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Universal transversal gates with color codes - a simplified approach

Aleksander Kubica, Michael E. Beverland

arXiv:1410.0069v1quant-ph

TL;DR

The paper addresses how topological color codes can support fault-tolerant universal computation without the overhead associated with some other approaches. It gives a simplified rigorous construction of color codes in arbitrary dimensions, derives transversal phase gates, and combines gauge fixing with transversal CNOT to obtain a universal set in three dimensions.

  • Problem

    Topological codes protect against local errors, but fault-tolerant universal computation can require substantial overhead, motivating color-code constructions with suitable logical gates.

  • Method

    The paper gives an explicit higher-dimensional color-code construction, analyzes transversal gates including R_d, and uses gauge fixing to switch between codes with different transversal gates.

  • Results

    In three dimensions, gauge fixing combines transversal H and CNOT with R3 to yield the fault-tolerant universal gate set {H, CNOT, R3}.

  • Takeaways & Limitations

    The construction provides a simplified route to universal fault-tolerant computation with color codes, and the presentation identifies Reed–Muller codes as members of the corresponding fractal color-code family.

Abstract

from arXiv · show

We provide a simplified, yet rigorous presentation of the ideas from Bombín's paper "Gauge Color Codes" [arXiv:1311.0879v3]. Our presentation is self-contained, and assumes only basic concepts from quantum error correction. We provide an explicit construction of a family of color codes in arbitrary dimensions and describe some of their crucial properties. Within this framework, we explicitly show how to transversally implement the generalized phase gate $R_n=\text{diag}(1,e^{2πi/2^n})$, which deviates from the method in "Gauge Color Codes", allowing an arguably simpler proof. We describe how to implement the Hadamard gate $H$ fault-tolerantly using code switching. In three dimensions, this yields, together with the transversal $CNOT$, a fault-tolerant universal gate set $\{H,CNOT,R_3\}$ without state-distillation.

I. INTRODUCTION

The paper presents a self-contained simplification of gauge-color-code ideas aimed at fault-tolerant universal computation with topological color codes. It introduces two-dimensional color codes and extends the construction to higher dimensions, emphasizing transversal gates and gauge fixing.

  • Motivation: Topological codes protect encoded information from local noise, but fault-tolerant universal computation can require substantial overhead.The paper motivates color codes as a route toward processing as well as storing quantum information.
  • Contributions: Gauge fixing switches between a stabilizer color code with transversal CNOT and R_d and a subsystem color code with transversal H.For d ≥ 3, the resulting set {H, CNOT, R_d} is universal.
  • Contributions: The presentation follows Bombín’s construction while providing a simplified and rigorous account of its main ideas.The paper focuses on explicit color-code constructions, transversal gates, and gauge fixing.
  • Two-dimensional construction: The paper constructs two-dimensional color codes from 3-valent lattices with three-colorable faces, placing qubits on vertices and X- and Z-type stabilizers on faces.Removing one vertex and its incident cells produces a code encoding one logical qubit.
  • Two-dimensional construction: The two-dimensional construction yields commuting stabilizers because face intersections contain an even number of vertices under the lattice’s valence and colorability conditions.The total number of vertices in the sphere tiling is also even before the single-vertex removal.

B. Color code with one logical qubit

A one-logical-qubit color code is obtained by removing one vertex and all incident edges and faces from the closed-lattice construction. The resulting code retains local stabilizer structure and has macroscopic distance under the stated geometric conditions.

  • Construction: Removing one vertex, its three incident edges, and its three incident faces produces a color code encoding one logical qubit.The resulting lattice has an odd number of physical qubits, |Q| ≡ 1 mod 2.
  • Construction: The removal discards stabilizer generators associated with the removed faces, so the remaining generators no longer obey the closed-lattice relations.The code has one more qubit than independent stabilizer generators.
  • Properties: When faces are geometrically local and contain few vertices, the resulting color code has low-weight local stabilizer generators and macroscopic distance.Under these conditions it is a topological stabilizer code.
  • Higher-dimensional extension: The higher-dimensional construction follows the same pattern by removing one vertex and all cells containing it from a sphere tiling.This construction is stated to encode only one logical qubit.

C. Transversal gates

The two-dimensional color code supports transversal H, CNOT, and R2. The R2 construction relies on bipartitioning the lattice and applying different phase powers to the two vertex subsets while preserving logical and stabilizer actions.

  • Transversal gates: Transversal gates are tensor products of single-qubit or corresponding-pair unitaries that preserve the code space and do not spread errors within a code block.This makes transversal gates fault-tolerant in the paper’s setting.
  • Transversal gates: The transversal gate set {H, CNOT, R2} generates the Clifford group for the two-dimensional color code.The paper verifies each gate through its conjugation action on logical Pauli operators and stabilizers.
  • Hadamard: Self-duality makes physical H transversal because it exchanges the logical X and Z operators while exchanging the corresponding stabilizer types.The same conjugation preserves the code’s stabilizer structure.
  • CNOT: Applying physical CNOT to corresponding qubits in two identical code blocks implements logical CNOT and preserves S ⊗ S.The verification uses the standard conjugation rules for X and Z on control and target blocks.
  • R2: The lattice is bipartite, so its vertices split into T and T^c; applying R2^k to T^c implements logical R2 when k ≡ |T| − |T^c| mod 4.The construction uses the fact that every face has equal numbers of vertices from the two subsets.
  • R2: Choosing k = |T| − |T^c| mod 4 gives k(|T| − |T^c|) ≡ 1 mod 4, yielding the required logical phase action while preserving stabilizers.The paper concludes that the resulting transversal unitary implements R2.

D. Dual lattice picture

The dual-lattice formulation places qubits on faces of a colorable simplicial lattice and stabilizers on vertices. In higher dimensions, the construction begins with a d-sphere tiled by d-simplices and removes one simplex to form the color-code lattice.

  • Dual lattice: The dual two-dimensional lattice has triangular faces and three-colorable vertices, conditions equivalent to the primal lattice’s 3-valence and three-colorable faces.The dual and primal descriptions define the same color code.
  • Dual lattice: In the dual picture, qubits occupy faces and X- and Z-type stabilizers are associated with vertices incident to those faces.Removing one face and its associated vertex stabilizers yields a code encoding one logical qubit.
  • Dual lattice: The bipartition of primal vertices becomes a bipartition of dual faces, with adjacent faces belonging to opposite subsets.This correspondence supports the phase-gate construction in the dual description.
  • Higher dimensions: In d dimensions, the construction starts from a d-sphere tiled by d-simplices whose vertices use d+1 colors, with adjacent vertices assigned different colors.The color-code lattice is formed by removing one d-simplex from the initial triangulation.
  • Simplicial structure: A d-simplex contains faces of dimensions from zero through d, including vertices, edges, triangles, and higher-dimensional simplices.The paper uses these simplicial faces to organize the higher-dimensional lattice construction.
  • Higher dimensions: The resulting lattice is a finite homogeneous simplicial d-complex assembled by gluing d-simplices along matching proper faces.Its boundary consists of simplices on the boundary of the resulting manifold.

B. Definition of color code

The paper defines d-dimensional color codes as CSS subsystem codes on a homogeneous, (d+1)-colorable simplicial complex, with qubits on d-simplices and gauge generators supported on lower-dimensional simplices. Intersection, disjoint-union, and even-support properties establish the code's structural and commutation properties.

  • Color codes are defined on a homogeneous simplicial d-complex obtained by triangulating the interior of a d-simplex.
  • The lattice must be (d + 1)-colorable, and such lattices can be obtained from colorable tilings of the d-sphere by removing one d-simplex.
  • Qubits occupy every d-simplex, while Q(δ) denotes the qubits on d-simplices containing a simplex δ.
  • The resulting color code is a CSS subsystem code whose X- and Z-type gauge generators are supported on x- and z-simplices with x + z ≤ d − 2.
  • The Intersection and Disjoint Union lemmas organize overlapping supports, while the Even Support lemma ensures relevant intersections contain an even number of qubits.These properties imply commutation of X- and Z-type stabilizer and gauge generators.
  • The d-simplices, and therefore the qubits, form a bipartite graph because every graph cycle is even.

IV. TRANSVERSAL GATES IN COLOR CODES

The paper introduces CSS subsystem codes through their gauge and stabilizer groups, then specifies a class encoding one logical qubit with global physical X and Z operators as bare logical operators. Its codewords are constructed using fixed gauge-qubit states and gauge-group actions.

  • A CSS subsystem code is specified by a gauge group G of Pauli operators, with stabilizer group S ⊆ G generated by gauge-group elements commuting with all of G.
  • The paper considers codes encoding one logical qubit whose bare logical X and Z operators are X(Q) and Z(Q).
  • These codes have an odd number of physical qubits because the global X and Z operators must anticommute.
  • Representatives of logical |0⟩ and |1⟩ are constructed using computational-basis states together with a fixed gauge-qubit state, and other codewords arise through gauge-group actions outside the stabilizer.

B. Transversal gates in subsystem codes

For suitable one-logical-qubit CSS subsystem codes, transversal gates are analyzed through their action on logical operators and gauge structure. The paper gives a sufficient support condition for implementing R_n and notes that CNOT and, under self-duality, Hadamard are transversal.

  • A physical unitary can be checked as a dressed logical gate by its conjugation action on logical X and Z together with code-space preservation.Preserving the gauge group is sufficient but not necessary for implementing a dressed logical gate.
  • CNOT is transversal between identical copies of a CSS subsystem code by applying CNOT to corresponding physical-qubit pairs.
  • For self-dual CSS subsystem codes, applying H to every physical qubit implements a dressed logical Hadamard gate.
  • The transversal implementation of R_n applies R_n^k on a subset T of qubits and its inverse on the complement T^c.
  • The support condition can be reduced to checking subsets of X-type gauge generators, making verification easier than checking every X-type gauge-group element.
  • If T satisfies the sufficient support conditions, R_n^k(T)R_n^{-k}(T^c) implements logical R_n, with k chosen by k(|T| − |T^c|) ≡ 1 mod 2^n.

C. Transversal implementation of Rn in color code

For the d-dimensional color code family, the paper chooses T and T^c from the bipartition of qubits and proves that their supports balance on lower-dimensional simplices. This establishes transversal implementation of R_n for n ≤ dim(L)/(x + 1), including R_d for a specific code family.

  • The color-code construction implements R_n transversally for any integer n ≤ dim(L)/(x + 1).
  • The physical operation applies R_n^k on T and its complement, where T and T^c are the two parts of the qubit bipartition.
  • Every (d−1)-simplex has one qubit in T and one in T^c, so lower-dimensional simplex supports contain equal numbers from the two parts.
  • The support-balance property implies the sufficient condition required for transversal R_n in the subsystem-code analysis.
  • In particular, the code CC_d(0, d−2) supports transversal R_d.

V. UNIVERSAL TRANSVERSAL GATES WITH COLOR CODES

Gauge fixing switches between related color or subsystem codes so that different transversal gates become available, yielding a fault-tolerant universal set in three dimensions.

  • Universal gate construction: The method uses gauge fixing to switch between codes and exploit their different transversally implementable gates.The section introduces the approach through two 15-qubit codes before generalizing it to color codes.
  • Universal gate construction: The 15-qubit stabilizer code CA implements R3 transversally, whereas the subsystem code CB does not satisfy the required conditions for R3.The distinction follows from the extra X-type gauge generators in CB.
  • Gauge fixing for H: H is transversal in CB but changes the gauge state, so H⊗15 is a dressed implementation rather than a valid direct implementation in CA.Applying H⊗15 maps |ψ⟩|gZ⟩ to (H|ψ⟩)|gX⟩, which lies outside CA.
  • Gauge fixing for H: Fault-tolerant H in CA is obtained by applying H⊗15, measuring the differing Z-type stabilizers, and correcting violated generators to restore the desired gauge state.Gauge fixing measures and sets the gauge qubits to a specified state.
  • Universal gate construction: Together with transversal CNOT and R3, the gauge-fixed Hadamard gives the fault-tolerant universal gate set {H, CNOT, R3}.The set is universal because {H, CNOT, Rn} is universal for integer n > 2.

B. Partial order of color codes

Color codes on a common lattice are partially ordered by gauge-group inclusion, enabling one-way codeword inclusion and gauge-fixing switches between comparable codes.

  • Partial order: A color code C exceeds C′ in the partial order when they encode the same logical qubits with identical bare logical Pauli operators and G ⊂ G′.Gauge-group inclusion implies the stabilizer group of C′ is contained in that of C.
  • Partial order: For codes CCL(x,z) and CCL(x′,z′), the gauge generators of the smaller simplices can be expressed through generators on larger simplices when x ≤ x′ and similarly for Z generators.This relation supplies the arrow structure in the family of color codes.
  • Switching codes: Switching from CCL(x,z) to a comparable larger code requires no operation because its codewords are already valid there.The reverse switch requires fixing the additional gauge qubits to an appropriate state.
  • Universal gate set: In three dimensions, CC3(0,0) ≺ CC3(0,1), allowing H from CC3(0,0) to complement transversal CNOT and R3 in CC3(0,1).This code relationship supports a universal gate set starting from CC3(0,1).

Appendix: Examples of color codes

The construction builds color-code lattices from colored simplicial complexes, while an explicit recursive family supplies examples in arbitrary dimensions but sacrifices topological-code properties.

  • Lattice construction: The lattice recipe starts with a colored d-simplex, constructs a color-preserving homogeneous simplicial complex K, embeds K in τ, and attaches complementary-color simplices.The resulting collection of simplices defines a lattice supporting a d-dimensional color code.
  • Lattice construction: Any homogeneous simplicial d-complex K that is (d + 1)-colorable can be used, and the remaining construction steps produce a lattice for a d-dimensional color code.The recipe therefore applies in every dimension d ≥ 2.
  • Scope and limitations: The explicit fractal construction produces families in arbitrary dimensions but lacks spatially local generators and macroscopic distance, so its codes are not topological stabilizer codes.Bombín’s separate constructions use triangular- or BCC-lattice pieces to obtain topological color codes.
  • Recursive families: The first recursive family member uses K equal to a d-simplex, and each subsequent member uses the preceding lattice as K.The first three two-dimensional members encode one logical qubit with 7, 13, and 19 physical qubits.

2. Quantum Reed-Muller codes as color codes

The paper identifies the quantum Reed-Muller code QRM(m) with the stabilizer color code CCm−1(0, m−3), establishing that QRM(3) and QRM(4) correspond to Steane’s and the 15-qubit Reed-Muller codes. The equivalence follows by matching physical qubits, logical Pauli operators, and stabilizer generators.

  • QRM(m) is the stabilizer color code CCm−1(0, m−3) constructed from an (m−1)-simplex.This identifies QRM(m) with the first member of the fractal color-code family in m−1 dimensions.
  • The equivalence is proved by identifying physical qubits and matching logical X and Z operators and X-type stabilizer generators.Matching the X-type generator matrices up to a column permutation completely specifies the stabilizer group because the Z-type matrix is dual.
  • The color-code construction has dimension m−1 and contains 2^m−1 physical (m−1)-simplices.The simplices are attached through the construction based on complementary-colored faces of the simplex.
  • The constructed code has 2^m−1 physical qubits, and its X-type generator matrix agrees with the Reed-Muller matrix up to relabeling.The proof shows that the columns are the nonzero binary vectors of length m, yielding the same code and logical operators.
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