Poking holes and cutting corners to achieve Clifford gates with the surface code

Benjamin J. Brown; Katharina Laubscher; Markus S. Kesselring; James R. Wootton

Source-linked AI summary

Poking holes and cutting corners to achieve Clifford gates with the surface code

Benjamin J. Brown, Katharina Laubscher, Markus S. Kesselring, James R. Wootton

arXiv:1609.04673v5Sep 15, 2016quant-phcond-mat.str-el

TL;DR

The paper addresses the resource cost and variety of fault-tolerant Clifford operations in the surface code. It unifies lattice-surgery and defect-based schemes through a planar-corner–twist correspondence, then uses code deformation and hybrid encodings to implement Clifford operations. The resulting constructions include single-qubit Clifford gates and the full Clifford group on planar-code architectures without reducing code distance.

Problem
Fault-tolerant quantum computation requires many physical qubits, motivating schemes that reduce resource costs while implementing the Clifford gates needed for universal computation.
Method
The paper builds a unified surface-code framework relating lattice surgery, holes, planar-code corners, twist defects, and code deformations, including hybrid logical encodings.
Results
The constructions implement single-qubit Clifford gates by exchanging planar-code twists and realize the full Clifford group using code deformations, including an implementation without decreasing code distance.
Takeaways & Limitations
Planar-code code deformations provide an alternative to ancilla-mediated completion of the Clifford group and support hybrid hole–twist schemes.

Abstract

from arXiv · show

The surface code is currently the leading proposal to achieve fault-tolerant quantum computation. Among its strengths are the plethora of known ways in which fault-tolerant Clifford operations can be performed, namely, by deforming the topology of the surface, by the fusion and splitting of codes and even by braiding engineered Majorana modes using twist defects. Here we present a unified framework to describe these methods, which can be used to better compare different schemes, and to facilitate the design of hybrid schemes. Our unification includes the identification of twist defects with the corners of the planar code. This identification enables us to perform single-qubit Clifford gates by exchanging the corners of the planar code via code deformation. We analyse ways in which different schemes can be combined, and propose a new logical encoding. We also show how all of the Clifford gates can be implemented with the planar code without loss of distance using code deformations, thus offering an attractive alternative to ancilla-mediated schemes to complete the Clifford group with lattice surgery.

I. INTRODUCTION

The introduction motivates low-overhead fault-tolerant computation and presents a unified surface-code framework linking lattice surgery, defect encodings, and code deformation. It proposes combining holes and twists while completing the Clifford group in two-dimensional architectures.

Quantum technologies are noise-sensitive, so error correction requires substantial physical-qubit resources and is essential for scalable quantum information processing.
Fault-tolerant schemes should minimize physical-qubit costs across both the chosen error-correcting code and the implementation of a universal gate set.
Topological quantum computation protects information using non-local degrees of freedom such as anyons, punctures, and twist defects, with gates implemented through braiding or lattice operations.
The paper unifies surface-code lattice surgery and defect encodings through a correspondence between planar-code corners and twist defects.
The framework targets the full Clifford group in two dimensions using lattice surgery and code deformation, while also examining hole–twist interactions and hybrid logical encodings.
Existing alternatives trade resource costs against engineering demands, including Y-state ancillas, higher-weight stabilizers, three-dimensional layouts, or reduced threshold error rates.

III. ENCODING QUBITS WITH THE SURFACE CODE

This section introduces stabilizer-code notation for the surface code, defining code spaces through stabilizers and logical operators while relating code distance to noise tolerance. It also explains that stabilizer multiplication can change logical-operator support without changing its action.

A stabilizer code encodes robust quantum states in the common +1 eigenspace of an Abelian stabilizer group within the Pauli group.
Logical operators commute with every stabilizer but lie outside the stabilizer group, generating logical Pauli operators with anticommutation for the same qubit.
Code distance d is the smallest support of a nontrivial logical operator and provides a first-order measure of the code’s ability to tolerate noise.
Multiplying a logical operator by a stabilizer preserves its action on all code states, allowing the operator’s support to be cleaned or relocated.

B. Anyons

The anyon framework describes topological excitations and their exchanges, which provide the physical language for the paper’s error-correction schemes. It distinguishes the surface code’s Abelian anyons from non-Abelian Ising anyons associated with Majorana modes.

Anyons are point-like quasiparticles confined to two dimensions, where restricted motion permits exotic exchange behavior.
The surface code uses the D(Z2) anyon model, containing e, m, ψ, and vacuum particles; e and m are self-antiparticles whose pairs fuse according to the model’s rules.
Exchanging e with m produces a nontrivial −1 phase, whereas exchanging identical e or m pairs is bosonic and produces a trivial phase.
The ψ particle combines e and m and has fermionic exchange behavior, acquiring a −1 phase upon exchange.
The Ising model contains non-Abelian σ anyons and fermionic ψ particles; fusing two σ anyons can yield either vacuum or a fermion.
σ anyons correspond to Majorana modes, whose parity and monodromy operations are equivalent to logical Pauli operations used in the work.

C. The Clifford group

The Clifford group consists of operations that map Pauli operators to Pauli operators under conjugation and is generated by phase, Hadamard, and controlled-not gates.

Clifford operations map elements of the Pauli group onto elements of the Pauli group under conjugation.
The Clifford group is generated by the single-qubit phase and Hadamard gates together with the two-qubit controlled-not gate.
The controlled-not gate has an operator expression involving tensor products of identity, Z, and X Pauli matrices.
The phase, Hadamard, and controlled-not gates are characterized by their respective actions on Pauli matrices.

D. The planar code

The planar code places qubits on a square lattice and encodes one logical qubit through stabilizers and string-like logical operators connecting boundaries of matching type.

The planar code uses an L × L square lattice with one qubit per vertex and has distance d = L.
The stabilizer group contains two face-operator types, with boundary stabilizers defined by adding faces around the lattice.
Boundary stabilizer choices define rough boundaries with black faces and smooth boundaries with white faces.
The code encodes one logical qubit, with Z extending between rough boundaries and X extending between smooth boundaries.
Logical-string paths can be deformed without changing their terminal boundaries by multiplying them by stabilizer operators.
String-like Pauli operators create electric or magnetic excitations in the bulk, whereas boundary-terminating logical strings are absorbed at rough or smooth boundaries.

E. Encoding logical qubits using holes

Introducing punctures increases the number of encoded qubits: a rough-boundary hole supports logical operators given by a cycle around the hole and a string connecting it to the lattice boundary.

A puncture removes qubits from the lattice and allows a single central hole to encode one logical qubit in an otherwise non-encoding rough-boundary lattice.
The hole’s logical X is a Pauli-X cycle enclosing the puncture, while logical Z is a Pauli-Z string connecting the puncture boundary to the lattice boundary.
The hole and lattice boundary both absorb electric charges, so the hole’s logical state records the parity of charges transferred between them.
Maintaining code distance d requires same-type holes to be separated by at least d, kept d from the lattice boundary, and given boundaries of length at least d.
Smooth-boundary holes absorb magnetic rather than electric particles, while this discussion uses only rough-boundary holes.

F. Encoding logical qubits with twist defects

Twist defects encode logical qubits through modified stabilizers and defect-line string operators, and they are equivalent to planar-code corners where rough and smooth boundaries meet.

Twist defects are introduced by removing qubits along defect lines, with twists at their endpoints and modified stabilizers along the lines.
Four twist defects collectively encode one qubit whose logical operators are represented by Pauli-string operators associated with the defects.
Twist defects mimic Ising anyons: one logical operator transports a fermion between twists, while another measures fermionic-charge parity.
Crossing a defect line changes red Pauli-Z strings to blue Pauli-X strings and vice versa, and a loop around one twist must wind twice to close.
Logical operators for four twists can be deformed to terminate at boundaries while preserving their anticommutation.
Twists at planar-code boundaries have logical operators identical to those of the planar code, identifying twists with corners where rough and smooth boundaries meet.

IV. LOGICAL OPERATIONS BY MANIPULATING CORNERS

Code deformation changes a surface code’s stabilizer group through carefully chosen measurements, allowing planar-code corners to move into the lattice bulk as twist defects. The procedure preserves encoded information by avoiding measurements that act as logical operators and by cleaning logical operators away from measured qubits.

Code deformation: Code deformation maps a stabilizer group S to a new group S′ by measuring elements of S′, enabling logical rotations.The deformation must avoid measuring logical information encoded by the original code.
Fault-tolerant deformation: Sequential low-weight measurements can deform the code while logical operators are cleaned away from the measured qubits.This supports fault-tolerant manipulation without directly measuring encoded information.
Code deformation: When a newly measured element anticommutes with existing stabilizers, those stabilizers are replaced by pairwise products, with a possible unitary correction for outcome −1.This updates the code space while accounting for the measurement outcome.
Moving corners: Single-qubit Pauli measurements along a defect line move a planar-code corner into the lattice bulk, with mostly Pauli-Y and some Pauli-X and Pauli-Z measurements.The measured line projects its qubits into a product state, disentangling them from the code.
Moving corners: The deformation modifies stabilizers near defect lines and corners, including weight-three, weight-four, and weight-five operators.Qubits projected into product states are removed from the code.

C. Single-qubit Clifford rotations on the planar code by code deformation

Treating planar-code corners as twist defects allows code deformation to exchange them and rotate logical Pauli operators. These exchanges generate the single-qubit Clifford group, while rotated lattice geometry limits the associated distance loss to a small constant.

Corner exchanges: Exchanging two adjacent corner twists maps Z → Y and Y → Z up to phases while leaving X invariant.The logical operator is deformed away from measured qubits and twist trajectories during the exchange.
Corner exchanges: The exchange action is equivalent up to phases to braiding two Ising anyons and realizes a square-root-of-X operation.The equivalence connects planar-code corner exchanges with twist-defect braiding.
Clifford generation: Exchanging twists on the left maps X → Y and Y → X while preserving Z, and combinations of exchanges yield S and H up to Pauli rotations.Because Pauli operations can be implemented transversally or tracked in the Pauli frame, these exchanges recover the single-qubit Clifford group.
Distance and geometry: On the square geometry, twist trajectories maintain separation of at least ∼L/2, giving code distance d ∼O(L/2) during the operations.The square geometry modifies the distance through the exchanges but keeps it proportional to lattice size.
Distance and geometry: A rotated lattice geometry permits corner exchanges with no more than a small constant loss in code distance.This geometry uses 2d2 qubits for distance d, compared with d2 for the square geometry.

D. Encoding two qubits on a single planar code

A single planar code can encode two logical qubits using six boundary twist defects. Exchanging twists within each encoding gives single-qubit Clifford operations, while exchanging central twists implements an entangling gate equivalent to CNOT up to local Clifford rotations.

Multi-qubit encoding: Adding corners to the planar code enables multiple logical qubits and entangling gates through corner braiding.The construction is presented as a way to combine single-qubit and two-qubit operations in one code.
Multi-qubit encoding: Six boundary twist defects encode two logical qubits, with three twists assigned to each qubit and logical operators X1, Z1, X2, and Z2.Each qubit’s single-qubit Clifford operations arise from exchanging its associated twists.
Entangling operation: Exchanging the two central twists performs an entangling operation between the two encoded qubits.The central pair is associated with the joint logical operator X1X2 under the even-parity fusion constraint.
Entangling operation: The central-twist exchange is equivalent to a controlled-not gate up to local Clifford rotations.The equivalence is established using methods analogous to those used for twist braiding.
Resource scaling: Both single-qubit and entangling operations can be performed on an L×2L square lattice with ∼2d2 physical qubits and code distance ∼d.The proposed exchanges can be arranged without decreasing the code distance.

V. ENTANGLING DIFFERENT TYPES OF LOGICAL QUBITS

The paper entangles a quadruple-twist qubit with a hole-pair qubit by deforming one hole around two twists. The deformation transforms logical operators as required for a controlled-not gate while preserving code distance under stated spacing conditions.

A. Braiding twists and holes: A hole-pair qubit can be entangled with a quadruple-twist qubit by moving one hole along a trajectory enclosing two twists.The four-twist qubit is the control and the hole-pair qubit is the target.
A. Braiding twists and holes: Hole motion is implemented by enlarging and shrinking the puncture through physical-qubit and stabilizer measurements without changing its topology.The prescription can be repeated to transport holes along prescribed paths.
A. Braiding twists and holes: The deformation maps ZT to ZCZT because the logical operator follows the hole and threads between twists as the hole crosses a defect line.Crossing the defect line changes the hole boundary type and the Pauli strings it can absorb.
A. Braiding twists and holes: The same process maps XC to XCXT, while leaving ZC and XT unchanged, satisfying the controlled-not transformation rules.The resulting string operator is equivalent to XCXT up to stabilizer multiplication.
A. Braiding twists and holes: The controlled-not operation preserves distance d when holes have width O(d/4), separation ∼d, and twists are mutually separated by d.A controlled-phase gate can also be obtained by deforming the hole around the other pair of twists.
A. Braiding twists and holes: The deformation and subsequent logical measurement can be interpreted as gauge fixing that teleports logical information between different encodings.The two logical qubits can be viewed as one logical qubit and a gauge qubit in this interpretation.

B. Entangling twist qubits

A hole-pair ancilla enables parity measurements between quadruple-twist qubits, which in turn implement controlled-not gates through a sequence of fault-tolerant measurements. These operations, combined with the paper’s other methods, generate the Clifford group.

B. Entangling twist qubits: A hole-pair ancilla can mediate parity measurements between two quadruple-twist qubits by sequentially entangling with both and then measuring the ancilla.The ancilla is used to transfer the parity information to a measurable logical degree of freedom.
B. Entangling twist qubits: The required controlled-not protocol measures ZCZA, followed by XAXT, after preparing the ancilla in the +1 eigenstate of XA.The ancilla is then measured in the computational basis.
B. Entangling twist qubits: The ZCZA measurement braids a hole around two twists of the ancilla and two twists of the control, transforming Xh into an operator equivalent to ZCZAXh.The equivalence holds up to stabilizers.
B. Entangling twist qubits: The XAXT measurement uses a high-weight string enclosing two horizontally aligned twists from both the ancilla and target qubits.An additional hole-pair qubit is used to measure this operator fault-tolerantly.
B. Entangling twist qubits: A final ZA measurement determines the Pauli correction, and these parity-measurement methods combine with earlier procedures to generate all Clifford gates.The scheme is presented as an alternative measurement-based route for entangling twist qubits.

VI. LATTICE SURGERY WITHIN THE TWIST FRAMEWORK

The twist framework identifies lattice surgery as a measurement-only realization of entangling operations involving planar-code corners viewed as Majorana modes. The paper also points toward extending this correspondence to other topological codes.

VI. LATTICE SURGERY WITHIN THE TWIST FRAMEWORK: Planar-code corners can be regarded as Majorana modes, making lattice surgery reminiscent of measurement-only entangling gates involving twist defects.The correspondence links the two descriptions within the same surface-code framework.
VI. LATTICE SURGERY WITHIN THE TWIST FRAMEWORK: Lattice surgery entangles planar codes through fault-tolerant logical parity measurements while allowing codes to remain separated when not interacting.This separation gives the architecture a modular character.
VI. LATTICE SURGERY WITHIN THE TWIST FRAMEWORK: The lattice-surgery construction uses ZCZA and XAXT measurements on three planar codes containing control, ancilla, and target qubits.The figure contrasts these codes with additional qubits used in the corresponding measurement-only scheme.
VI. LATTICE SURGERY WITHIN THE TWIST FRAMEWORK: The paper proposes exploring the twist interpretation of lattice surgery in other topological models, including the color code.Such extensions are presented as directions for finding low-resource fault-tolerant schemes.
VII. A HYBRID ENCODING SCHEME: Hybrid qubits combine holes and twist defects, support ancilla-free entangling operations, and provide one non-trivial single-qubit Clifford operation.Their logical operators use one rough hole, one smooth hole, and a pair of twist defects.

A. A hybrid qubit

Hybrid qubits combine holes and twist defects to provide complementary logical operations and enable transitions among hole-pair, twist-quadruple, and hybrid encodings. The paper leaves optimal physical-to-logical encoding rates open.

A. A hybrid qubit: A hybrid qubit uses one rough hole, one smooth hole, and a pair of twist defects, with logical operators constrained by the defect line.The Z operator connects the two holes through a twist defect.
A. A hybrid qubit: Exchanging the smooth hole of one hybrid qubit with the rough hole of another entangles the two hybrid qubits.The operation uses the distinct boundary types of the holes.
A. A hybrid qubit: Braiding the two holes or exchanging the two twist defects implements the single-qubit Clifford operation B1.These operations provide alternative realizations within the hybrid encoding.
A. A hybrid qubit: The optimal rate for encoding logical qubits into physical qubits remains an open problem.The paper suggests combined puncture-and-dislocation encodings could improve known rates but does not calculate the optimum.
A. A hybrid qubit: Code deformation and logical measurements can switch fault-tolerantly among hybrid, twist-quadruple, and hole-pair encodings.The procedures braid holes around twists and use string or loop measurements to teleport the logical information.
A. A hybrid qubit: The three encodings have complementary capabilities: twist qubits provide all single-qubit Clifford operations, whereas hole-pair schemes support other operations with different resource requirements.The paper frames code switching as a way to exploit these complementary properties.