Permutation-invariant codes: a numerical study and qudit constructions

• URL: http://arxiv.org/abs/2603.10981v1
• Date: Wed, 11 Mar 2026 17:10:44 GMT
• Title: Permutation-invariant codes: a numerical study and qudit constructions
• Abstract summary: We investigate Permutation-Invariant (PI) quantum error-correcting codes encoding a logical qudit of dimension $mathrmd_mathrmL$ in PI states.<n>We investigate numerically both qubit ($mathrmd_mathrmL = mathrmd_mathrmP = 2$) and qudit ($mathrmd_mathrmL > 2$ or $mathrmd_mathrmP > 2$) PI codes.
• Score: 0.0
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: We investigate Permutation-Invariant (PI) quantum error-correcting codes encoding a logical qudit of dimension $\mathrm{d}_\mathrm{L}$ in PI states using physical qudits of dimension $\mathrm{d}_\mathrm{P}$. We extend the Knill--Laflamme (KL) conditions for $d-1$ deletion errors from qubits to qudits and investigate numerically both qubit ($\mathrm{d}_\mathrm{L} = \mathrm{d}_\mathrm{P} = 2$) and qudit ($\mathrm{d}_\mathrm{L} > 2$ or $\mathrm{d}_\mathrm{P} > 2$) PI codes. We analyze the scaling of the block length $n$ in terms of the code distance $d$, and compare to existing families of PI codes due to Ouyang, Aydin--Alekseyev--Barg (AAB) and Pollatsek--Ruskai (PR). Our three main findings are: (i) We conjecture that qubit PI codes correcting up to $d-1$ deletion errors have block length $n(d) \geq (3d^2 + 1) / 4$, which implies an upper bound $d \leq \sqrt{12n-3}/3$ on their code distance, and that PR codes can saturate this bound. (ii) For qudit PI codes encoding a single qudit we numerically observe that increasing $\mathrm{d}_\mathrm{P}$ results in $n$ monotonically decreasing and approaching the quantum Singleton bound $n(d) \geq 2d-1$. (iii) We propose a semi-analytic extension of the qubit AAB construction to qudits that finds explicit solutions by solving a linear program. Our results therefore provide key insights into lower bounds on the block length scaling of both qubit and qudit PI codes, and demonstrate the benefit of increased physical local dimension in the context of PI codes.

Related papers

• Faster quantum linear system solver beyond the condition number [50.84794327094274]
  We present two quantum algorithms that produce the normalized solution $|xrangle$ of linear system $Ax=| b rangle$ to accuracy $$ with complexity independent of the condition number $=lVert A-1rVert$.<n>Our filtering-based solver is extremely simple with a favorable runtime prefactor.
  arXiv  Detail & Related papers  (2026-07-08T17:49:40Z)
• Discrepancy for Random Linear Codes [10.105438043325357]
  We prove that random linear codes have nearly optimal discrepancy properties in a broad range of regimes.<n>First, random linear codes match unstructured random codes for list-decoding from errors above capacity.<n>Second, over prime fields, random linear codes match unstructured random codes for zero-error list-recovery above capacity.
  arXiv  Detail & Related papers  (2026-06-23T12:01:54Z)
• A Mathematical Theory of Top-$k$ Sparse Attention via Total Variation Distance [7.014801584517052]
  We develop a unified framework certified Top-$k$ attention truncation that quantifies error at both the distribution and output levels.<n>We show that the total-variation distance coincides with the discarded softmax tail mass and satisfies $mathrmTV(P,hat P)=1-e-mathrmTV(P,hat P)=1-e-mathrmTV(P,hat P)$.
  arXiv  Detail & Related papers  (2025-12-08T15:36:41Z)
• Transversal Gates in Nonadditive Quantum Codes [3.980076328494117]
  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$.
  arXiv  Detail & Related papers  (2025-04-29T15:18:33Z)
• 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)
• Horizon-free Reinforcement Learning in Adversarial Linear Mixture MDPs [72.40181882916089]
  We show that our algorithm achieves an $tildeObig((d+log (|mathcalS|2 |mathcalA|))sqrtKbig)$ regret with full-information feedback, where $d$ is the dimension of a known feature mapping linearly parametrizing the unknown transition kernel of the MDP, $K$ is the number of episodes, $|mathcalS|$ and $|mathcalA|$ are the cardinalities of the state and action spaces
  arXiv  Detail & Related papers  (2023-05-15T05:37:32Z)
• Performance Analysis of Quantum CSS Error-Correcting Codes via MacWilliams Identities [9.69910104594168]
  We analyze the performance of stabilizer codes, one of the most important classes for practical implementations. We introduce a novel approach that combines the knowledge of WE with a logical operator analysis. For larger codes our bound provides $rho_mathrmL approx 1215 rho4$ and $rho_mathrmL approx 663 rho5$ for the $[85,1,7]]$ and the $[181,1,10]]$ surface codes.
  arXiv  Detail & Related papers  (2023-05-02T10:19:02Z)
• Low-degree learning and the metric entropy of polynomials [44.99833362998488]
  We prove that any (deterministic or randomized) algorithm which learns $mathscrF_nd$ with $L$-accuracy $varepsilon$ requires at least $Omega(sqrtvarepsilon)2dlog n leq log mathsfM(mathscrF_n,d,|cdot|_L,varepsilon) satisfies the two-sided estimate $$c (1-varepsilon)2dlog
  arXiv  Detail & Related papers  (2022-03-17T23:52:08Z)
• Threshold Phenomena in Learning Halfspaces with Massart Noise [56.01192577666607]
  We study the problem of PAC learning halfspaces on $mathbbRd$ with Massart noise under Gaussian marginals. Our results qualitatively characterize the complexity of learning halfspaces in the Massart model.
  arXiv  Detail & Related papers  (2021-08-19T16:16:48Z)
• Taking a hint: How to leverage loss predictors in contextual bandits? [63.546913998407405]
  We study learning in contextual bandits with the help of loss predictors. We show that the optimal regret is $mathcalO(minsqrtT, sqrtmathcalETfrac13)$ when $mathcalE$ is known.
  arXiv  Detail & Related papers  (2020-03-04T07:36:38Z)
• Fixed-Support Wasserstein Barycenters: Computational Hardness and Fast Algorithm [100.11971836788437]
  We study the fixed-support Wasserstein barycenter problem (FS-WBP) We develop a provably fast textitdeterministic variant of the celebrated iterative Bregman projection (IBP) algorithm, named textscFastIBP.
  arXiv  Detail & Related papers  (2020-02-12T03:40:52Z)

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.