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Lattice Surgery with a Twist: Simplifying Clifford Gates of Surface Codes

Daniel Litinski, Felix von Oppen

arXiv:1709.02318v2quant-phcond-mat.mes-hall

TL;DR

Surface codes support local stabilizer measurements but lack transversal single-qubit Clifford gates. This scheme tracks Clifford gates classically and uses twist-based lattice surgery for long-range multi-target CNOTs, eliminating Clifford time overhead and giving CNOT time overhead that scales as O(log s).

  • Problem

    Surface codes are compatible with local two-dimensional hardware but lack transversal single-qubit Clifford gates.

    Source: §1 · p. 1

  • Method

    The scheme tracks single-qubit Clifford gates classically on planar surface codes and uses twist-based lattice surgery for inter-qubit CNOTs.

    Source: §1 · pp. 1–2, §2 · p. 2

  • Results

    The scheme eliminates the time overhead of logical single-qubit Clifford gates and provides long-range multi-target CNOTs with O(log s) time and O(s log s) space overhead.

    Source: §3.2 · p. 5, §5 · p. 9

  • Takeaways & Limitations

    With distilled magic states, the scheme supports fault-tolerant non-Clifford gates, while Pauli-product measurements are the hardware operations that must be performed explicitly.

    Source: §1 · p. 2, §5 · p. 10

    Open questions include efficient decoding of wide, long, and double-sided qubits, survival-time estimates, and implementation in a concrete physical system.

    Source: §5 · p. 10

Abstract

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We present a planar surface-code-based scheme for fault-tolerant quantum computation which eliminates the time overhead of single-qubit Clifford gates, and implements long-range multi-target CNOT gates with a time overhead that scales only logarithmically with the control-target separation. This is done by replacing hardware operations for single-qubit Clifford gates with a classical tracking protocol. Inter-qubit communication is added via a modified lattice surgery protocol that employs twist defects of the surface code. The long-range multi-target CNOT gates facilitate magic state distillation, which renders our scheme fault-tolerant and universal.

1 Introduction

Quantum error correction can extend physical-qubit survival times, and surface codes suit local two-dimensional hardware but lack transversal single-qubit Clifford gates. The paper introduces edge tracking for planar surface codes and twist-based lattice surgery for long-range multi-target CNOTs.

1 Introduction

  • Quantum error correction encodes information in logical qubits made from many physical qubits, while topological codes use spatially local measurements compatible with two-dimensional hardware.Surface codes use four-qubit stabilizers for syndrome readout, compared with six-qubit operators in color codes.Source: §1 · p. 1
  • Surface codes lack transversal single-qubit Clifford gates, unlike color codes, and existing surface-code approaches use multi-step H measurements or distilled ancillas for S gates.Twist braiding can also be implemented by classical tracking, motivating a tracking-based alternative.Source: §1 · p. 1
  • The scheme applies classical tracking to planar surface codes instead of using twist-based encodings, calling the protocol edge tracking.Twists reappear in lattice-surgery protocols involving the logical YL operator.Source: §1 · pp. 1–2
  • Twist-based lattice surgery provides long-range multi-target CNOTs between edge-tracked qubits, which are useful for magic state distillation and fault-tolerant T gates.The T gate completes the universal gate set.Source: §1 · p. 2

2 Edge Tracking

Edge tracking replaces logical H and S gates with classical updates to the logical operators represented by the surface-code edges. Wide qubits make the logical operators accessible from one side for lattice surgery, at the cost of more physical qubits.

2 Edge Tracking

  • A surface-code qubit uses physical qubits on a square lattice, with face operators defining X- and Z-type stabilizers.Qubits can be initialized in logical |0L⟩ or |+L⟩ states and read out in the XL or ZL basis using physical-qubit measurements and classical error correction.Source: §2 · p. 2
  • Classical edge tracking replaces single-qubit Clifford gates by updating which logical Pauli operators the physical edge strings encode.H maps XL to ZL and ZL to XL; S maps XL to YL while leaving ZL unchanged.Source: §2 · p. 2
  • Wide qubits place X and Z edges on the same side, making XL, YL, and ZL accessible there for lattice surgery at the cost of more physical qubits.The illustrated H then S updates leave Xedge = ZL and Zedge = YL.Source: §2 · pp. 2–3

3 Lattice surgery with a twist

Lattice surgery builds CNOTs from logical parity measurements, then extends them to long-range, edge-tracked, and multi-target operations. The long-range protocol has O(log s) time overhead, while twist defects appear in surgery configurations for certain parity measurements.

3.1 Nearest-neighbor CNOT

  • Lattice surgery implements CNOT through successive logical Z⊗Z and X⊗X parity measurements using local stabilizer measurements, followed by ancilla readout and Pauli correction.The stabilizer products yield the logical parity outcomes; the resulting Clifford corrections can be handled by edge tracking.Source: §3.1 · pp. 3–4, Figure 4 · p. 3, §3 · p. 3

3.2 Long-range CNOT

  • For separated control and target qubits, two |+⟩ ancillas relay the ZL⊗ZL parity to an ancilla that then participates in the X⊗X measurement with the target.The long ancilla is measured in the X basis at the end, and its outcome determines a Z^m correction on the control.Source: §3.2 · p. 4, Figure 5 · p. 4
  • Long-range CNOT time overhead scales as O(log s), while maintaining fidelity requires ancilla width O(log s), giving O(s log s) space overhead.The required number of syndrome-measurement rounds also scales as O(log s) and depends on measurement fidelity.Source: §3.2 · pp. 4–5

3.3 CNOT between edge-tracked qubits

  • Edge-tracked targets can require surgery across different edge types, including stabilizer configurations that measure XL⊗XL when the target encodes its logical operators on different edges.For a target with Xedge=ZL and Zedge=YL, the measurement includes a Y-basis stabilizer on a qubit shared by both edges.Source: §3.3 · p. 6

3.4 Connection to twist defects

  • The modified surgeries can measure XL⊗XL, ZL⊗XL, or YL⊗XL; the YL measurement uses a five-qubit operator associated with a twist defect.One configuration is a dislocation line terminated by a five-qubit twist defect.Source: §3.3 · p. 6, §3.4 · p. 7

3.5 Long-range multi-target CNOT

  • Simultaneously initialized ancillas provide long-range multi-target CNOTs with the same time overhead as a single CNOT, supporting protocols such as magic state distillation.In the illustrated arrangement, ancillas 1, 3, and 5 serve three targets while ancillas 2 and 4 bridge distances.Source: §3.5 · pp. 7–8

4 2D arrangement of logical qubits

Double-sided qubits reduce the space overhead for two-dimensional arrangements, while block separation governs connectivity and space use. The magic-state-distillation example illustrates how multi-target CNOTs support a 15-to-1 conversion.

4 2D arrangement of logical qubits

  • Double-sided qubits reduce the space overhead from about 2d^2 to about d^2 physical qubits per logical qubit, but complicate initialization and readout.Lattice surgery enables initialization and readout in any Pauli basis; wide qubits remain an option when fast initialization and readout are needed.Source: §4 · p. 8
  • Greater separation between qubit blocks allows more ancillas and simultaneous multi-target CNOTs, increasing connectivity at the cost of space overhead.The example has three simultaneous CNOTs using two parallel ancilla lanes.Source: §4 · p. 8

4.1 Example: Magic state distillation

  • The 15-to-1 distillation example converts 15 faulty magic states into one better state using 34 CNOT gates grouped into five multi-target CNOTs.Each multi-target CNOT requires two code cycles for its lattice-surgery parity measurements, including repetitions for stabilizer measurement errors.Source: §4.1 · pp. 8–9
  • The five multi-target CNOTs can be implemented by partitioning the long ancilla qubit and applying the protocol in Fig. 7.The conversion's space overhead depends on magic-state code distances and ancilla width.Source: §4.1 · p. 9

5 Conclusion

The conclusion presents edge tracking and twist-based lattice surgery as a route to fault-tolerant universal computation with surface codes. It also identifies trade-offs against color codes, circuit-optimization goals, and unresolved implementation questions.

5 Conclusion

  • Edge tracking removes the time overhead of logical single-qubit Clifford gates, while twist-based lattice surgery provides long-range multi-target CNOTs with time overhead O(log s) and space overhead O(s log s).Surface-code qubits require about d^2 physical qubits per logical qubit and mostly weight-four stabilizers; connectivity depends on qubit-block separation and space overhead.Source: §5 · p. 9
  • Magic state distillation together with the scheme enables fault-tolerant universal quantum computation.The scheme supports full 2D connectivity between logical qubits, with connectivity governed by their separation.Source: §5 · p. 9
  • Color codes offer fast readout in all Pauli bases and lower space overhead, whereas surface codes require only weight-four stabilizer measurements except at twist defects.The text favors color-code schemes when higher-weight stabilizer measurements are not substantially harder, and surface-code schemes in settings such as superconducting qubits where they are difficult.Source: §5 · p. 9
  • A layer of Clifford gates followed by single-qubit measurements can be compressed into measurements of nonlocal Pauli products, with no preceding gate operations.The conclusion describes computation using resource-state distillation and Pauli-product measurements via multi-target CNOTs on edge-tracked qubits; Fig. 12 illustrates measuring a Pauli product with an ancilla and multi-target CNOT.Source: Figure 12 · p. 9, §5 · p. 10
  • Circuit optimization under the stated hardware assumptions targets fewer resource states and fewer layers of Pauli-product measurements.These are identified as the operations that cannot be relegated to a classical computer.Source: §5 · p. 10
  • Efficient decoding of wide, long, and double-sided qubits, survival-time estimates, and implementation in a concrete physical system remain open questions.The scheme may also be adapted to twist-based triangle codes to avoid triangle reorientation and further reduce surface-code space overhead.Source: §5 · p. 10

A Stabilizer measurements in concrete implementations

The appendix describes hardware-dependent ways to measure stabilizers in the twist-based surgery scheme, including operators acting on up to five qubits.

A Stabilizer measurements in concrete implementations

A.1 Majorana-based qubits

The appendix explains how local Majorana-operator measurements in tetron networks can implement stabilizer and twist-operator measurements.

A.1 Majorana-based qubits

  • Majorana-based qubits measure local Majorana products, which correspond to local Pauli products and can therefore measure the scheme's stabilizers.The tetron example encodes a qubit in the doubly degenerate ground-state space of four Majorana zero modes in a fixed parity sector.Source: §A.1 · p. 11
  • In a square tetron lattice, tunnel couplings form closed paths whose Majorana products are measured through spectroscopy of semiconductor quantum-dot energy levels.Each Majorana operator along the path is involved once in the virtual tunneling process.Source: §A.1 · p. 11
  • The described tunnel-coupling configurations can measure X and Z stabilizers, dislocation operators, and twist operators.The twist-operator path uses additional Majoranas in a fixed parity sector as coherent links.Source: §A.1 · p. 11

A.2 Non-topological qubits

For ancilla-mediated stabilizer readout, the section derives gate-ordering constraints that prevent hook errors from reducing effective code distance and gives a five-step schedule for a twist-based lattice-surgery configuration.

A.2 Non-topological qubits

  • Ancilla measurement qubits interact with each data qubit in a stabilizer through up to five successive two-qubit gates before readout.This readout yields the corresponding Pauli product.Source: §A.2 · p. 11
  • A measurement-qubit error can propagate into correlated data-qubit hook errors, with two data-qubit errors the worst case for effective distance.Gate ordering is chosen so a distance-d logical operator requires at least d physical errors.Source: §A.2 · pp. 11–12
  • Stabilizer orientations vary across the double-sided qubit: Z and X checks take Z shapes in designated regions and N shapes outside them.This arrangement prevents logical strings from forming with fewer than d physical errors.Source: §A.2 · p. 12, Figure 15 · p. 12
  • Each data qubit's four assigned two-qubit gates must occupy distinct time steps, chosen from five because the twist check has five qubits.The five-step assignment accommodates the largest check operator.Source: §A.2 · p. 12, Figure 15 · p. 12
  • Neighboring stabilizers' gate orders must preserve the intended measurement sequence, avoiding back-action that would produce random outcomes.The two CNOTs must be ordered consistently relative to the neighboring stabilizer's gates on each shared edge.Source: §A.2 · pp. 12, 14, Figure 15 · p. 12
  • A five-step gate ordering satisfies the three readout constraints for a configuration with bulk stabilizers, a dislocation line, and a twist defect.The configuration corresponds to twist-based lattice surgery between a double-sided and a standard rectangular qubit.Source: Figure 15 · p. 12, Figure 16 · p. 13, §A.2 · p. 14

B Magic state distillation protocol

The protocol lays out five multi-target CNOTs for 15-to-1 magic state distillation. It specifies inverted edge encodings and distinguishes the shown Z-parity measurements from subsequent X-parity measurements.

B Magic state distillation protocol

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