arXiv subjects

Anand Natarajan

Publications and source records attributed to Anand Natarajan.

31 records · Page 2Linked to original sources

arXiv2022

Quantum soundness of testing tensor codes

A locally testable code is an error-correcting code that admits very efficient probabilistic tests of membership. Tensor codes provide a simple family of combinatorial constructions of locally testable codes that generalize the family of Reed-Muller codes. The natural test for tensor codes, the axis-parallel line vs. point test, plays an essential role in constructions of probabilistically checkable proofs. We analyze the axis-parallel line vs. point test as a two-prover game and show that the test is sound against quantum provers sharing entanglement. Our result implies the quantum-soundness of the low individual degree test, which is an essential component of the MIP* = RE theorem. Our proof also generalizes to the infinite-dimensional commuting-operator model of quantum provers.

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arXiv2022

MIP*=RE

We show that the class MIP* of languages that can be decided by a classical verifier interacting with multiple all-powerful quantum provers sharing entanglement is equal to the class RE of recursively enumerable languages. Our proof builds upon the quantum low-degree test of (Natarajan and Vidick, FOCS 2018) and the classical low-individual degree test of (Ji, et al., 2020) by integrating recent developments from (Natarajan and Wright, FOCS 2019) and combining them with the recursive compression framework of (Fitzsimons et al., STOC 2019). An immediate byproduct of our result is that there is an efficient reduction from the Halting Problem to the problem of deciding whether a two-player nonlocal game has entangled value $1$ or at most $1/2$. Using a known connection, undecidability of the entangled value implies a negative answer to Tsirelson's problem: we show, by providing an explicit example, that the closure $C_{qa}$ of the set of quantum tensor product correlations is strictly included in the set $C_{qc}$ of quantum commuting correlations. Following work of (Fritz, Rev. Math. Phys. 2012) and (Junge et al., J. Math. Phys. 2011) our results provide a refutation of Connes' embedding conjecture from the theory of von Neumann algebras.

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arXiv2022

Quantum search-to-decision reductions and the state synthesis problem

It is a useful fact in classical computer science that many search problems are reducible to decision problems; this has led to decision problems being regarded as the $\textit{de facto}$ computational task to study in complexity theory. In this work, we explore search-to-decision reductions for quantum search problems, wherein a quantum algorithm makes queries to a classical decision oracle to output a desired quantum state. In particular, we focus on search-to-decision reductions for $\mathsf{QMA}$, and show that there exists a quantum polynomial-time algorithm that can generate a witness for a $\mathsf{QMA}$ problem up to inverse polynomial precision by making one query to a $\mathsf{PP}$ decision oracle. We complement this result by showing that $\mathsf{QMA}$-search does $\textit{not}$ reduce to $\mathsf{QMA}$-decision in polynomial-time, relative to a quantum oracle. We also explore the more general $\textit{state synthesis problem}$, in which the goal is to efficiently synthesize a target state by making queries to a classical oracle encoding the state. We prove that there exists a classical oracle with which any quantum state can be synthesized to inverse polynomial precision using only one oracle query and to inverse exponential precision using two oracle queries. This answers an open question of Aaronson from 2016, who presented a state synthesis algorithm that makes $O(n)$ queries to a classical oracle to prepare an $n$-qubit state, and asked if the query complexity could be made sublinear.

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arXiv2020

Two-player entangled games are NP-hard

We show that the maximum success probability of players sharing quantum entanglement in a two-player game with classical questions of logarithmic length and classical answers of constant length is NP-hard to approximate to within constant factors. As a corollary, the inclusion $\mathrm{NEXP}\subseteq\mathrm{MIP}^*$, first shown in [IV12] with three provers, holds with two provers only. The proof is based on a simpler, improved analysis of the low-degree test Raz and Safra (STOC'97) against two entangled provers.

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arXiv2020

Quantum soundness of the classical low individual degree test

Low degree tests play an important role in classical complexity theory, serving as basic ingredients in foundational results such as $\mathsf{MIP} = \mathsf{NEXP}$ [BFL91] and the PCP theorem [AS98,ALM+98]. Over the last ten years, versions of these tests which are sound against quantum provers have found increasing applications to the study of nonlocal games and the complexity class~$\mathsf{MIP}^*$. The culmination of this line of work is the result $\mathsf{MIP}^* = \mathsf{RE}$ [arXiv:2001.04383]. One of the key ingredients in the first reported proof of $\mathsf{MIP}^* = \mathsf{RE}$ is a two-prover variant of the low degree test, initially shown to be sound against multiple quantum provers in [arXiv:1302.1242]. Unfortunately a mistake was recently discovered in the latter result, invalidating the main result of [arXiv:1302.1242] as well as its use in subsequent works, including [arXiv:2001.04383]. We analyze a variant of the low degree test called the low individual degree test. Our main result is that the two-player version of this test is sound against quantum provers. This soundness result is sufficient to re-derive several bounds on~$\mathsf{MIP}^*$ that relied on [arXiv:1302.1242], including $\mathsf{MIP}^* = \mathsf{RE}$.

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arXiv2019

NEEXP in MIP*

We study multiprover interactive proof systems. The power of classical multiprover interactive proof systems, in which the provers do not share entanglement, was characterized in a famous work by Babai, Fortnow, and Lund (Computational Complexity 1991), whose main result was the equality MIP = NEXP. The power of quantum multiprover interactive proof systems, in which the provers are allowed to share entanglement, has proven to be much more difficult to characterize. The best known lower-bound on MIP* is NEXP, due to Ito and Vidick (FOCS 2012). As for upper bounds, MIP* could be as large as RE, the class of recursively enumerable languages. The main result of this work is the inclusion of NEEXP (nondeterministic doubly exponential time) in MIP*. This is an exponential improvement over the prior lower bound and shows that proof systems with entangled provers are at least exponentially more powerful than classical provers. In our protocol the verifier delegates a classical, exponentially large MIP protocol for NEEXP to two entangled provers: the provers obtain their exponentially large questions by measuring their shared state, and use a classical PCP to certify the correctness of their exponentially-long answers. For the soundness of our protocol, it is crucial that each player should not only sample its own question correctly but also avoid performing measurements that would reveal the other player's sampled question. We ensure this by commanding the players to perform a complementary measurement, relying on the Heisenberg uncertainty principle to prevent the forbidden measurements from being performed.

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arXiv2018

Limitations of semidefinite programs for separable states and entangled games

Semidefinite programs (SDPs) are a framework for exact or approximate optimization that have widespread application in quantum information theory. We introduce a new method for using reductions to construct integrality gaps for SDPs. These are based on new limitations on the sum-of-squares (SoS) hierarchy in approximating two particularly important sets in quantum information theory, where previously no $ω(1)$-round integrality gaps were known: the set of separable (i.e. unentangled) states, or equivalently, the $2 \rightarrow 4$ norm of a matrix, and the set of quantum correlations; i.e. conditional probability distributions achievable with local measurements on a shared entangled state. In both cases no-go theorems were previously known based on computational assumptions such as the Exponential Time Hypothesis (ETH) which asserts that 3-SAT requires exponential time to solve. Our unconditional results achieve the same parameters as all of these previous results (for separable states) or as some of the previous results (for quantum correlations). In some cases we can make use of the framework of Lee-Raghavendra-Steurer (LRS) to establish integrality gaps for any SDP, not only the SoS hierarchy. Our hardness result on separable states also yields a dimension lower bound of approximate disentanglers, answering a question of Watrous and Aaronson et al. These results can be viewed as limitations on the monogamy principle, the PPT test, the ability of Tsirelson-type bounds to restrict quantum correlations, as well as the SDP hierarchies of Doherty-Parrilo-Spedalieri, Navascues-Pironio-Acin and Berta-Fawzi-Scholz.

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arXiv2018

Low-degree testing for quantum states, and a quantum entangled games PCP for QMA

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 $n\geq2$, we give a test in which questions from the verifier are $O(\log n)$ bits long, and answers are $\mathrm{poly}(\log\log n)$ bits long. We show that for any constant $\varepsilon\geq0$, any strategy that succeeds with probability at least $1-\varepsilon$ in the test must use a state that is within distance $O(\varepsilon^c)$ 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 $\mathbb{F}_q$, 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 constructions of PCPs of proximity introduced by Ben-Sasson et al. (CCC'05), and crucially relies 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 Hamiltonian quantum PCP conjecture.

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arXiv2018

Algorithms, Bounds, and Strategies for Entangled XOR Games

We study the complexity of computing the commuting-operator value $ω^*$ of entangled XOR games with any number of players. We introduce necessary and sufficient criteria for an XOR game to have $ω^* = 1$, and use these criteria to derive the following results: 1. An algorithm for symmetric games that decides in polynomial time whether $ω^* = 1$ or $ω^* < 1$, a task that was not previously known to be decidable, together with a simple tensor-product strategy that achieves value 1 in the former case. The only previous candidate algorithm for this problem was the Navascués-Pironio-Acín (also known as noncommutative Sum of Squares or ncSoS) hierarchy, but no convergence bounds were known. 2. A family of games with three players and with $ω^* < 1$, where it takes doubly exponential time for the ncSoS algorithm to witness this (in contrast with our algorithm which runs in polynomial time). 3. A family of games achieving a bias difference $2(ω^* - ω)$ arbitrarily close to the maximum possible value of $1$ (and as a consequence, achieving an unbounded bias ratio), answering an open question of Briët and Vidick. 4. Existence of an unsatisfiable phase for random (non-symmetric) XOR games: that is, we show that there exists a constant $C_k^{\text{unsat}}$ depending only on the number $k$ of players, such that a random $k$-XOR game over an alphabet of size $n$ has $ω^* < 1$ with high probability when the number of clauses is above $C_k^{\text{unsat}} n$. 5. A lower bound of $Ω(n \log(n)/\log\log(n))$ on the number of levels in the ncSoS hierarchy required to detect unsatisfiability for most random 3-XOR games. This is in contrast with the classical case where the $n$-th level of the sum-of-squares hierarchy is equivalent to brute-force enumeration of all possible solutions.

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arXiv2017

An improved semidefinite programming hierarchy for testing entanglement

We present a stronger version of the Doherty-Parrilo-Spedalieri (DPS) hierarchy of approximations for the set of separable states. Unlike DPS, our hierarchy converges exactly at a finite number of rounds for any fixed input dimension. This yields an algorithm for separability testing which is singly exponential in dimension and polylogarithmic in accuracy. Our analysis makes use of tools from algebraic geometry, but our algorithm is elementary and differs from DPS only by one simple additional collection of constraints.

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arXiv2016

Robust self-testing of many-qubit states

We introduce a simple two-player test which certifies that the players apply tensor products of Pauli $σ_X$ and $σ_Z$ observables on the tensor product of $n$ EPR pairs. The test has constant robustness: any strategy achieving success probability within an additive $\varepsilon$ of the optimal must be $\mathrm{poly}(\varepsilon)$-close, in the appropriate distance measure, to the honest $n$-qubit strategy. The test involves $2n$-bit questions and $2$-bit answers. The key technical ingredient is a quantum version of the classical linearity test of Blum, Luby, and Rubinfeld. As applications of our result we give (i) the first robust self-test for $n$ EPR pairs; (ii) a quantum multiprover interactive proof system for the local Hamiltonian problem with a constant number of provers and classical questions and answers, and a constant completeness-soundness gap independent of system size; (iii) a robust protocol for delegated quantum computation.

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arXiv2016

The Parallel-Repeated Magic Square Game is Rigid

We show that the $n$-round parallel repetition of the Magic Square game of Mermin and Peres is rigid, in the sense that for any entangled strategy succeeding with probability $1 -\varepsilon$, the players' shared state is $O(\mathrm{poly}(n\varepsilon))$-close to $2n$ EPR pairs under a local isometry. Furthermore, we show that, under local isometry, the players' measurements in said entangled strategy must be $O(\mathrm{poly}(n\varepsilon))$ close to the "ideal" strategy when acting on the shared state.

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arXiv2015

Constant-Soundness Interactive Proofs for Local Hamiltonians

$ \newcommand{\Xlin}{\mathcal{X}} \newcommand{\Zlin}{\mathcal{Z}} \newcommand{\C}{\mathbb{C}} $We give a quantum multiprover interactive proof system for the local Hamiltonian problem in which there is a constant number of provers, questions are classical of length polynomial in the number of qubits, and answers are of constant length. The main novelty of our protocol is that the gap between completeness and soundness is directly proportional to the promise gap on the (normalized) ground state energy of the Hamiltonian. This result can be interpreted as a concrete step towards a quantum PCP theorem giving entangled-prover interactive proof systems for QMA-complete problems. The key ingredient is a quantum version of the classical linearity test of Blum, Luby, and Rubinfeld, where the function $f:\{0,1\}^n\to\{0,1\}$ is replaced by a pair of functions $\Xlin, \Zlin:\{0,1\}^n\to \text{Obs}_d(\C)$, the set of $d$-dimensional Hermitian matrices that square to identity. The test enforces that (i) each function is exactly linear, $\Xlin(a)\Xlin(b)=\Xlin(a+b)$ and $\Zlin(a) \Zlin(b)=\Zlin(a+b)$, and (ii) the two functions are approximately complementary, $\Xlin(a)\Zlin(b)\approx (-1)^{a\cdot b} \Zlin(b)\Xlin(a)$.

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