Abstract

Error probability distribution associated with a given Clifford measurement circuit is described exactly in terms of the circuit error-equivalence group, or the circuit subsystem code previously introduced by Bacon, Flammia, Harrow, and Shi. This gives a prescription for maximum-likelihood decoding with a given measurement circuit. Marginal distributions for subsets of circuit errors are also analyzed; these generate a family of related asymmetric LDPC codes of varying degeneracy. More generally, such a family is associated with any quantum code. Implications for decoding highly-degenerate quantum codes are discussed.

Highlights

  • Quantum computation offers exponential algorithmic speed-up for some classically hard problems

  • A decoder should be designed for the specific syndrome measurement circuit, as it must be aware of the associated correlations between the errors[3,4,5]—at sufficiently high error probabilities such correlations are present even in fault-tolerant (FT) circuits designed to prevent a single fault from affecting multiple data qubits

  • A quantum [[n, k, d]] stabilizer code is a 2k-dimensional subspace specified as a common +1 eigenspace of all operators in an Abelian stabilizer group S ⊂ Pn, −11 ∈ S, where the n-qubit Pauli group Pn is generated by tensor products of single-qubit Pauli operators

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Summary

INTRODUCTION

Quantum computation offers exponential algorithmic speed-up for some classically hard problems. A parameterization of the probability distribution of correlated quantum errors in terms of a spin model has been recently considered by Chubb and Flammia[5]. They describe how such a formalism can be used for maximum-likelihood (ML) decoding in the presence of circuit-level noise. The goal of this work is to give an explicit numerically efficient algorithm for analyzing error correlations resulting from a given qubit-based Clifford measurement circuit, and for constructing decoders optimized for such a circuit.

Quantum codes
Bernoulli distribution
ERROR CORRELATIONS IN A CLIFFORD MEASUREMENT CIRCUIT
Circuit subsystem code
MARGINAL DISTRIBUTIONS FOR CORRELATED ERRORS
Generator matrix and the coupling coefficients
Transformation for a parity check matrix
Marginal distribution for error equivalence classes
Independent X and Z errors
A more general Pauli channel
Decoding based on circuit EEG
Generator-based decoding via marginal distributions
Parity-based decoding via marginal distributions
DISCUSSION
Rank of the circuit-EEG generator matrix
Rank of the circuit stabilizer group
Full Text
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