Local automorphisms speed up quantum code syndrome measurement scheduling
Local Automorphism-Aware Syndrome Compilation for General Quantum LDPC Codes
Information Theory
Summary
Measuring errors in quantum computers quickly and correctly is important for keeping them working well. The authors figured out a way to schedule these measurements more efficiently by using symmetrical patterns in the quantum code graphs. Their method, called LocalASC, reduces complexity and finds the shortest measurement sequences for certain quantum error-correcting codes. This helps improve how quantum codes are implemented in practice.
What this means in practice
- •For quantum hardware engineers: Create optimal sequences of two-qubit gates for error detection in quantum processors using local automorphism-based scheduling.
- •For quantum software developers: Develop efficient syndrome-extraction routines for error-correcting codes with irregular check weights using the LocalASC method.
Authors
Eugenio Durazo Rocha, Olai Å. Mostad, Hsuan-Yin Lin, Eirik Rosnes
Abstract
Low-depth syndrome extraction for Calderbank-Shor-Steane (CSS) quantum low-density parity-check codes can be formulated as a proper ordered edge-coloring problem subject to quantum parity constraints. A proper edge-coloring of the CSS Tanner graph ensures that each data or ancilla qubit participates in at most one two-qubit gate per layer, but does not guarantee a valid interleaving of the X- and Z-check measurements as for every overlapping X/Z check pair, the number of shared data qubits on which the X interaction precedes the Z interaction must be even. The minimum number of colors in a proper ordered edge-coloring satisfying the quantum parity constraints equals the minimum two-qubit depth when each stabilizer check is measured with a single ancilla. We introduce local automorphism-aware syndrome compilation (LocalASC), which reduces the constraint system to edge-orbit variables under a subgroup of the Tanner graph automorphisms and lifts each feasible orbit assignment to the full graph. Although the $6$-layer degree lower bound is unattainable for the published weight-$6$ IBM bivariate bicycle codes, we show that this is not universal among two-block CSS codes. Among code instances for which the maximum check weight equals the maximum Tanner graph degree, LocalASC finds depth-optimal syndrome-extraction schedules for several two-block CSS codes with odd component weights, including instances with unequal odd weights. We also obtain lower-bound-saturating syndrome-extraction schedules for several quantum Tanner codes satisfying the same degree condition. To obtain the subgroups used by LocalASC without computing the full automorphism group of the Tanner graph, we construct translation subgroups for two-block group-algebra CSS codes over abelian groups. For quantum Tanner codes, we give conditions under which square-complex symmetries extend to Tanner graph automorphisms.