Bond dimension

Coherent runs need a matrix product state (MPS), with bond dimension χ set by MaestroBackend(max_bond_dimension=) (default 32). Too small a χ truncates the state, which shows up as extra detection events, the same direction as the coherent effect being studied. So χ has to be checked, not assumed.

Checking χ

1. Noiseless check. Run with simulation_type=MatrixProductState and noise_type='none'. Every detector must stay silent, so any detection event is truncation error. It needs no reference and a few dozen shots. It relies on the pinned type: with fixed_backend=False, Maestro would sample this Clifford circuit on Pauli frames and the check would always pass.

2. Coherent convergence. Coherent noise needs more χ than the noiseless check. Double χ until the result is unchanged within its error bars.

import maestro
from deltakit_maestro import MaestroBackend

MPS = maestro.SimulationType.MatrixProductState

for chi in (4, 8, 16, 32):
    mps = MaestroBackend(simulation_type=MPS, max_bond_dimension=chi)
    dets = mps.simulate(noisy, 200, noise_type='none', seed=1).detection_events
    print(chi, dets.any(axis=1).mean())   # shots with a detection event: 0 once χ is enough

for chi in (8, 16, 32, 64):
    backend = MaestroBackend(max_bond_dimension=chi)
    dets = backend.simulate(noisy, 400, noise_type='coherent', seed=1).detection_events
    print(chi, dets.mean())               # raise χ until this stops changing

On the rotated d=3 code at p=0.01, the noiseless circuit is exact from χ=8, but the coherent detection-event rate is 0.294 at χ=8, 0.272 at χ=16, and 0.263 at χ=32 and 64, against 0.236 for its Pauli twirl. At χ=8 truncation adds about as much as the coherent effect itself:

Both checks on the rotated d=3 code, SI1000(0.01), 3 rounds. From the coherent noise notebook.

example_bond_dimension.py runs both checks on a rotated and an unrotated code.

Which codes are practical

Only the rotated d=3 code is practical. Checked with the noiseless test above (3 rounds, SI1000(0.01) circuits):

Code	Qubits	Smallest exact χ	Cost
Rotated planar d=3	17	8	~1 ms/shot
Unrotated planar d=3	25	> 64	~0.9 s/shot at χ=64, still 75% of shots wrong
Unrotated toric d=3	36	> 32	~1 s/shot at χ=32, every shot wrong
Rotated planar d=5	49	> 32	~1.5 s/shot at χ=32, every shot wrong

These noiseless runs have no noise to spread over cores, so they take as long on one. Renumbering the qubits row by row from their coordinates did not help; it made the rotated d=3 code need χ=64.