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The surface code with a twist

Theodore J. Yoder, Isaac H. Kim

arXiv:1612.04795v2quant-ph

TL;DR

Patch-based schemes had not yet fully explored the computational advantages of twist defects. This paper develops triangle-code patches with a modified, weight-four twist and lattice-surgery methods; they provide full Clifford operations without state distillation for S, while using fewer qubits than rotated surface-code patches. The authors also relate triangle codes to surface-code patches and describe a small-code route to logical tomography.

  • Problem

    Twist defects enable full Clifford operations without state distillation in defect-based layouts, but their advantages for patch-based computation had been little explored.

    Source: §1 · p. 1

  • Method

    The authors develop planar triangle-code patches with a central weight-four twist and use lattice surgery to relate them to surface-code patches.

    Source: §1 · pp. 1–2, §2 · p. 2

  • Results

    Triangle codes support full Clifford operations without state distillation for S and use 3d^2/4 + 1/4 data qubits per logical qubit versus d^2 for the rotated surface code.

    Source: §1 · p. 2

  • Takeaways & Limitations

    Triangle codes and their surgery methods add a tool for planar fault tolerance; future processors might use tessellations of triangles rather than grids.

    Source: §6 · p. 11

    For the simple all-Z logical measurement, omitting the central mixed-type stabilizer leaves an undetectable X-error string of weight (d + 1)/2 < d.

    Source: §3 · p. 5

Abstract

from arXiv · show

The surface code is one of the most successful approaches to topological quantum error-correction. It boasts the smallest known syndrome extraction circuits and correspondingly largest thresholds. Defect-based logical encodings of a new variety called twists have made it possible to implement the full Clifford group without state distillation. Here we investigate a patch-based encoding involving a modified twist. In our modified formulation, the resulting codes, called triangle codes for the shape of their planar layout, have only weight-four checks and relatively simple syndrome extraction circuits that maintain a high, near surface-code-level threshold. They also use 25% fewer physical qubits per logical qubit than the surface code. Moreover, benefiting from the twist, we can implement all Clifford gates by lattice surgery without the need for state distillation. By a surgical transformation to the surface code, we also develop a scheme of doing all Clifford gates on surface code patches in an atypical planar layout, though with less qubit efficiency than the triangle code. Finally, we remark that logical qubits encoded in triangle codes are naturally amenable to logical tomography, and the smallest triangle code can demonstrate high-pseudothreshold fault-tolerance to depolarizing noise using just 13 physical qubits.

1 Introduction

The paper adapts twist defects to patch-based computation, targeting full Clifford operations without state distillation for S while retaining surface-code-like error-correction features. Triangle codes also offer lower qubit use and support logical tomography with a small patch.

1 Introduction

  • Twist defects support fault-tolerant full Clifford operations without state distillation for S, but their patch-based computational advantages had received little attention.Existing hole- and patch-based surface-code strategies use state distillation for S.Source: §1 · p. 1
  • A central twist yields a planar triangle-code patch for full Clifford computation by local operations; lattice surgery also gives a method for S on surface-code patches in a nonstandard layout.The triangle-code formulation reduces the twist check to weight four.Source: §1 · p. 2
  • Triangle codes use 3d^2/4 + 1/4 data qubits per logical qubit versus d^2 for the rotated surface code, with near-surface-code thresholds and simple syndrome extraction.Their extraction circuits use minimal timesteps amortized over many rounds and nearly the minimal number of ancillas.Source: §1 · p. 2
  • At distance 3, logical initialization, measurement, and an order-3 gate support triangle-code logical tomography with 13 physical qubits including ancillas.The order-3 gate cyclically permutes the Paulis up to a permutation.Source: §1 · p. 2
  • Figure 1 transforms a two-twist dislocation code into an asymmetric triangle code using inner and outer CNOT layers followed by a single-qubit Clifford.The Clifford removes a Pauli Y from one stabilizer.Source: Figure 1 · p. 2

2 Constructing and error-correcting triangle codes

Triangle codes are constructed from dislocation codes by lattice surgery, yielding a triangular patch with simplified checks. The section describes their logical operators, surface-code conversion, threshold estimates, and syndrome-extraction trade-offs.

2 Constructing and error-correcting triangle codes

  • Lattice surgery from dislocation codes produces triangle codes with stabilizer generators of at most weight four and fewer physical qubits.The general layout places qubits on integer lattices in the xy, yz, and xz planes, with at least one coordinate zero; its planar projection fits within a triangle.Source: §2 · pp. 2–3
  • Triangle-code stabilizers are weight-four plaquettes or weight-two boundary loops, including mixed-type plaquettes centered at the origin.Logical X, Y, and Z operators can lie along the boundary, crossing the x, y, and z axes, respectively.Source: §2 · p. 3
  • Each triangle code encodes one logical qubit, and symmetric codes with r = s = t = (d + 1)/2 have logical-operator weights d_x = s + t − 1, d_y = r + t − 1, and d_z = r + s − 1.The symmetric family is called distance-d.Source: §2 · p. 3
  • Numerical phenomenological-noise estimates compare bit-flip and bit-phase-flip storage errors with ideal or noisy syndrome measurements; reported code-family differences are appreciable only for noisy measurements with bit-phase flips.The estimates compare triangle codes with rotated surface codes.Source: Table 1 · p. 3, §2 · p. 4
  • A d x d surface code is a d x 1 x d triangle code, and the conversion procedure from a surface-code patch has effective distance d.The conversion extends the surface-code patch from a corner and measures triangle-code stabilizers O(d) times.Source: §2 · p. 3, Figure 3 · p. 3
  • Full-distance syndrome extraction uses two qubits beyond one per check and amortizes to six timesteps; 2-CATs detect central hook errors, while a one-ancilla design may lose one distance unit.For distance three, the authors develop a specialized circuit using one ancilla per check.Source: §2 · p. 4

3 Initialization and measurement

The non-CSS structure complicates triangle-code initialization and measurement, and qubitwise-only measurement can fail to retain maximum distance when it omits stabilizer information. The paper presents three alternatives for full-distance initialization and measurement.

3 Initialization and measurement

  • Qubitwise Z-basis measurement can omit the central mixed-type stabilizer, leaving an undetectable weight-(d+1)/2 error that defeats maximum-distance logical measurement.More generally, omitting any stabilizer from a complete set can leave a sub-distance error that anticommutes with the measured logical operator.Source: §3 · p. 5
  • Three schemes achieve full-distance initialization and measurement: code conversion (CC), basis-state conversion (BC), and CAT states (CS).Each approach yields a different computation scheme, with tradeoffs among them.Source: §3 · p. 5
  • CC extends a d × d surface-code patch to a distance-2d+1 triangle code, whose X-, Y-, or Z-basis transversal measurement has effective distance d.Surface-code initialization procedures also provide effective distance d.Source: §3 · p. 5
  • BC converts a d × d surface-code basis state by stopping bottom-loop measurements and measuring triangle stabilizers for O(d) cycles, retaining effective distance d regardless of new-qubit initialization.Logical measurement reverses the extension and measures all qubits in the Z-basis.Source: §3 · pp. 5–6, Figure 6 · p. 6
  • CS uses a row of d ancillas to create and verify a d-CAT state, but repeated logical measurement and initialization take O(d^2) time.The paper describes CS as the most qubit-efficient, but least time-efficient, of the three approaches.Source: §3 · pp. 5–6

4 Planar, nearest-neighbor computation with triangles

The paper lays out three planar computation designs using triangle-code patches, with different space and gate-time efficiencies. These layouts support Clifford operations, including a triangular-layout scheme for surface-code patches.

4 Planar, nearest-neighbor computation with triangles

  • The three designs vary in how efficiently they store data and perform gates, compared by qubits per logical qubit and Clifford gate times.Qubit counts include ancillas needed for gates.Source: §4 · p. 6, Table 2 · p. 6
  • In the CC design, surface-code patches hold data between gates, while larger triangular patches enable single-qubit Clifford gates through code conversion and 1-bit teleportation.The described operations include S†, H S† H, and H.Source: §4 · p. 6
  • On surface codes, a triangular patch layout supports the whole Clifford group, at a not-insignificant cost of 3d2 + O(d) physical qubits per logical.Source: §4 · p. 6
  • The BC design uses neighboring empty ancilla patches for initialization and measurement; its CNOT requires two ancillas and correctly oriented data patches, with reorientation and CNOT each taking O(d) time.The tessellated layout leaves room to reorient any data patch, though not necessarily in parallel with nearby patches.Source: §4 · pp. 6–7, Figure 11 · p. 8
  • The CS design places d ancillas between triangles to make CAT states, enabling initialization and measurement without intruding on neighboring patches, but ancilla preparation and measurement for CNOT and reorientation take O(d^2) time.The layout needs one ancillary patch per data patch for arbitrary reorientation and neighboring-patch CNOTs.Source: Figure 7 · p. 7, §4 · p. 8

5 The smallest triangle code, code comparison, and logical tomography

The distance-3 triangle code compares favorably with small fault-tolerant designs in qubit use and pseudothreshold, and the paper describes circuits for extraction, measurement, initialization, and tomography. It leaves open whether interwoven extraction extends to full distance at larger code sizes.

5 The smallest triangle code, code comparison, and logical tomography

  • The smallest fully fault-tolerant triangle code uses 13 qubits, compared with 17 for the smallest fully fault-tolerant surface code.The authors report comparable pseudothresholds to the surface code and better ones than other small designs such as the color code.Source: §5 · p. 8
  • Interwoven plaquette-and-loop extraction reduces the distance-3 circuit to six ancillas and eight timesteps, while a degree-4 implementation uses a depth-2 ancilla-gate cascade that lowers the threshold.The proposed circuit can perform two subsequent extractions in 15 timesteps.Source: §5 · p. 8, Figure 13 · p. 9
  • The paper leaves open whether interwoven extraction achieves full distance for larger triangle codes; the two-qubit saving also matters less in the large-distance limit.The interwoven circuit is given for distance 3.Source: §5 · pp. 8–9
  • Localized plaquette ancillas support logical X, Y, and Z measurement after at most three repeats; measurement pseudothresholds are lower than gate values but can be raised by adding one qubit.The paper attributes the lower measurement pseudothresholds to slightly greater circuit depth from measuring logical Z.Source: §5 · p. 9
  • The depth-2 circuit prepares logical |0⟩ non-fault-tolerantly, with postselection on trivial syndrome and logical-Z measurements used to guarantee fault-tolerance.Similar circuits prepare logical X- and Y-eigenstates.Source: §5 · p. 9, Figure 14 · p. 10
  • Logical process tomography uses Pauli-eigenstate preparation and Pauli-basis measurement, while transversal order-3 Clifford operations cyclically permute the logical Paulis.The described gate applies H or SH on specified qubits and SH on the central qubit, rotating the code by 120 degrees.Source: §5 · pp. 9–10
  • Triangle-code designs save two or four qubits over the surface code with pseudothreshold drops of 3% or 14%, respectively, in the reported distance-3 comparison.The comparison covers small designs with fewer than 20 qubits, low connectivity, and no postselection to create syndrome-extraction ancillas.Source: §5 · p. 9, Table 3 · p. 10

6 Conclusion

The triangle code and its lattice-surgery methods add a tool for planar fault-tolerant computation, while the twist-based approach also yields a surgery for implementing S on surface-code patches without state distillation. The authors suggest exploring multi-twist patches and triangle tessellations as future directions.

6 Conclusion

  • Triangle codes and their syndrome-extraction circuits and lattice-surgery methods add another tool for fault-tolerant quantum computing, whose ideal planar scheme remains uncertain.The authors emphasize the nascent stage of fault-tolerant quantum-computing experiments.Source: §6 · p. 11
  • The triangle-code-and-twist approach gives a simple surgery for implementing S on surface-code patches without state distillation.The authors suggest triangle tessellations as a possible way to expedite logical gates, even if triangle codes are not the resting code of a fault-tolerant architecture.Source: §6 · p. 11
  • A proposed generalization uses eight twists on a closed surface to encode two logical qubits at distance d.The example is a cube-like surface with d qubits per edge; determining the properties and usefulness of such patches is left for future research.Source: §6 · p. 11

A Memory threshold

The section establishes analytic memory-threshold lower bounds and reports numerical estimates for triangle and rotated codes under ideal and noisy syndrome measurement. It also explains the decoding-graph approach and how correlated errors constrain the proof and recovery strategy.

A Memory threshold

  • Numerical estimates model data-storage and syndrome-bit noise rather than circuit-level syndrome extraction, using ideal or d+1-round noisy measurements and matching-based decoding.The noisy model flips syndrome bits during the first d rounds and includes a final error-free syndrome measurement.Source: A Memory threshold · pp. 11–13
  • The proof maps errors to edges in a single decoding graph and treats logical errors as paths between boundary edges on different sides of the triangle.For syndrome-measurement errors, copies of the graph are stacked with vertical edges between layers; triangle codes use one graph because they are not CSS.Source: A Memory threshold · pp. 11–12
  • For uncorrelated X and Z errors, recovery minimizes the number of edges; correlated errors require a modified recovery criterion.The original criterion minimizes edge count rather than the number of affected qubits.Source: A Memory threshold · p. 12
  • The analytic argument establishes thresholds of at least 0.7% for uncorrelated X and Z errors and 0.019% for depolarizing noise.For correlated errors, the proof threshold is about 36 times lower than for uncorrelated errors.Source: A Memory threshold · pp. 11–12
  • Noisy-syndrome thresholds are near 2.6% for triangle codes and 3.2% for rotated codes; ideal-syndrome thresholds are near 10% for both.The noisy-measurement simulations use odd distances 3, 5, 7, and 9; the ideal-measurement plots include odd distances 3 to 21.Source: Figure 17 · p. 13, Figure 16 · p. 13
  • Triangle-code logical error rates are generally much lower under bit-flip noise and close under bit-phase-flip noise, consistent with lowest-weight logical-operator counts.The triangle code has fewer lowest-weight operators for bit or bit-and-phase errors, but many more when correlated errors are allowed.Source: A Memory threshold · p. 13, Table 5 · p. 13

B Effective distance of syndrome extraction

The appendix argues that the triangle-code syndrome-extraction circuit has effective distance d, addressing a dangerous hook-error case at the central twist with two 2-CAT checks. The proof builds on full-distance surface-code extraction and first establishes a d−1 bound for the triangle code.

B Effective distance of syndrome extraction

  • In the surface-code circuit, hooks are oriented so each faulty component affects at most one column of X errors or one row of Z errors, requiring at least d faults for either logical operator.The corresponding logical Y requires at least 2d−1 faults because X and Z errors occupy disconnected decoding graphs.Source: B Effective distance of syndrome… · pp. 13–14, Figure 18 · p. 14
  • Triangle-code layers use surface-code patches with full-distance extraction, and staggering measurements between planes can only make an undetected logical error harder to place.Ignoring the staggered timing, the appendix argues for effective distance d of the extraction.Source: B Effective distance of syndrome… · p. 14
  • Two 2-CAT checks on central plaquettes detect the dangerous hooks at the twist and suffice for effective distance d.The mixed-type stabilizer ray must terminate in a twist at the origin; two hook faults there can otherwise create a weight-three error.Source: B Effective distance of syndrome… · p. 14
  • Restricting logical Y to codes x and z disconnects their X and Z decoding graphs, so it requires at least (d−1)/2 + (d+1)/2 − 1 = d−1 faults, establishing effective distance d−1.For code x, at least (d−1)/2 faults are required; for code z, at least (d+1)/2.Source: B Effective distance of syndrome… · p. 14

C exREC pseudothresholds

The section describes fault-tolerant syndrome-extraction designs and rigorous exREC pseudothreshold calculations for small codes. The calculations enumerate up to two faults, with the CDT criteria allowing incoming errors to be ignored when bounding logical failure.

C exREC pseudothresholds

  • Pseudothresholds are bounded by enumerating sets of at most two faults under an independent depolarizing circuit-noise model.The bounds are intended for small experiments; fault-set probabilities multiply the failure probabilities of faulty components by the success probabilities of all others.Source: C exREC pseudothresholds · pp. 14, 16
  • The exREC designs are fault-tolerant and satisfy the CDT criteria, so incoming errors do not affect the calculated logical failure rate.For distance-3 codes, a single fault in the exREC must not cause a logical error.Source: C exREC pseudothresholds · p. 15
  • The 5-qubit code uses one repeatedly prepared 3-CAT ancilla set for all stabilizers, yielding an 8-qubit completely fault-tolerant design.Verified 4-CAT extraction is unsuitable for this small design because it requires postselection.Source: C exREC pseudothresholds · p. 15, Figure 20 · p. 16
  • The smallest color-code design uses decoded 2-CATs to detect hook errors, while adding ancillas can increase parallelism and threshold at the cost of planarity.Each face has two ancillas, which are prepared and reprepared to measure both X- and Z-type stabilizers.Source: C exREC pseudothresholds · pp. 15–16, Figure 21 · p. 16
  • The Bacon-Shor design measures weight-two gauge operators with eight ancillas, extracting all X-type gauges before all Z-type gauges.Weight-two measurements cannot introduce more than weight-one data errors, making this procedure naturally fault-tolerant.Source: C exREC pseudothresholds · p. 16, Figure 22 · p. 17
  • Because error correction may conditionally require a third syndrome extraction, success probabilities are split into four cases according to whether each bracketing EC uses two or three extractions.Measurement exRECs replace the trailing EC with a measurement EC that repeats measurement syndrome extraction, adding another round if the first two disagree.Source: C exREC pseudothresholds · p. 17
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