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arXiv:2207.03562 · Quantum

Finite-rate sparse quantum codes aplenty

Maxime Tremblay, Guillaume Duclos-Cianci, Stefanos Kourtis

Published Apr 20, 2023• 3 citations
arXivHTMLPDF

Abstract

We introduce a methodology for generating random multi-qubit stabilizer codes based on solving a constraint satisfaction problem (CSP) on random bipartite graphs. This framework allows us to enforce stabilizer commutation, X/Z balancing, finite rate, sparsity, and maximum-degree constraints simultaneously in a CSP that we can then solve numerically. Using a state-of-the-art CSP solver, we obtain convincing evidence for the existence of a satisfiability threshold. Furthermore, the extent of the satisfiable phase increases with the number of qubits. In that phase, finding sparse codes becomes an easy problem. Moreover, we observe that the sparse codes found in the satisfiable phase practically achieve the channel capacity for erasure noise. Our results show that intermediate-size finite-rate sparse quantum codes are easy to find, while also demonstrating a flexible methodology for generating good codes with custom properties. We therefore establish a complete and customizable pipeline for random quantum code discovery.

DOI
10.22331/q-2023-04-20-985

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Quantum low-density parity-check (qLDPC) codes can encode multiple logical qubits using sparse parity checks, yet searching for useful finite-length instances remains a challenging design problem because code performance must be optimized while satisfying practical constraints. Motivated by recent advances in artificial-intelligence agents for scientific discovery, we develop a multi-agent framework for discovering practical qLDPC codes. The framework combines specialist proposal and review, persistent scientific memory, long-horizon evolution of executable programs, and deterministic construction and evaluation within a closed-loop search. These programs instantiate coset-orbit balanced-product codes, providing a search space that includes bicycle and lifted-product constructions as well as non-normal subgroup actions. To incorporate practical constraints, we restrict the search to binary CSS codes with block length n≤400n\leq400n≤400 and overall weight w≤10w\leq10w≤10. Within this regime, the framework discovers codes with leading or competitive rate--distance performance in every weight class considered, with representative instances including [[288,16,18]][[288,16,18]][[288,16,18]] at w=7w=7w=7, [[288,18,18]][[288,18,18]][[288,18,18]] at w=9w=9w=9, and [[234,28,18]][[234,28,18]][[234,28,18]] at w=10w=10w=10. The search also uncovers structurally distinct, high-performing constructions, including a [[336,12,≤24]][[336,12,\leq24]][[336,12,≤24]] candidate and a [[368,18,16]][[368,18,16]][[368,18,16]] code, both of which are genuine balanced-product constructions with non-normal subgroup actions. When evaluated under code-capacity depolarizing noise using a common BP-OSD decoding protocol, the discovered codes also exhibit low logical failure rates. Together, these results provide hardware-relevant finite-length candidates for further experimental evaluation and show how structured agentic search can contribute to scientific discovery.