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Fault-tolerant quantum error correction for Steane's seven-qubit color code with few or no extra qubits

Ben W. Reichardt

arXiv:1804.06995v1quant-ph

TL;DR

Fault-tolerant correction for Steane’s code must balance qubit overhead, syndrome-extraction speed, and geometric constraints. The paper extends flagged extraction so syndrome qubits flag one another and shows that two code blocks can enable correction with no extra qubits. These constructions provide parallel, qubit-efficient alternatives across several codes, with specific gate-count and locality benefits.

  • Problem

    Existing flagged extraction uses only two extra qubits but measures one syndrome at a time, while faster methods require substantially more qubits; this matters when idle, initialization, or measurement errors are high.

  • Method

    The paper couples syndrome qubits so they reciprocally flag correlated errors, extracts multiple syndromes in parallel, and uses two code blocks to free qubits for correction.

  • Results

    The constructions extract three Steane syndromes in parallel with three extra qubits, use local gates, and achieve zero-extra-qubit correction with at least two Steane code blocks.

  • Takeaways & Limitations

    Parallel flagged extraction offers faster, more qubit-efficient fault-tolerant alternatives for Steane’s code and related color, erasure, and Hamming codes.

  • Takeaways & Limitations

    The no-extra-qubit construction requires at least two code blocks, and the two-block method uses more CNOT gates than two-qubit flagged correction.

Abstract

from arXiv · show

Steane's seven-qubit quantum code is a natural choice for fault-tolerance experiments because it is small and just two extra qubits are enough to correct errors. However, the two-qubit error-correction technique, known as "flagged" syndrome extraction, works slowly, measuring only one syndrome at a time. This is a disadvantage in experiments with high qubit rest error rates. We extend the technique to extract multiple syndromes at once, without needing more qubits. Qubits for different syndromes can flag errors in each other. This gives equally fast and more qubit-efficient alternatives to Steane's error-correction method, and also conforms to planar geometry constraints. We further show that Steane's code and some others can be error-corrected with no extra qubits, provided there are at least two code blocks. The rough idea is that two seven-qubit codewords can be temporarily joined into a twelve-qubit code, freeing two qubits for flagged syndrome measurement.

I. INTRODUCTION

The paper develops qubit-efficient, parallel flagged syndrome extraction for Steane’s code, addressing locality and high idle-error concerns. It also shows that fault-tolerant correction can require no additional qubits when at least two code blocks are available.

  • I. INTRODUCTION: Steane’s [[7, 1, 3]] code corrects any one-qubit error, but prior fault-tolerant methods require two to seven extra qubits.Flagged syndrome extraction reduced the overhead to two extra qubits.
  • I. INTRODUCTION: One-extra-qubit-per-plaquette extraction uses 10 total qubits and measures all six syndromes in two measurement rounds.This design targets fixed planar geometries and experiments where initialization or measurement is slow.
  • I. INTRODUCTION: Multiple syndrome qubits can reciprocally flag correlated errors, so no extra qubit is dedicated solely to flagging.The technique extracts two syndromes at once and extends to three syndromes in parallel.
  • I. INTRODUCTION: The approach also supports parallel extraction for other color and Hamming codes, including shared-flag extraction for the [[5, 1, 3]] and [[15, 7, 3]] codes.For the [[15, 7, 3]] code, two syndromes can use 17 instead of 20 CNOT gates.
  • I. INTRODUCTION: With at least two Steane code blocks, fault-tolerant correction needs zero extra qubits by temporarily joining the blocks into a [[12, 2, 3]] code.The construction can instead use one code block to flag errors in the other, though it uses more CNOT gates than two-qubit flagged correction.
  • I. INTRODUCTION: For Steane’s code, three syndromes can be extracted in parallel with three extra qubits, fewer than the seven extra qubits required by Steane’s method.The circuit uses local gates and is useful when qubit initialization or measurement is slow.

A. Steane code parallel syndrome extraction

The Steane code can extract two syndromes in parallel using one syndrome to flag errors from the other, while preserving fault-tolerant correction under the stated single-fault conditions.

  • A. Steane code parallel syndrome extraction: Two syndromes are extracted at once, with one syndrome serving as a flag for the other.The circuit uses three such two-syndrome circuits, measuring all six syndromes when a nontrivial syndrome appears.
  • A. Steane code parallel syndrome extraction: With no faults and input errors of weight at most one, correction maps the state to the closest codeword.This is the first simplified extended-rectangle condition for the distance-three perfect CSS code.
  • A. Steane code parallel syndrome extraction: With a perfect input and at most one fault, the output X and Z error components each have weight at most one.X and Z faults can be analyzed separately under the stated fault-tolerance conditions.
  • A. Steane code parallel syndrome extraction: A single fault can spread into a weight-two data error, so the circuit uses additional gates to catch these correlated faults.The relevant X and Z fault locations can otherwise produce errors equivalent to uncorrectable weight-two errors.
  • A. Steane code parallel syndrome extraction: When a syndrome is nontrivial, all possible data errors are distinguishable and an appropriate correction can be applied.The circuit’s marked failure locations yield distinguishable error cases after the syndrome measurements.

2. Two dual syndromes in parallel, X4,5,6,7 and Z4,5,6,7

The approach also supports local parallel extraction of three Steane syndromes, with a special correction rule for one ambiguous fault case and fewer qubits than Steane’s one-shot method.

  • 3. Three syndromes in parallel: Three of the six syndromes can be extracted in parallel while the syndrome qubits flag each other using geometrically local gates.The circuit is local according to the geometry used for the one-extra-qubit-per-plaquette layout.
  • 3. Three syndromes in parallel: The possible errors triggered by a nontrivial X syndrome are distinguishable, but two X-error cases require a special correction rule.The exceptional case occurs only with −, −Z measurements.
  • 3. Three syndromes in parallel: Figure 8 compares logical error rates for parallel and sequential syndrome extraction under varying resting-qubit error rates.The simulated error-detection procedure conditions on no detected errors, so its overhead can be substantial.
  • 3. Three syndromes in parallel: The circuit uses one initialization round, six CNOT rounds, and one measurement round when CNOTs on different qubits run in parallel.It uses three extra qubits, fewer than the seven required by Steane’s method for one-shot measurement of three syndromes.

B. Parallel syndrome extraction for the [[4, 2, 2]] code

The paper develops parallel, fault-tolerant syndrome extraction for several codes, including the [[4, 2, 2]] color code and the [[15, 7, 3]] Hamming code, under qubit and geometry constraints.

  • [[4, 2, 2]] color code: The [[4, 2, 2]] color code uses two extra qubits to extract both syndromes simultaneously, with one syndrome flagging the other.The circuit is geometrically local.
  • [[4, 2, 2]] color code: No single fault, including at shaded swap locations, causes an undetectable logical error in the [[4, 2, 2]] circuit.Faults that propagate to logical operators are detected by the opposite syndrome measurement.
  • Other codes: The [[5, 1, 3]] code extracts two syndromes at once using a third extra qubit as a shared flag.The authors did not find a fault-tolerant circuit without that third extra qubit.
  • [[15, 7, 3]] Hamming code: The [[15, 7, 3]] Hamming-code circuits fault-tolerantly extract two syndromes at once with a shared flag.One circuit exploits syndrome overlap and uses 17 CNOT gates.
  • [[15, 7, 3]] Hamming code: For the dual-stabilizer circuit, X components help distinguish errors whose Z components are inequivalent but indistinguishable from the measured information alone.The circuit measures Z8...15 and X8...15.

III. COLOR CODE SYNDROME EXTRACTION WITHOUT FLAGS

The paper studies unflagged syndrome extraction for color codes, showing fault-tolerant procedures for planar layouts and larger codes formed by joining Steane blocks.

  • Larger color codes: For the [[16, 4, 3]] color code, one extra qubit per plaquette suffices for fault-tolerant unflagged syndrome extraction.The resulting correlated errors are detectable and distinguishable from weight-one errors.
  • Larger color codes: Joining four Steane code blocks produces [[22, 4, 3]] and [[16, 4, 3]] color codes.The paper presents these constructions as larger color codes formed from block graphs.
  • [[12, 2, 3]] color code: The [[12, 2, 3]] code is formed by joining two Steane [[7, 1, 3]] code blocks.The paper also gives circuits for extracting syndrome pairs.
  • [[12, 2, 3]] color code: For the [[12, 2, 3]] color code, the planar layout with six extra qubits measures weight-four stabilizers in parallel while flagging each other.The central weight-six stabilizers are measured in parallel using two enclosed qubits.
  • [[12, 2, 3]] color code: Fault-tolerant error correction remains possible for the sparser connectivity pattern of Fig. 12(c).This layout requires careful unflagged measurement of weight-four stabilizers.
  • [[12, 2, 3]] color code: In the sparse layout, left and right plaquette Z stabilizers cannot be measured together because their correlated errors are indistinguishable.Only the horizontal CNOT order avoids individually uncorrectable correlated errors, but it leaves this cross-plaquette ambiguity.

V. STEANE CODE ERROR CORRECTION WITH NO EXTRA QUBITS

The paper constructs fault-tolerant error correction for two Steane code blocks without extra qubits by using interactions between the blocks to provide syndrome-extraction resources.

  • Construction: Combining two Steane code blocks frees two qubits into which two syndromes can be measured, requiring no extra qubits.The construction is inspired by joining the blocks into a [[12, 2, 3]] color code, although the actual procedure uses one block to flag errors in the other.
  • Procedure: The procedure repeats six syndrome-extraction rounds while switching the code blocks and rotating each by one notch.The qubit reindexing is assumed perfect for presentation.
  • Procedure: Without errors, the six rounds provide all twelve syndromes.The procedure restarts when any measurement is nontrivial and uses the syndromes to diagnose and correct the error.
  • Fault tolerance: A single circuit fault can create correlated errors across the two blocks, so correction rules must account for cross-block correlations.Treating the blocks separately could turn an error equivalent to X1X into a logical X error.
  • Fault-tolerance claim: Claim 1 states that faultless correction returns outputs to the code space and corrects errors whose components lie in SX and SZ.The claim also bounds the output error components after at most one fault.
  • Practical scope: The method does not generally include arbitrary pairs of one-qubit errors because some such pairs are indistinguishable from inequivalent errors.The authors accept this because two one-qubit errors are generally a second-order event.
  • Fault-tolerance proof: The proof uses SX and SZ as induction invariants, showing that single faults and subsequent rounds preserve the allowed error sets.The allowed errors have distinguishable syndromes for faultless correction.
  • Practical scope: The construction is probably not useful in a 14-qubit experiment but becomes useful with a larger number of logical qubits.The paper envisions arranging Steane codewords in a two-dimensional lattice.

VI. ERROR-CORRECTING OTHER CODES WITHOUT EXTRA QUBITS

The no-extra-qubit construction extends to some other CSS codes but does not appear to work for the non-CSS [[5, 1, 3]] code.

  • Scope: The method for two Steane codewords extends to some other CSS codes.The paper does not claim extension to all codes.
  • Scope: The analogous construction does not seem to work for the non-CSS [[5, 1, 3]] code because a ZX fault after a CNOT is not caught and is uncorrectable.This identifies a code-structure-dependent limitation of the extension.

A. [[n, n −2, 2]] erasure code

For even n, two [[n, n −2, 2]] erasure-code blocks support fault-tolerant syndrome extraction without extra qubits, although the circuit uses substantially more CNOTs than two-extra-qubit flagged correction.

  • For even n, the [[n, n −2, 2]] erasure code is a single-plaquette color code with stabilizers X⊗n and Z⊗n.
  • A circuit on two code blocks extracts the syndromes of Z⊗n ⊗1 and 1 ⊗X⊗n without extra qubits.
  • The procedure is fault tolerant because no single fault can produce an undetectable logical error.For example, a single X fault cannot propagate to an error with even weight on both blocks.
  • 6n −2 CNOTs are required, substantially more than the 2(n + 2) CNOTs used by the two-extra-qubit flagged procedure.

B. [[15, 7, 3]] Hamming code

For two [[15, 7, 3]] Hamming-code blocks, the construction repeatedly switches and permutes blocks while extracting syndromes, ultimately obtaining all 16 code syndromes for correction.

  • The circuit uses 46 CNOT gates to extract syndromes for Z8,...,15 ⊗1 and 1 ⊗X8,...,15, then reinitialize those stabilizers.The red gates perform syndrome extraction; black gates formally commute and cancel.
  • The error-correction procedure repeats eight times, switching the code blocks and permuting qubits by σ.
  • The eight rounds give all 16 code syndromes; if any measurement is nontrivial, all syndromes are measured to diagnose and correct the error.
  • The induction invariants SX and SZ track propagated X and Z errors, with SX defined from Fig. 15 and SZ obtained by swapping blocks and replacing X with Z.
  • Any X fault in the circuit propagates to an error in SX, whose errors have distinct Z syndromes for subsequent correction.
  • An SX error missed by one measured syndrome is mapped by σ to another SX error that can be detected in a later round.

VII. STEANE CODE ERROR CORRECTION WITH ONE EXTRA QUBIT

Using one extra qubit, the method first measures parity between corresponding syndromes of two Steane blocks, then separately measures their syndromes when an error is detected.

  • The circuit uses one extra qubit to measure the parity of corresponding syndromes, such as Z4,5,6,7 ⊗Z4,5,6,7.
  • A single Z fault can produce paired Z errors on the two code blocks, including errors of weight two or more.
  • These paired errors are detectable by syndrome parities and distinguishable through separate block syndromes.
  • The scheme uses eight CNOTs instead of 22, but enlarges the error-correction rectangle and misses some simultaneous weight-one errors, such as Xα ⊗Xα.It therefore fails AGP fault-tolerance conditions 0 and 0′, while remaining suitable for one encoding level.

VIII. HIGHER-DISTANCE CODES

The paper focuses on distance-three codes and leaves extension of parallel syndrome extraction to higher-distance color codes as an open problem under geometric constraints.

  • The paper focuses on distance-three codes capable of correcting a weight-one error.
  • Flagged syndrome extraction has been studied for the distance-five [[17, 1, 5]] and [[19, 1, 5]] color codes.
  • Whether the parallel syndrome-extraction techniques extend to these higher-distance codes remains an open problem.
  • The extension should also be evaluated for how well it handles geometrical interaction constraints.
  • A [[16, 4, 4]] color code is another moderate-sized higher-distance color code shown in Fig. 17.Its two eight-qubit plaquettes consist of qubits incident to the green and blue regions.
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