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arXiv:2609.26735

Good Quantum Locally Testable Codes from Product Expansion

Mitali Bafna, Anqi Li, Quynh T. Nguyen

quant-phcs.CCcs.ITSubmitted Sep 22, 2026• Announced Sep 23, 2026• 0 citations
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Abstract

We construct quantum locally testable codes (LTCs) with constant rate, distance, soundness and locality under a variant of a product expansion conjecture of Bafna and Vyas about Reed-Solomon codes. In particular, we use the high-dimensional expansion framework of Dinur, Lin and Vidick for constructing quantum LTCs, instantiated with the non-Abelian cubical complexes of Rungtanapirom, Stix and Vdovina. Our code is obtained by equipping the complex with carefully chosen Reed-Solomon local codes whose symmetries are compatible with those of the complex.

Related papers

Expansion of High-Dimensional Cubical Complexes: with Application to Quantum Locally Testable Codes

Irit Dinur, Ting-Chun Lin, Thomas Vidick

We introduce a high-dimensional cubical complex, for any dimension$t \in \mathbb{N}$, and apply it to the design of quantum locally testable codes. Our complex is a natural generalization of the constructions by Panteleev and Kalachev and by Dinur et. al of a square complex (case$t=2$), which have been applied to the design of classical locally testable codes (LTC) and quantum low-density parity check codes (qLDPC) respectively. We turn the geometric (cubical) complex into a chain complex by relying on constant-sized local codes$h_{1}, \ldots,h_{t}$as gadgets. A recent result of Panteleev and Kalachev on existence of tuples of codes that are product expanding enables us to prove lower bounds on the cycle and co-cycle expansion of our chain complex. For$t=4$our construction gives a new family of “almost-good” quantum LTCs - with constant relative rate, inverse-polylogarithmic relative distance and soundness, and constant-size parity checks. Both the distance of the quantum code and its local testability are proven directly from the cycle and co-cycle expansion of our chain complex.

Annual Symposium on Foundations of Computer Science•Oct 27, 2024•4 citations

Good Quantum Locally Testable Codes from Lossless Cubical Complexes

Itay Cohen, Itai Leigh, Assaf Reiner +2 more

Sipser and Spielman constructed LDPC codes from either bipartite \emph{spectral} expanders or one-sided \emph{lossless} expanders. In higher dimensions, \emph{spectral} expansion similarly played a central role in the constructions of asymptotically good classical LTCs and qLDPC codes by Dinur, Evra, Livne, Lubotzky, and Mozes and by Panteleev and Kalachev. Alternatively, Lin and Hsieh constructed classical LTCs and qLDPC codes from two-dimensional \emph{lossless} cubical complexes. In this work we develop the higher-dimensional \emph{lossless} approach. We do not construct the required high-dimensional lossless cubical complexes; rather, we investigate what their existence would imply. We associate with a high-dimensional cubical complex a \emph{level chain complex}, whose chain groups are supported on the level sets of the Boolean cube rather than on its cells. Our main technical contribution is a clean local-to-global theorem for this structure: suitable one-dimensional lossless expansion in the directional graphs implies small-set coboundary expansion of the global level complex. As a consequence, sufficiently imbalanced, two-sided lossless four-dimensional cubical complexes give rise to asymptotically good quantum locally testable codes. We expect the local-to-global principle developed here to have further applications.

quant-pharXiv:2610.00525•Oct 2, 2026•0 citations

Expansion of higher-dimensional cubical complexes with application to quantum locally testable codes

Irit Dinur, Ting-Chun Lin, Thomas Vidick

We introduce a high-dimensional cubical complex, for any dimension t>0, and apply it to the design of quantum locally testable codes. Our complex is a natural generalization of the constructions by Panteleev and Kalachev and by Dinur et. al of a square complex (case t=2), which have been applied to the design of classical locally testable codes (LTC) and quantum low-density parity check codes (qLDPC) respectively. We turn the geometric (cubical) complex into a chain complex by relying on constant-sized local codes $h_1,\ldots,h_t$ as gadgets. A recent result of Panteleev and Kalachev on existence of tuples of codes that are product expanding enables us to prove lower bounds on the cycle and co-cycle expansion of our chain complex. For t=4 our construction gives a new family of "almost-good" quantum LTCs -- with constant relative rate, inverse-polylogarithmic relative distance and soundness, and constant-size parity checks. Both the distance of the quantum code and its local testability are proven directly from the cycle and co-cycle expansion of our chain complex.

arXiv:2402.07476•Feb 12, 2024•1 citation

Asymptotically Good Quantum Locally Testable Codes

William Gay, Fernando Granha Jeronimo

We construct explicit families of asymptotically good quantum locally testable codes over qubits. More precseily, we construct explicit quantum LDPC CSS codes over qubits having constant rate, constant relative distance and constant weight local testers with constant soundness.

quant-pharXiv:2609.20780•Sep 18, 2026•0 citations

Quantum Locally Testable Codes

Dorit Aharonov, Lior Eldar

We initiate the study of quantum locally testable codes ($\text{qLTC}$s). Classical $\text{LTC}$s are very important in computational complexity. These codes are defined as the linear subspace satisfying a set of local constraints, with the additional requirement that their soundness, $R(\delta)$, which is the probability that a randomly chosen constraint is violated, is proportional to the proximity $\delta$, where $\delta n$ is the distance of a word from the code. Excellent $\text{LTC}$s exist in the classical world, and they are tightly related to the celebrated $\text{PCP}$ (probabilistically checkable proof) theorem. In quantum complexity, quantum error correcting codes provide central examples in the study of the illusive behavior of multiparticle entanglement, and they have played a crucial role in many computational complexity results. We provide a definition of the quantum analogue of $\text{LTC}$s and motivate it by connecting its central notions in the study of both entanglement and quantum Hamiltonian complexity. A natural question is whether such codes exist, and how good can their soundness be. To the best of our knowledge all quantum codes known today exhibit poor soundness. Moreover, we show that the soundness of $\text{CSS}$ codes (which are commonly used quantum codes defined by two classical codes) is governed by the minimal soundness of the two classical codes; in the most natural $\text{CSS}$ code we examined as a candidate $\text{qLTC}$, namely, the Reed--Muller code, there is a tradeoff between the parameters of the two codes, which prevents the resulting quantum code from being $\text{qLTC}$. These facts seem to suggest a more general phenomenon, by which the soundness of $\text{qLTC}$s is inherently restricted due to multiparticle entanglement. Our main technical contribution consists of two complementary results regarding $\text{qLTC}$s which are stabilizer codes (denoted $\text{sLTC}$s). We first prove a surprising, inherently quantum property of $\text{sLTC}$s. For small constant values of proximity, the better the local expansion of the interaction graph of the constraints, the less sound the $\text{sLTC}$ becomes. This stands in sharp contrast to the classical setting. The complementary, more intuitive result also holds (and is actually much more involved technically to prove in the quantum case): an upper bound on the soundness when the code is defined on bad local expanders. Together we arrive at a quantum upper bound on the soundness of $\text{sLTC}$s set on any graph, which does not hold in the classical case. Many open questions are raised regarding what possible parameters are achievable for $\text{qLTC}$s, and their relation to other objects of interest in quantum information theory. In the appendix we also define a quantum analogue of $\text{PCP}$s of proximity ($\text{PCPP}$s) and point out that the result of [E. Ben-Sasson et al., SIAM J. Comput., 36 (2006), pp. 889--974] by which $\text{PCPP}$s imply $\text{LTC}$s with related parameters carries over to the $\text{sLTC}$s. This creates a first link between $\text{qLTC}$s and quantum $\text{PCP}$s [D. Aharonov, I. Arad, and T. Vidick, ACM SIGACT News Archive, 44 (2013), pp. 47--79].

SIAM Journal on Computing•Jan 1, 2015•35 citations

Quantum Locally Testable Code with Constant Soundness

Andrew W. Cross, Zhiyang He, Anand Natarajan +2 more

In this paper, we present two constructions of quantum locally testable codes (QLTC) with constant soundness. In the first approach, we introduce an operation called check product, and show how this operation gives rise to QLTCs of constant soundness, constant rate, and distance scaling with locality. In the second approach, we consider hypergraph product of a quantum code and a classical repetition code, and observe a special case in which the soundness of component codes is preserved. This insight leads us to construct QLTCs of constant soundness, scalable rate and distance, and constant average locality. Our work marks a step towards constructing QLTCs of high soundness and distance, which would give a different construction to the No Low-Energy Trivial States (NLTS) theorem.

arXiv:2209.11405•Quantum•Oct 18, 2024•3 citations
We construct quantum locally testable codes (LTCs) with constant rate, distance, soundness and locality under a variant of a product expansion conjecture of Bafna and Vyas about Reed-Solomon codes. In particular, we use the high-dimensional expansion framework of Dinur, Lin and Vidick for constructing quantum LTCs, instantiated with the non-Abelian cubical complexes of Rungtanapirom, Stix and Vdovina. Our code is obtained by equipping the complex with carefully chosen Reed-Solomon local codes whose symmetries are compatible with those of the complex.
We introduce a high-dimensional cubical complex, for any dimensiont∈Nt \in \mathbb{N}t∈N, and apply it to the design of quantum locally testable codes. Our complex is a natural generalization of the constructions by Panteleev and Kalachev and by Dinur et. al of a square complex (caset=2t=2t=2), which have been applied to the design of classical locally testable codes (LTC) and quantum low-density parity check codes (qLDPC) respectively. We turn the geometric (cubical) complex into a chain complex by relying on constant-sized local codesh1,…,hth_{1}, \ldots,h_{t}h1​,…,ht​as gadgets. A recent result of Panteleev and Kalachev on existence of tuples of codes that are product expanding enables us to prove lower bounds on the cycle and co-cycle expansion of our chain complex. Fort=4t=4t=4our construction gives a new family of “almost-good” quantum LTCs - with constant relative rate, inverse-polylogarithmic relative distance and soundness, and constant-size parity checks. Both the distance of the quantum code and its local testability are proven directly from the cycle and co-cycle expansion of our chain complex.
Sipser and Spielman constructed LDPC codes from either bipartite \emph{spectral} expanders or one-sided \emph{lossless} expanders. In higher dimensions, \emph{spectral} expansion similarly played a central role in the constructions of asymptotically good classical LTCs and qLDPC codes by Dinur, Evra, Livne, Lubotzky, and Mozes and by Panteleev and Kalachev. Alternatively, Lin and Hsieh constructed classical LTCs and qLDPC codes from two-dimensional \emph{lossless} cubical complexes. In this work we develop the higher-dimensional \emph{lossless} approach. We do not construct the required high-dimensional lossless cubical complexes; rather, we investigate what their existence would imply. We associate with a high-dimensional cubical complex a \emph{level chain complex}, whose chain groups are supported on the level sets of the Boolean cube rather than on its cells. Our main technical contribution is a clean local-to-global theorem for this structure: suitable one-dimensional lossless expansion in the directional graphs implies small-set coboundary expansion of the global level complex. As a consequence, sufficiently imbalanced, two-sided lossless four-dimensional cubical complexes give rise to asymptotically good quantum locally testable codes. We expect the local-to-global principle developed here to have further applications.
We introduce a high-dimensional cubical complex, for any dimension t>0, and apply it to the design of quantum locally testable codes. Our complex is a natural generalization of the constructions by Panteleev and Kalachev and by Dinur et. al of a square complex (case t=2), which have been applied to the design of classical locally testable codes (LTC) and quantum low-density parity check codes (qLDPC) respectively. We turn the geometric (cubical) complex into a chain complex by relying on constant-sized local codes h1,…,hth_1,\ldots,h_th1​,…,ht​ as gadgets. A recent result of Panteleev and Kalachev on existence of tuples of codes that are product expanding enables us to prove lower bounds on the cycle and co-cycle expansion of our chain complex. For t=4 our construction gives a new family of "almost-good" quantum LTCs -- with constant relative rate, inverse-polylogarithmic relative distance and soundness, and constant-size parity checks. Both the distance of the quantum code and its local testability are proven directly from the cycle and co-cycle expansion of our chain complex.
We initiate the study of quantum locally testable codes (qLTC\text{qLTC}qLTCs). Classical LTC\text{LTC}LTCs are very important in computational complexity. These codes are defined as the linear subspace satisfying a set of local constraints, with the additional requirement that their soundness, R(δ)R(\delta)R(δ), which is the probability that a randomly chosen constraint is violated, is proportional to the proximity δ\deltaδ, where δn\delta nδn is the distance of a word from the code. Excellent LTC\text{LTC}LTCs exist in the classical world, and they are tightly related to the celebrated PCP\text{PCP}PCP (probabilistically checkable proof) theorem. In quantum complexity, quantum error correcting codes provide central examples in the study of the illusive behavior of multiparticle entanglement, and they have played a crucial role in many computational complexity results. We provide a definition of the quantum analogue of LTC\text{LTC}LTCs and motivate it by connecting its central notions in the study of both entanglement and quantum Hamiltonian complexity. A natural question is whether such codes exist, and how good can their soundness be. To the best of our knowledge all quantum codes known today exhibit poor soundness. Moreover, we show that the soundness of CSS\text{CSS}CSS codes (which are commonly used quantum codes defined by two classical codes) is governed by the minimal soundness of the two classical codes; in the most natural CSS\text{CSS}CSS code we examined as a candidate qLTC\text{qLTC}qLTC, namely, the Reed--Muller code, there is a tradeoff between the parameters of the two codes, which prevents the resulting quantum code from being qLTC\text{qLTC}qLTC. These facts seem to suggest a more general phenomenon, by which the soundness of qLTC\text{qLTC}qLTCs is inherently restricted due to multiparticle entanglement. Our main technical contribution consists of two complementary results regarding qLTC\text{qLTC}qLTCs which are stabilizer codes (denoted sLTC\text{sLTC}sLTCs). We first prove a surprising, inherently quantum property of sLTC\text{sLTC}sLTCs. For small constant values of proximity, the better the local expansion of the interaction graph of the constraints, the less sound the sLTC\text{sLTC}sLTC becomes. This stands in sharp contrast to the classical setting. The complementary, more intuitive result also holds (and is actually much more involved technically to prove in the quantum case): an upper bound on the soundness when the code is defined on bad local expanders. Together we arrive at a quantum upper bound on the soundness of sLTC\text{sLTC}sLTCs set on any graph, which does not hold in the classical case. Many open questions are raised regarding what possible parameters are achievable for qLTC\text{qLTC}qLTCs, and their relation to other objects of interest in quantum information theory. In the appendix we also define a quantum analogue of PCP\text{PCP}PCPs of proximity (PCPP\text{PCPP}PCPPs) and point out that the result of [E. Ben-Sasson et al., SIAM J. Comput., 36 (2006), pp. 889--974] by which PCPP\text{PCPP}PCPPs imply LTC\text{LTC}LTCs with related parameters carries over to the sLTC\text{sLTC}sLTCs. This creates a first link between qLTC\text{qLTC}qLTCs and quantum PCP\text{PCP}PCPs [D. Aharonov, I. Arad, and T. Vidick, ACM SIGACT News Archive, 44 (2013), pp. 47--79].
In this paper, we present two constructions of quantum locally testable codes (QLTC) with constant soundness. In the first approach, we introduce an operation called check product, and show how this operation gives rise to QLTCs of constant soundness, constant rate, and distance scaling with locality. In the second approach, we consider hypergraph product of a quantum code and a classical repetition code, and observe a special case in which the soundness of component codes is preserved. This insight leads us to construct QLTCs of constant soundness, scalable rate and distance, and constant average locality. Our work marks a step towards constructing QLTCs of high soundness and distance, which would give a different construction to the No Low-Energy Trivial States (NLTS) theorem.