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Non-Clifford and parallelizable fault-tolerant logical gates on constant and almost-constant rate homological quantum LDPC codes via higher symmetries

Guanyu Zhu, Shehryar Sikander, Elia Portnoy, Andrew W. Cross, Benjamin J. Brown

TL;DR

<p>We develop a topologically grounded framework for parallel fault-tolerant quantum computation using homological qLDPC codes defined on 3-manifolds. A transversal T gate on 3D color codes implements collective CCZ gates on triplets of logical qubits when their X-membranes have a $Z_2$ triple intersection, a topological invariant tied to cup products in the corresponding TQFT; transversal S gates on codimension-1 submanifolds yield parallelizable logical CZ gates as 1-form symmetries. The authors construct three families of 3D codes with constant or almost-constant rates—quasi-hyperbolic, homological fibre-bundle, and Torelli mapping-torus codes—and present a generic, constant-overhead scheme for a parallelizable universal gate set using CCZ/Toffoli, Hadamard, and measurements. A operator-valued cochain formalism, rooted in a $(Z_2)^3$ gauge theory, underpins the CCZ/CZ gate structure and connects to higher-form symmetries in topological quantum field theory. The work also develops a framework to compute triple-intersection invariants, maps code-gate interactions to interaction hypergraphs, and discusses strategies to back-engineer manifolds for desired parallelizability, offering a promising route toward scalable, fault-tolerant quantum computation with LDPC codes.</p>

Abstract

We study parallel fault-tolerant quantum computing for families of homological quantum low-density parity-check (LDPC) codes defined on 3-manifolds with constant or almost-constant encoding rate. We derive generic formula for a transversal $T$ gate of color codes on general 3-manifolds, which acts as collective non-Clifford logical CCZ gates on any triplet of logical qubits with their logical-$X$ membranes having a $\mathbb{Z}_2$ triple intersection at a single point. The triple intersection number is a topological invariant, which also arises in the path integral of the emergent higher symmetry operator in a topological quantum field theory: the $\mathbb{Z}_2^3$ gauge theory. Moreover, the transversal $S$ gate of the color code corresponds to a higher-form symmetry supported on a codimension-1 submanifold, giving rise to exponentially many addressable and parallelizable logical CZ gates. A construction of constant-depth circuits of the above logical gates via cup product cohomology operation is also presented for three copies of identical toric codes on arbitrary 3-manifolds. We have developed a generic formalism to compute the triple intersection invariants for 3-manifolds. We further develop three types of LDPC codes supporting such logical gates: (1) A quasi-hyperbolic code from the product of 2D hyperbolic surface and a circle, with almost-constant rate $k/n=O(1/\log(n))$ and $O(\log(n))$ distance; (2) A homological fibre bundle code with $O(1/\log^{\frac{1}{2}}(n))$ rate and $O(\log^{\frac{1}{2}}(n))$ distance; (3) A specific family of 3D hyperbolic codes: the Torelli mapping torus code, constructed from mapping tori of a pseudo-Anosov element in the Torelli subgroup, which has constant rate while the distance scaling is currently unknown. We then show a generic constant-overhead scheme for applying a parallelizable universal gate set with the aid of logical-$X$ measurements.

Non-Clifford and parallelizable fault-tolerant logical gates on constant and almost-constant rate homological quantum LDPC codes via higher symmetries

TL;DR

<p>We develop a topologically grounded framework for parallel fault-tolerant quantum computation using homological qLDPC codes defined on 3-manifolds. A transversal T gate on 3D color codes implements collective CCZ gates on triplets of logical qubits when their X-membranes have a triple intersection, a topological invariant tied to cup products in the corresponding TQFT; transversal S gates on codimension-1 submanifolds yield parallelizable logical CZ gates as 1-form symmetries. The authors construct three families of 3D codes with constant or almost-constant rates—quasi-hyperbolic, homological fibre-bundle, and Torelli mapping-torus codes—and present a generic, constant-overhead scheme for a parallelizable universal gate set using CCZ/Toffoli, Hadamard, and measurements. A operator-valued cochain formalism, rooted in a gauge theory, underpins the CCZ/CZ gate structure and connects to higher-form symmetries in topological quantum field theory. The work also develops a framework to compute triple-intersection invariants, maps code-gate interactions to interaction hypergraphs, and discusses strategies to back-engineer manifolds for desired parallelizability, offering a promising route toward scalable, fault-tolerant quantum computation with LDPC codes.</p>

Abstract

We study parallel fault-tolerant quantum computing for families of homological quantum low-density parity-check (LDPC) codes defined on 3-manifolds with constant or almost-constant encoding rate. We derive generic formula for a transversal gate of color codes on general 3-manifolds, which acts as collective non-Clifford logical CCZ gates on any triplet of logical qubits with their logical- membranes having a triple intersection at a single point. The triple intersection number is a topological invariant, which also arises in the path integral of the emergent higher symmetry operator in a topological quantum field theory: the gauge theory. Moreover, the transversal gate of the color code corresponds to a higher-form symmetry supported on a codimension-1 submanifold, giving rise to exponentially many addressable and parallelizable logical CZ gates. A construction of constant-depth circuits of the above logical gates via cup product cohomology operation is also presented for three copies of identical toric codes on arbitrary 3-manifolds. We have developed a generic formalism to compute the triple intersection invariants for 3-manifolds. We further develop three types of LDPC codes supporting such logical gates: (1) A quasi-hyperbolic code from the product of 2D hyperbolic surface and a circle, with almost-constant rate and distance; (2) A homological fibre bundle code with rate and distance; (3) A specific family of 3D hyperbolic codes: the Torelli mapping torus code, constructed from mapping tori of a pseudo-Anosov element in the Torelli subgroup, which has constant rate while the distance scaling is currently unknown. We then show a generic constant-overhead scheme for applying a parallelizable universal gate set with the aid of logical- measurements.
Paper Structure (29 sections, 7 theorems, 191 equations, 32 figures)

This paper contains 29 sections, 7 theorems, 191 equations, 32 figures.

Table of Contents

  1. Introduction
  2. Non-Clifford and parallelizable logical gates of color codes on 3D manifolds
  3. Color codes and toric codes on 3D manifold
  4. Logical non-Clifford gates and the triple-intersection invariant
  5. Parallelizable logical Clifford gates
  6. Interaction hypergraph and hypergraph magic state injection
  7. Logical gates for three identical copies of toric codes on 3D manifold via constant-depth circuits and the connection to emergent $k$-form symmetries in TQFT
  8. Operator-valued cochain formalism
  9. Logical CCZ as a 0-form symmetry
  10. Logical CZ gates as 1-form symmetries
  11. Scaling of the code parameters for homological quantum codes defined on manifolds
  12. Review of hyperbolic geometry and codes
  13. Betti number and encoding rate scaling in three and higher dimensions
  14. Code construction and logical gate structure for 3-manifolds
  15. Quasi-hyperbolic codes
  16. ...and 14 more sections

Key Result

Lemma 1

$sys_1(\mathcal{M}^D_h; \mathbb{Z}_2) \ge 2R \ge c' \log vol(\mathcal{M}^D_h)$ for some constant $c'>0$.

Figures (32)

  • Figure 1: Summary of the connection between various subjects in this paper.
  • Figure 2: A color-code lattice from a triangulation of manifold $\mathcal{M}^3$ via a fattening procedure. We show a single tetrahedron of $\mathcal{M}^3$. We construct a color-code lattice with four-colourable vertices from an arbitrary cellulation by placing a vertex of colour $j$ onto each of the $j$-cells $\sigma_j$ of manifold $\mathcal{M}^3$. The $\sigma_j$ cells are each assigned a color-code vertex $v$ that is coloured red, blue, green, and yellow for $j = 0, 1, 2, 3$, respectively. We show a single qubit, associated to a tetrahedron $\Delta$ in purple. The figure shows it is adjacent to exactly one vertex of each colour.
  • Figure 3: The intersection property which guarantees a non-trivial logical CCZ action.
  • Figure 4: (a) The base interaction hypergraph (intersection hypergraph) of a particular 3-manifold. (b) The corresponding (full) interaction hypergraph of the color codes defined on the same 3-manifold. Red, green and blue vertices represent the logical-$X$ operators belonging to the three copies of toric codes respectively.
  • Figure 5: Correspondence between an interaction hypergraph of a color code on a 3-manifold in (a) with a logical 3-uniform hypergraph magic state in (b) injected via a transversal $T$ gate acted on the entire code with each logical qubit initialized in state $\overline{\ket{+}}$. The number of logical CCZ gates in the quantum circuit in (b) equaling the number hyperedges in the corresponding hypergraph shown in (a) defines the complexity of the hypergraph state.
  • ...and 27 more figures

Theorems & Definitions (11)

  • Lemma 1
  • proof
  • Theorem 1
  • Theorem 2
  • proof
  • Theorem 3
  • Theorem 4
  • Corollary 4.1
  • proof
  • Theorem 5
  • ...and 1 more