Transversal Gates in Nonadditive Quantum Codes

• URL: http://arxiv.org/abs/2504.20847v1
• Date: Tue, 29 Apr 2025 15:18:33 GMT
• Title: Transversal Gates in Nonadditive Quantum Codes
• Authors: Chao Zhang, Zipeng Wu, Shilin Huang, Bei Zeng,
• Abstract summary: We search codes with specified groups by parametrizing logical subspaces on the Stiefel manifold.<n>Applying this method, we uncover a new $((6,2,3))$ code admitting a $Zbigl(tfrac2pi5bigr)$ gate.<n>Several new $(7,2,3))$ codes realize the binary icosahedral group $2I$.
• Score: 3.980076328494117
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: Transversal gates play a crucial role in suppressing error propagation in fault-tolerant quantum computation, yet they are intrinsically constrained: any nontrivial code encoding a single logical qubit admits only a finite subgroup of $\mathrm{SU}(2)$ as its transversal operations. We introduce a systematic framework for searching codes with specified transversal groups by parametrizing their logical subspaces on the Stiefel manifold and minimizing a composite loss that enforces both the Knill-Laflamme conditions and a target transversal-group structure. Applying this method, we uncover a new $((6,2,3))$ code admitting a transversal $Z\bigl(\tfrac{2\pi}{5}\bigr)$ gate (transversal group $\mathrm{C}_{10}$), the smallest known distance $3$ code supporting non-Clifford transversal gates, as well as several new $((7,2,3))$ codes realizing the binary icosahedral group $2I$. We further propose the \emph{Subset-Sum-Linear-Programming} (SS-LP) construction for codes with transversal \emph{diagonal} gates, which dramatically shrinks the search space by reducing to integer partitions subject to linear constraints. In a more constrained form, the method also applies directly to the binary-dihedral groups $\mathrm{BD}_{2m}$. Specializing to $n=7$, the SS-LP method yields codes for all $\mathrm{BD}_{2m}$ with $2m\le 36$, including the first $((7,2,3))$ examples supporting transversal $T$ gate ($\mathrm{BD}_{16}$) and $\sqrt{T}$ gate ($\mathrm{BD}_{32}$), improving on the previous smallest examples $((11,2,3))$ and $((19,2,3))$. Extending the SS-LP approach to $((8,2,3))$, we construct new codes for $2m>36$, including one supporting a transversal $T^{1/4}$ gate ($\mathrm{BD}_{64}$). These results reveal a far richer landscape of nonadditive codes than previously recognized and underscore a deeper connection between quantum error correction and the algebraic constraints on transversal gate groups.

Related papers

• Guessing Efficiently for Constrained Subspace Approximation [49.83981776254246]
  We introduce a general framework for constrained subspace approximation.<n>We show it provides new algorithms for partition-constrained subspace approximation with applications to $k$-means clustering, and projected non-negative matrix factorization.
  arXiv  Detail & Related papers  (2025-04-29T15:56:48Z)
• Quantum Codes with Addressable and Transversal Non-Clifford Gates [8.194994143531677]
  We study codes that support gates which induce $textitaddressable$ logical gates.<n>We develop a formalism for constructing quantum codes with $textitaddressable and $ell neq 2$ gates.
  arXiv  Detail & Related papers  (2025-02-03T22:24:34Z)
• The Communication Complexity of Approximating Matrix Rank [50.6867896228563]
  We show that this problem has randomized communication complexity $Omega(frac1kcdot n2log|mathbbF|)$. As an application, we obtain an $Omega(frac1kcdot n2log|mathbbF|)$ space lower bound for any streaming algorithm with $k$ passes.
  arXiv  Detail & Related papers  (2024-10-26T06:21:42Z)
• Asymptotically Good Quantum Codes with Transversal Non-Clifford Gates [23.22566380210149]
  We construct quantum codes that support $CCZ$ gates over qudits of arbitrary prime power dimension $q$. The only previously known construction with such linear dimension and distance required a growing alphabet size $q$.
  arXiv  Detail & Related papers  (2024-08-17T16:54:51Z)
• S-FABLE and LS-FABLE: Fast approximate block-encoding algorithms for unstructured sparse matrices [0.0]
  The Fast Approximate BLock-Lazy algorithm (FABLE) is a technique to block-encode arbitrary $Ntimes N$ dense matrices into quantum circuits. We describe two modifications of FABLE to efficiently encode sparse matrices.
  arXiv  Detail & Related papers  (2024-01-08T20:57:16Z)
• Near-Optimal Regret Bounds for Multi-batch Reinforcement Learning [54.806166861456035]
  We study the episodic reinforcement learning (RL) problem modeled by finite-horizon Markov Decision Processes (MDPs) with constraint on the number of batches. We design a computational efficient algorithm to achieve near-optimal regret of $tildeO(sqrtSAH3Kln (1/delta))$tildeO(cdot) hides logarithmic terms of $(S,A,H,K)$ in $K$ episodes. Our technical contribution are two-fold: 1) a near-optimal design scheme to explore
  arXiv  Detail & Related papers  (2022-10-15T09:22:22Z)
• Algebraic Aspects of Boundaries in the Kitaev Quantum Double Model [77.34726150561087]
  We provide a systematic treatment of boundaries based on subgroups $Ksubseteq G$ with the Kitaev quantum double $D(G)$ model in the bulk. The boundary sites are representations of a $*$-subalgebra $Xisubseteq D(G)$ and we explicate its structure as a strong $*$-quasi-Hopf algebra. As an application of our treatment, we study patches with boundaries based on $K=G$ horizontally and $K=e$ vertically and show how these could be used in a quantum computer
  arXiv  Detail & Related papers  (2022-08-12T15:05:07Z)
• Divisible Codes for Quantum Computation [0.6445605125467572]
  Divisible codes are defined by the property that codeword weights share a common divisor greater than one. This paper explores how they can be used to protect quantum information as it is transformed by logical gates.
  arXiv  Detail & Related papers  (2022-04-27T20:18:51Z)
• Climbing the Diagonal Clifford Hierarchy [0.6445605125467572]
  We introduce a method of synthesizing codes that realize a target logical diagonal gate at some level $l$ in the Clifford hierarchy. The method combines three basic operations: concatenation, removal of $Z$-stabilizers, and addition of $X$-stabilizers. For the coherent noise model, we describe how to switch between computation and storage of intermediate results in a decoherence-free subspace.
  arXiv  Detail & Related papers  (2021-10-22T17:08:18Z)
• Quantum double aspects of surface code models [77.34726150561087]
  We revisit the Kitaev model for fault tolerant quantum computing on a square lattice with underlying quantum double $D(G)$ symmetry. We show how our constructions generalise to $D(H)$ models based on a finite-dimensional Hopf algebra $H$.
  arXiv  Detail & Related papers  (2021-06-25T17:03:38Z)
• $Q$-learning with Logarithmic Regret [60.24952657636464]
  We prove that an optimistic $Q$-learning enjoys a $mathcalOleft(fracSAcdot mathrmpolyleft(Hright)Delta_minlogleft(SATright)right)$ cumulative regret bound, where $S$ is the number of states, $A$ is the number of actions, $H$ is the planning horizon, $T$ is the total number of steps, and $Delta_min$ is the minimum sub-optimality gap.
  arXiv  Detail & Related papers  (2020-06-16T13:01:33Z)

This list is automatically generated from the titles and abstracts of the papers in this site.

This site does not guarantee the quality of this site (including all information) and is not responsible for any consequences.