✓ Z(f2) kept (exact)
✓ X(f3) kept (exact)
✓ Z(f3) kept (exact)
✓ X(f4) kept (exact)
✓ Z(f4) kept (exact)
✓ X(f5) kept (exact)
✓ Z(f5) kept (exact)
✓ X(f6) kept (exact)
✓ Z(f6) kept (exact)
✓ X(f7) kept (exact)
✓ Z(f7) kept (exact)
✓ X(f8) kept (exact)
✓ Z(f8) kept (exact)
✓ X̄_AX̄_BZ̄_C measured (measure ⊕ 1 record)
✓ X̄_A kept (exact)
✓ X̄_B kept (exact)
✓ Z̄_C kept (exact)
✓ Z̄_AZ̄_B kept (exact)
✓ Z̄_BX̄_C kept (exact)

a resident probe ion in |+⟩ takes a CX with each X̄-support ion and a CZ with each Z̄-support ion as the ring brings it past its dock, then is read in X: a controlled-P̄ measurement of the product, repeated 3 times
not fault-tolerant at circuit level (the user does not ask for circuit distance)

what was checked for ppm.XZX

✓ X(d0) kept (exact)
✓ X(d1) kept (exact)
✓ Z(d1) kept (exact)
✓ X(d2) kept (exact)
✓ Z(d2) kept (exact)
✓ X(d3) kept (exact)
✓ Z(d3)·Z(d0) kept (exact)
✓ X(d4) kept (exact)
✓ Z(d4) kept (exact)
✓ X(d5) kept (exact)
✓ Z(d5) kept (exact)
✓ X(d6) kept (exact)
✓ Z(d6)·Z(d0) kept (exact)
✓ X(d7) kept (exact)
✓ Z(d7) kept (exact)
✓ X(d8) kept (exact)
✓ Z(d8) kept (exact)
✓ X(e0)·Z(d0) kept (exact)
✓ Z(e0) kept (exact)
✓ X(e1)·Z(d0) kept (exact)
✓ Z(e1) kept (exact)
✓ X(e2)·Z(d0) kept (exact)
✓ Z(e2) kept (exact)
✓ X(e3) kept (exact)
✓ Z(e3) kept (exact)
✓ X(e4) kept (exact)
✓ Z(e4) kept (exact)
✓ X(e5) kept (exact)
✓ Z(e5) kept (exact)
✓ X(e6) kept (exact)
✓ Z(e6) kept (exact)
✓ X(e7) kept (exact)
✓ Z(e7) kept (exact)
✓ X(e8) kept (exact)
✓ Z(e8) kept (exact)
✓ X(f0) kept (exact)
✓ Z(f0)·Z(d0) kept (exact)
✓ X(f1) kept (exact)
✓ Z(f1) kept (exact)
✓ X(f2) kept (exact)
✓ Z(f2) kept (exact)
✓ X(f3) kept (exact)
✓ Z(f3)·Z(d0) kept (exact)
✓ X(f4) kept (exact)
✓ Z(f4) kept (exact)
✓ X(f5) kept (exact)
✓ Z(f5) kept (exact)
✓ X(f6) kept (exact)
✓ Z(f6)·Z(d0) kept (exact)
✓ X(f7) kept (exact)
✓ Z(f7) kept (exact)
✓ X(f8) kept (exact)
✓ Z(f8) kept (exact)
✓ X̄_AZ̄_BX̄_C measured (measure ⊕ 1 record)
✓ X̄_A kept (exact)
✓ Z̄_B kept (exact)
✓ X̄_C kept (exact)
✓ Z̄_AX̄_B kept (exact)
✓ X̄_BZ̄_C kept (exact)

a resident probe ion in |+⟩ takes a CX with each X̄-support ion and a CZ with each Z̄-support ion as the ring brings it past its dock, then is read in X: a controlled-P̄ measurement of the product, repeated 3 times
not fault-tolerant at circuit level (the user does not ask for circuit distance)

what was checked for ppm.XZZ

The circuit read off the hardware program (3 R, 6 H, 18 CZ, 9 CX, 3 M) was run once on a Choi state — every ion entangled with a reference — with every measurement outcome kept symbolic. 59 stabilizer flows hold, and their rank 54 is the 54 a complete specification needs, so no other channel satisfies them.

✓ X(d0) kept (exact)
✓ X(d1) kept (exact)
✓ Z(d1) kept (exact)
✓ X(d2) kept (exact)
✓ Z(d2) kept (exact)
✓ X(d3) kept (exact)
✓ Z(d3)·Z(d0) kept (exact)
✓ X(d4) kept (exact)
✓ Z(d4) kept (exact)
✓ X(d5) kept (exact)
✓ Z(d5) kept (exact)
✓ X(d6) kept (exact)
✓ Z(d6)·Z(d0) kept (exact)
✓ X(d7) kept (exact)
✓ Z(d7) kept (exact)
✓ X(d8) kept (exact)
✓ Z(d8) kept (exact)
✓ X(e0)·Z(d0) kept (exact)
✓ Z(e0) kept (exact)
✓ X(e1)·Z(d0) kept (exact)
✓ Z(e1) kept (exact)
✓ X(e2)·Z(d0) kept (exact)
✓ Z(e2) kept (exact)
✓ X(e3) kept (exact)
✓ Z(e3) kept (exact)
✓ X(e4) kept (exact)
✓ Z(e4) kept (exact)
✓ X(e5) kept (exact)
✓ Z(e5) kept (exact)
✓ X(e6) kept (exact)
✓ Z(e6) kept (exact)
✓ X(e7) kept (exact)
✓ Z(e7) kept (exact)
✓ X(e8) kept (exact)
✓ Z(e8) kept (exact)
✓ X(f0)·Z(d0) kept (exact)
✓ Z(f0) kept (exact)
✓ X(f1)·Z(d0) kept (exact)
✓ Z(f1) kept (exact)
✓ X(f2)·Z(d0) kept (exact)
✓ Z(f2) kept (exact)
✓ X(f3) kept (exact)
✓ Z(f3) kept (exact)
✓ X(f4) kept (exact)
✓ Z(f4) kept (exact)
✓ X(f5) kept (exact)
✓ Z(f5) kept (exact)
✓ X(f6) kept (exact)
✓ Z(f6) kept (exact)
✓ X(f7) kept (exact)
✓ Z(f7) kept (exact)
✓ X(f8) kept (exact)
✓ Z(f8) kept (exact)
✓ X̄_AZ̄_BZ̄_C measured (measure ⊕ 1 record)
✓ X̄_A kept (exact)
✓ Z̄_B kept (exact)
✓ Z̄_C kept (exact)
✓ Z̄_AX̄_B kept (exact)
✓ X̄_BX̄_C kept (exact)

a resident probe ion in |+⟩ takes a CX with each X̄-support ion and a CZ with each Z̄-support ion as the ring brings it past its dock, then is read in X: a controlled-P̄ measurement of the product, repeated 3 times
not fault-tolerant at circuit level (the user does not ask for circuit distance)

what was checked for ppm.ZXX

✓ Z(d0) kept (exact)
✓ X(d1)·X(d0) kept (exact)
✓ Z(d1) kept (exact)
✓ X(d2)·X(d0) kept (exact)
✓ Z(d2) kept (exact)
✓ X(d3) kept (exact)
✓ Z(d3) kept (exact)
✓ X(d4) kept (exact)
✓ Z(d4) kept (exact)
✓ X(d5) kept (exact)
✓ Z(d5) kept (exact)
✓ X(d6) kept (exact)
✓ Z(d6) kept (exact)
✓ X(d7) kept (exact)
✓ Z(d7) kept (exact)
✓ X(d8) kept (exact)
✓ Z(d8) kept (exact)
✓ X(e0) kept (exact)
✓ Z(e0)·X(d0) kept (exact)
✓ X(e1) kept (exact)
✓ Z(e1) kept (exact)
✓ X(e2) kept (exact)
✓ Z(e2) kept (exact)
✓ X(e3) kept (exact)
✓ Z(e3)·X(d0) kept (exact)
✓ X(e4) kept (exact)
✓ Z(e4) kept (exact)
✓ X(e5) kept (exact)
✓ Z(e5) kept (exact)
✓ X(e6) kept (exact)
✓ Z(e6)·X(d0) kept (exact)
✓ X(e7) kept (exact)
✓ Z(e7) kept (exact)
✓ X(e8) kept (exact)
✓ Z(e8) kept (exact)
✓ X(f0) kept (exact)
✓ Z(f0)·X(d0) kept (exact)
✓ X(f1) kept (exact)
✓ Z(f1) kept (exact)
✓ X(f2) kept (exact)
✓ Z(f2) kept (exact)
✓ X(f3) kept (exact)
✓ Z(f3)·X(d0) kept (exact)
✓ X(f4) kept (exact)
✓ Z(f4) kept (exact)
✓ X(f5) kept (exact)
✓ Z(f5) kept (exact)
✓ X(f6) kept (exact)
✓ Z(f6)·X(d0) kept (exact)
✓ X(f7) kept (exact)
✓ Z(f7) kept (exact)
✓ X(f8) kept (exact)
✓ Z(f8) kept (exact)
✓ Z̄_AX̄_BX̄_C measured (measure ⊕ 1 record)
✓ Z̄_A kept (exact)
✓ X̄_B kept (exact)
✓ X̄_C kept (exact)
✓ X̄_AZ̄_B kept (exact)
✓ Z̄_BZ̄_C kept (exact)

a resident probe ion in |+⟩ takes a CX with each X̄-support ion and a CZ with each Z̄-support ion as the ring brings it past its dock, then is read in X: a controlled-P̄ measurement of the product, repeated 3 times
not fault-tolerant at circuit level (the user does not ask for circuit distance)

what was checked for ppm.ZXZ

The circuit read off the hardware program (3 R, 6 H, 9 CX, 18 CZ, 3 M) was run once on a Choi state — every ion entangled with a reference — with every measurement outcome kept symbolic. 59 stabilizer flows hold, and their rank 54 is the 54 a complete specification needs, so no other channel satisfies them.

✓ Z(d0) kept (exact)
✓ X(d1)·X(d0) kept (exact)
✓ Z(d1) kept (exact)
✓ X(d2)·X(d0) kept (exact)
✓ Z(d2) kept (exact)
✓ X(d3) kept (exact)
✓ Z(d3) kept (exact)
✓ X(d4) kept (exact)
✓ Z(d4) kept (exact)
✓ X(d5) kept (exact)
✓ Z(d5) kept (exact)
✓ X(d6) kept (exact)
✓ Z(d6) kept (exact)
✓ X(d7) kept (exact)
✓ Z(d7) kept (exact)
✓ X(d8) kept (exact)
✓ Z(d8) kept (exact)
✓ X(e0) kept (exact)
✓ Z(e0)·X(d0) kept (exact)
✓ X(e1) kept (exact)
✓ Z(e1) kept (exact)
✓ X(e2) kept (exact)
✓ Z(e2) kept (exact)
✓ X(e3) kept (exact)
✓ Z(e3)·X(d0) kept (exact)
✓ X(e4) kept (exact)
✓ Z(e4) kept (exact)
✓ X(e5) kept (exact)
✓ Z(e5) kept (exact)
✓ X(e6) kept (exact)
✓ Z(e6)·X(d0) kept (exact)
✓ X(e7) kept (exact)
✓ Z(e7) kept (exact)
✓ X(e8) kept (exact)
✓ Z(e8) kept (exact)
✓ X(f0)·X(d0) kept (exact)
✓ Z(f0) kept (exact)
✓ X(f1)·X(d0) kept (exact)
✓ Z(f1) kept (exact)
✓ X(f2)·X(d0) kept (exact)
✓ Z(f2) kept (exact)
✓ X(f3) kept (exact)
✓ Z(f3) kept (exact)
✓ X(f4) kept (exact)
✓ Z(f4) kept (exact)
✓ X(f5) kept (exact)
✓ Z(f5) kept (exact)
✓ X(f6) kept (exact)
✓ Z(f6) kept (exact)
✓ X(f7) kept (exact)
✓ Z(f7) kept (exact)
✓ X(f8) kept (exact)
✓ Z(f8) kept (exact)
✓ Z̄_AX̄_BZ̄_C measured (measure ⊕ 1 record)
✓ Z̄_A kept (exact)
✓ X̄_B kept (exact)
✓ Z̄_C kept (exact)
✓ X̄_AZ̄_B kept (exact)
✓ Z̄_BX̄_C kept (exact)

a resident probe ion in |+⟩ takes a CX with each X̄-support ion and a CZ with each Z̄-support ion as the ring brings it past its dock, then is read in X: a controlled-P̄ measurement of the product, repeated 3 times
not fault-tolerant at circuit level (the user does not ask for circuit distance)

what was checked for ppm.ZZX

✓ Z(d0) kept (exact)
✓ X(d1)·X(d0) kept (exact)
✓ Z(d1) kept (exact)
✓ X(d2)·X(d0) kept (exact)
✓ Z(d2) kept (exact)
✓ X(d3) kept (exact)
✓ Z(d3) kept (exact)
✓ X(d4) kept (exact)
✓ Z(d4) kept (exact)
✓ X(d5) kept (exact)
✓ Z(d5) kept (exact)
✓ X(d6) kept (exact)
✓ Z(d6) kept (exact)
✓ X(d7) kept (exact)
✓ Z(d7) kept (exact)
✓ X(d8) kept (exact)
✓ Z(d8) kept (exact)
✓ X(e0)·X(d0) kept (exact)
✓ Z(e0) kept (exact)
✓ X(e1)·X(d0) kept (exact)
✓ Z(e1) kept (exact)
✓ X(e2)·X(d0) kept (exact)
✓ Z(e2) kept (exact)
✓ X(e3) kept (exact)
✓ Z(e3) kept (exact)
✓ X(e4) kept (exact)
✓ Z(e4) kept (exact)
✓ X(e5) kept (exact)
✓ Z(e5) kept (exact)
✓ X(e6) kept (exact)
✓ Z(e6) kept (exact)
✓ X(e7) kept (exact)
✓ Z(e7) kept (exact)
✓ X(e8) kept (exact)
✓ Z(e8) kept (exact)
✓ X(f0) kept (exact)
✓ Z(f0)·X(d0) kept (exact)
✓ X(f1) kept (exact)
✓ Z(f1) kept (exact)
✓ X(f2) kept (exact)
✓ Z(f2) kept (exact)
✓ X(f3) kept (exact)
✓ Z(f3)·X(d0) kept (exact)
✓ X(f4) kept (exact)
✓ Z(f4) kept (exact)
✓ X(f5) kept (exact)
✓ Z(f5) kept (exact)
✓ X(f6) kept (exact)
✓ Z(f6)·X(d0) kept (exact)
✓ X(f7) kept (exact)
✓ Z(f7) kept (exact)
✓ X(f8) kept (exact)
✓ Z(f8) kept (exact)
✓ Z̄_AZ̄_BX̄_C measured (measure ⊕ 1 record)
✓ Z̄_A kept (exact)
✓ Z̄_B kept (exact)
✓ X̄_C kept (exact)
✓ X̄_AX̄_B kept (exact)
✓ X̄_BZ̄_C kept (exact)

a resident probe ion in |+⟩ takes a CX with each X̄-support ion and a CZ with each Z̄-support ion as the ring brings it past its dock, then is read in X: a controlled-P̄ measurement of the product, repeated 3 times
not fault-tolerant at circuit level (the user does not ask for circuit distance)

what was checked for ppm.ZZZ

The circuit read off the hardware program (3 R, 6 H, 27 CZ, 3 M) was run once on a Choi state — every ion entangled with a reference — with every measurement outcome kept symbolic. 59 stabilizer flows hold, and their rank 54 is the 54 a complete specification needs, so no other channel satisfies them.

✓ Z(d0) kept (exact)
✓ X(d1)·X(d0) kept (exact)
✓ Z(d1) kept (exact)
✓ X(d2)·X(d0) kept (exact)
✓ Z(d2) kept (exact)
✓ X(d3) kept (exact)
✓ Z(d3) kept (exact)
✓ X(d4) kept (exact)
✓ Z(d4) kept (exact)
✓ X(d5) kept (exact)
✓ Z(d5) kept (exact)
✓ X(d6) kept (exact)
✓ Z(d6) kept (exact)
✓ X(d7) kept (exact)
✓ Z(d7) kept (exact)
✓ X(d8) kept (exact)
✓ Z(d8) kept (exact)
✓ X(e0)·X(d0) kept (exact)
✓ Z(e0) kept (exact)
✓ X(e1)·X(d0) kept (exact)
✓ Z(e1) kept (exact)
✓ X(e2)·X(d0) kept (exact)
✓ Z(e2) kept (exact)
✓ X(e3) kept (exact)
✓ Z(e3) kept (exact)
✓ X(e4) kept (exact)
✓ Z(e4) kept (exact)
✓ X(e5) kept (exact)
✓ Z(e5) kept (exact)
✓ X(e6) kept (exact)
✓ Z(e6) kept (exact)
✓ X(e7) kept (exact)
✓ Z(e7) kept (exact)
✓ X(e8) kept (exact)
✓ Z(e8) kept (exact)
✓ X(f0)·X(d0) kept (exact)
✓ Z(f0) kept (exact)
✓ X(f1)·X(d0) kept (exact)
✓ Z(f1) kept (exact)
✓ X(f2)·X(d0) kept (exact)
✓ Z(f2) kept (exact)
✓ X(f3) kept (exact)
✓ Z(f3) kept (exact)
✓ X(f4) kept (exact)
✓ Z(f4) kept (exact)
✓ X(f5) kept (exact)
✓ Z(f5) kept (exact)
✓ X(f6) kept (exact)
✓ Z(f6) kept (exact)
✓ X(f7) kept (exact)
✓ Z(f7) kept (exact)
✓ X(f8) kept (exact)
✓ Z(f8) kept (exact)
✓ Z̄_AZ̄_BZ̄_C measured (measure ⊕ 1 record)
✓ Z̄_A kept (exact)
✓ Z̄_B kept (exact)
✓ Z̄_C kept (exact)
✓ X̄_AX̄_B kept (exact)
✓ X̄_BX̄_C kept (exact)

a resident probe ion in |+⟩ takes a CX with each X̄-support ion and a CZ with each Z̄-support ion as the ring brings it past its dock, then is read in X: a controlled-P̄ measurement of the product, repeated 3 times
not fault-tolerant at circuit level (the user does not ask for circuit distance)

what was checked for ppm.XXXX

The circuit read off the hardware program (3 R, 6 H, 36 CX, 3 M) was run once on a Choi state — every ion entangled with a reference — with every measurement outcome kept symbolic. 79 stabilizer flows hold, and their rank 72 is the 72 a complete specification needs, so no other channel satisfies them.

✓ X(d0) kept (exact)
✓ X(d1) kept (exact)
✓ Z(d1) kept (exact)
✓ X(d2) kept (exact)
✓ Z(d2) kept (exact)
✓ X(d3) kept (exact)
✓ Z(d3)·Z(d0) kept (exact)
✓ X(d4) kept (exact)
✓ Z(d4) kept (exact)
✓ X(d5) kept (exact)
✓ Z(d5) kept (exact)
✓ X(d6) kept (exact)
✓ Z(d6)·Z(d0) kept (exact)
✓ X(d7) kept (exact)
✓ Z(d7) kept (exact)
✓ X(d8) kept (exact)
✓ Z(d8) kept (exact)
✓ X(e0) kept (exact)
✓ Z(e0)·Z(d0) kept (exact)
✓ X(e1) kept (exact)
✓ Z(e1) kept (exact)
✓ X(e2) kept (exact)
✓ Z(e2) kept (exact)
✓ X(e3) kept (exact)
✓ Z(e3)·Z(d0) kept (exact)
✓ X(e4) kept (exact)
✓ Z(e4) kept (exact)
✓ X(e5) kept (exact)
✓ Z(e5) kept (exact)
✓ X(e6) kept (exact)
✓ Z(e6)·Z(d0) kept (exact)
✓ X(e7) kept (exact)
✓ Z(e7) kept (exact)
✓ X(e8) kept (exact)
✓ Z(e8) kept (exact)
✓ X(f0) kept (exact)
✓ Z(f0)·Z(d0) kept (exact)
✓ X(f1) kept (exact)
✓ Z(f1) kept (exact)
✓ X(f2) kept (exact)
✓ Z(f2) kept (exact)
✓ X(f3) kept (exact)
✓ Z(f3)·Z(d0) kept (exact)
✓ X(f4) kept (exact)
✓ Z(f4) kept (exact)
✓ X(f5) kept (exact)
✓ Z(f5) kept (exact)
✓ X(f6) kept (exact)
✓ Z(f6)·Z(d0) kept (exact)
✓ X(f7) kept (exact)
✓ Z(f7) kept (exact)
✓ X(f8) kept (exact)
✓ Z(f8) kept (exact)
✓ X(g0) kept (exact)
✓ Z(g0)·Z(d0) kept (exact)
✓ X(g1) kept (exact)
✓ Z(g1) kept (exact)
✓ X(g2) kept (exact)
✓ Z(g2) kept (exact)
✓ X(g3) kept (exact)
✓ Z(g3)·Z(d0) kept (exact)
✓ X(g4) kept (exact)
✓ Z(g4) kept (exact)
✓ X(g5) kept (exact)
✓ Z(g5) kept (exact)
✓ X(g6) kept (exact)
✓ Z(g6)·Z(d0) kept (exact)
✓ X(g7) kept (exact)
✓ Z(g7) kept (exact)
✓ X(g8) kept (exact)
✓ Z(g8) kept (exact)
✓ X̄_AX̄_BX̄_CX̄_D measured (measure ⊕ 1 record)
✓ X̄_A kept (exact)
✓ X̄_B kept (exact)
✓ X̄_C kept (exact)
✓ X̄_D kept (exact)
✓ Z̄_AZ̄_B kept (exact)
✓ Z̄_BZ̄_C kept (exact)
✓ Z̄_CZ̄_D kept (exact)

a resident probe ion in |+⟩ takes a CX with each X̄-support ion and a CZ with each Z̄-support ion as the ring brings it past its dock, then is read in X: a controlled-P̄ measurement of the product, repeated 3 times
not fault-tolerant at circuit level (the user does not ask for circuit distance)

what was checked for ppm.XXXZ

The circuit read off the hardware program (3 R, 6 H, 27 CX, 9 CZ, 3 M) was run once on a Choi state — every ion entangled with a reference — with every measurement outcome kept symbolic. 79 stabilizer flows hold, and their rank 72 is the 72 a complete specification needs, so no other channel satisfies them.

✓ X(d0) kept (exact)
✓ X(d1) kept (exact)
✓ Z(d1) kept (exact)
✓ X(d2) kept (exact)
✓ Z(d2) kept (exact)
✓ X(d3) kept (exact)
✓ Z(d3)·Z(d0) kept (exact)
✓ X(d4) kept (exact)
✓ Z(d4) kept (exact)
✓ X(d5) kept (exact)
✓ Z(d5) kept (exact)
✓ X(d6) kept (exact)
✓ Z(d6)·Z(d0) kept (exact)
✓ X(d7) kept (exact)
✓ Z(d7) kept (exact)
✓ X(d8) kept (exact)
✓ Z(d8) kept (exact)
✓ X(e0) kept (exact)
✓ Z(e0)·Z(d0) kept (exact)
✓ X(e1) kept (exact)
✓ Z(e1) kept (exact)
✓ X(e2) kept (exact)
✓ Z(e2) kept (exact)
✓ X(e3) kept (exact)
✓ Z(e3)·Z(d0) kept (exact)
✓ X(e4) kept (exact)
✓ Z(e4) kept (exact)
✓ X(e5) kept (exact)
✓ Z(e5) kept (exact)
✓ X(e6) kept (exact)
✓ Z(e6)·Z(d0) kept (exact)
✓ X(e7) kept (exact)
✓ Z(e7) kept (exact)
✓ X(e8) kept (exact)
✓ Z(e8) kept (exact)
✓ X(f0) kept (exact)
✓ Z(f0)·Z(d0) kept (exact)
✓ X(f1) kept (exact)
✓ Z(f1) kept (exact)
✓ X(f2) kept (exact)
✓ Z(f2) kept (exact)
✓ X(f3) kept (exact)
✓ Z(f3)·Z(d0) kept (exact)
✓ X(f4) kept (exact)
✓ Z(f4) kept (exact)
✓ X(f5) kept (exact)
✓ Z(f5) kept (exact)
✓ X(f6) kept (exact)
✓ Z(f6)·Z(d0) kept (exact)
✓ X(f7) kept (exact)
✓ Z(f7) kept (exact)
✓ X(f8) kept (exact)
✓ Z(f8) kept (exact)
✓ X(g0)·Z(d0) kept (exact)
✓ Z(g0) kept (exact)
✓ X(g1)·Z(d0) kept (exact)
✓ Z(g1) kept (exact)
✓ X(g2)·Z(d0) kept (exact)
✓ Z(g2) kept (exact)
✓ X(g3) kept (exact)
✓ Z(g3) kept (exact)
✓ X(g4) kept (exact)
✓ Z(g4) kept (exact)
✓ X(g5) kept (exact)
✓ Z(g5) kept (exact)
✓ X(g6) kept (exact)
✓ Z(g6) kept (exact)
✓ X(g7) kept (exact)
✓ Z(g7) kept (exact)
✓ X(g8) kept (exact)
✓ Z(g8) kept (exact)
✓ X̄_AX̄_BX̄_CZ̄_D measured (measure ⊕ 1 record)
✓ X̄_A kept (exact)
✓ X̄_B kept (exact)
✓ X̄_C kept (exact)
✓ Z̄_D kept (exact)
✓ Z̄_AZ̄_B kept (exact)
✓ Z̄_BZ̄_C kept (exact)
✓ Z̄_CX̄_D kept (exact)

a resident probe ion in |+⟩ takes a CX with each X̄-support ion and a CZ with each Z̄-support ion as the ring brings it past its dock, then is read in X: a controlled-P̄ measurement of the product, repeated 3 times
not fault-tolerant at circuit level (the user does not ask for circuit distance)

what was checked for ppm.XXZX

✓ X(d0) kept (exact)
✓ X(d1) kept (exact)
✓ Z(d1) kept (exact)
✓ X(d2) kept (exact)
✓ Z(d2) kept (exact)
✓ X(d3) kept (exact)
✓ Z(d3)·Z(d0) kept (exact)
✓ X(d4) kept (exact)
✓ Z(d4) kept (exact)
✓ X(d5) kept (exact)
✓ Z(d5) kept (exact)
✓ X(d6) kept (exact)
✓ Z(d6)·Z(d0) kept (exact)
✓ X(d7) kept (exact)
✓ Z(d7) kept (exact)
✓ X(d8) kept (exact)
✓ Z(d8) kept (exact)
✓ X(e0) kept (exact)
✓ Z(e0)·Z(d0) kept (exact)
✓ X(e1) kept (exact)
✓ Z(e1) kept (exact)
✓ X(e2) kept (exact)
✓ Z(e2) kept (exact)
✓ X(e3) kept (exact)
✓ Z(e3)·Z(d0) kept (exact)
✓ X(e4) kept (exact)
✓ Z(e4) kept (exact)
✓ X(e5) kept (exact)
✓ Z(e5) kept (exact)
✓ X(e6) kept (exact)
✓ Z(e6)·Z(d0) kept (exact)
✓ X(e7) kept (exact)
✓ Z(e7) kept (exact)
✓ X(e8) kept (exact)
✓ Z(e8) kept (exact)
✓ X(f0)·Z(d0) kept (exact)
✓ Z(f0) kept (exact)
✓ X(f1)·Z(d0) kept (exact)
✓ Z(f1) kept (exact)
✓ X(f2)·Z(d0) kept (exact)
✓ Z(f2) kept (exact)
✓ X(f3) kept (exact)
✓ Z(f3) kept (exact)
✓ X(f4) kept (exact)
✓ Z(f4) kept (exact)
✓ X(f5) kept (exact)
✓ Z(f5) kept (exact)
✓ X(f6) kept (exact)
✓ Z(f6) kept (exact)
✓ X(f7) kept (exact)
✓ Z(f7) kept (exact)
✓ X(f8) kept (exact)
✓ Z(f8) kept (exact)
✓ X(g0) kept (exact)
✓ Z(g0)·Z(d0) kept (exact)
✓ X(g1) kept (exact)
✓ Z(g1) kept (exact)
✓ X(g2) kept (exact)
✓ Z(g2) kept (exact)
✓ X(g3) kept (exact)
✓ Z(g3)·Z(d0) kept (exact)
✓ X(g4) kept (exact)
✓ Z(g4) kept (exact)
✓ X(g5) kept (exact)
✓ Z(g5) kept (exact)
✓ X(g6) kept (exact)
✓ Z(g6)·Z(d0) kept (exact)
✓ X(g7) kept (exact)
✓ Z(g7) kept (exact)
✓ X(g8) kept (exact)
✓ Z(g8) kept (exact)
✓ X̄_AX̄_BZ̄_CX̄_D measured (measure ⊕ 1 record)
✓ X̄_A kept (exact)
✓ X̄_B kept (exact)
✓ Z̄_C kept (exact)
✓ X̄_D kept (exact)
✓ Z̄_AZ̄_B kept (exact)
✓ Z̄_BX̄_C kept (exact)
✓ X̄_CZ̄_D kept (exact)

a resident probe ion in |+⟩ takes a CX with each X̄-support ion and a CZ with each Z̄-support ion as the ring brings it past its dock, then is read in X: a controlled-P̄ measurement of the product, repeated 3 times
not fault-tolerant at circuit level (the user does not ask for circuit distance)

what was checked for ppm.XXZZ

The circuit read off the hardware program (3 R, 6 H, 18 CX, 18 CZ, 3 M) was run once on a Choi state — every ion entangled with a reference — with every measurement outcome kept symbolic. 79 stabilizer flows hold, and their rank 72 is the 72 a complete specification needs, so no other channel satisfies them.

✓ X(d0) kept (exact)
✓ X(d1) kept (exact)
✓ Z(d1) kept (exact)
✓ X(d2) kept (exact)
✓ Z(d2) kept (exact)
✓ X(d3) kept (exact)
✓ Z(d3)·Z(d0) kept (exact)
✓ X(d4) kept (exact)
✓ Z(d4) kept (exact)
✓ X(d5) kept (exact)
✓ Z(d5) kept (exact)
✓ X(d6) kept (exact)
✓ Z(d6)·Z(d0) kept (exact)
✓ X(d7) kept (exact)
✓ Z(d7) kept (exact)
✓ X(d8) kept (exact)
✓ Z(d8) kept (exact)
✓ X(e0) kept (exact)
✓ Z(e0)·Z(d0) kept (exact)
✓ X(e1) kept (exact)
✓ Z(e1) kept (exact)
✓ X(e2) kept (exact)
✓ Z(e2) kept (exact)
✓ X(e3) kept (exact)
✓ Z(e3)·Z(d0) kept (exact)
✓ X(e4) kept (exact)
✓ Z(e4) kept (exact)
✓ X(e5) kept (exact)
✓ Z(e5) kept (exact)
✓ X(e6) kept (exact)
✓ Z(e6)·Z(d0) kept (exact)
✓ X(e7) kept (exact)
✓ Z(e7) kept (exact)
✓ X(e8) kept (exact)
✓ Z(e8) kept (exact)
✓ X(f0)·Z(d0) kept (exact)
✓ Z(f0) kept (exact)
✓ X(f1)·Z(d0) kept (exact)
✓ Z(f1) kept (exact)
✓ X(f2)·Z(d0) kept (exact)
✓ Z(f2) kept (exact)
✓ X(f3) kept (exact)
✓ Z(f3) kept (exact)
✓ X(f4) kept (exact)
✓ Z(f4) kept (exact)
✓ X(f5) kept (exact)
✓ Z(f5) kept (exact)
✓ X(f6) kept (exact)
✓ Z(f6) kept (exact)
✓ X(f7) kept (exact)
✓ Z(f7) kept (exact)
✓ X(f8) kept (exact)
✓ Z(f8) kept (exact)
✓ X(g0)·Z(d0) kept (exact)
✓ Z(g0) kept (exact)
✓ X(g1)·Z(d0) kept (exact)
✓ Z(g1) kept (exact)
✓ X(g2)·Z(d0) kept (exact)
✓ Z(g2) kept (exact)
✓ X(g3) kept (exact)
✓ Z(g3) kept (exact)
✓ X(g4) kept (exact)
✓ Z(g4) kept (exact)
✓ X(g5) kept (exact)
✓ Z(g5) kept (exact)
✓ X(g6) kept (exact)
✓ Z(g6) kept (exact)
✓ X(g7) kept (exact)
✓ Z(g7) kept (exact)
✓ X(g8) kept (exact)
✓ Z(g8) kept (exact)
✓ X̄_AX̄_BZ̄_CZ̄_D measured (measure ⊕ 1 record)
✓ X̄_A kept (exact)
✓ X̄_B kept (exact)
✓ Z̄_C kept (exact)
✓ Z̄_D kept (exact)
✓ Z̄_AZ̄_B kept (exact)
✓ Z̄_BX̄_C kept (exact)
✓ X̄_CX̄_D kept (exact)

a resident probe ion in |+⟩ takes a CX with each X̄-support ion and a CZ with each Z̄-support ion as the ring brings it past its dock, then is read in X: a controlled-P̄ measurement of the product, repeated 3 times
not fault-tolerant at circuit level (the user does not ask for circuit distance)

what was checked for ppm.XZXX

✓ X(d0) kept (exact)
✓ X(d1) kept (exact)
✓ Z(d1) kept (exact)
✓ X(d2) kept (exact)
✓ Z(d2) kept (exact)
✓ X(d3) kept (exact)
✓ Z(d3)·Z(d0) kept (exact)
✓ X(d4) kept (exact)
✓ Z(d4) kept (exact)
✓ X(d5) kept (exact)
✓ Z(d5) kept (exact)
✓ X(d6) kept (exact)
✓ Z(d6)·Z(d0) kept (exact)
✓ X(d7) kept (exact)
✓ Z(d7) kept (exact)
✓ X(d8) kept (exact)
✓ Z(d8) kept (exact)
✓ X(e0)·Z(d0) kept (exact)
✓ Z(e0) kept (exact)
✓ X(e1)·Z(d0) kept (exact)
✓ Z(e1) kept (exact)
✓ X(e2)·Z(d0) kept (exact)
✓ Z(e2) kept (exact)
✓ X(e3) kept (exact)
✓ Z(e3) kept (exact)
✓ X(e4) kept (exact)
✓ Z(e4) kept (exact)
✓ X(e5) kept (exact)
✓ Z(e5) kept (exact)
✓ X(e6) kept (exact)
✓ Z(e6) kept (exact)
✓ X(e7) kept (exact)
✓ Z(e7) kept (exact)
✓ X(e8) kept (exact)
✓ Z(e8) kept (exact)
✓ X(f0) kept (exact)
✓ Z(f0)·Z(d0) kept (exact)
✓ X(f1) kept (exact)
✓ Z(f1) kept (exact)
✓ X(f2) kept (exact)
✓ Z(f2) kept (exact)
✓ X(f3) kept (exact)
✓ Z(f3)·Z(d0) kept (exact)
✓ X(f4) kept (exact)
✓ Z(f4) kept (exact)
✓ X(f5) kept (exact)
✓ Z(f5) kept (exact)
✓ X(f6) kept (exact)
✓ Z(f6)·Z(d0) kept (exact)
✓ X(f7) kept (exact)
✓ Z(f7) kept (exact)
✓ X(f8) kept (exact)
✓ Z(f8) kept (exact)
✓ X(g0) kept (exact)
✓ Z(g0)·Z(d0) kept (exact)
✓ X(g1) kept (exact)
✓ Z(g1) kept (exact)
✓ X(g2) kept (exact)
✓ Z(g2) kept (exact)
✓ X(g3) kept (exact)
✓ Z(g3)·Z(d0) kept (exact)
✓ X(g4) kept (exact)
✓ Z(g4) kept (exact)
✓ X(g5) kept (exact)
✓ Z(g5) kept (exact)
✓ X(g6) kept (exact)
✓ Z(g6)·Z(d0) kept (exact)
✓ X(g7) kept (exact)
✓ Z(g7) kept (exact)
✓ X(g8) kept (exact)
✓ Z(g8) kept (exact)
✓ X̄_AZ̄_BX̄_CX̄_D measured (measure ⊕ 1 record)
✓ X̄_A kept (exact)
✓ Z̄_B kept (exact)
✓ X̄_C kept (exact)
✓ X̄_D kept (exact)
✓ Z̄_AX̄_B kept (exact)
✓ X̄_BZ̄_C kept (exact)
✓ Z̄_CZ̄_D kept (exact)

a resident probe ion in |+⟩ takes a CX with each X̄-support ion and a CZ with each Z̄-support ion as the ring brings it past its dock, then is read in X: a controlled-P̄ measurement of the product, repeated 3 times
not fault-tolerant at circuit level (the user does not ask for circuit distance)

what was checked for ppm.XZXZ

✓ X(d0) kept (exact)
✓ X(d1) kept (exact)
✓ Z(d1) kept (exact)
✓ X(d2) kept (exact)
✓ Z(d2) kept (exact)
✓ X(d3) kept (exact)
✓ Z(d3)·Z(d0) kept (exact)
✓ X(d4) kept (exact)
✓ Z(d4) kept (exact)
✓ X(d5) kept (exact)
✓ Z(d5) kept (exact)
✓ X(d6) kept (exact)
✓ Z(d6)·Z(d0) kept (exact)
✓ X(d7) kept (exact)
✓ Z(d7) kept (exact)
✓ X(d8) kept (exact)
✓ Z(d8) kept (exact)
✓ X(e0)·Z(d0) kept (exact)
✓ Z(e0) kept (exact)
✓ X(e1)·Z(d0) kept (exact)
✓ Z(e1) kept (exact)
✓ X(e2)·Z(d0) kept (exact)
✓ Z(e2) kept (exact)
✓ X(e3) kept (exact)
✓ Z(e3) kept (exact)
✓ X(e4) kept (exact)
✓ Z(e4) kept (exact)
✓ X(e5) kept (exact)
✓ Z(e5) kept (exact)
✓ X(e6) kept (exact)
✓ Z(e6) kept (exact)
✓ X(e7) kept (exact)
✓ Z(e7) kept (exact)
✓ X(e8) kept (exact)
✓ Z(e8) kept (exact)
✓ X(f0) kept (exact)
✓ Z(f0)·Z(d0) kept (exact)
✓ X(f1) kept (exact)
✓ Z(f1) kept (exact)
✓ X(f2) kept (exact)
✓ Z(f2) kept (exact)
✓ X(f3) kept (exact)
✓ Z(f3)·Z(d0) kept (exact)
✓ X(f4) kept (exact)
✓ Z(f4) kept (exact)
✓ X(f5) kept (exact)
✓ Z(f5) kept (exact)
✓ X(f6) kept (exact)
✓ Z(f6)·Z(d0) kept (exact)
✓ X(f7) kept (exact)
✓ Z(f7) kept (exact)
✓ X(f8) kept (exact)
✓ Z(f8) kept (exact)
✓ X(g0)·Z(d0) kept (exact)
✓ Z(g0) kept (exact)
✓ X(g1)·Z(d0) kept (exact)
✓ Z(g1) kept (exact)
✓ X(g2)·Z(d0) kept (exact)
✓ Z(g2) kept (exact)
✓ X(g3) kept (exact)
✓ Z(g3) kept (exact)
✓ X(g4) kept (exact)
✓ Z(g4) kept (exact)
✓ X(g5) kept (exact)
✓ Z(g5) kept (exact)
✓ X(g6) kept (exact)
✓ Z(g6) kept (exact)
✓ X(g7) kept (exact)
✓ Z(g7) kept (exact)
✓ X(g8) kept (exact)
✓ Z(g8) kept (exact)
✓ X̄_AZ̄_BX̄_CZ̄_D measured (measure ⊕ 1 record)
✓ X̄_A kept (exact)
✓ Z̄_B kept (exact)
✓ X̄_C kept (exact)
✓ Z̄_D kept (exact)
✓ Z̄_AX̄_B kept (exact)
✓ X̄_BZ̄_C kept (exact)
✓ Z̄_CX̄_D kept (exact)

a resident probe ion in |+⟩ takes a CX with each X̄-support ion and a CZ with each Z̄-support ion as the ring brings it past its dock, then is read in X: a controlled-P̄ measurement of the product, repeated 3 times
not fault-tolerant at circuit level (the user does not ask for circuit distance)

what was checked for ppm.XZZX

The circuit read off the hardware program (3 R, 6 H, 18 CZ, 18 CX, 3 M) was run once on a Choi state — every ion entangled with a reference — with every measurement outcome kept symbolic. 79 stabilizer flows hold, and their rank 72 is the 72 a complete specification needs, so no other channel satisfies them.

✓ X(d0) kept (exact)
✓ X(d1) kept (exact)
✓ Z(d1) kept (exact)
✓ X(d2) kept (exact)
✓ Z(d2) kept (exact)
✓ X(d3) kept (exact)
✓ Z(d3)·Z(d0) kept (exact)
✓ X(d4) kept (exact)
✓ Z(d4) kept (exact)
✓ X(d5) kept (exact)
✓ Z(d5) kept (exact)
✓ X(d6) kept (exact)
✓ Z(d6)·Z(d0) kept (exact)
✓ X(d7) kept (exact)
✓ Z(d7) kept (exact)
✓ X(d8) kept (exact)
✓ Z(d8) kept (exact)
✓ X(e0)·Z(d0) kept (exact)
✓ Z(e0) kept (exact)
✓ X(e1)·Z(d0) kept (exact)
✓ Z(e1) kept (exact)
✓ X(e2)·Z(d0) kept (exact)
✓ Z(e2) kept (exact)
✓ X(e3) kept (exact)
✓ Z(e3) kept (exact)
✓ X(e4) kept (exact)
✓ Z(e4) kept (exact)
✓ X(e5) kept (exact)
✓ Z(e5) kept (exact)
✓ X(e6) kept (exact)
✓ Z(e6) kept (exact)
✓ X(e7) kept (exact)
✓ Z(e7) kept (exact)
✓ X(e8) kept (exact)
✓ Z(e8) kept (exact)
✓ X(f0)·Z(d0) kept (exact)
✓ Z(f0) kept (exact)
✓ X(f1)·Z(d0) kept (exact)
✓ Z(f1) kept (exact)
✓ X(f2)·Z(d0) kept (exact)
✓ Z(f2) kept (exact)
✓ X(f3) kept (exact)
✓ Z(f3) kept (exact)
✓ X(f4) kept (exact)
✓ Z(f4) kept (exact)
✓ X(f5) kept (exact)
✓ Z(f5) kept (exact)
✓ X(f6) kept (exact)
✓ Z(f6) kept (exact)
✓ X(f7) kept (exact)
✓ Z(f7) kept (exact)
✓ X(f8) kept (exact)
✓ Z(f8) kept (exact)
✓ X(g0) kept (exact)
✓ Z(g0)·Z(d0) kept (exact)
✓ X(g1) kept (exact)
✓ Z(g1) kept (exact)
✓ X(g2) kept (exact)
✓ Z(g2) kept (exact)
✓ X(g3) kept (exact)
✓ Z(g3)·Z(d0) kept (exact)
✓ X(g4) kept (exact)
✓ Z(g4) kept (exact)
✓ X(g5) kept (exact)
✓ Z(g5) kept (exact)
✓ X(g6) kept (exact)
✓ Z(g6)·Z(d0) kept (exact)
✓ X(g7) kept (exact)
✓ Z(g7) kept (exact)
✓ X(g8) kept (exact)
✓ Z(g8) kept (exact)
✓ X̄_AZ̄_BZ̄_CX̄_D measured (measure ⊕ 1 record)
✓ X̄_A kept (exact)
✓ Z̄_B kept (exact)
✓ Z̄_C kept (exact)
✓ X̄_D kept (exact)
✓ Z̄_AX̄_B kept (exact)
✓ X̄_BX̄_C kept (exact)
✓ X̄_CZ̄_D kept (exact)

a resident probe ion in |+⟩ takes a CX with each X̄-support ion and a CZ with each Z̄-support ion as the ring brings it past its dock, then is read in X: a controlled-P̄ measurement of the product, repeated 3 times
not fault-tolerant at circuit level (the user does not ask for circuit distance)

what was checked for ppm.XZZZ

The circuit read off the hardware program (3 R, 6 H, 27 CZ, 9 CX, 3 M) was run once on a Choi state — every ion entangled with a reference — with every measurement outcome kept symbolic. 79 stabilizer flows hold, and their rank 72 is the 72 a complete specification needs, so no other channel satisfies them.

✓ X(d0) kept (exact)
✓ X(d1) kept (exact)
✓ Z(d1) kept (exact)
✓ X(d2) kept (exact)
✓ Z(d2) kept (exact)
✓ X(d3) kept (exact)
✓ Z(d3)·Z(d0) kept (exact)
✓ X(d4) kept (exact)
✓ Z(d4) kept (exact)
✓ X(d5) kept (exact)
✓ Z(d5) kept (exact)
✓ X(d6) kept (exact)
✓ Z(d6)·Z(d0) kept (exact)
✓ X(d7) kept (exact)
✓ Z(d7) kept (exact)
✓ X(d8) kept (exact)
✓ Z(d8) kept (exact)
✓ X(e0)·Z(d0) kept (exact)
✓ Z(e0) kept (exact)
✓ X(e1)·Z(d0) kept (exact)
✓ Z(e1) kept (exact)
✓ X(e2)·Z(d0) kept (exact)
✓ Z(e2) kept (exact)
✓ X(e3) kept (exact)
✓ Z(e3) kept (exact)
✓ X(e4) kept (exact)
✓ Z(e4) kept (exact)
✓ X(e5) kept (exact)
✓ Z(e5) kept (exact)
✓ X(e6) kept (exact)
✓ Z(e6) kept (exact)
✓ X(e7) kept (exact)
✓ Z(e7) kept (exact)
✓ X(e8) kept (exact)
✓ Z(e8) kept (exact)
✓ X(f0)·Z(d0) kept (exact)
✓ Z(f0) kept (exact)
✓ X(f1)·Z(d0) kept (exact)
✓ Z(f1) kept (exact)
✓ X(f2)·Z(d0) kept (exact)
✓ Z(f2) kept (exact)
✓ X(f3) kept (exact)
✓ Z(f3) kept (exact)
✓ X(f4) kept (exact)
✓ Z(f4) kept (exact)
✓ X(f5) kept (exact)
✓ Z(f5) kept (exact)
✓ X(f6) kept (exact)
✓ Z(f6) kept (exact)
✓ X(f7) kept (exact)
✓ Z(f7) kept (exact)
✓ X(f8) kept (exact)
✓ Z(f8) kept (exact)
✓ X(g0)·Z(d0) kept (exact)
✓ Z(g0) kept (exact)
✓ X(g1)·Z(d0) kept (exact)
✓ Z(g1) kept (exact)
✓ X(g2)·Z(d0) kept (exact)
✓ Z(g2) kept (exact)
✓ X(g3) kept (exact)
✓ Z(g3) kept (exact)
✓ X(g4) kept (exact)
✓ Z(g4) kept (exact)
✓ X(g5) kept (exact)
✓ Z(g5) kept (exact)
✓ X(g6) kept (exact)
✓ Z(g6) kept (exact)
✓ X(g7) kept (exact)
✓ Z(g7) kept (exact)
✓ X(g8) kept (exact)
✓ Z(g8) kept (exact)
✓ X̄_AZ̄_BZ̄_CZ̄_D measured (measure ⊕ 1 record)
✓ X̄_A kept (exact)
✓ Z̄_B kept (exact)
✓ Z̄_C kept (exact)
✓ Z̄_D kept (exact)
✓ Z̄_AX̄_B kept (exact)
✓ X̄_BX̄_C kept (exact)
✓ X̄_CX̄_D kept (exact)

a resident probe ion in |+⟩ takes a CX with each X̄-support ion and a CZ with each Z̄-support ion as the ring brings it past its dock, then is read in X: a controlled-P̄ measurement of the product, repeated 3 times
not fault-tolerant at circuit level (the user does not ask for circuit distance)

what was checked for ppm.ZXXX

✓ Z(d0) kept (exact)
✓ X(d1)·X(d0) kept (exact)
✓ Z(d1) kept (exact)
✓ X(d2)·X(d0) kept (exact)
✓ Z(d2) kept (exact)
✓ X(d3) kept (exact)
✓ Z(d3) kept (exact)
✓ X(d4) kept (exact)
✓ Z(d4) kept (exact)
✓ X(d5) kept (exact)
✓ Z(d5) kept (exact)
✓ X(d6) kept (exact)
✓ Z(d6) kept (exact)
✓ X(d7) kept (exact)
✓ Z(d7) kept (exact)
✓ X(d8) kept (exact)
✓ Z(d8) kept (exact)
✓ X(e0) kept (exact)
✓ Z(e0)·X(d0) kept (exact)
✓ X(e1) kept (exact)
✓ Z(e1) kept (exact)
✓ X(e2) kept (exact)
✓ Z(e2) kept (exact)
✓ X(e3) kept (exact)
✓ Z(e3)·X(d0) kept (exact)
✓ X(e4) kept (exact)
✓ Z(e4) kept (exact)
✓ X(e5) kept (exact)
✓ Z(e5) kept (exact)
✓ X(e6) kept (exact)
✓ Z(e6)·X(d0) kept (exact)
✓ X(e7) kept (exact)
✓ Z(e7) kept (exact)
✓ X(e8) kept (exact)
✓ Z(e8) kept (exact)
✓ X(f0) kept (exact)
✓ Z(f0)·X(d0) kept (exact)
✓ X(f1) kept (exact)
✓ Z(f1) kept (exact)
✓ X(f2) kept (exact)
✓ Z(f2) kept (exact)
✓ X(f3) kept (exact)
✓ Z(f3)·X(d0) kept (exact)
✓ X(f4) kept (exact)
✓ Z(f4) kept (exact)
✓ X(f5) kept (exact)
✓ Z(f5) kept (exact)
✓ X(f6) kept (exact)
✓ Z(f6)·X(d0) kept (exact)
✓ X(f7) kept (exact)
✓ Z(f7) kept (exact)
✓ X(f8) kept (exact)
✓ Z(f8) kept (exact)
✓ X(g0) kept (exact)
✓ Z(g0)·X(d0) kept (exact)
✓ X(g1) kept (exact)
✓ Z(g1) kept (exact)
✓ X(g2) kept (exact)
✓ Z(g2) kept (exact)
✓ X(g3) kept (exact)
✓ Z(g3)·X(d0) kept (exact)
✓ X(g4) kept (exact)
✓ Z(g4) kept (exact)
✓ X(g5) kept (exact)
✓ Z(g5) kept (exact)
✓ X(g6) kept (exact)
✓ Z(g6)·X(d0) kept (exact)
✓ X(g7) kept (exact)
✓ Z(g7) kept (exact)
✓ X(g8) kept (exact)
✓ Z(g8) kept (exact)
✓ Z̄_AX̄_BX̄_CX̄_D measured (measure ⊕ 1 record)
✓ Z̄_A kept (exact)
✓ X̄_B kept (exact)
✓ X̄_C kept (exact)
✓ X̄_D kept (exact)
✓ X̄_AZ̄_B kept (exact)
✓ Z̄_BZ̄_C kept (exact)
✓ Z̄_CZ̄_D kept (exact)

a resident probe ion in |+⟩ takes a CX with each X̄-support ion and a CZ with each Z̄-support ion as the ring brings it past its dock, then is read in X: a controlled-P̄ measurement of the product, repeated 3 times
not fault-tolerant at circuit level (the user does not ask for circuit distance)

what was checked for ppm.ZXXZ

✓ Z(d0) kept (exact)
✓ X(d1)·X(d0) kept (exact)
✓ Z(d1) kept (exact)
✓ X(d2)·X(d0) kept (exact)
✓ Z(d2) kept (exact)
✓ X(d3) kept (exact)
✓ Z(d3) kept (exact)
✓ X(d4) kept (exact)
✓ Z(d4) kept (exact)
✓ X(d5) kept (exact)
✓ Z(d5) kept (exact)
✓ X(d6) kept (exact)
✓ Z(d6) kept (exact)
✓ X(d7) kept (exact)
✓ Z(d7) kept (exact)
✓ X(d8) kept (exact)
✓ Z(d8) kept (exact)
✓ X(e0) kept (exact)
✓ Z(e0)·X(d0) kept (exact)
✓ X(e1) kept (exact)
✓ Z(e1) kept (exact)
✓ X(e2) kept (exact)
✓ Z(e2) kept (exact)
✓ X(e3) kept (exact)
✓ Z(e3)·X(d0) kept (exact)
✓ X(e4) kept (exact)
✓ Z(e4) kept (exact)
✓ X(e5) kept (exact)
✓ Z(e5) kept (exact)
✓ X(e6) kept (exact)
✓ Z(e6)·X(d0) kept (exact)
✓ X(e7) kept (exact)
✓ Z(e7) kept (exact)
✓ X(e8) kept (exact)
✓ Z(e8) kept (exact)
✓ X(f0) kept (exact)
✓ Z(f0)·X(d0) kept (exact)
✓ X(f1) kept (exact)
✓ Z(f1) kept (exact)
✓ X(f2) kept (exact)
✓ Z(f2) kept (exact)
✓ X(f3) kept (exact)
✓ Z(f3)·X(d0) kept (exact)
✓ X(f4) kept (exact)
✓ Z(f4) kept (exact)
✓ X(f5) kept (exact)
✓ Z(f5) kept (exact)
✓ X(f6) kept (exact)
✓ Z(f6)·X(d0) kept (exact)
✓ X(f7) kept (exact)
✓ Z(f7) kept (exact)
✓ X(f8) kept (exact)
✓ Z(f8) kept (exact)
✓ X(g0)·X(d0) kept (exact)
✓ Z(g0) kept (exact)
✓ X(g1)·X(d0) kept (exact)
✓ Z(g1) kept (exact)
✓ X(g2)·X(d0) kept (exact)
✓ Z(g2) kept (exact)
✓ X(g3) kept (exact)
✓ Z(g3) kept (exact)
✓ X(g4) kept (exact)
✓ Z(g4) kept (exact)
✓ X(g5) kept (exact)
✓ Z(g5) kept (exact)
✓ X(g6) kept (exact)
✓ Z(g6) kept (exact)
✓ X(g7) kept (exact)
✓ Z(g7) kept (exact)
✓ X(g8) kept (exact)
✓ Z(g8) kept (exact)
✓ Z̄_AX̄_BX̄_CZ̄_D measured (measure ⊕ 1 record)
✓ Z̄_A kept (exact)
✓ X̄_B kept (exact)
✓ X̄_C kept (exact)
✓ Z̄_D kept (exact)
✓ X̄_AZ̄_B kept (exact)
✓ Z̄_BZ̄_C kept (exact)
✓ Z̄_CX̄_D kept (exact)

a resident probe ion in |+⟩ takes a CX with each X̄-support ion and a CZ with each Z̄-support ion as the ring brings it past its dock, then is read in X: a controlled-P̄ measurement of the product, repeated 3 times
not fault-tolerant at circuit level (the user does not ask for circuit distance)

what was checked for ppm.ZXZX

✓ Z(d0) kept (exact)
✓ X(d1)·X(d0) kept (exact)
✓ Z(d1) kept (exact)
✓ X(d2)·X(d0) kept (exact)
✓ Z(d2) kept (exact)
✓ X(d3) kept (exact)
✓ Z(d3) kept (exact)
✓ X(d4) kept (exact)
✓ Z(d4) kept (exact)
✓ X(d5) kept (exact)
✓ Z(d5) kept (exact)
✓ X(d6) kept (exact)
✓ Z(d6) kept (exact)
✓ X(d7) kept (exact)
✓ Z(d7) kept (exact)
✓ X(d8) kept (exact)
✓ Z(d8) kept (exact)
✓ X(e0) kept (exact)
✓ Z(e0)·X(d0) kept (exact)
✓ X(e1) kept (exact)
✓ Z(e1) kept (exact)
✓ X(e2) kept (exact)
✓ Z(e2) kept (exact)
✓ X(e3) kept (exact)
✓ Z(e3)·X(d0) kept (exact)
✓ X(e4) kept (exact)
✓ Z(e4) kept (exact)
✓ X(e5) kept (exact)
✓ Z(e5) kept (exact)
✓ X(e6) kept (exact)
✓ Z(e6)·X(d0) kept (exact)
✓ X(e7) kept (exact)
✓ Z(e7) kept (exact)
✓ X(e8) kept (exact)
✓ Z(e8) kept (exact)
✓ X(f0)·X(d0) kept (exact)
✓ Z(f0) kept (exact)
✓ X(f1)·X(d0) kept (exact)
✓ Z(f1) kept (exact)
✓ X(f2)·X(d0) kept (exact)
✓ Z(f2) kept (exact)
✓ X(f3) kept (exact)
✓ Z(f3) kept (exact)
✓ X(f4) kept (exact)
✓ Z(f4) kept (exact)
✓ X(f5) kept (exact)
✓ Z(f5) kept (exact)
✓ X(f6) kept (exact)
✓ Z(f6) kept (exact)
✓ X(f7) kept (exact)
✓ Z(f7) kept (exact)
✓ X(f8) kept (exact)
✓ Z(f8) kept (exact)
✓ X(g0) kept (exact)
✓ Z(g0)·X(d0) kept (exact)
✓ X(g1) kept (exact)
✓ Z(g1) kept (exact)
✓ X(g2) kept (exact)
✓ Z(g2) kept (exact)
✓ X(g3) kept (exact)
✓ Z(g3)·X(d0) kept (exact)
✓ X(g4) kept (exact)
✓ Z(g4) kept (exact)
✓ X(g5) kept (exact)
✓ Z(g5) kept (exact)
✓ X(g6) kept (exact)
✓ Z(g6)·X(d0) kept (exact)
✓ X(g7) kept (exact)
✓ Z(g7) kept (exact)
✓ X(g8) kept (exact)
✓ Z(g8) kept (exact)
✓ Z̄_AX̄_BZ̄_CX̄_D measured (measure ⊕ 1 record)
✓ Z̄_A kept (exact)
✓ X̄_B kept (exact)
✓ Z̄_C kept (exact)
✓ X̄_D kept (exact)
✓ X̄_AZ̄_B kept (exact)
✓ Z̄_BX̄_C kept (exact)
✓ X̄_CZ̄_D kept (exact)

a resident probe ion in |+⟩ takes a CX with each X̄-support ion and a CZ with each Z̄-support ion as the ring brings it past its dock, then is read in X: a controlled-P̄ measurement of the product, repeated 3 times
not fault-tolerant at circuit level (the user does not ask for circuit distance)

what was checked for ppm.ZXZZ

The circuit read off the hardware program (3 R, 6 H, 9 CX, 27 CZ, 3 M) was run once on a Choi state — every ion entangled with a reference — with every measurement outcome kept symbolic. 79 stabilizer flows hold, and their rank 72 is the 72 a complete specification needs, so no other channel satisfies them.

✓ Z(d0) kept (exact)
✓ X(d1)·X(d0) kept (exact)
✓ Z(d1) kept (exact)
✓ X(d2)·X(d0) kept (exact)
✓ Z(d2) kept (exact)
✓ X(d3) kept (exact)
✓ Z(d3) kept (exact)
✓ X(d4) kept (exact)
✓ Z(d4) kept (exact)
✓ X(d5) kept (exact)
✓ Z(d5) kept (exact)
✓ X(d6) kept (exact)
✓ Z(d6) kept (exact)
✓ X(d7) kept (exact)
✓ Z(d7) kept (exact)
✓ X(d8) kept (exact)
✓ Z(d8) kept (exact)
✓ X(e0) kept (exact)
✓ Z(e0)·X(d0) kept (exact)
✓ X(e1) kept (exact)
✓ Z(e1) kept (exact)
✓ X(e2) kept (exact)
✓ Z(e2) kept (exact)
✓ X(e3) kept (exact)
✓ Z(e3)·X(d0) kept (exact)
✓ X(e4) kept (exact)
✓ Z(e4) kept (exact)
✓ X(e5) kept (exact)
✓ Z(e5) kept (exact)
✓ X(e6) kept (exact)
✓ Z(e6)·X(d0) kept (exact)
✓ X(e7) kept (exact)
✓ Z(e7) kept (exact)
✓ X(e8) kept (exact)
✓ Z(e8) kept (exact)
✓ X(f0)·X(d0) kept (exact)
✓ Z(f0) kept (exact)
✓ X(f1)·X(d0) kept (exact)
✓ Z(f1) kept (exact)
✓ X(f2)·X(d0) kept (exact)
✓ Z(f2) kept (exact)
✓ X(f3) kept (exact)
✓ Z(f3) kept (exact)
✓ X(f4) kept (exact)
✓ Z(f4) kept (exact)
✓ X(f5) kept (exact)
✓ Z(f5) kept (exact)
✓ X(f6) kept (exact)
✓ Z(f6) kept (exact)
✓ X(f7) kept (exact)
✓ Z(f7) kept (exact)
✓ X(f8) kept (exact)
✓ Z(f8) kept (exact)
✓ X(g0)·X(d0) kept (exact)
✓ Z(g0) kept (exact)
✓ X(g1)·X(d0) kept (exact)
✓ Z(g1) kept (exact)
✓ X(g2)·X(d0) kept (exact)
✓ Z(g2) kept (exact)
✓ X(g3) kept (exact)
✓ Z(g3) kept (exact)
✓ X(g4) kept (exact)
✓ Z(g4) kept (exact)
✓ X(g5) kept (exact)
✓ Z(g5) kept (exact)
✓ X(g6) kept (exact)
✓ Z(g6) kept (exact)
✓ X(g7) kept (exact)
✓ Z(g7) kept (exact)
✓ X(g8) kept (exact)
✓ Z(g8) kept (exact)
✓ Z̄_AX̄_BZ̄_CZ̄_D measured (measure ⊕ 1 record)
✓ Z̄_A kept (exact)
✓ X̄_B kept (exact)
✓ Z̄_C kept (exact)
✓ Z̄_D kept (exact)
✓ X̄_AZ̄_B kept (exact)
✓ Z̄_BX̄_C kept (exact)
✓ X̄_CX̄_D kept (exact)

a resident probe ion in |+⟩ takes a CX with each X̄-support ion and a CZ with each Z̄-support ion as the ring brings it past its dock, then is read in X: a controlled-P̄ measurement of the product, repeated 3 times
not fault-tolerant at circuit level (the user does not ask for circuit distance)

what was checked for ppm.ZZXX

✓ Z(d0) kept (exact)
✓ X(d1)·X(d0) kept (exact)
✓ Z(d1) kept (exact)
✓ X(d2)·X(d0) kept (exact)
✓ Z(d2) kept (exact)
✓ X(d3) kept (exact)
✓ Z(d3) kept (exact)
✓ X(d4) kept (exact)
✓ Z(d4) kept (exact)
✓ X(d5) kept (exact)
✓ Z(d5) kept (exact)
✓ X(d6) kept (exact)
✓ Z(d6) kept (exact)
✓ X(d7) kept (exact)
✓ Z(d7) kept (exact)
✓ X(d8) kept (exact)
✓ Z(d8) kept (exact)
✓ X(e0)·X(d0) kept (exact)
✓ Z(e0) kept (exact)
✓ X(e1)·X(d0) kept (exact)
✓ Z(e1) kept (exact)
✓ X(e2)·X(d0) kept (exact)
✓ Z(e2) kept (exact)
✓ X(e3) kept (exact)
✓ Z(e3) kept (exact)
✓ X(e4) kept (exact)
✓ Z(e4) kept (exact)
✓ X(e5) kept (exact)
✓ Z(e5) kept (exact)
✓ X(e6) kept (exact)
✓ Z(e6) kept (exact)
✓ X(e7) kept (exact)
✓ Z(e7) kept (exact)
✓ X(e8) kept (exact)
✓ Z(e8) kept (exact)
✓ X(f0) kept (exact)
✓ Z(f0)·X(d0) kept (exact)
✓ X(f1) kept (exact)
✓ Z(f1) kept (exact)
✓ X(f2) kept (exact)
✓ Z(f2) kept (exact)
✓ X(f3) kept (exact)
✓ Z(f3)·X(d0) kept (exact)
✓ X(f4) kept (exact)
✓ Z(f4) kept (exact)
✓ X(f5) kept (exact)
✓ Z(f5) kept (exact)
✓ X(f6) kept (exact)
✓ Z(f6)·X(d0) kept (exact)
✓ X(f7) kept (exact)
✓ Z(f7) kept (exact)
✓ X(f8) kept (exact)
✓ Z(f8) kept (exact)
✓ X(g0) kept (exact)
✓ Z(g0)·X(d0) kept (exact)
✓ X(g1) kept (exact)
✓ Z(g1) kept (exact)
✓ X(g2) kept (exact)
✓ Z(g2) kept (exact)
✓ X(g3) kept (exact)
✓ Z(g3)·X(d0) kept (exact)
✓ X(g4) kept (exact)
✓ Z(g4) kept (exact)
✓ X(g5) kept (exact)
✓ Z(g5) kept (exact)
✓ X(g6) kept (exact)
✓ Z(g6)·X(d0) kept (exact)
✓ X(g7) kept (exact)
✓ Z(g7) kept (exact)
✓ X(g8) kept (exact)
✓ Z(g8) kept (exact)
✓ Z̄_AZ̄_BX̄_CX̄_D measured (measure ⊕ 1 record)
✓ Z̄_A kept (exact)
✓ Z̄_B kept (exact)
✓ X̄_C kept (exact)
✓ X̄_D kept (exact)
✓ X̄_AX̄_B kept (exact)
✓ X̄_BZ̄_C kept (exact)
✓ Z̄_CZ̄_D kept (exact)

a resident probe ion in |+⟩ takes a CX with each X̄-support ion and a CZ with each Z̄-support ion as the ring brings it past its dock, then is read in X: a controlled-P̄ measurement of the product, repeated 3 times
not fault-tolerant at circuit level (the user does not ask for circuit distance)

what was checked for ppm.ZZXZ

✓ Z(d0) kept (exact)
✓ X(d1)·X(d0) kept (exact)
✓ Z(d1) kept (exact)
✓ X(d2)·X(d0) kept (exact)
✓ Z(d2) kept (exact)
✓ X(d3) kept (exact)
✓ Z(d3) kept (exact)
✓ X(d4) kept (exact)
✓ Z(d4) kept (exact)
✓ X(d5) kept (exact)
✓ Z(d5) kept (exact)
✓ X(d6) kept (exact)
✓ Z(d6) kept (exact)
✓ X(d7) kept (exact)
✓ Z(d7) kept (exact)
✓ X(d8) kept (exact)
✓ Z(d8) kept (exact)
✓ X(e0)·X(d0) kept (exact)
✓ Z(e0) kept (exact)
✓ X(e1)·X(d0) kept (exact)
✓ Z(e1) kept (exact)
✓ X(e2)·X(d0) kept (exact)
✓ Z(e2) kept (exact)
✓ X(e3) kept (exact)
✓ Z(e3) kept (exact)
✓ X(e4) kept (exact)
✓ Z(e4) kept (exact)
✓ X(e5) kept (exact)
✓ Z(e5) kept (exact)
✓ X(e6) kept (exact)
✓ Z(e6) kept (exact)
✓ X(e7) kept (exact)
✓ Z(e7) kept (exact)
✓ X(e8) kept (exact)
✓ Z(e8) kept (exact)
✓ X(f0) kept (exact)
✓ Z(f0)·X(d0) kept (exact)
✓ X(f1) kept (exact)
✓ Z(f1) kept (exact)
✓ X(f2) kept (exact)
✓ Z(f2) kept (exact)
✓ X(f3) kept (exact)
✓ Z(f3)·X(d0) kept (exact)
✓ X(f4) kept (exact)
✓ Z(f4) kept (exact)
✓ X(f5) kept (exact)
✓ Z(f5) kept (exact)
✓ X(f6) kept (exact)
✓ Z(f6)·X(d0) kept (exact)
✓ X(f7) kept (exact)
✓ Z(f7) kept (exact)
✓ X(f8) kept (exact)
✓ Z(f8) kept (exact)
✓ X(g0)·X(d0) kept (exact)
✓ Z(g0) kept (exact)
✓ X(g1)·X(d0) kept (exact)
✓ Z(g1) kept (exact)
✓ X(g2)·X(d0) kept (exact)
✓ Z(g2) kept (exact)
✓ X(g3) kept (exact)
✓ Z(g3) kept (exact)
✓ X(g4) kept (exact)
✓ Z(g4) kept (exact)
✓ X(g5) kept (exact)
✓ Z(g5) kept (exact)
✓ X(g6) kept (exact)
✓ Z(g6) kept (exact)
✓ X(g7) kept (exact)
✓ Z(g7) kept (exact)
✓ X(g8) kept (exact)
✓ Z(g8) kept (exact)
✓ Z̄_AZ̄_BX̄_CZ̄_D measured (measure ⊕ 1 record)
✓ Z̄_A kept (exact)
✓ Z̄_B kept (exact)
✓ X̄_C kept (exact)
✓ Z̄_D kept (exact)
✓ X̄_AX̄_B kept (exact)
✓ X̄_BZ̄_C kept (exact)
✓ Z̄_CX̄_D kept (exact)

a resident probe ion in |+⟩ takes a CX with each X̄-support ion and a CZ with each Z̄-support ion as the ring brings it past its dock, then is read in X: a controlled-P̄ measurement of the product, repeated 3 times
not fault-tolerant at circuit level (the user does not ask for circuit distance)

what was checked for ppm.ZZZX

✓ Z(d0) kept (exact)
✓ X(d1)·X(d0) kept (exact)
✓ Z(d1) kept (exact)
✓ X(d2)·X(d0) kept (exact)
✓ Z(d2) kept (exact)
✓ X(d3) kept (exact)
✓ Z(d3) kept (exact)
✓ X(d4) kept (exact)
✓ Z(d4) kept (exact)
✓ X(d5) kept (exact)
✓ Z(d5) kept (exact)
✓ X(d6) kept (exact)
✓ Z(d6) kept (exact)
✓ X(d7) kept (exact)
✓ Z(d7) kept (exact)
✓ X(d8) kept (exact)
✓ Z(d8) kept (exact)
✓ X(e0)·X(d0) kept (exact)
✓ Z(e0) kept (exact)
✓ X(e1)·X(d0) kept (exact)
✓ Z(e1) kept (exact)
✓ X(e2)·X(d0) kept (exact)
✓ Z(e2) kept (exact)
✓ X(e3) kept (exact)
✓ Z(e3) kept (exact)
✓ X(e4) kept (exact)
✓ Z(e4) kept (exact)
✓ X(e5) kept (exact)
✓ Z(e5) kept (exact)
✓ X(e6) kept (exact)
✓ Z(e6) kept (exact)
✓ X(e7) kept (exact)
✓ Z(e7) kept (exact)
✓ X(e8) kept (exact)
✓ Z(e8) kept (exact)
✓ X(f0)·X(d0) kept (exact)
✓ Z(f0) kept (exact)
✓ X(f1)·X(d0) kept (exact)
✓ Z(f1) kept (exact)
✓ X(f2)·X(d0) kept (exact)
✓ Z(f2) kept (exact)
✓ X(f3) kept (exact)
✓ Z(f3) kept (exact)
✓ X(f4) kept (exact)
✓ Z(f4) kept (exact)
✓ X(f5) kept (exact)
✓ Z(f5) kept (exact)
✓ X(f6) kept (exact)
✓ Z(f6) kept (exact)
✓ X(f7) kept (exact)
✓ Z(f7) kept (exact)
✓ X(f8) kept (exact)
✓ Z(f8) kept (exact)
✓ X(g0) kept (exact)
✓ Z(g0)·X(d0) kept (exact)
✓ X(g1) kept (exact)
✓ Z(g1) kept (exact)
✓ X(g2) kept (exact)
✓ Z(g2) kept (exact)
✓ X(g3) kept (exact)
✓ Z(g3)·X(d0) kept (exact)
✓ X(g4) kept (exact)
✓ Z(g4) kept (exact)
✓ X(g5) kept (exact)
✓ Z(g5) kept (exact)
✓ X(g6) kept (exact)
✓ Z(g6)·X(d0) kept (exact)
✓ X(g7) kept (exact)
✓ Z(g7) kept (exact)
✓ X(g8) kept (exact)
✓ Z(g8) kept (exact)
✓ Z̄_AZ̄_BZ̄_CX̄_D measured (measure ⊕ 1 record)
✓ Z̄_A kept (exact)
✓ Z̄_B kept (exact)
✓ Z̄_C kept (exact)
✓ X̄_D kept (exact)
✓ X̄_AX̄_B kept (exact)
✓ X̄_BX̄_C kept (exact)
✓ X̄_CZ̄_D kept (exact)

a resident probe ion in |+⟩ takes a CX with each X̄-support ion and a CZ with each Z̄-support ion as the ring brings it past its dock, then is read in X: a controlled-P̄ measurement of the product, repeated 3 times
not fault-tolerant at circuit level (the user does not ask for circuit distance)

what was checked for ppm.ZZZZ

The circuit read off the hardware program (3 R, 6 H, 36 CZ, 3 M) was run once on a Choi state — every ion entangled with a reference — with every measurement outcome kept symbolic. 79 stabilizer flows hold, and their rank 72 is the 72 a complete specification needs, so no other channel satisfies them.

✓ Z(d0) kept (exact)
✓ X(d1)·X(d0) kept (exact)
✓ Z(d1) kept (exact)
✓ X(d2)·X(d0) kept (exact)
✓ Z(d2) kept (exact)
✓ X(d3) kept (exact)
✓ Z(d3) kept (exact)
✓ X(d4) kept (exact)
✓ Z(d4) kept (exact)
✓ X(d5) kept (exact)
✓ Z(d5) kept (exact)
✓ X(d6) kept (exact)
✓ Z(d6) kept (exact)
✓ X(d7) kept (exact)
✓ Z(d7) kept (exact)
✓ X(d8) kept (exact)
✓ Z(d8) kept (exact)
✓ X(e0)·X(d0) kept (exact)
✓ Z(e0) kept (exact)
✓ X(e1)·X(d0) kept (exact)
✓ Z(e1) kept (exact)
✓ X(e2)·X(d0) kept (exact)
✓ Z(e2) kept (exact)
✓ X(e3) kept (exact)
✓ Z(e3) kept (exact)
✓ X(e4) kept (exact)
✓ Z(e4) kept (exact)
✓ X(e5) kept (exact)
✓ Z(e5) kept (exact)
✓ X(e6) kept (exact)
✓ Z(e6) kept (exact)
✓ X(e7) kept (exact)
✓ Z(e7) kept (exact)
✓ X(e8) kept (exact)
✓ Z(e8) kept (exact)
✓ X(f0)·X(d0) kept (exact)
✓ Z(f0) kept (exact)
✓ X(f1)·X(d0) kept (exact)
✓ Z(f1) kept (exact)
✓ X(f2)·X(d0) kept (exact)
✓ Z(f2) kept (exact)
✓ X(f3) kept (exact)
✓ Z(f3) kept (exact)
✓ X(f4) kept (exact)
✓ Z(f4) kept (exact)
✓ X(f5) kept (exact)
✓ Z(f5) kept (exact)
✓ X(f6) kept (exact)
✓ Z(f6) kept (exact)
✓ X(f7) kept (exact)
✓ Z(f7) kept (exact)
✓ X(f8) kept (exact)
✓ Z(f8) kept (exact)
✓ X(g0)·X(d0) kept (exact)
✓ Z(g0) kept (exact)
✓ X(g1)·X(d0) kept (exact)
✓ Z(g1) kept (exact)
✓ X(g2)·X(d0) kept (exact)
✓ Z(g2) kept (exact)
✓ X(g3) kept (exact)
✓ Z(g3) kept (exact)
✓ X(g4) kept (exact)
✓ Z(g4) kept (exact)
✓ X(g5) kept (exact)
✓ Z(g5) kept (exact)
✓ X(g6) kept (exact)
✓ Z(g6) kept (exact)
✓ X(g7) kept (exact)
✓ Z(g7) kept (exact)
✓ X(g8) kept (exact)
✓ Z(g8) kept (exact)
✓ Z̄_AZ̄_BZ̄_CZ̄_D measured (measure ⊕ 1 record)
✓ Z̄_A kept (exact)
✓ Z̄_B kept (exact)
✓ Z̄_C kept (exact)
✓ Z̄_D kept (exact)
✓ X̄_AX̄_B kept (exact)
✓ X̄_BX̄_C kept (exact)
✓ X̄_CX̄_D kept (exact)

a resident probe ion in |+⟩ takes a CX with each X̄-support ion and a CZ with each Z̄-support ion as the ring brings it past its dock, then is read in X: a controlled-P̄ measurement of the product, repeated 3 times
not fault-tolerant at circuit level (the user does not ask for circuit distance)

Logical Readout

the post office

A block comes in and every data ion is measured in one basis; the logical outcome is the parity of a logical operator's support. The ions go back to a depot.

Logical readout (d=3) · `read_d3`

four measurement docks on a 12-slot ring; a block's ions are read in parallel batches and leave for a depot

op	hardware	logic	fault distance	time
`read.Z` measure every data ion in Z; Z̄ is the parity of a row	✓ 25 rules	✓ 10/10 flows, rank 9/9	✓ d ≥ 3 (9 faults)	10.8 ms 0 two-qubit	watch · studio
`read.X` measure every data ion in X; X̄ is the parity of a column	✓ 25 rules	✓ 10/10 flows, rank 9/9	✓ d ≥ 3 (36 faults)	10.8 ms 0 two-qubit	watch · studio

what was checked for read.Z

The circuit read off the hardware program (9 M) was run once on a Choi state — every ion entangled with a reference — with every measurement outcome kept symbolic. 10 stabilizer flows hold, and their rank 9 is the 9 a complete specification needs, so no other channel satisfies them.

✓ Z(d0) measured (measure ⊕ 1 record)
✓ Z(d1) measured (measure ⊕ 1 record)
✓ Z(d2) measured (measure ⊕ 1 record)
✓ Z(d3) measured (measure ⊕ 1 record)
✓ Z(d4) measured (measure ⊕ 1 record)
✓ Z(d5) measured (measure ⊕ 1 record)
✓ Z(d6) measured (measure ⊕ 1 record)
✓ Z(d7) measured (measure ⊕ 1 record)
✓ Z(d8) measured (measure ⊕ 1 record)
✓ Z̄ measured (measure ⊕ 3 records)

Fault distance: the op between ideal encoders and decoders, 9 fault locations, 4 detectors, 1 observables — no single fault and no pair of faults flips a logical observable without a detector firing.

what was checked for read.X

The circuit read off the hardware program (9 H, 9 M) was run once on a Choi state — every ion entangled with a reference — with every measurement outcome kept symbolic. 10 stabilizer flows hold, and their rank 9 is the 9 a complete specification needs, so no other channel satisfies them.

✓ X(d0) measured (measure ⊕ 1 record)
✓ X(d1) measured (measure ⊕ 1 record)
✓ X(d2) measured (measure ⊕ 1 record)
✓ X(d3) measured (measure ⊕ 1 record)
✓ X(d4) measured (measure ⊕ 1 record)
✓ X(d5) measured (measure ⊕ 1 record)
✓ X(d6) measured (measure ⊕ 1 record)
✓ X(d7) measured (measure ⊕ 1 record)
✓ X(d8) measured (measure ⊕ 1 record)
✓ X̄ measured (measure ⊕ 3 records)

Fault distance: the op between ideal encoders and decoders, 36 fault locations, 4 detectors, 1 observables — no single fault and no pair of faults flips a logical observable without a detector firing.

Magic State Factory

the factory

Fifteen noisy T gates in, one better T state out: the 15-to-1 Reed-Muller protocol on resident ions.

15-to-1 magic-state factory · `msf15`

14 resident ions on a 20-slot ring with one gate dock; a visiting ion becomes the output of each run

op	hardware	logic	fault distance	time
`produce` 15-to-1 distillation; the visitor leaves holding the output state	✓ 25 rules	✓ state vector: T†\|+⟩ exact	✓ 1–2 T errors caught; 35 triples: 35p³	194.7 ms 66 two-qubit	watch · studio

what was checked for produce

carries out rm15.circ operation by operation (119 operations, each naming its line); accepts with probability 1 and outputs T†|+⟩ exactly; every 1 or 2 T-gate Z errors detected; 35 undetected triples, all logical: output error 35p³.

66 CX, 15 T, 14 syndrome measurements at one dock trap, gate by gate
checked: with ideal gates the 14 outcomes are all 0 and the visitor holds T†|+⟩; a Z error after any 1 or 2 of the 15 T gates always shows in the outcomes, and exactly 35 of the 455 triples do not (output error 35p³)
not checked: errors on the Clifford gates, which this protocol does not protect against (distillation assumes Cliffords much better than T)

Rotation Source

the lens shop

A fresh ion comes in and leaves in a small rotation state, RZn|+⟩; a pharmacy encodes it, and a Pauli-product measurement against that block turns the program's blocks by half the angle -- rotations finer than T, repeated until they succeed.

Rotation source · `rot1`

a fresh ion comes in, is put in a small rotation state RZn|+⟩ at the dock, and leaves for a pharmacy to encode

op	hardware	logic	fault distance	time
`rz3` one ion in RZ3\|+⟩	✓ 25 rules	✓ state vector: RZ3\|+⟩ exact	by design one ion, not encoded	1.1 ms 0 two-qubit
`rz3dg` one ion in RZ3†\|+⟩	✓ 25 rules	✓ state vector: RZ3†\|+⟩ exact	by design one ion, not encoded	1.1 ms 0 two-qubit
`rz4` one ion in RZ4\|+⟩	✓ 25 rules	✓ state vector: RZ4\|+⟩ exact	by design one ion, not encoded	1.1 ms 0 two-qubit
`rz4dg` one ion in RZ4†\|+⟩	✓ 25 rules	✓ state vector: RZ4†\|+⟩ exact	by design one ion, not encoded	1.1 ms 0 two-qubit
`rz5` one ion in RZ5\|+⟩	✓ 25 rules	✓ state vector: RZ5\|+⟩ exact	by design one ion, not encoded	1.1 ms 0 two-qubit
`rz5dg` one ion in RZ5†\|+⟩	✓ 25 rules	✓ state vector: RZ5†\|+⟩ exact	by design one ion, not encoded	1.1 ms 0 two-qubit
`rz6` one ion in RZ6\|+⟩	✓ 25 rules	✓ state vector: RZ6\|+⟩ exact	by design one ion, not encoded	1.1 ms 0 two-qubit
`rz6dg` one ion in RZ6†\|+⟩	✓ 25 rules	✓ state vector: RZ6†\|+⟩ exact	by design one ion, not encoded	1.1 ms 0 two-qubit
`rz7` one ion in RZ7\|+⟩	✓ 25 rules	✓ state vector: RZ7\|+⟩ exact	by design one ion, not encoded	1.1 ms 0 two-qubit
`rz7dg` one ion in RZ7†\|+⟩	✓ 25 rules	✓ state vector: RZ7†\|+⟩ exact	by design one ion, not encoded	1.1 ms 0 two-qubit
`rz8` one ion in RZ8\|+⟩	✓ 25 rules	✓ state vector: RZ8\|+⟩ exact	by design one ion, not encoded	1.1 ms 0 two-qubit
`rz8dg` one ion in RZ8†\|+⟩	✓ 25 rules	✓ state vector: RZ8†\|+⟩ exact	by design one ion, not encoded	1.1 ms 0 two-qubit
`rz9` one ion in RZ9\|+⟩	✓ 25 rules	✓ state vector: RZ9\|+⟩ exact	by design one ion, not encoded	1.1 ms 0 two-qubit
`rz9dg` one ion in RZ9†\|+⟩	✓ 25 rules	✓ state vector: RZ9†\|+⟩ exact	by design one ion, not encoded	1.1 ms 0 two-qubit
`rz10` one ion in RZ10\|+⟩	✓ 25 rules	✓ state vector: RZ10\|+⟩ exact	by design one ion, not encoded	1.1 ms 0 two-qubit
`rz10dg` one ion in RZ10†\|+⟩	✓ 25 rules	✓ state vector: RZ10†\|+⟩ exact	by design one ion, not encoded	1.1 ms 0 two-qubit

what was checked for rz3

the ion that leaves holds RZ3|+⟩ exactly.

reset, H and RZ3 on one ion at the dock: it leaves holding (|0⟩ + e^{iπ/8}|1⟩)/√2, the resource of a Pauli-product rotation by π/16
checked as a state vector; not fault-tolerant (an injection, distance 1)

what was checked for rz3dg

the ion that leaves holds RZ3†|+⟩ exactly.

reset, H and RZ3_DAG on one ion at the dock: it leaves holding (|0⟩ + e^{-iπ/8}|1⟩)/√2, the resource of a Pauli-product rotation by -π/16
checked as a state vector; not fault-tolerant (an injection, distance 1)

what was checked for rz4

the ion that leaves holds RZ4|+⟩ exactly.

reset, H and RZ4 on one ion at the dock: it leaves holding (|0⟩ + e^{iπ/16}|1⟩)/√2, the resource of a Pauli-product rotation by π/32
checked as a state vector; not fault-tolerant (an injection, distance 1)

what was checked for rz4dg

the ion that leaves holds RZ4†|+⟩ exactly.

reset, H and RZ4_DAG on one ion at the dock: it leaves holding (|0⟩ + e^{-iπ/16}|1⟩)/√2, the resource of a Pauli-product rotation by -π/32
checked as a state vector; not fault-tolerant (an injection, distance 1)

what was checked for rz5

the ion that leaves holds RZ5|+⟩ exactly.

reset, H and RZ5 on one ion at the dock: it leaves holding (|0⟩ + e^{iπ/32}|1⟩)/√2, the resource of a Pauli-product rotation by π/64
checked as a state vector; not fault-tolerant (an injection, distance 1)

what was checked for rz5dg

the ion that leaves holds RZ5†|+⟩ exactly.

reset, H and RZ5_DAG on one ion at the dock: it leaves holding (|0⟩ + e^{-iπ/32}|1⟩)/√2, the resource of a Pauli-product rotation by -π/64
checked as a state vector; not fault-tolerant (an injection, distance 1)

what was checked for rz6

the ion that leaves holds RZ6|+⟩ exactly.

reset, H and RZ6 on one ion at the dock: it leaves holding (|0⟩ + e^{iπ/64}|1⟩)/√2, the resource of a Pauli-product rotation by π/128
checked as a state vector; not fault-tolerant (an injection, distance 1)

what was checked for rz6dg

the ion that leaves holds RZ6†|+⟩ exactly.

reset, H and RZ6_DAG on one ion at the dock: it leaves holding (|0⟩ + e^{-iπ/64}|1⟩)/√2, the resource of a Pauli-product rotation by -π/128
checked as a state vector; not fault-tolerant (an injection, distance 1)

what was checked for rz7

the ion that leaves holds RZ7|+⟩ exactly.

reset, H and RZ7 on one ion at the dock: it leaves holding (|0⟩ + e^{iπ/128}|1⟩)/√2, the resource of a Pauli-product rotation by π/256
checked as a state vector; not fault-tolerant (an injection, distance 1)

what was checked for rz7dg

the ion that leaves holds RZ7†|+⟩ exactly.

reset, H and RZ7_DAG on one ion at the dock: it leaves holding (|0⟩ + e^{-iπ/128}|1⟩)/√2, the resource of a Pauli-product rotation by -π/256
checked as a state vector; not fault-tolerant (an injection, distance 1)

what was checked for rz8

the ion that leaves holds RZ8|+⟩ exactly.

reset, H and RZ8 on one ion at the dock: it leaves holding (|0⟩ + e^{iπ/256}|1⟩)/√2, the resource of a Pauli-product rotation by π/512
checked as a state vector; not fault-tolerant (an injection, distance 1)

what was checked for rz8dg

the ion that leaves holds RZ8†|+⟩ exactly.

reset, H and RZ8_DAG on one ion at the dock: it leaves holding (|0⟩ + e^{-iπ/256}|1⟩)/√2, the resource of a Pauli-product rotation by -π/512
checked as a state vector; not fault-tolerant (an injection, distance 1)

what was checked for rz9

the ion that leaves holds RZ9|+⟩ exactly.

reset, H and RZ9 on one ion at the dock: it leaves holding (|0⟩ + e^{iπ/512}|1⟩)/√2, the resource of a Pauli-product rotation by π/1024
checked as a state vector; not fault-tolerant (an injection, distance 1)

what was checked for rz9dg

the ion that leaves holds RZ9†|+⟩ exactly.

reset, H and RZ9_DAG on one ion at the dock: it leaves holding (|0⟩ + e^{-iπ/512}|1⟩)/√2, the resource of a Pauli-product rotation by -π/1024
checked as a state vector; not fault-tolerant (an injection, distance 1)

what was checked for rz10

the ion that leaves holds RZ10|+⟩ exactly.

reset, H and RZ10 on one ion at the dock: it leaves holding (|0⟩ + e^{iπ/1024}|1⟩)/√2, the resource of a Pauli-product rotation by π/2048
checked as a state vector; not fault-tolerant (an injection, distance 1)

what was checked for rz10dg

the ion that leaves holds RZ10†|+⟩ exactly.

reset, H and RZ10_DAG on one ion at the dock: it leaves holding (|0⟩ + e^{-iπ/1024}|1⟩)/√2, the resource of a Pauli-product rotation by -π/2048
checked as a state vector; not fault-tolerant (an injection, distance 1)

Magic State Injection

the pharmacy

A physical magic state and fresh ions come in; a logical block holding that state goes out. Not fault-tolerant by design: distillation is what suppresses its errors.

Magic-state injection (d=3) · `inj_d3`

the magic ion and 8 fresh ions ride past 8 resident ancillas for one round and leave as a code block

op	hardware	logic	fault distance	time
`inject` encode the magic ion's state into a fresh block	✓ 25 rules	✓ 10/10 flows, rank 10/10	by design d = 1	30.7 ms 24 two-qubit	watch · studio
`inject.Y` prepare \|Ȳ⟩ by injecting S\|+⟩ on the corner ion	✓ 25 rules	✓ 9/9 flows, rank 9/9	by design d = 1	30.7 ms 24 two-qubit	watch · studio

what was checked for inject

The circuit read off the hardware program (16 R, 10 H, 24 CX, 8 M) was run once on a Choi state — every ion entangled with a reference — with every measurement outcome kept symbolic. 10 stabilizer flows hold, and their rank 10 is the 10 a complete specification needs, so no other channel satisfies them.

✓ X(m) → X̄ (frame)
✓ Z(m) → Z̄ (frame)
✓ X0 holds (frame ⊕ 1 record)
✓ X1 holds (frame ⊕ 1 record)
✓ X2 holds (frame ⊕ 1 record)
✓ X3 holds (frame ⊕ 1 record)
✓ Z0 holds (frame ⊕ 1 record)
✓ Z1 holds (frame)
✓ Z2 holds (frame ⊕ 1 record)
✓ Z3 holds (frame)

Fault distance: the op between ideal encoders and decoders, 414 fault locations, 10 detectors, 2 observables — 1 fault(s) suffice: instruction 34: reset leaves d6 in |1>. injection is not fault-tolerant: its logical error rate is O(p), removed by distillation downstream.

the magic ion becomes code qubit 0; d3, d6 start in |+⟩ and the rest of the fresh ions in |0⟩ (qubit 0 is on both Z̄ = row 0 and X̄ = column 0)
not fault-tolerant: one fault on the magic ion before the round changes the encoded state undetectably, which is why injected states are distilled

what was checked for inject.Y

The circuit read off the hardware program (17 R, 11 H, 1 S, 24 CX, 8 M) was run once on a Choi state — every ion entangled with a reference — with every measurement outcome kept symbolic. 9 stabilizer flows hold, and their rank 9 is the 9 a complete specification needs, so no other channel satisfies them.

✓ Ȳ = +1 (frame)
✓ X0 holds (frame ⊕ 1 record)
✓ X1 holds (frame ⊕ 1 record)
✓ X2 holds (frame ⊕ 1 record)
✓ X3 holds (frame ⊕ 1 record)
✓ Z0 holds (frame ⊕ 1 record)
✓ Z1 holds (frame)
✓ Z2 holds (frame ⊕ 1 record)
✓ Z3 holds (frame)

Fault distance: the op between ideal encoders and decoders, 421 fault locations, 10 detectors, 1 observables — 1 fault(s) suffice: instruction 34: reset leaves d6 in |1>. a state injection, fault-tolerant to distance 1 by design: the |Ȳ⟩ it prepares carries an O(p) logical error, as every injection does.

the corner ion is reset, H and S here becomes code qubit 0; d3, d6 start in |+⟩ and the rest of the fresh ions in |0⟩ (qubit 0 is on both Z̄ = row 0 and X̄ = column 0)
not fault-tolerant: one fault on the magic ion before the round changes the encoded state undetectably, which is why injected states are distilled
this is the resource of the logical S̄ gadget: measure Z̄Z̄ between the data block and this one by lattice surgery, read this one out in X, and the correction is Z̄ when the two outcomes disagree -- a Pauli frame, kept in the classical memory

Decoder

the control room

The classical processor (FPGA or GPU) wired to every place that measures: syndrome bits come in, and after a stated latency it says which correction to keep -- an update to a block's Pauli frame, not a pulse on an ion.

Lookup decoder (surface code, d = 3) · `dec_d3`

a classical processor wired to every place that measures: syndrome bits in, an update to a block's Pauli frame out, after a stated latency

op	hardware	logic	fault distance	time
`decode` decode one window and emit a frame update	modeled latency from the literature	✓ 136 entries, 16 syndrome bits	✓ corrects every one of the 400 single faults	1 µs	watch

what was checked for decode

a 16-bit lookup table with 136 entries corrects every one of the 400 single faults: each syndrome has one logical effect.

one table read; a distance-3 surface-code window is a 16-bit syndrome with 136 entries (qccd.gadget.logic.decode builds and verifies it); sub-µs table read; an FPGA belief-propagation iteration is 24 ns (IBM real-time decoder, arXiv:2510.21600)
the wire from a measuring place charges 5.0 µs: measurement result to a conditional branch on the control system: 5 µs on a trapped-ion QCCD machine (AQT M-ACTION, arXiv:2101.11390)
the correction is a Pauli-frame update, never a pulse: it is written into the classical memory place, which the sign-off then reads as a declared frame

Classical Memory

the archive

Holds the classical state of the computation: each block's Pauli frame and Clifford frame, and every logical outcome. Corrections kept in software are kept HERE, and a conditional operation reads its guard from here.

Classical memory (frames and outcomes) · `arch1`

holds every logical block's Pauli frame and Clifford frame and every logical outcome: the corrections this machine keeps in software rather than applying to ions

op	hardware	logic	time
`record` write one logical outcome	modeled latency from the literature	no quantum content	0.1 µs	watch
`update` apply a decoder's frame update	modeled latency from the literature	no quantum content	0.1 µs	watch
`resolve` read the frame back to answer one question	modeled latency from the literature	no quantum content	0.2 µs	watch

what was checked for record

the outcome is kept as the parity of the records the measuring gadget's own flow report names for it -- the archive stores what was measured, not a guess
writing one Pauli-frame update into the classical memory

what was checked for update

px ^= a, pz ^= b on one block: the correction is applied to the frame, and the ions are left alone (Pauli frame tracking)
writing one Pauli-frame update into the classical memory

what was checked for resolve

the answer travels to the place that waits for it on the decision wire (2.0 µs: the guard travelling from the memory to the place that waits for it, on the decision wire)
reading the frame back to answer one question (a guard, a basis)

Ion Depot

the station

Supplies bundles of fresh ions and takes used ones back.

Ion depot · `depot9`

a loading rail: fresh ions out (1, 8 or 9 at a time), used ions in

op	hardware	logic	time
`load.1` 1 freshly loaded ion out	✓ 25 rules	no quantum content	305 µs 0 two-qubit	watch · studio
`load.8` 8 freshly loaded ions out	✓ 25 rules	no quantum content	935 µs 0 two-qubit	watch · studio
`load.9` 9 freshly loaded ions out	✓ 25 rules	no quantum content	1.0 ms 0 two-qubit	watch · studio
`dump.9` nine used ions in, out of the computation	✓ 25 rules	no quantum content	725 µs 0 two-qubit	watch · studio

what was checked for load.1

a depot's ions carry no quantum promise: every place that takes fresh ions resets them before use (their logic checks start from an arbitrary input state)

what was checked for load.8

a depot's ions carry no quantum promise: every place that takes fresh ions resets them before use (their logic checks start from an arbitrary input state)

what was checked for load.9

a depot's ions carry no quantum promise: every place that takes fresh ions resets them before use (their logic checks start from an arbitrary input state)

what was checked for dump.9

a depot's ions carry no quantum promise: every place that takes fresh ions resets them before use (their logic checks start from an arbitrary input state)

Roads and Junctions

the road

Conveyor channels and junctions: ions travel between places on them.

Junction (bundles of 9) · `junc9`

a degree-4 junction with one site on each arm; bundles cross ion by ion

op	hardware	logic	fault distance	time
`pass.N.E` a bundle of 9 from N to E	✓ 25 rules	✓ 18/18 flows, rank 18/18	—	2.5 ms 0 two-qubit
`pass.N.S` a bundle of 9 from N to S	✓ 25 rules	✓ 18/18 flows, rank 18/18	—	2.5 ms 0 two-qubit
`pass.N.W` a bundle of 9 from N to W	✓ 25 rules	✓ 18/18 flows, rank 18/18	—	2.5 ms 0 two-qubit
… and 33 more passes: every pair of arms, bundles of 1, 8 and 9 ions

Classical Data Bus

the wire

Classical messages: syndromes to the decoder, frame updates to the archive, guards back to the places. Drawn dashed and dark -- these are bits with a latency, not ions on a conveyor.

The classical half

A fault-tolerant machine is half classical, and that half is just as finite: a decoder sitting somewhere, wired to every place that measures, and a memory holding what it decides. Both are gadgets on the same map — a control floor below the last street — and the wires between them are drawn dashed and dark, because bits are not ions.

Syndromes travel on wires. A place's classical port sends its check outcomes to the decoder 5 µs later (the measurement-to-branch latency measured on a trapped-ion machine, AQT M-ACTION, arXiv:2101.11390); the decoder's lookup takes 1 µs and the write into the memory 0.1 µs, so a decision is ready 6.1 µs after the last measurement it depends on — the reaction time. Every message is checked for causality (G10) and the decoder for keeping up (G11).
The decoder is verified, not asserted. Its lookup table is built from the syndrome-extraction gadget's own fault experiment: every single fault is grouped by the syndrome it produces, and the table is a function exactly when no two faults with the same syndrome differ in their logical effect. That is the proof that it corrects every single fault.
Corrections are kept in software. Every block carries a Pauli frame X̄^a Z̄^b whose bits are parities of logical outcomes. A logical Pauli is a line in the archive and touches no ion; a transversal CNOT copies the frame across; the S̄ gadget conjugates it and adds its own correction. Every readout the frame anticommutes with is flipped when the archive reports it.
And the sign-off allows exactly those. The flat check reads the frames the archive holds and nothing else: drop one outcome from a correction, or empty the archive, and the schedule stops matching the algorithm, and a test says so.
Classically controlled execution. if m: s q0 is a dynamic circuit: the ions travel and the place reserves the op either way — the worst case, so the schedule stays static — and only the pulses depend on a bit measured while the machine runs, delivered to that place's ctl port before it may start. A condition is exact: the parity of a constant and some measured outcomes. Each branch of the program is signed off on its own, and a guard on a place that cannot be told is refused with that reason.
Why it has to exist. A T gate by teleportation needs an S̄ correction, and S̄ is Clifford, not Pauli: no frame can hold it, so the S̄ gadget itself must run — and only when the surgery outcome says so (t_gate). Applying a correction rather than recording it also hands the next instruction a known state instead of a note: teleport_fix reads out r = 0 in every branch, where the same schedule without the correction only promises r ⊕ m₂ = 0.

Logical Algorithm

Places compose into algorithms: a program of logical instructions is scheduled on the town, blocks travel from door to door, and the whole schedule — every ion, every gate — is flattened and checked against the algorithm's ideal circuit. All algorithms, and how to write one →

Bell pair by transversal CNOT signed off

|+0> -> CNOT -> |00>+|11>: the two Z readouts always agree

prep q0 X
prep q1 Z
cx q0 q1
se q0 1
se q1 1
decode q0 q1      # the rounds so far: each place sends its syndrome down its wire
read q0 Z -> a
read q1 Z -> b
decode            # and the readouts' own: now a and b are results

✓ a=Z̄[q0] ⊕ b=Z̄[q1] = 0
✓ no one or two of 1,798 fault locations flip a result unseen

2blocks

163.2 msmachine time

915gadget ops

24/25machine rules

Bell pair by lattice surgery signed off

|++> -> measure Z̄Z̄ -> a Bell pair up to the outcome m: a xor b = m

prep q0 X
prep q1 X
zz q0 q1 -> m
decode q0 q1      # m is a result once the surgery's syndrome is decoded
read q0 Z -> a
read q1 Z -> b
decode

✓ m=Z̄Z̄[q0,q1] ⊕ a=Z̄[q0] ⊕ b=Z̄[q1] = 0
✓ no one or two of 4,058 fault locations flip a result unseen

2blocks

299.9 msmachine time

719gadget ops

24/25machine rules

A transversal CNOT that fires only when the bit says so signed off

Classically controlled hardware, in its plainest form: "these ions do this in that gadget only if the condition is triggered". A logical block is prepared in |+-bar> and read out in Z: one honestly random bit. The transversal-CNOT workshop is then told whether to fire. The ions travel to the workshop either way and the place reserves the same slot either way -- the worst case, so the schedule stays static -- and only the pulses depend on the bit. What the two branches promise is different, and both are checked: with no CNOT, q0 is still |+-bar> so u = 0 and v is random; with the CNOT, the blocks are in a Bell state, so u and v are each random but u + v = 0. A sign-off that ignored the guard could not get both right.

prep q0 X
prep q1 Z
prep a  X
read a  Z    -> m
decode a           # the guard below reads m: it must be decoded first
if m: cx q0 q1
read q0 X    -> u
read q1 X    -> v
decode

✓ [m=0] u=X̄[q0] = 0
✓ [m=1] u=X̄[q0] ⊕ v=X̄[q1] = 0
✓ no one or two of 1,362 fault locations flip a result unseen

3blocks

132.1 msmachine time

809gadget ops

24/25machine rules

Logical Paulis kept in the classical memory signed off

Corrections this machine keeps in software. A logical Pauli is a line in the classical memory, not an operation on ions: the archive holds it and flips every later outcome it anticommutes with. Both readouts must come out -1.

prep q0 Z
x    q0
se   q0 1
read q0 Z -> m
decode q0
prep q1 X
z    q1
read q1 X -> n
decode q1

✓ m=Z̄[q0] = 1
✓ n=X̄[q1] = 1
✓ no one or two of 1,290 fault locations flip a result unseen

2blocks

121.7 msmachine time

347gadget ops

24/25machine rules

GHZ state on three blocks signed off

|+00> -> CNOT q0->q1, CNOT q0->q2: all three Z readouts agree

prep q0 X
prep q1 Z
prep q2 Z
cx q0 q1
cx q0 q2
se q0 1
decode q0 q1 q2
read q0 Z -> a
read q1 Z -> b
read q2 Z -> c
decode

✓ a=Z̄[q0] ⊕ b=Z̄[q1] = 0
✓ a=Z̄[q0] ⊕ c=Z̄[q2] = 0
✓ no one or two of 1,951 fault locations flip a result unseen

3blocks

178.0 msmachine time

1122gadget ops

24/25machine rules

A logical magic state signed off

15-to-1 distillation, injection into a d = 3 block, three syndrome rounds, and a night in a logical zone

inject t
se t 3
decode t
store t

✓ X̄[t] → X̄[t]
✓ Z̄[t] → Z̄[t]
distance: the algorithm injects a state (inject), and injection is not fault-tolerant by design (distance 1)

1blocks

293.3 msmachine time

272gadget ops

24/25machine rules

Teleporting a magic state signed off

the factory distills a T state, the pharmacy encodes it into block t, and the surgery Bell pair carries it to block b: any input state arrives, up to the frame

inject t
prep a X
prep b X
zz a b -> m1
xx t a -> m2
read t Z -> m3
read a Z -> m4
decode              # every outcome so far becomes a result
se b 3
decode b

✓ X̄[t] → X̄[b] ⊕ m2=X̄X̄[t,a]
✓ Z̄[t] → Z̄[b] ⊕ m1=Z̄Z̄[a,b] ⊕ m3=Z̄[t] ⊕ m4=Z̄[a]
distance: the algorithm injects a state (inject), and injection is not fault-tolerant by design (distance 1)

3blocks

508.0 msmachine time

1278gadget ops

24/25machine rules

Two logical S gates make a Z signed off

Two logical S gates make a Z, and Z on |+> is |->: reading X must give -1 every time. Each S is the S-bar gadget: a |Y-bar> block is prepared at the pharmacy (an injection), measured against the data block by Z-bar Z-bar lattice surgery, and read out in X; the correction is Z-bar when the two outcomes disagree, and it is kept in the archive as a Pauli frame -- no ion is ever touched by it.

prep q0 X
s    q0
se   q0 1
s    q0
decode q0      # both S gadgets' rounds, their |Y> blocks' included
read q0 X -> m
decode q0

✓ m=X̄[q0] = 1
distance: the algorithm injects a state (inject.Y), and injection is not fault-tolerant by design (distance 1)

3blocks

542.0 msmachine time

1186gadget ops

24/25machine rules

Shor's algorithm: the order of 7 modulo 15 signed off

Shor's algorithm, fault-tolerant: the order of 7 modulo 15. Order finding for a = 7, N = 15 by iterative phase estimation (Kitaev): one control block per bit of the phase, each prepared, used, corrected and read before the next -- the form Monz et al. ran on trapped ions (Science 351:1068, 2016). U multiplies the work register by 7 modulo 15; its eigenphases are s/4, so three bits m3 m2 m1 of the phase 0.m3m2m1 hold s exactly: m1 = 0 always, and m3 m2 is s, uniformly 0, 1, 2 or 3. An odd s gives the order r = 4, and gcd(7^2 - 1, 15) = 3, gcd(7^2 + 1, 15) = 5. Every logical qubit is a d = 3 surface-code block; every gate is a verified place or one of the standard gates (include "standard"), proved to be its contract in every branch. The controlled multiplications are compiled for the values the register can hold when they act (1 before iteration 2; 1 or 4 before iteration 3), as every experimental Shor has been; the two controlled swaps of iteration 3 are two Toffolis, fourteen T gates from the factories. The inverse quantum Fourier transform is semiclassical: each control is rotated back by the phase the earlier bits fix (S̄† when m1 or m2 says so, T̄† when m1 does) before it is read in X. the work register holds 1 = |0001⟩, b0 its least significant bit iteration 1: U^4 = 1 (7^4 = 2401 = 1 mod 15), so the control picks up no phase iteration 2: U^2 multiplies by 4 (7^2 = 49 = 4 mod 15); on the register's 1 that is 1 -> 4 iteration 3: U multiplies by 7, on 1 and 4 (where iteration 2 left the register): 1 -> 7, 4 -> 13. Rotate the four bits right by one -- on these two values two controlled swaps do it -- then flip all four.

include "standard"
floorplan "shor"
prep b0 Z
x    b0
prep b1 Z
prep b2 Z
prep b3 Z
prep c1 X
read c1 X -> m1
decode c1
prep c2 X
cx   c2 b0
cx   c2 b2
if m1: sdg c2
read c2 X -> m2
decode c2
prep c3 X
cswap c3 b0 b3
cswap c3 b1 b2
cx   c3 b0
cx   c3 b1
cx   c3 b2
cx   c3 b3
if m2: sdg c3
if m1: tdg c3
read c3 X -> m3
decode

189 instructions, 15 magic states: signed off in parts, as a program this size must be
✓ every gate it calls is its contract in every branch: sdg, h, t, tdg, ccx, cswap
✓ the floorplan and the schedule obey every machine rule (25)
✓ with every call as its contract, the outcomes: m1=0 m2=0 m3=0 with probability 0.25; m1=0 m2=0 m3=1 with probability 0.25; m1=0 m2=1 m3=0 with probability 0.25; m1=0 m2=1 m3=1 with probability 0.25

51blocks

11.11 smachine time

27559gadget ops

25/25machine rules

A logical T by teleportation, with the S correction only when needed signed off

A logical T by gate teleportation -- the reason classically controlled hardware exists. A distilled magic state is measured against the data block by Z-bar Z-bar surgery and then read out. The outcomes decide the correction: the X readout asks for a Z-bar, which is a Pauli and can be kept in the archive as a frame, but the surgery outcome asks for an S-bar, which is NOT a Pauli -- no frame can hold it, so the S-bar gadget itself must run, and only when the bit says so. That is a dynamic circuit: the hardware does something different depending on a measurement taken while it runs. Every one of the four outcome branches is signed off separately, and each must implement the same channel from the magic block's state to q0. The factory makes T-dagger|+> (msf15's own state-vector check), and gate teleportation with this correction table needs T|+>: X-bar on the magic block turns one into the other (X T-dagger|+> is T|+> up to a phase), and costs nothing -- it is a line in the archive. Without it every branch was T-dagger or S T, never T: the flat sign-off cannot see that, because it treats the injected state as a free input; checking the gate's contract does, and the standard gates' `t` is this program.

prep   q0 X
inject t
x      t           # the factory's T-dagger|+> becomes T|+>, in software
zz     q0 t -> m
read   t X   -> k
decode             # both corrections read an outcome: decode them first
if m:  s q0        # the Clifford correction: the S-bar gadget runs only when m = 1
if k:  z q0        # the Pauli correction: a line in the archive, no ions touched
decode q0          # the S-bar gadget's own rounds

✓ [m=0, k=0] X̄[t] → X̄[q0]
✓ [m=0, k=0] Z̄[t] → Z̄[q0]
✓ [m=1, k=0] X̄[t] → Ȳ[q0]
✓ [m=1, k=0] Z̄[t] → Z̄[q0]
✓ [m=0, k=1] X̄[t] → X̄[q0]
✓ [m=0, k=1] Z̄[t] → Z̄[q0]
✓ [m=1, k=1] X̄[t] → Ȳ[q0]
✓ [m=1, k=1] Z̄[t] → Z̄[q0]
distance: the algorithm injects a state (inject, inject.Y), and injection is not fault-tolerant by design (distance 1)

3blocks

684.6 msmachine time

1020gadget ops

24/25machine rules

Teleportation by lattice surgery signed off

t in |+>; a Bell pair (a, b) by Z̄Z̄ surgery; X̄X̄ surgery on (t, a) and two Z readouts complete a Bell measurement: b holds the state of t, up to the recorded frame

prep t X
prep a X
prep b X
zz a b -> m1
xx t a -> m2
read t Z -> m3
read a Z -> m4
decode              # every outcome so far becomes a result
se b 1
decode b

✓ 1 → X̄[b] ⊕ m2=X̄X̄[t,a]
✓ no one or two of 8,116 fault locations flip a result unseen

3blocks

536.3 msmachine time

1239gadget ops

24/25machine rules

Teleportation whose correction is applied, not just recorded signed off

Teleportation that hands back a known state, not a state plus a note. `teleport.alg` moves |+-bar> from t to b and the sign-off reports what b holds up to a recorded outcome: "1 -> X-bar[b] + m2". That note has to be carried by everything downstream. Here the machine applies the correction instead: the archive holds a Z-bar on b exactly when the X-bar X-bar surgery said so, and the final readout of b is then deterministic in every branch -- r = 0, whatever the Bell measurement happened to give. That is what classically controlled execution buys: the state is where it should be, and no bookkeeping travels with it.

prep t X
prep a X
prep b X
zz   a b  -> m1
xx   t a  -> m2
read t Z  -> m3
read a Z  -> m4
decode              # the correction below reads m2
if m2: z b
read b X  -> r
decode b

✓ [m2=0] r=X̄[b] = 0
✓ [m2=1] r=X̄[b] = 0
✓ no one or two of 7,752 fault locations flip a result unseen

3blocks

547.7 msmachine time

1214gadget ops

24/25machine rules

How the checks work

Stabilizer flows. A place's promise is a list of flows: the value P had on the ions before equals the value Q has after, up to named measurement records. The circuit runs once on a stabilizer tableau whose signs are symbolic in the measurement record, starting from a Choi state (every ion Bell-paired with a reference), so one run covers every input and every outcome. A spec must also be complete — as many independent flows as the channel has degrees of freedom — so nothing but the intended channel can satisfy it.
Fault distance. The op is embedded between ideal encoders and decoders. A record parity is a detector if it involves no logical measurement and an observable if it does, which the reference qubits make canonical. Every single fault and every pair of faults of the standard circuit-level model is pushed through at once, as the bits of Pauli frames.
The T factory is the one non-Clifford place: a sparse state vector shows it accepts with probability 1 and outputs T†|+⟩ exactly, and Pauli frames show that every one or two T-gate errors are caught while exactly 35 triples are not (output error 35p³).
Composition. An algorithm's schedule is flattened ion by ion and simulated once. Its decoded outcomes must satisfy every relation of the ideal circuit and carry the same number of random bits, every input must reach every surviving block, and no one or two faults anywhere in the schedule may flip a result unseen.
Cross-checked. The tableau agrees with stim on thousands of random flows, every place's fault distance agrees with stim's own search, and mutated programs — a CNOT turned around, a hook-unsafe order, a missing reset — fail.

What is established, and what is not

Established: every place op above passes the hardware rules; every quantum op computes exactly its specification; every error-correcting op has fault distance 3 on the d = 3 surface code. Injection and the rotation source are not fault-tolerant by design, and neither is the Pauli-product bench, whose distance is not computed. Every algorithm in the Logical Algorithm section computes its ideal circuit, and the Clifford ones keep distance 3 as a whole schedule.
Not established: logical error rates (no noise is sampled; distance is counted), decoding (the records are shown to determine the answer; no decoder is run), Clifford errors inside the factory (the protocol assumes them rare), crosstalk from measuring one ion beside another, and throughput — the town runs one instruction at a time.

Beta at scale: hardware only

The first version of this tool composed bivariate-bicycle blocks at scale: 9,720 ions in 96 blocks of [[72,12,6]], and a four-block demo. Their operations pass the hardware rules, but their logic is not verified: the syndrome rounds are contact schedules without H layers or CNOT orientation, a parity is taken once by one bare ancilla, and the T-gate correction is tracked classically. They stay to show the scale the hierarchy handles, not as correct programs.

The demo reports G7 failed, and that is the demo working: its programme asks for a transversal H on a bivariate-bicycle block, which does not preserve that code's stabilizers, so the synthesizer refuses the instruction — for the same reason LogicQ's own checker refuses it — and G7 reports that one instruction of the programme was never realised. A refused instruction is meant to be visible.

The design document · every place in this library is checked twice, as hardware and as logic.

The places

The places

Logical Preparation

Logical preparation (d=3) · prep_d3

Syndrome Extraction

Syndrome extraction (d=3) · se_d3

Logical Zone

Logical zone (1 block) · zone_d3

Logical Memory

Logical memory (d=3) · mem_d3

Logical Operation

Transversal CNOT (d=3) · tcx_d3

Lattice Surgery

Lattice surgery (d=3) · ls_d3

Pauli Product Measurement

Pauli-product bench (d=3) · ppm_d3

Logical Readout

Logical readout (d=3) · read_d3

Magic State Factory

15-to-1 magic-state factory · msf15

Rotation Source

Rotation source · rot1

Magic State Injection

Magic-state injection (d=3) · inj_d3

Decoder

Lookup decoder (surface code, d = 3) · dec_d3

Classical Memory

Classical memory (frames and outcomes) · arch1

Ion Depot

Ion depot · depot9

Roads and Junctions

Junction (bundles of 9) · junc9

Classical Data Bus

The classical half

Logical Algorithm

Bell pair by transversal CNOT signed off

Bell pair by lattice surgery signed off

A transversal CNOT that fires only when the bit says so signed off

Logical Paulis kept in the classical memory signed off

GHZ state on three blocks signed off

A logical magic state signed off

Teleporting a magic state signed off

Two logical S gates make a Z signed off

Shor's algorithm: the order of 7 modulo 15 signed off

A logical T by teleportation, with the S correction only when needed signed off

Teleportation by lattice surgery signed off

Teleportation whose correction is applied, not just recorded signed off

How the checks work

What is established, and what is not

Beta at scale: hardware only

Logical preparation (d=3) · `prep_d3`

Syndrome extraction (d=3) · `se_d3`

Logical zone (1 block) · `zone_d3`

Logical memory (d=3) · `mem_d3`

Transversal CNOT (d=3) · `tcx_d3`

Lattice surgery (d=3) · `ls_d3`

Pauli-product bench (d=3) · `ppm_d3`

Logical readout (d=3) · `read_d3`

15-to-1 magic-state factory · `msf15`

Rotation source · `rot1`

Magic-state injection (d=3) · `inj_d3`

Lookup decoder (surface code, d = 3) · `dec_d3`

Classical memory (frames and outcomes) · `arch1`

Ion depot · `depot9`

Junction (bundles of 9) · `junc9`