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Quantum computing by color-code lattice surgery

Andrew J. Landahl, Ciaran Ryan-Anderson

arXiv:1407.5103v1quant-ph

TL;DR

Fault-tolerant lattice surgery had not been extended to color codes, despite their resource advantages over surface codes. The paper develops a universal color-code lattice-surgery scheme and improves surface-code methods, finding lower qubit use with equal or lower time and favorable suppression under sufficiently low noise and demanding targets.

  • Problem

    Existing lattice-surgery methods lacked an extension to color codes, despite color codes and lattice surgery each offering qubit-efficiency advantages.

    Source: §I · p. 1

  • Method

    The paper develops universal fault-tolerant quantum operations using color-code lattice surgery and compares their resource overheads with improved surface-code methods.

    Source: §I · pp. 1–2

  • Results

    Color-code lattice surgery uses approximately half the qubits and the same time or less per code distance, and uses fewer qubits for equal suppression when noise and suppression demands are sufficiently low and high, respectively.

    Source: A. Overhead per code distance · p. 8, B. Overhead per desired level of error… · p. 10, §V · p. 11

  • Takeaways & Limitations

    Color-code lattice surgery offers a resource-efficient route to fault-tolerant computation, including single-step transversal Hadamard and phase gates.

    Source: §V · p. 11

    Color codes have a lower accuracy threshold against circuit-level depolarizing noise than surface codes, which can require a larger code distance for equal suppression.

    Source: B. Overhead per desired level of error… · pp. 8–9

Abstract

from arXiv · show

We demonstrate how to use lattice surgery to enact a universal set of fault-tolerant quantum operations with color codes. Along the way, we also improve existing surface-code lattice-surgery methods. Lattice-surgery methods use fewer qubits and the same time or less than associated defect-braiding methods. Furthermore, per code distance, color-code lattice surgery uses approximately half the qubits and the same time or less than surface-code lattice surgery. Color-code lattice surgery can also implement the Hadamard and phase gates in a single transversal step---much faster than surface-code lattice surgery can. Against uncorrelated circuit-level depolarizing noise, color-code lattice surgery uses fewer qubits to achieve the same degree of fault-tolerant error suppression as surface-code lattice surgery when the noise rate is low enough and the error suppression demand is high enough.

I. INTRODUCTION

The paper motivates color-code lattice surgery as a resource-efficient route to fault-tolerant computation, addressing the missing extension of surface-code lattice surgery to color codes. It introduces triangular 4.8.8 color codes, syndrome-extraction choices, and the circuit-level noise setting used for comparison.

I. INTRODUCTION

  • Color-code lattice surgery fills the gap in extending lattice-surgery methods from surface codes to color codes, motivated by their qubit efficiency.The paper focuses on triangular 4.8.8 color codes and develops methods for universal fault-tolerant quantum computation.Source: §I · p. 1
  • Color codes provide transversal logical Hadamard and phase gates in one parallelized step, unlike surface codes on any lattice.Surface-code alternatives require multistep deformation for Hadamard and lengthy magic-state teleportation for phase.Source: §I · p. 1
  • Color codes have a lower estimated depolarizing circuit-level accuracy threshold than surface codes, but can still use fewer qubits at sufficiently low error rates and stringent suppression demands.The cited estimates are 0.143% for color codes versus 0.502(1)% to 1.140(1)% for surface codes.Source: §I · p. 1
  • 30 qubits implement the distance-three color-code CNOT with one syndrome qubit per face, or 22 with a single mobile syndrome qubit.The improved surface-code method uses 39 or 28 qubits under the same two syndrome-allocation choices.Source: §I · p. 2

II. BACKGROUND

  • Triangular 4.8.8 color codes encode one logical qubit using data qubits and face-based X- and Z-type stabilizer checks.Logical X and Z representatives can be chosen along the triangle’s sides.Source: §II · p. 2
  • Syndrome extraction trades qubits against speed and robustness, ranging from one mobile syndrome qubit to multiple qubits per face or data qubit.The paper generally compares one syndrome qubit per face with one per check.Source: §II · p. 2
  • The study focuses on circuit-level noise, where data qubits, syndrome qubits, and individual gates can fail, while abstracting hardware-specific gate details.An iid depolarizing model is described as a commonly used surrogate when hardware-informed error models are unavailable.Source: §II · pp. 2–3

III. UNIVERSAL GATE SET

This section defines a universal encoded gate set implemented through color-code lattice surgery under a local physical gate basis. Most operations become increasingly fault tolerant with code distance, while T|+⟩ preparation requires separate fidelity improvement.

III. UNIVERSAL GATE SET

  • Color-code lattice surgery implements a universal set of fault-tolerant encoded operations on triangular 4.8.8 color codes.The constructions use the notation of Ref. and assume nearest-neighbor CNOTs between data and ancilla qubits.Source: §III · p. 3
  • Pauli corrections can be tracked classically in a Pauli frame instead of being physically applied to data qubits.This follows because the selected operations are stabilizer operations together with T|+⟩ preparation.Source: §III · p. 3
  • All fault-tolerant constructions except T|+⟩ preparation become exponentially more fault tolerant as code distance increases.Magic-state distillation can improve the fidelity of T|+⟩ preparations using higher-fidelity operations from the rest of the gate set.Source: §III · p. 3

A. The identity gate I

The identity and computational-basis state preparations use repeated syndrome extraction and classical decoding to establish fault-tolerant encoded states. The |+⟩ preparation mirrors |0⟩ in the complementary Pauli basis.

A. The identity gate I

  • Fault-tolerant identity uses quantum error correction for d syndrome rounds followed by classical decoding to infer a corrective action.The method can use decoders described in Refs. [5] [16] [27] [28] [30].Source: A. The identity gate I · p. 3

B. Preparation of |0⟩and |+⟩states

  • Encoded |0⟩ preparation initializes every data qubit in |0⟩, then measures the syndrome d times and decodes the results.The resulting checks include the color-code Z checks and the encoded Z operator.Source: B. Preparation of |0⟩and |+⟩states · p. 3
  • Encoded |+⟩ preparation follows the same procedure as |0⟩ preparation but initializes physical qubits in +1 X eigenstates.This swaps the relevant preparation basis while retaining repeated syndrome measurement and decoding.Source: B. Preparation of |0⟩and |+⟩states · p. 3

C. Measurement MZ and MX

Logical measurements are performed destructively by measuring all data qubits in the corresponding Pauli basis and classically correcting the outcomes. Triangular color codes additionally support transversal Hadamard and phase gates.

C. Measurement MZ and MX

  • Encoded Z and X measurements each measure every data qubit once in the corresponding basis, followed by classical error correction.Both are destructive; nondestructive versions use Steane’s ancilla-coupled measurement method [21].Source: C. Measurement MZ and MX · pp. 3–4

D. Phase and Hadamard gates (S and H)

  • The transversal Hadamard is fault-tolerant because strong CSS structure swaps X- and Z-checks, while triangular geometry makes logical X and Z exchange as well.The construction applies specifically to triangular color codes, not all color codes on compact surfaces.Source: D. Phase and Hadamard gates (S and H) · p. 4
  • The phase gate S is transversal for 2D color codes, using S or S† according to whether the 4.8.8 code distance is 1 or 3 modulo 4.The allocation rule is S for distance 1 mod 4 and S† for distance 3 mod 4.Source: D. Phase and Hadamard gates (S and H) · p. 4

E. The CNOT gate

The color-code CNOT is implemented by fault-tolerant lattice-surgery measurements that reproduce a measurement-based CNOT circuit. The construction uses MXX and MZZ fusions, repeated syndrome extraction, splitting, and destructive ancilla measurement, with parallelization reducing its depth.

E. The CNOT gate

  • MXX and MZZ lattice-surgery measurements provide the missing operations needed to construct the encoded CNOT.The method follows either of two equivalent measurement-based CNOT circuits.Source: E. The CNOT gate · p. 4, Figure 2 · p. 4, Figure 3 · p. 4
  • The MXX and MZZ measurements connect adjacent triangular color codes by measuring interface checks whose lighter-colored outcomes determine the joint-measurement result.The pattern is shown for code distances 3 and 5 and generalizes to arbitrary distances.Source: E. The CNOT gate · p. 4, Figure 4 · p. 4, Figure 5 · p. 5
  • The fault-tolerant CNOT uses an ancilla prepared in Z = +1 eigenstates, followed by d-round syndrome extraction and sequential target-ancilla and control-ancilla fusions.Each fusion and subsequent separation is accompanied by d rounds of checks and fault-tolerant error correction.Source: E. The CNOT gate · p. 5
  • After the second fusion is split, the ancilla is destructively measured in the X basis and the result is classically error-corrected while control and target checks continue fault-tolerantly.This completes the measurement-based CNOT procedure.Source: E. The CNOT gate · p. 5, Figure 10 · p. 6, Figure 11 · p. 6
  • The unoptimized sequence takes 5d syndrome-extraction rounds, while parallelizing preparation, splitting, and destructive measurement reduces the depth to 3d.The intermediate reductions are 4d + 1 and 3d + 1 rounds before the final optimization.Source: E. The CNOT gate · p. 7

F. Preparation of T|+⟩states

The paper prepares fault-tolerant encoded T|+⟩ states by injecting a physical T|+⟩ qubit into a triangular 4.8.8 color code. Ancillary stabilizer resources and persistent checks suppress preparation errors while keeping the syndrome-extraction count independent of code distance.

F. Preparation of T|+⟩states

  • Code injection embeds a physical T|+⟩ qubit into triangular 4.8.8 color codes at distances d = 3, 5, and 7.The injection layouts are illustrated for each of these distances.Source: F. Preparation of T|+⟩states · p. 7, Figure 14 · p. 7, Figure 15 · p. 7
  • The initial step prepares an ancillary distance-(d−1) color-code stabilizer state, with two additional Bell pairs when d ≡ 3 mod 4.For d > 3, the Bell pairs are prepared by post-selection to O(p^2) error; d = 3 uses a separate three-Bell-state procedure.Source: F. Preparation of T|+⟩states · p. 7
  • The injection step extends the physical qubit’s Pauli operators into distance-d logical operators by introducing new boundary checks while other checks continue to be measured.The new checks accept the obtained syndrome values as correct, and the green boundary checks cease to be measured.Source: F. Preparation of T|+⟩states · pp. 7–8
  • Persistent checks detect the relevant single- and two-qubit faults, allowing three rounds of syndrome decoding to suppress errors to O(p^2) for d > 3.The d = 3 case can instead use post-selection over the full injection process.Source: F. Preparation of T|+⟩states · p. 8
  • Six syndrome-extraction rounds suffice for state injection, while the resulting error is O(p) with a multiplicative constant independent of code distance.Encoded magic-state distillation can further reduce the error after injection.Source: F. Preparation of T|+⟩states · p. 8

A. Overhead per code distance

The resource comparison evaluates color-code and improved surface-code lattice surgery using tabulated qubit counts, depths, and error scalings. Color-code methods achieve comparable error suppression with approximately half as many qubits and similar or lower time overhead.

A. Overhead per code distance

B. Overhead per desired level of error suppression

Color codes can require larger distances because of their lower accuracy threshold, reducing their nominal qubit advantage. Under the paper’s scaling assumptions, they nevertheless use fewer qubits at sufficiently low noise and sufficiently large distances.

B. Overhead per desired level of error suppression

C. Overhead for small logical CNOT gates

The small-CNOT comparison catalogs qubit overheads across color-code and surface-code methods under several syndrome-qubit allocations. It includes both the paper’s lattice-surgery methods and prior alternatives.

C. Overhead for small logical CNOT gates

  • At small distances, transversal methods are included despite their poor fit to local two-dimensional processing because nonlocal movement or processing may be available.A trapped-ion distance-three color-code demonstration illustrates that setting, but it was not fault-tolerant because syndrome measurements were not repeated into auxiliary qubits.Source: C. Overhead for small logical CNOT gates · pp. 10–11
  • The comparison considers one roving syndrome qubit, one per face, and one per check; for color codes, one per check equals two per face.Transversal methods also include an intermediate allocation of one syndrome qubit per two faces.Source: C. Overhead for small logical CNOT gates · p. 11
  • Table III compares qubit overheads for color-code and surface-code transversal and lattice-surgery CNOT methods, including the earlier surface-code method of Ref. [12].The table covers small code distances and uses the low end of the syndrome-allocation spectrum.Source: C. Overhead for small logical CNOT gates · p. 10, Table III · p. 11

V. CONCLUSIONS

Color-code lattice surgery offers a resource-efficient route to fault-tolerant quantum computation. Per code distance, it uses about half the qubits as surface-code lattice surgery with equal or lower time.

V. CONCLUSIONS

  • Color-code lattice surgery uses approximately half as many qubits as surface-code lattice surgery and performs logical operations in the same time or less.The conclusion also states that it uses fewer qubits and no more time than defect-based spacetime braiding for both code families.Source: §V · p. 11
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