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Quantum error correction with only two extra qubits

Rui Chao, Ben W. Reichardt

arXiv:1705.02329v1quant-ph

TL;DR

Fault-tolerant quantum error correction remains difficult to test on small devices because existing schemes use substantial qubit overhead. The paper introduces flagged syndrome-extraction procedures using two ancilla qubits, enabling fault-tolerant correction across distance-three codes. The scheme reaches seven total qubits for the [[5,1,3]] code and 17 physical qubits for protecting seven encoded qubits with the [[15,7,3]] code.

  • Problem

    Existing fault-tolerance schemes use substantial qubit overhead, limiting tests of fault tolerance on small- and medium-scale devices.

  • Method

    The paper adds flags to syndrome-extraction circuits to catch faults that can produce correlated data errors, using only two extra qubits.

  • Results

    7 total qubits suffice for the [[5,1,3]] code, and 17 physical qubits suffice to protect seven encoded qubits with the [[15,7,3]] Hamming code.

  • Takeaways & Limitations

    The procedures make substantial fault-tolerance tests possible on quantum computers with fewer than twenty qubits.

Abstract

from arXiv · show

Noise rates in quantum computing experiments have dropped dramatically, but reliable qubits remain precious. Fault-tolerance schemes with minimal qubit overhead are therefore essential. We introduce fault-tolerant error-correction procedures that use only two ancilla qubits. The procedures are based on adding "flags" to catch the faults that can lead to correlated errors on the data. They work for various distance-three codes. In particular, our scheme allows one to test the [[5,1,3]] code, the smallest error-correcting code, using only seven qubits total. Our techniques also apply to the [[7,1,3]] and [[15,7,3]] Hamming codes, thus allowing to protect seven encoded qubits on a device with only 17 physical qubits.

I. INTRODUCTION

The paper addresses the high qubit overhead of fault-tolerant schemes by introducing flagged syndrome extraction using only two ancilla qubits. The approach enables small-device tests of distance-three codes, including seven-qubit and 17-qubit examples.

  • Motivation: Fault-tolerance schemes use substantial physical-qubit overhead, making them difficult to test on small- and medium-scale devices.The [[4,1,2]] code detects but cannot correct errors, while the smallest previously known correcting scheme used ten qubits.
  • Contribution: Two-ancilla flagged procedures reduce the overhead for fault-tolerant syndrome extraction across several distance-three codes.Flags catch faults that can produce correlated data errors, provided syndromes are extracted in a careful order.
  • Results: 7 total qubits suffice to test the [[5, 1, 3]] code, while 17 physical qubits suffice to protect seven encoded qubits with the [[15, 7, 3]] code.The scheme also uses ten total qubits with the [[8, 3, 3]] code.
  • Implications: The procedures extend fault-tolerance testing to devices with fewer than twenty qubits.Arbitrary encoded Clifford operations use two extra qubits, while universal operations use four extra qubits and 19 total.

II. TWO-QUBIT FAULT-TOLERANT ERROR CORRECTION FOR THE [[5, 1, 3]] CODE

The [[5,1,3]] procedure replaces vulnerable single-ancilla extraction with a flagged circuit that detects correlated errors and enables their syndrome-based correction. Its fault-tolerance argument covers no-fault operation and any single faulty gate.

  • Problem: A single ancilla fault in the unflagged circuit can propagate into a weight-two data error and induce a logical error after miscorrection.The example fault produces IIZXI, which can be incorrectly corrected into the logical error IIZXZ ∼X.
  • Flagged extraction: The flagged circuit extracts the same XZZXI syndrome while using a flag ancilla to detect faults that can create correlated data errors.Relevant failures produce a |−⟩ flag outcome, and the resulting seven distinct data errors have distinct nontrivial syndromes.
  • Fault-tolerance argument: With no faults, the procedure corrects the data to the codespace.This is the first case in the paper’s fault-tolerance argument.
  • Fault-tolerance argument: With at most one faulty gate, trivial syndromes and flags leave at most a weight-one error, while nontrivial outcomes trigger sufficient recovery.No correction is applied when all outcomes are trivial; flagged or nontrivial outcomes invoke further syndrome extraction.
  • Correction procedure: When a flag is raised, subsequent unflagged syndrome extraction distinguishes and corrects the possible correlated error.If no flag is raised but the syndrome is nontrivial, all four syndromes are still extracted before correction because a data fault may occur mid-extraction.

III. TWO-QUBIT FAULT-TOLERANT ERROR CORRECTION FOR HAMMING CODES

The Hamming codes form a family of distance-three quantum codes with a shared structure. Their general form supports the paper’s two-ancilla fault-tolerant syndrome-extraction approach.

  • Hamming-code family: Hamming codes are [[2^r−1, 2^r−1−2r, 3]] self-dual perfect CSS codes for r = 3, 4, 5, … .The family’s structure underlies the later constructions for the [[7,1,3]] and [[15,7,3]] codes.

A. [[7, 1, 3]] Steane code

For Steane’s [[7,1,3]] code, two-ancilla flagged circuits extract stabilizer syndromes fault tolerantly. Flag-triggered errors are distinguishable by their X syndromes.

  • Code structure: The [[7,1,3]] Steane code has stabilizers and logical operators specified in the paper’s construction.The code is the r = 3 member of the Hamming-code family.
  • Flagged extraction: The two-ancilla circuit extracts the IIIZZZZ syndrome while flagging single-fault data errors of weight ≥2.Analogous circuits handle the other stabilizers, yielding a two-qubit fault-tolerant error-correction procedure.
  • Error correction: Flag-triggered errors are distinguishable by their X syndromes, enabling their correction.This is the key decoding property stated for the Steane-code construction.

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

The [[15, 7, 3]] Hamming code uses two-ancilla flagged circuits to fault-tolerantly extract syndromes, with gate ordering making possible errors distinguishable by their syndromes.

  • B. [[15, 7, 3]] Hamming code: The [[15, 7, 3]] code has four X and four Z stabilizers defined by binary parity checks.Its columns represent the numbers 1 through 2^4−1 in binary, as for Hamming codes.
  • B. [[15, 7, 3]] Hamming code: A two-ancilla flagged circuit fault-tolerantly extracts the first Z syndrome for the operator Z_{8,...,15}.If the flag triggers, the possible Z components include a sequence of distinguishable multi-qubit errors.
  • B. [[15, 7, 3]] Hamming code: The possible flagged errors are distinguishable by their X syndromes, provided the CNOT gate order is chosen appropriately.The ordering of the CNOT gates is explicitly important for this discrimination.
  • B. [[15, 7, 3]] Hamming code: The section includes stabilizer data for related [[8, 3, 3]], [[10, 4, 3]], and [[11, 5, 3]] codes, plus a single-ancilla circuit for preparing encoded |07⟩.These materials place the Hamming-code construction alongside other distance-three-code implementations.

C. General Hamming codes

The flagged construction generalizes to Hamming codes of the form [[2^r−1, 2^r−1−2r, 3]], whose syndromes can be extracted fault-tolerantly with two qubits.

  • C. General Hamming codes: Syndromes for the [[2^r−1, 2^r−1−2r, 3]] Hamming code can be fault-tolerantly extracted with two qubits.This is stated as the section’s general claim.
  • C. General Hamming codes: The proof permutes the final 2^r−1 qubits so that flagged faults produce possible Z errors with distinct syndromes.The permutation is constructed using powers of x modulo a primitive polynomial over GF(2).
  • C. General Hamming codes: A degree-(r−1) primitive polynomial generates distinct remainders q_j(x)=x^j mod p for the construction.The cumulative-sum remainders are also shown to be distinct.

IV. TWO-QUBIT FAULT-TOLERANT ERROR CORRECTION FOR OTHER DISTANCE-THREE CODES

The two-qubit flagged procedure extends beyond Hamming codes to several other distance-three codes, provided their stabilizers admit suitable error-distinguishing permutations or replacements.

  • IV. TWO-QUBIT FAULT-TOLERANT ERROR CORRECTION FOR OTHER DISTANCE-THREE CODES: The flagged procedure requires distinct, nontrivial syndromes for errors that can arise from faults during measurement of a stabilizer operator.For CSS codes, distinguishing the relevant I and Z error cases is sufficient.
  • IV. TWO-QUBIT FAULT-TOLERANT ERROR CORRECTION FOR OTHER DISTANCE-THREE CODES: The required property was verified for suitable qubit permutations of every stabilizer generator in the [[10, 4, 3]] and [[11, 5, 3]] codes.These codes are presented alongside the [[8, 3, 3]] code and other distance-three codes.
  • IV. TWO-QUBIT FAULT-TOLERANT ERROR CORRECTION FOR OTHER DISTANCE-THREE CODES: For the [[8, 3, 3]] code, replacement stabilizers satisfy the desired property when simple permutations do not.The syndrome extraction order 1, 2, 3, 6, 4, 7 is given for one replacement stabilizer.
  • IV. TWO-QUBIT FAULT-TOLERANT ERROR CORRECTION FOR OTHER DISTANCE-THREE CODES: Two ancilla qubits suffice to fault-tolerantly extract syndromes and apply error correction for all these codes.The flagged circuit is illustrated as an addition to a syndrome-extraction circuit.

V. TWO-QUBIT FAULT-TOLERANT ERROR DETECTION FOR [[n, n −2, 2]] CODES

Flagging also yields fault-tolerant error detection for even-length [[n, n−2, 2]] codes, while simulations compare the two-qubit procedure with Shor- and Steane-style correction.

  • V. TWO-QUBIT FAULT-TOLERANT ERROR DETECTION FOR [[n, n −2, 2]] CODES: The flagging approach applies to the even-n [[n, n−2, 2]] error-detecting code with stabilizers X^⊗n and Z^⊗n.The code encodes n−2 logical qubits using the stated logical operators.
  • V. TWO-QUBIT FAULT-TOLERANT ERROR DETECTION FOR [[n, n −2, 2]] CODES: A single-ancilla syndrome-extraction circuit is not fault tolerant because certain Z faults create undetectable logical errors.Adding a flag makes any single fault either detectable or harmless to the data.
  • V. TWO-QUBIT FAULT-TOLERANT ERROR DETECTION FOR [[n, n −2, 2]] CODES: The ancilla can be interpreted as encoded into a two-qubit Z-error-detecting code that detects single Z faults propagating to the data.The interpretation uses stabilizer XX and logical operators XI and ZZ.
  • V. TWO-QUBIT FAULT-TOLERANT ERROR DETECTION FOR [[n, n −2, 2]] CODES: Figure 6 compares logical error rates for two-qubit, Shor-style, and Steane-style correction across the [[5, 1, 3]], [[7, 1, 3]], and [[15, 7, 3]] codes.Rates are divided by p^2 to expose leading-order coefficients, with p and 7p shown for pseudo-threshold assessment.
  • V. TWO-QUBIT FAULT-TOLERANT ERROR DETECTION FOR [[n, n −2, 2]] CODES: Single-ancilla fault-tolerant circuits are also provided for initialization and projective measurement.These procedures are given in an appendix.

VI. SIMULATIONS AND CONCLUSION

The paper evaluates two-qubit fault-tolerant error correction under depolarizing noise and discusses flag-based extensions that trade qubit overhead against circuit complexity. Simulations compare the two-qubit, Shor-style, and Steane-style procedures, while the conclusion identifies higher-distance codes as a future target.

  • VI. SIMULATIONS AND CONCLUSION: At least 10^6 consecutive error-correction rounds were simulated for each CNOT failure rate p under a standard depolarizing noise model.The model covered one- and two-qubit operations.
  • VI. SIMULATIONS AND CONCLUSION: For the [[15, 7, 3]] code, Steane-style error correction performed better than the Shor-style and two-qubit procedures in the simulations.
  • VI. SIMULATIONS AND CONCLUSION: Overlapping flags can localize Z faults, while shared flags reduce resources when extracting multiple syndromes.The [[7, 1, 3]] shared-flag circuit uses four qubits and 15 CNOT gates, compared with seven qubits and 25 CNOT gates for Steane-style extraction with ancilla decoding.
  • VI. SIMULATIONS AND CONCLUSION: The shared-flag [[7, 1, 3]] construction requires verification before correction because syndrome 001 can correspond to multiple errors.
  • VI. SIMULATIONS AND CONCLUSION: A natural next problem is extending the flag technique to medium-size codes with higher distance.

Appendix A: Stephens-Yoder-Kim space-optimized syndrome extraction

This appendix describes space-optimized syndrome extraction for distance-three codes, using paired data-qubit couplings and corrections to remain fault tolerant. It also applies flag-based procedures to encoded-state preparation and destructive logical-X measurement.

  • Appendix A: Stephens-Yoder-Kim space-optimized syndrome extraction: For w = 4, a Z measurement catches correlated X errors before ancilla-data coupling and X-basis ancilla measurements.
  • Appendix A: Stephens-Yoder-Kim space-optimized syndrome extraction: For a distance-three code, max{3, ⌈w/2⌉} ancilla qubits suffice to fault-tolerantly extract a weight-w stabilizer’s syndrome.Unlike the two-qubit schemes, this method can also be deterministic.
  • Appendix A: Stephens-Yoder-Kim space-optimized syndrome extraction: The space-optimized method couples two data qubits to each ancilla and applies corrections based on ancilla measurement outcomes.The construction is illustrated for three ancillas and for w = 10.
  • Appendix A: Stephens-Yoder-Kim space-optimized syndrome extraction: For the [[5, 1, 3]] code, a fault raised by a flag during logical-X measurement allows the procedure to return a previously measured syndrome.With at most one fault, the earlier syndrome must have been correct.
  • Appendix A: Stephens-Yoder-Kim space-optimized syndrome extraction: The [[15, 7, 3]] encoded |07⟩ preparation circuit is fault tolerant only when X and Z faults are considered separately.A combined ZX fault can produce a weight-two error whose X and Z components each have weight one.

Appendix D: Fault-tolerant state preparation and measurement for the [[n, n −2, 2]] codes

The appendix gives one-ancilla state-preparation and projective-measurement circuits for [[n, n −2, 2]] codes. It also describes direct encoded-state preparation and repeated or two-ancilla parity measurements.

  • Appendix D: Fault-tolerant state preparation and measurement for the [[n, n −2, 2]] codes: One-qubit fault-tolerant circuits are provided for state preparation and projective measurement in [[n, n −2, 2]] codes.
  • Appendix D: Fault-tolerant state preparation and measurement for the [[n, n −2, 2]] codes: Encoded states |+j 0n−2−j⟩ can be prepared directly instead of applying targeted logical Hadamard gates to encoded |0n−2⟩.
  • Appendix D: Fault-tolerant state preparation and measurement for the [[n, n −2, 2]] codes: Projective measurement of Zj is performed twice using one ancilla, with differing results indicating an error.
  • Appendix D: Fault-tolerant state preparation and measurement for the [[n, n −2, 2]] codes: Other parity measurements use analogous circuits, including two-ancilla circuits, and symmetric circuits apply to logical X operators.
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