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Fault-Tolerance of "Bad" Quantum Low-Density Parity Check Codes
Alexey A. Kovalev, Leonid P. Pryadko
TL;DR
Finite-rate quantum LDPC codes can have relative distance that vanishes with blocklength, while their fault-tolerance properties and general bounds remain little explored. The paper analyzes errors by separating them into small independent graph clusters and establishes positive error thresholds for power-law-distance code families. For QHPCs, its estimates give a lower threshold than toric codes but fewer required physical qubits at large blocklength.
Problem
Finite-rate quantum LDPC codes can have vanishing relative distance, and their parameters and fault tolerance are largely unexplored despite the need for quantum error correction.
Source: pp. 1–2Method
The analysis separates errors into small independent clusters on a graph to study decoding thresholds for quantum and classical LDPC families.
Source: pp. 1–2Results
Power-law-distance LDPC families have positive thresholds for erasure and depolarizing errors, with asymptotically certain recovery below those thresholds.
Source: pp. 3–4Takeaways & Limitations
A finite threshold is a necessary condition for fault tolerance, and the estimates indicate QHPCs use fewer physical qubits than toric-code copies at large blocklength despite a lower threshold.
Source: p. 4For subsystem codes, the finite-threshold guarantee applies only when the actual stabilizer group has limited-weight generators.
Source: p. 4
Abstract
from arXiv · showhide
We discuss error-correction properties for families of quantum low-density parity check (LDPC) codes with relative distance that tends to zero in the limit of large blocklength. In particular, we show that any family of LDPC codes, quantum or classical, where distance scales as a positive power of the block length, $d \propto n^\alpha$, $\alpha>0$, can correct all errors with certainty if the error rate per (qu)bit is sufficiently small. We specifically analyze the case of LDPC version of the quantum hypergraph-product codes recently suggested by Tillich and Z\'emor. These codes are a finite-rate generalization of the toric codes, and, for sufficiently large quantum computers, offer an advantage over the toric codes.