We prove a deterministic exponential time upper bound for Quantum Merlin-Arthur games with k unentangled provers. This is the first non-trivial upper bound of QMA(k) better than NEXP and can be considered an exponential improvement, unless EXP=NEXP. The key ideas of our proof are to use perturbation theory to reduce the QMA(2)-complete Separable Sparse Hamiltonian problem to a variant of the Separable Local Hamiltonian problem with an exponentially small promise gap, and then to decide this instance using epsilon-net methods. Our results imply an exponential time algorithm for the Pure State N-Representability problem in quantum chemistry, which is in QMA(2), but is not known to be in QMA. We also discuss the implications of our results on the Best Separable State problem.
We prove a deterministic exponential time upper bound for Quantum Merlin-Arthur games with k unentangled provers. This is the first non-trivial upper bound of QMA(k) better than NEXP and can be considered an exponential improvement, unless EXP=NEXP. The key ideas of our proof are to use perturbation theory to reduce the QMA(2)-complete Separable Sparse Hamiltonian problem to a variant of the Separable Local Hamiltonian problem with an exponentially small promise gap, and then to decide this instance using epsilon-net methods. Our results imply an exponential time algorithm for the Pure State N-Representability problem in quantum chemistry, which is in QMA(2), but is not known to be in QMA. We also discuss the implications of our results on the Best Separable State problem.
This paper studies multiple-proof quantum Merlin-Arthur (QMA) proof systems in the setting when the completeness-soundness gap is small. Small means that we only lower-bound the gap with an inverse-exponential function of the input length, or with an even smaller function. Using the protocol of Blier and Tapp [arXiv:0709.0738], we show that in this case the proof system has the same expressive power as non-deterministic exponential time (NEXP). Since single-proof QMA proof systems, with the same bound on the gap, have expressive power at most exponential time (EXP), we get a separation between single and multi-prover proof systems in the 'small-gap setting', under the assumption that EXP is not equal to NEXP. This implies, among others, the nonexistence of certain operators called disentanglers (defined by Aaronson et al. [arXiv:0804.0802]), with good approximation parameters. We also show that in this setting the proof system has the same expressive power if we restrict the verifier to be able to perform only Bell-measurements, i.e., using a BellQMA verifier. This is not known to hold in the usual setting, when the gap is bounded by an inverse-polynomial function of the input length. To show this we use the protocol of Chen and Drucker [arXiv:1011.0716]. The only caveat here is that we need at least a linear amount of proofs to achieve the power of NEXP, while in the previous setting two proofs were enough. We also study the case when the proof-lengths are only logarithmic in the input length and observe that in some cases the expressive power decreases. However, we show that it doesn't decrease further if we make the proof lengths to be even shorter.
We find a modification to QMA where having one quantum proof is strictly less powerful than having two unentangled proofs, assuming EXP = NEXP. This gives a new route to prove QMA(2) = NEXP that overcomes the primary drawback of a recent approach [arXiv:2402.18790 , arXiv:2306.13247] (QIP 2024). Our modification endows each proof with a form of *multipartite* unentanglement: after tracing out one register, a small number of qubits are separable from the rest of the state.
Stoquasticity, originating in sign-problem-free physical systems, gives rise to StoqMA, introduced by Bravyi, Bessen, and Terhal (2006), a quantum-inspired intermediate class between MA and AM. Unentanglement similarly gives rise to QMA(2), introduced by Kobayashi, Matsumoto, and Yamakami (CJTCS 2009), which generalizes QMA to two unentangled proofs and still has only the trivial NEXP upper bound. In this work, we initiate a systematic study of the power of unentanglement without destructive interference via StoqMA(2), the class of unentangled stoquastic Merlin--Arthur proof systems. Beyond its complexity-theoretic interest, StoqMA(2) is connected to the optimality of non-negative tensor optimization algorithms. We highlight:
1. NP⊆StoqMA(2) with O(n)-qubit proofs and completenes 1−2−polylog(n). Conversely, the Sum-of-Squares algorithm of Barak, Kelner, and Steurer (STOC 2014) gives an exponential-time upper bound for StoqMA(2). Our tightened analysis shows the optimality of our protocol and the BKS algorithm under ETH.
2. For logarithmic-size proofs, NP⊆StoqMA(2)log with completeness 1−O(n−2) and vanishing gap, while StoqMA(2)log⊆MA. Consequently, quantum-inspired randomness enables exponentially shorter unentangled proofs even under the assumption MA=NP.
3. For StoqMA(2)1, the parameter dependence in the general ETH-optimal time bound can be exponentially improved, or the bound achieved simultaneously with polynomial space.
Our lower bounds are obtained by stoquastizing the short-proof QMA(2) protocols using distribution testing techniques. Our upper bounds for the nearly perfect completeness case are proved via our rectangular closure testing framework.
The class QMA(k), introduced by Kobayashi et al., consists of all languages that can be verified using k unentangled quantum proofs. Many of the simplest questions about this class have remained embarrassingly open: for example, can we give any evidence that k quantum proofs are more powerful than one? Can we show any upper bound on QMA(k), besides the trivial NEXP? Does QMA(k) = QMA(2) for k ≥ 2? Can QMA(k) protocols be amplified to exponentially small error? In this paper, we make progress on all of the above questions. • We give a protocol by which a verifier can be convinced that a 3SAT formula of size n is satisfiable, with constant soundness, given Õ (√n) unentangled quantum witnesses with O (log n) qubits each. Our protocol relies on Dinur’s version of the PCP Theorem and is inherently non-relativizing. • We show that assuming the famous Additivity Conjecture from quantum information theory, any QMA(2) protocol can be amplified to exponentially small error, and QMA(k) = QMA(2) for all k ≥ 2. • We give evidence that QMA(2) ⊆ PSPACE, by showing that this would follow from “strong amplification” of QMA(2) protocols. • We prove the nonexistence of “perfect disentanglers” for simulating multiple Merlins with one.
The class QMA(k), introduced by Kobayashi et al., consists of all languages that can be verified using k unentangled quantum proofs. Many of the simplest questions about this class have remained embarrassingly open: for example, can we give any evidence that k quantum proofs are more powerful than one? Can we show any upper bound on QMA(k), besides the trivial NEXP? Does QMA(k)=QMA(2) for kges2? Can QMA(k) protocols be amplified to exponentially small error? In this paper, we make progress on all of the above questions. *We give a protocol by which a verifier can be convinced that a 3SAT formula of size n is satisfiable, with constant soundness, given O tilde(radicn) unentangled quantum witnesses with O(log n) qubits each. Our protocol relies on Dinur's version of the PCP Theorem and is inherently non-relativizing. *We show that assuming the famous Additivity Conjecture from quantum information theory, any QMA(2) protocol can be amplified to exponentially small error, and QMA(k)=QMA(2) for all kges=2. *We give evidence that QMA(2) sube PSPACE, by showing that this would follow from "strong amplification" of QMA(2) protocols. *We prove the nonexistence of "perfect disentanglers" for simulating multiple Merlins with one.
Interactive proof systems form an important complexity model that has been central to many prominent results in computational complexity theory, such as those on probabilistically checkable proofs, hardness of approximation, and fundamental cryptographic primitives. In this thesis, we study the quantum-enhanced version of interactive proof systems, in which each party has access to quantum computing resources. We focus on its power and limitations, which lead us to the following results. (*) We provide an alternative and conceptually simpler proof of Jain et al.’s recent breakthrough result, which demonstrates that the expressive power of quantum interactive proof systems coincides with that of its classical counterpart, and therefore PSPACE. (*) We determine the complexity of several variants of quantum competing-prover interactive proof systems, which introduce zero-sum games into interactive proof systems. In particular, we prove that the complexity class of two-turn quantum refereed games coincides with PSPACE, answering an important open problem of Jain et al.’s. Our result suggests that a more general model called double quantum interactive proof systems coincides with PSPACE, which subsumes and unifies all previous results on short refereed games. (*) We contribute to the study of two-prover quantum Merlin-Arthur games (QMA(2)), which exhibit an intriguing tradeoff between an intrinsic quantum ingredient, entanglement, and computational power. We prove a PSPACE upper bound for a variant of QMA(2) that is to date the most general one known in PSPACE. We also provide a quasi-polynomial time algorithm for optimizing linear functions over separable states, giving an alternative to the one proposed by Brandao et al. Our main technical contribution behind the above results is the equilibrium value method for obtaining space-efficient algorithms for a class of optimization problems that arise naturally in quantum computation. We also contribute to QMA-complete problems, where we demonstrate that the “Non-Identity Check” problem remains QMA-complete on circuits of poly-logarithmic depths, improving upon polynomial depths from previous results.