Sparse Quantum Codes from Quantum Circuits

TL;DR

This paper introduces a method to convert quantum circuits into sparse quantum subsystem codes with constant-weight generators, maintaining code distance and qubit count, and achieving near-optimal parameters for local codes.

Contribution

The authors present a general construction that transforms stabilizer codes into sparse subsystem codes with constant-weight generators, preserving code parameters and enabling spatial locality.

Findings

Achieves subsystem codes with constant-weight generators and high code distance.

Maps arbitrary stabilizer codes into new codes with the same distance and qubit count.

Nearly saturates bounds for spatially local codes in multiple dimensions.

Abstract

We describe a general method for turning quantum circuits into sparse quantum subsystem codes. The idea is to turn each circuit element into a set of low-weight gauge generators that enforce the input-output relations of that circuit element. Using this prescription, we can map an arbitrary stabilizer code into a new subsystem code with the same distance and number of encoded qubits but where all the generators have constant weight, at the cost of adding some ancilla qubits. With an additional overhead of ancilla qubits, the new code can also be made spatially local. Applying our construction to certain concatenated stabilizer codes yields families of subsystem codes with constant-weight generators and with minimum distance $d = n^{1-\epsilon}$, where $\epsilon = O(1/\sqrt{\log n})$. For spatially local codes in $D$ dimensions we nearly saturate a bound due to Bravyi and Terhal and achieve $d = n^{1-\epsilon-1/D}$. Previously the best code distance achievable with constant-weight generators in any dimension, due to Freedman, Meyer and Luo, was $O(\sqrt{n\log n})$ for a stabilizer code.