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Surface code quantum computing by lattice surgery
Dominic Horsman, Austin G. Fowler, Simon Devitt, Rodney Van Meter
TL;DR
Planar surface codes reduce qubit requirements but previously depended on transversal interactions that conflict with strictly two-dimensional nearest-neighbour architectures. The paper introduces lattice surgery, which couples planar code surfaces through splitting and merging while supporting universal operations. It demonstrates encoded entanglement and a distance-3 CNOT using 53 physical qubits, while noting that resource savings are comparatively modest.
Problem
Planar codes use fewer physical qubits but their transversal two-qubit gates are unsuitable for systems restricted to two-dimensional nearest-neighbour interactions.
Method
The paper introduces lattice surgery, deforming and combining planar code surfaces through cutting, stitching, splitting, and merging operations.
Results
The method supports universal operations including magic-state injection and demonstrates a distance-3 planar CNOT using 53 physical qubits.
Takeaways & Limitations
Lattice surgery preserves nearest-neighbour structure while enabling planar-code coupling and reducing resources for logical operations and distributed entanglement.
Takeaways & Limitations
The paper characterizes the qubit-resource savings as comparatively modest.
Abstract
from arXiv · showhide
In recent years, surface codes have become a leading method for quantum error correction in theoretical large scale computational and communications architecture designs. Their comparatively high fault-tolerant thresholds and their natural 2-dimensional nearest neighbour (2DNN) structure make them an obvious choice for large scale designs in experimentally realistic systems. While fundamentally based on the toric code of Kitaev, there are many variants, two of which are the planar- and defect- based codes. Planar codes require fewer qubits to implement (for the same strength of error correction), but are restricted to encoding a single qubit of information. Interactions between encoded qubits are achieved via transversal operations, thus destroying the inherent 2DNN nature of the code. In this paper we introduce a new technique enabling the coupling of two planar codes without transversal operations, maintaining the 2DNN of the encoded computer. Our lattice surgery technique comprises splitting and merging planar code surfaces, and enables us to perform universal quantum computation (including magic state injection) while removing the need for braided logic in a strictly 2DNN design, and hence reduces the overall qubit resources for logic operations. Those resources are further reduced by the use of a rotated lattice for the planar encoding. We show how lattice surgery allows us to distribute encoded GHZ states in a more direct (and overhead friendly) manner, and how a demonstration of an encoded CNOT between two distance 3 logical states is possible with 53 physical qubits, half of that required in any other known construction in 2D.
1. Introduction
Surface codes offer fault-tolerant quantum error correction with 2D nearest-neighbour structure, but planar codes previously required transversal gates for interactions. The paper introduces lattice surgery to couple planar codes while preserving locality and reducing resources.
- Motivation: Surface codes encode logical qubits in entangled states of many physical qubits, with code distance controlling error suppression.Below the threshold, increasing physical-qubit count can exponentially suppress logical errors.
- Motivation: Planar codes use fewer qubits than defect-based codes but previously required transversal gates between separate code surfaces.This requirement made planar encodings problematic for systems restricted to two-dimensional nearest-neighbour interactions.
- Contribution: Lattice surgery couples planar code surfaces by deforming and combining them through cutting and stitching operations.The technique is introduced to address the locality problem without transversal operations.
- Capabilities: The method supports magic-state injection, a direct Hadamard construction, planar–defect conversion, and entangled Bell and GHZ states.These operations extend lattice surgery beyond a single two-qubit coupling primitive.
- Result: 53 physical qubits suffice for the smallest described lattice-surgery CNOT between two distance-3 planar qubits.The construction is reported as half the physical-qubit requirement of the smallest known defect-encoded CNOT operation.
2. Surface codes
Surface codes use a two-dimensional lattice of data and syndrome qubits to protect logical information. Logical qubits can be encoded with planar boundaries or defects, while conventional interactions rely on transversal gates or defect braiding.
- Lattice and error correction: Surface-code lattices contain data qubits for computation and syndrome qubits that are repeatedly measured to detect errors.Syndrome qubits occupy plaquette centres and measure four-party stabilizers during error-correction rounds.
- Lattice and error correction: Syndrome changes identify endpoints of local error chains, and minimum-weight perfect matching infers likely corrective operations.Corrections are applied to classical measurement data rather than directly to physical qubits.
- Logical encodings: Planar encoding uses rough and smooth boundaries to create one logical qubit, whereas defect-based encoding removes stabilizers and commonly uses double defects.Logical operators are defined by the resulting boundary or defect structure.
- Logical encodings: Code distance is the length of the smallest undetectable error chain and determines the code’s ability to correct physical errors.A distance-3 surface code can detect and correct a single physical error.
- Logical operations: Planar CNOTs were performed transversally between corresponding physical qubits, while defect-based CNOTs use defect braiding.Lattice surgery removes transversal two-qubit operations and introduces split and merge operations for nearest-neighbour computation.
3. Lattice surgery
Lattice surgery couples planar code surfaces through merging and splitting, using stabilizer measurements and error-correction rounds to preserve fault tolerance while changing the number and arrangement of logical qubits.
- Core operations: Lattice surgery introduces discontinuous deformations that merge or split planar code lattices instead of continuously deforming them.Merging joins surfaces; splitting cuts joint stabilizers to create additional boundaries.
- Lattice merging: Two logical surfaces are merged by preparing intermediate qubits in |0⟩ and performing d rounds of error correction over the combined surface.The intermediate qubits form the connection between the separately stabilized code surfaces.
- Lattice merging: A rough merge measures the joint logical operator XLXL and maps two input logical qubits to one post-merge logical qubit.The measurement outcome determines the resulting state and required classical correction; corrections are tracked through later measurement interpretation.
- Lattice merging: Merge truth tables implement XOR in the computational basis for rough merges and XOR in the Hadamard basis for smooth merges.Smooth merging prepares intermediate qubits in |+⟩ before measuring the joint operators.
- Fault tolerance: For a distance d code, d error-correction rounds are required for fault-tolerant merging, while the merge preserves the original code distance in this configuration.The merged operation is fault-tolerant and preserves the code space, but it is non-unitary because two logical qubits become one.
- Lattice splitting: Smooth splitting measures an intermediate row in the X basis, producing two separately stabilized surfaces whose logical X states remain entangled.A symmetric split can halve code distance; achieving two distance-d surfaces requires an initial d × 2d surface.
4. Universal gate operations with lattice surgery
Lattice surgery implements universal planar-code computation by merging and splitting surfaces, including fault-tolerant CNOTs, magic-state injection, and Hadamard operations.
- Lattice surgery constructs a universal gate set from a logical CNOT and arbitrary single-qubit rotations using magic-state distillation and injection.
- 4.1. The CNOT gate: The CNOT layout uses control and target surfaces that merge and split with an intermediate |INT⟩=|+⟩L surface.
- 4.3. The Hadamard gate: A transversal Hadamard rotates the planar surface, while expansion, error correction, and Z-basis contraction restore the desired orientation fault-tolerantly.
- 4.2. State injection: State injection prepares α|000⟩+β|111⟩, stabilizes it into a distance-3 logical state, then merges additional |0⟩ qubits to reach distance 4 or higher.
5. Relationship to defect qubits
Planar and defect-based logical qubits can be converted into one another by measuring, enlarging, and removing selected lattice regions.
- A defect-based qubit near a lattice edge can be extruded into a separate planar qubit, connecting the two surface-code encodings.
- For a smooth double defect, Z-basis measurements isolate the defect, the defects are enlarged, and unstabilized qubits are removed to form a planar qubit.
- The extraction procedure also works in reverse, allowing planar and defect-based qubits to be interchanged.
6. Resource use of the planar code
Planar codes reduce physical-qubit area relative to defect-based codes, especially for individual qubits and small or medium-scale operations, though scalable layouts narrow the large-scale advantage.
- A distance-d planar qubit uses approximately 2d^2 physical qubits, versus approximately 6d^2 for a double-defect qubit.
- A lattice-surgery planar CNOT reduces leading-order physical-qubit use by around 6 times relative to the scalable double-defect construction.
- Scalable planar layouts reserve blank surfaces for CNOT operations, leaving only a quarter of available surfaces for logical data qubits.
- For large-scale quantum computers, planar and defect implementations have very little difference in overall resource requirements.
- Single-defect implementations usually use 1.5 to 2 times the qubits of planar implementations, making the difference significant at smaller scales.
7. Small scale experiments on the planar code
The section develops rotated planar lattices and lattice-surgery operations for small-scale experiments, including encoded entanglement and a low-resource logical CNOT. Rotated encoding preserves code distance while reducing physical-qubit requirements.
- The rotated lattice: A rotated lattice reduces physical-qubit requirements for a planar surface while retaining the original error-correction strength.The shortest logical-operator paths remain the same length after rotation.
- The rotated lattice: A distance-3 rotated planar qubit uses 9 data qubits and 8 measured syndromes.The encoding can use either 8 syndrome qubits or an equivalent measurement arrangement described in the paper.
- Creating entangled states: Lattice splitting generates encoded Bell pairs and higher-dimensional GHZ states without requiring more complicated two-qubit logical operators.Smooth splitting produces computational-basis GHZ states, while rough splitting can produce Hadamard-basis GHZ states.
- Logical CNOT: The rotated lattice reduces the smallest single-defect logical CNOT from 143 to 104 physical qubits.The comparison is between the standard and rotated implementations of the single-defect construction.
- Logical CNOT: 53 physical qubits implement the lattice-surgery CNOT between two distance-3 logical qubits.The construction uses 33 data qubits and 20 syndrome qubits, halving the smallest known CNOT implementation cited in the paper.
8. Conclusions
The paper concludes that lattice surgery couples planar-code logical qubits while preserving the 2D nearest-neighbour structure, fault tolerance, and universality. It also supports resource-efficient entanglement and logical operations, with particular relevance to near-term experiments and distributed settings.
- Conclusions: Lattice surgery couples multiple planar-code logical qubits without transversal protocols or breaking the 2D nearest-neighbour error-correction structure.The method uses deforming and combining planar code surfaces.
- Conclusions: Qubit savings are comparatively modest but remain advantageous for short- to medium-term experiments and prototype systems.The conclusion frames the strongest resource benefit around smaller implementations rather than large-scale systems.
- Conclusions: Boundary-only interactions reduce the number of distributed Bell states needed for logical operations between distributed planar qubits.This relaxes requirements on the repeater network connecting distributed systems.
- Conclusions: Encoded Bell and GHZ states and the smallest logical operation correcting one arbitrary error are identified as early experimental targets for 50–100-qubit systems.The paper presents these operations alongside its universal gate set.
Appendix A. Stabilizer description of merging for a distance-2 code
The appendix gives a stabilizer-level account of merging two distance-2 code surfaces, tracking measurement outcomes and the resulting stabilized merged state.
- Initial stabilizers: The merge begins with two logical planar surfaces and an intermediate qubit prepared in |0⟩.The initial state is expressed as a tensor product of the two logical states and the intermediate physical qubit.
- Initial stabilizers: Stabilizers are written as combinations of terms corresponding to the four logical-state components before the merge.The appendix explicitly analyzes the αα′|0⟩_L ⊗ |0⟩ ⊗ |0⟩_L component.
- Join measurements: Measuring stabilizers across the join merges the surfaces and introduces measurement-dependent signs into the resulting stabilizer representation.The appendix identifies outcomes m and m′ for the cross-join measurements.
- Merged state: The resulting stabilizers describe the fully stabilized new surface together with its logical operator state.The appendix rewrites the merged state using the stabilizer representations introduced earlier.
Appendix B. Stabilizer description of splitting for a distance-2 code
The appendix gives a stabilizer description of splitting a distance-2 code surface into two planar surfaces, including the pre-split stabilizers and the measurement producing the post-split state.
- Splitting construction: The appendix specifies the complete stabilizer set for smoothly splitting one code surface into two distance-2 planar surfaces.The operation is associated with the surface-splitting construction shown in Figure B1.
- Pre-split state: The stabilizers before splitting are listed as the starting description of the single surface.The appendix presents the pre-split stabilizer operators before introducing the measurement step.
- Measurement step: Measuring the intermediate qubit in the X basis, with outcome m, produces the first post-split stabilizer term.The outcome determines the sign appearing in the resulting state description.
- Post-split state: The post-split stabilizers are organized into complete sets for the first and second surfaces.The notation [S1(2)] denotes the complete stabilizer set for the first (second) surface after splitting.