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SIAM Journal on Computing

Quantum Locally Testable Codes

Dorit Aharonov, Lior Eldar

Published Jan 1, 2015• 35 citations
Publisher

Abstract

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].

DOI
10.1137/140975498

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We initiate the study of quantum Locally Testable Codes (qLTCs). We provide a definition together with a simplification, denoted sLTCs, for the special case of stabilizer codes, together with some basic results using those definitions. The most crucial parameter of such codes is their soundness, R(δ)R(δ)R(δ), namely, the probability that a randomly chosen constraint is violated as a function of the distance of a word from the code (δδδ, the relative distance from the code, is called the proximity). We then proceed to study limitations on qLTCs. In our first main result we prove a surprising, inherently quantum, property of sLTCs: for small values of proximity, the better the small-set expansion of the interaction graph of the constraints, the less sound the qLTC becomes. This phenomenon, which can be attributed to monogamy of entanglement, stands in sharp contrast to the classical setting. The complementary, more intuitive, result also holds: an upper bound on the soundness when the code is defined on poor small-set expanders (a bound which turns out to be far more difficult to show in the quantum case). Together we arrive at a quantum upper-bound on the soundness of stabilizer qLTCs set on any graph, which does not hold in the classical case. Many open questions are raised regarding what possible parameters are achievable for qLTCs. In the appendix we also define a quantum analogue of PCPs of proximity (PCPPs) and point out that the result of Ben-Sasson et. al. by which PCPPs imply LTCs with related parameters, carries over to the sLTCs. This creates a first link between qLTCs and quantum PCPs.
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.
In this paper, we prove a lower bound on the soundness of quantum locally testable codes under the distance balancing construction of Evra et al. arXiv:2004.07935 [quant-ph]. Our technical contribution is that the new soundness of the quantum code is at least the old soundness divided by the classical code length (up to a constant factor). This allows us to use any classical code with independent checks when distance balancing, where previously only the repetition code had been considered for qLTCs. By using a good classical LDPC code, we are able to grow the dimension of the hypersphere product codes arXiv:1608.05089 [quant-ph] and the hemicubic codes arXiv:1911.03069 [quant-ph] while maintaining their distance and locality, but at the expense of soundness. From this, and also by distance balancing a chain complex of Cross et al. arXiv:2209.11405 [cs.IT], we obtain quantum locally testable codes of new parameters.
The study of multiprover interactive proof systems, of locally testable codes, and of property testing are deeply linked, conceptually if not formally, through their role in the proof of the PCP theorem in complexity theory. Recently there has been substantial progress on an analogous research programme in quantum complexity theory. Two years ago we characterized the power of multiprover interactive proof systems with provers sharing entanglement, showing that MIP^* = RE [Ji et al., 2020], a hugely surprising increase in power from the classical result MIP = NEXP of [Babai et al., 1991]. The following year Panteleev and Kalachev gave the first construction of quantum low-density parity-check codes (QLDPC) [Panteleev and Kalachev, 2022], thus marking a major step towards the possible realization of good quantum locally testable codes - the classical analogue of which was only constructed quite recently [Dinur et al., 2022]. And finally, less than a year ago Anshu, Breuckmann and Nirkhe used facts evidenced in the construction of good decoders for the new QLDPC codes to resolve the NLTS conjecture [Anshu et al., 2022], widely viewed as a crucial step on the way to a possible quantum PCP theorem. In the talk I will survey these results, making an effort to motivate and present them to the non-expert. I will explain the connections between them and point to where, in my opinion, our understanding is currently lacking. Along the way I will highlight a number of open problems whose resolution could lead to further progress on one of the most important research programmes in quantum complexity theory.
In this work, we continue the search for quantum locally testable codes (qLTCs) of new parameters by presenting three constructions that can make new qLTCs from old. The first analyses the soundness of a quantum code under Hastings’ weight reduction construction for qLDPC codes to give a weight reduction procedure for qLTCs. Secondly, we describe a novel ‘soundness amplification’ procedure for qLTCs which can increase the soundness of any qLTC to a constant while preserving its distance and dimension, with an impact only felt on its locality. Finally, we apply the AEL distance amplification construction to the case of qLTCs for the first time which can turn a high-distance qLTC into one with linear distance, at the expense of other parameters. These constructions can be used on as-yet undiscovered qLTCs to obtain new parameters, but we also find a number of present applications to prove the existence of codes in previously unknown parameter regimes. In particular, applications of these operations to the hypersphere product code and the hemicubic code yield many previously unknown parameters. In addition, applications of all three results are described to an upcoming work.
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.