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Simplified circuit-level decoding using Knill error correction

Ewan Murphy, Subhayan Sahu, Michael Vasmer

TL;DR

It is shown analytically and numerically that the time-constrained decoding problem for Knill error correction can be solved using the same decoder used for the simpler code-capacity noise model, illustrating that Knill error correction may alleviate the stringent requirements on classical control required for building a large-scale quantum computer.

Abstract

Quantum error correction will likely be essential for building a large-scale quantum computer, but it comes with significant requirements at the level of classical control software. In particular, a quantum error-correcting code must be supplemented with a fast and accurate classical decoding algorithm. Standard techniques for measuring the parity-check operators of a quantum error-correcting code involve repeated measurements, which both increases the amount of data that needs to be processed by the decoder, and changes the nature of the decoding problem. Knill error correction is a technique that replaces repeated syndrome measurements with a single round of measurements, but requires an auxiliary logical Bell state. Here, we provide a theoretical and numerical investigation into Knill error correction from the perspective of decoding. We give a self-contained description of the protocol, prove its fault tolerance under locally decaying (circuit-level) noise, and numerically benchmark its performance for quantum low-density parity-check codes. We show analytically and numerically that the time-constrained decoding problem for Knill error correction can be solved using the same decoder used for the simpler code-capacity noise model, illustrating that Knill error correction may alleviate the stringent requirements on classical control required for building a large-scale quantum computer.

Simplified circuit-level decoding using Knill error correction

TL;DR

It is shown analytically and numerically that the time-constrained decoding problem for Knill error correction can be solved using the same decoder used for the simpler code-capacity noise model, illustrating that Knill error correction may alleviate the stringent requirements on classical control required for building a large-scale quantum computer.

Abstract

Quantum error correction will likely be essential for building a large-scale quantum computer, but it comes with significant requirements at the level of classical control software. In particular, a quantum error-correcting code must be supplemented with a fast and accurate classical decoding algorithm. Standard techniques for measuring the parity-check operators of a quantum error-correcting code involve repeated measurements, which both increases the amount of data that needs to be processed by the decoder, and changes the nature of the decoding problem. Knill error correction is a technique that replaces repeated syndrome measurements with a single round of measurements, but requires an auxiliary logical Bell state. Here, we provide a theoretical and numerical investigation into Knill error correction from the perspective of decoding. We give a self-contained description of the protocol, prove its fault tolerance under locally decaying (circuit-level) noise, and numerically benchmark its performance for quantum low-density parity-check codes. We show analytically and numerically that the time-constrained decoding problem for Knill error correction can be solved using the same decoder used for the simpler code-capacity noise model, illustrating that Knill error correction may alleviate the stringent requirements on classical control required for building a large-scale quantum computer.
Paper Structure (14 sections, 2 theorems, 19 equations, 7 figures, 1 algorithm)

This paper contains 14 sections, 2 theorems, 19 equations, 7 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Knill error correction
  4. Fault tolerance with Knill error correction
  5. Numerical results
  6. Simulating the composition of fault tolerant protocols
  7. Simulations
  8. Discussion
  9. Proofs
  10. Definitions
  11. Proof of Lemma \ref{['lem:noisy_bell_measurement']}
  12. Modularised circuit simulations
  13. Circuit-level error model
  14. Compressed Knill error correction

Key Result

Lemma 1

Error propagation with transversal Bell measurement circuit. Assume that the blocks $D$ and $A$ are subjected to LD noise with rate $p_1$ and $p_2$ respectively. The transversal CNOT gate $F \subseteq \{1,\dots,n\}$ has LD error with rate $p_3$, and the single-qubit measurements is LD with rate $p_

Figures (7)

  • Figure 1: Knill error correction. For a code block $D$ encoded in a general $[\![n,k,d]\!]$ stabilizer code, two auxiliary blocks $A$ and $B$ are initialized in $k$ logical Bell pairs. The shaded region shows the circuit for transversal Bell measurement on $D \otimes A$. The measurement outcomes are passed to a decoder $\mathcal{D}$, which deduces a classical recovery $\mathcal{R}$. The corrected logical operator measurements are then computed and used to determine the logical Pauli correction applied to $B$, completing a logical teleportation from $D$ to $B$.
  • Figure 2: Clifford fault tolerant protocols can be defined by: a physical circuit, a (classical) function that processes the measurement data sampled from this circuit, and a set of Pauli corrections that are applied based on the function.
  • Figure 3: Circuit for the fault tolerant preparation of logical Bell states used in the simulations. The syndrome data from the $d$ rounds of $X$ and $Z$ stabilizer measurement are decoded separately using the offline decoder.
  • Figure 4: Numerical simulations comparing the performance of the same decoder for different noise models and error correction gadgets. (a,c,e) Lifted product (LP) codes decoded using BP. We observe results consistent with a non-zero threshold in (a) and (c). (b,d,f) Surface codes decoded using MWPM. We observe results consistent with a non-zero threshold in all cases. Error bars show 95% confidence intervals calculated using the Agresti-Coull method brown2001. (g) Summary of decoders used in the simulations.
  • Figure 5: Circuit-level error model for state preparation and stabilizer measurement.
  • ...and 2 more figures

Theorems & Definitions (7)

  • Lemma 1
  • Lemma 2
  • proof
  • Definition 1
  • Definition 2
  • Definition 3
  • proof