We initiate the study of quantum locally testable codes (
qLTCs). Classical
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), which is the probability that a randomly chosen constraint is violated, is proportional to the proximity
δ, where
δn is the distance of a word from the code. Excellent
LTCs exist in the classical world, and they are tightly related to the celebrated
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
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 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 code we examined as a candidate
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. These facts seem to suggest a more general phenomenon, by which the soundness of
qLTCs is inherently restricted due to multiparticle entanglement. Our main technical contribution consists of two complementary results regarding
qLTCs which are stabilizer codes (denoted
sLTCs). We first prove a surprising, inherently quantum property of
sLTCs. For small constant values of proximity, the better the local expansion of the interaction graph of the constraints, the less sound the
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
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
qLTCs, and their relation to other objects of interest in quantum information theory. In the appendix we also define a quantum analogue of
PCPs of proximity (
PCPPs) and point out that the result of [E. Ben-Sasson et al., SIAM J. Comput., 36 (2006), pp. 889--974] by which
PCPPs imply
LTCs with related parameters carries over to the
sLTCs. This creates a first link between
qLTCs and quantum
PCPs [D. Aharonov, I. Arad, and T. Vidick, ACM SIGACT News Archive, 44 (2013), pp. 47--79].
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), namely, the probability that a randomly chosen constraint is violated as a function of the distance of a word from the code (
δ, 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.
We show that the NP-Complete language 3Sat has a PCPverifier that makes two queries to a proof of almost-linear size and achieves sub-constant probability of error o(1). The verifier performs only projection tests, meaning that the answer to the first query determines at most one accepting answer to the second query.Previously, by the parallel repetition theorem, there were PCP Theorems with two-query projection tests, but only (arbitrarily small) constant error and polynomial size.There were also PCP Theorems with sub-constant error andalmost-linear size, but a constant number of queries that is larger than 2.As a corollary, we obtain a host of new results. In particular, our theorem improves many of the hardness of approximation results that are proved using the parallel repetition theorem. A partial list includes the following:(1) 3Sat cannot be efficiently approximated to withina factor of 7/8+o(1), unless P = NP. This holds even under almost-linear reductions. Previously, the best knownNP-hardness factor was 7/8+epsilon for any constant epsilonGt0, under polynomial reductions.(2) 3Lin cannot be efficiently approximated to withina factor of 1/2+o(1), unless P = NP. This holdseven under almost-linear reductions. Previously, the best known NP-hardness factor was 1/2+epsilon for any constant epsilonGt0, under polynomial reductions.(3) A PCP Theorem with amortized query complexity 1 + o(1)and amortized free bit complexity o(1). Previously, the best known amortized query complexity and free bit complexity were 1+epsilon and epsilon, respectively, for any constant epsilon Gt 0.One of the new ideas that we use is a new technique for doing the composition step in the (classical) proof of the PCP Theorem, without increasing the number of queries to the proof. We formalize this as a composition of new objects that we call Locally Decode/Reject Codes (LDRC). The notion of LDRC was implicit in several previous works, and we make it explicit in this work. We believe that the formulation of LDRCs and their construction are of independent interest.
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.
We show that given an explicit description of a multiplayer game, with a classical verifier and a constant number of players, it is QMA-hard, under randomized reductions, to distinguish between the cases when the players have a strategy using entanglement that succeeds with probability 1 in the game, or when no such strategy succeeds with probability larger than 1/2. This proves the “games quantum PCP conjecture” of Fitzsimons and the second author (ITCS'15), albeit under randomized reductions. The core component in our reduction is a construction of a family of two-player games for testing n-qubit maximally entangled states. For any integer n ≥ 2, we give such a game in which questions from the verifier are O(log n) bits long, and answers are poly(loglogn) bits long. We show that for any constant ε ≥ 0, any strategy that succeeds with probability at least 1 - ε in the test must use a state that is within distance δ(ε) = O(εc) from a state that is locally equivalent to a maximally entangled state on n qubits, for some universal constant c > 0. The construction is based on the classical plane-vs-point test for multivariate low-degree polynomials of Raz and Safra (STOC'97). We extend the classical test to the quantum regime by executing independent copies of the test in the generalized Pauli X and Z bases over Fq, where q is a sufficiently large prime power, and combine the two through a test for the Pauli twisted commutation relations. Our main complexity-theoretic result is obtained by combining this family of games with techniques from the classical PCP literature. More specifically, we use constructions of PCPs of proximity introduced by Ben-Sasson et al. (CCC'05), and crucially rely on a linear property of such PCPs. Another consequence of our results is a deterministic reduction from the games quantum PCP conjecture to a suitable formulation of the constraint satisfaction quantum PCP conjecture.