Quantum vs Classical Proofs and Subset Verification

We study the ability of efficient quantum verifiers to decide properties of exponentially large subsets given either a classical or quantum witness. We develop a general framework that can be used to prove that QCMA machines, with only classical witnesses, cannot verify certain properties of subsets given implicitly via an oracle. We use this framework to prove an oracle separation between QCMA and QMA using an "in-place" permutation oracle, making the first progress on this question since Aaronson and Kuperberg in 2007. We also use the framework to prove a particularly simple standard oracle separation between QCMA and AM.Comment: 23 pages, presentation and notation clarified, small errors fixe

Pseudorandom generators and the BQP vs. PH problem

It is a longstanding open problem to devise an oracle relative to which BQP does not lie in the Polynomial-Time Hierarchy (PH). We advance a natural conjecture about the capacity of the Nisan-Wigderson pseudorandom generator [NW94] to fool AC_0, with MAJORITY as its hard function. Our conjecture is essentially that the loss due to the hybrid argument (which is a component of the standard proof from [NW94]) can be avoided in this setting. This is a question that has been asked previously in the pseudorandomness literature [BSW03]. We then make three main contributions: (1) We show that our conjecture implies the existence of an oracle relative to which BQP is not in the PH. This entails giving an explicit construction of unitary matrices, realizable by small quantum circuits, whose row-supports are "nearly-disjoint." (2) We give a simple framework (generalizing the setting of Aaronson [A10]) in which any efficiently quantumly computable unitary gives rise to a distribution that can be distinguished from the uniform distribution by an efficient quantum algorithm. When applied to the unitaries we construct, this framework yields a problem that can be solved quantumly, and which forms the basis for the desired oracle. (3) We prove that Aaronson's "GLN conjecture" [A10] implies our conjecture; our conjecture is thus formally easier to prove. The GLN conjecture was recently proved false for depth greater than 2 [A10a], but it remains open for depth 2. If true, the depth-2 version of either conjecture would imply an oracle relative to which BQP is not in AM, which is itself an outstanding open problem. Taken together, our results have the following interesting interpretation: they give an instantiation of the Nisan-Wigderson generator that can be broken by quantum computers, but not by the relevant modes of classical computation, if our conjecture is true.Comment: Updated in light of counterexample to the GLN conjectur

The Power of Quantum Fourier Sampling

A line of work initiated by Terhal and DiVincenzo and Bremner, Jozsa, and Shepherd, shows that quantum computers can efficiently sample from probability distributions that cannot be exactly sampled efficiently on a classical computer, unless the PH collapses. Aaronson and Arkhipov take this further by considering a distribution that can be sampled efficiently by linear optical quantum computation, that under two feasible conjectures, cannot even be approximately sampled classically within bounded total variation distance, unless the PH collapses. In this work we use Quantum Fourier Sampling to construct a class of distributions that can be sampled by a quantum computer. We then argue that these distributions cannot be approximately sampled classically, unless the PH collapses, under variants of the Aaronson and Arkhipov conjectures. In particular, we show a general class of quantumly sampleable distributions each of which is based on an "Efficiently Specifiable" polynomial, for which a classical approximate sampler implies an average-case approximation. This class of polynomials contains the Permanent but also includes, for example, the Hamiltonian Cycle polynomial, and many other familiar #P-hard polynomials. Although our construction, unlike that proposed by Aaronson and Arkhipov, likely requires a universal quantum computer, we are able to use this additional power to weaken the conjectures needed to prove approximate sampling hardness results

The Power of Unentanglement

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? 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 m is satisfiable, with constant soundness, given Õ (√m) unentangled quantum witnesses with O(log m) qubits each. Our protocol relies on the existence of very short PCPs. * We show that assuming a weak version of the 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 prove the nonexistence of "perfect disentanglers" for simulating multiple Merlins with one

A Complete Characterization of Unitary Quantum Space

Motivated by understanding the power of quantum computation with restricted number of qubits, we give two complete characterizations of unitary quantum space bounded computation. First we show that approximating an element of the inverse of a well-conditioned efficiently encoded 2^k(n) x 2^k(n) matrix is complete for the class of problems solvable by quantum circuits acting on O(k(n)) qubits with all measurements at the end of the computation. Similarly, estimating the minimum eigenvalue of an efficiently encoded Hermitian 2^k(n) x 2^k(n) matrix is also complete for this class. In the logspace case, our results improve on previous results of Ta-Shma by giving new space-efficient quantum algorithms that avoid intermediate measurements, as well as showing matching hardness results. Additionally, as a consequence we show that preciseQMA, the version of QMA with exponentially small completeness-soundess gap, is equal to PSPACE. Thus, the problem of estimating the minimum eigenvalue of a local Hamiltonian to inverse exponential precision is PSPACE-complete, which we show holds even in the frustration-free case. Finally, we can use this characterization to give a provable setting in which the ability to prepare the ground state of a local Hamiltonian is more powerful than the ability to prepare PEPS states. Interestingly, by suitably changing the parameterization of either of these problems we can completely characterize the power of quantum computation with simultaneously bounded time and space