Quantum proofs can be verified using only single qubit measurements

QMA (Quantum Merlin Arthur) is the class of problems which, though potentially hard to solve, have a quantum solution which can be verified efficiently using a quantum computer. It thus forms a natural quantum version of the classical complexity class NP (and its probabilistic variant MA, Merlin-Arthur games), where the verifier has only classical computational resources. In this paper, we study what happens when we restrict the quantum resources of the verifier to the bare minimum: individual measurements on single qubits received as they come, one-by-one. We find that despite this grave restriction, it is still possible to soundly verify any problem in QMA for the verifier with the minimum quantum resources possible, without using any quantum memory or multiqubit operations. We provide two independent proofs of this fact, based on measurement based quantum computation and the local Hamiltonian problem, respectively. The former construction also applies to QMA$_1$, i.e., QMA with one-sided error.Comment: 7 pages, 1 figur

Quantum superiority for verifying NP-complete problems with linear optics

Demonstrating quantum superiority for some computational task will be a milestone for quantum technologies and would show that computational advantages are possible not only with a universal quantum computer but with simpler physical devices. Linear optics is such a simpler but powerful platform where classically-hard information processing tasks, such as Boson Sampling, can be in principle implemented. In this work, we study a fundamentally different type of computational task to achieve quantum superiority using linear optics, namely the task of verifying NP-complete problems. We focus on a protocol by Aaronson et al. (2008) that uses quantum proofs for verification. We show that the proof states can be implemented in terms of a single photon in an equal superposition over many optical modes. Similarly, the tests can be performed using linear-optical transformations consisting of a few operations: a global permutation of all modes, simple interferometers acting on at most four modes, and measurement using single-photon detectors. We also show that the protocol can tolerate experimental imperfections.Comment: 10 pages, 6 figures, minor corrections, results unchange

Merlin-Arthur with efficient quantum Merlin and quantum supremacy for the second level of the Fourier hierarchy

We introduce a simple sub-universal quantum computing model, which we call the Hadamard-classical circuit with one-qubit (HC1Q) model. It consists of a classical reversible circuit sandwiched by two layers of Hadamard gates, and therefore it is in the second level of the Fourier hierarchy. We show that output probability distributions of the HC1Q model cannot be classically efficiently sampled within a multiplicative error unless the polynomial-time hierarchy collapses to the second level. The proof technique is different from those used for previous sub-universal models, such as IQP, Boson Sampling, and DQC1, and therefore the technique itself might be useful for finding other sub-universal models that are hard to classically simulate. We also study the classical verification of quantum computing in the second level of the Fourier hierarchy. To this end, we define a promise problem, which we call the probability distribution distinguishability with maximum norm (PDD-Max). It is a promise problem to decide whether output probability distributions of two quantum circuits are far apart or close. We show that PDD-Max is BQP-complete, but if the two circuits are restricted to some types in the second level of the Fourier hierarchy, such as the HC1Q model or the IQP model, PDD-Max has a Merlin-Arthur system with quantum polynomial-time Merlin and classical probabilistic polynomial-time Arthur.Comment: 30 pages, 4 figure

Stronger Methods of Making Quantum Interactive Proofs Perfectly Complete

This paper presents stronger methods of achieving perfect completeness in quantum interactive proofs. First, it is proved that any problem in QMA has a two-message quantum interactive proof system of perfect completeness with constant soundness error, where the verifier has only to send a constant number of halves of EPR pairs. This in particular implies that the class QMA is necessarily included by the class QIP_1(2) of problems having two-message quantum interactive proofs of perfect completeness, which gives the first nontrivial upper bound for QMA in terms of quantum interactive proofs. It is also proved that any problem having an $m$-message quantum interactive proof system necessarily has an $(m+1)$-message quantum interactive proof system of perfect completeness. This improves the previous result due to Kitaev and Watrous, where the resulting system of perfect completeness requires $m+2$ messages if not using the parallelization result.Comment: 41 pages; v2: soundness parameters improved, correction of a minor error in Lemma 23, and removal of the sentences claiming that our techniques are quantumly nonrelativizin

Testing product states, quantum Merlin-Arthur games and tensor optimisation

We give a test that can distinguish efficiently between product states of n quantum systems and states which are far from product. If applied to a state psi whose maximum overlap with a product state is 1-epsilon, the test passes with probability 1-Theta(epsilon), regardless of n or the local dimensions of the individual systems. The test uses two copies of psi. We prove correctness of this test as a special case of a more general result regarding stability of maximum output purity of the depolarising channel. A key application of the test is to quantum Merlin-Arthur games with multiple Merlins, where we obtain several structural results that had been previously conjectured, including the fact that efficient soundness amplification is possible and that two Merlins can simulate many Merlins: QMA(k)=QMA(2) for k>=2. Building on a previous result of Aaronson et al, this implies that there is an efficient quantum algorithm to verify 3-SAT with constant soundness, given two unentangled proofs of O(sqrt(n) polylog(n)) qubits. We also show how QMA(2) with log-sized proofs is equivalent to a large number of problems, some related to quantum information (such as testing separability of mixed states) as well as problems without any apparent connection to quantum mechanics (such as computing injective tensor norms of 3-index tensors). As a consequence, we obtain many hardness-of-approximation results, as well as potential algorithmic applications of methods for approximating QMA(2) acceptance probabilities. Finally, our test can also be used to construct an efficient test for determining whether a unitary operator is a tensor product, which is a generalisation of classical linearity testing.Comment: 44 pages, 1 figure, 7 appendices; v6: added references, rearranged sections, added discussion of connections to classical CS. Final version to appear in J of the AC

Quantum Pseudorandomness and Classical Complexity

We construct a quantum oracle relative to which BQP = QMA but cryptographic pseudorandom quantum states and pseudorandom unitary transformations exist, a counterintuitive result in light of the fact that pseudorandom states can be "broken" by quantum Merlin-Arthur adversaries. We explain how this nuance arises as the result of a distinction between algorithms that operate on quantum and classical inputs. On the other hand, we show that some computational complexity assumption is needed to construct pseudorandom states, by proving that pseudorandom states do not exist if BQP = PP. We discuss implications of these results for cryptography, complexity theory, and quantum tomography

Reliable quantum certification for photonic quantum technologies

A major roadblock for large-scale photonic quantum technologies is the lack of practical reliable certification tools. We introduce an experimentally friendly - yet mathematically rigorous - certification test for experimental preparations of arbitrary m-mode pure Gaussian states, pure non-Gaussian states generated by linear-optical circuits with n-boson Fock-basis states as inputs, and states of these two classes subsequently post-selected with local measurements on ancillary modes. The protocol is efficient in m and the inverse post-selection success probability for all Gaussian states and all mentioned non-Gaussian states with constant n. We follow the mindset of an untrusted prover, who prepares the state, and a skeptic certifier, with classical computing and single-mode homodyne-detection capabilities only. No assumptions are made on the type of noise or capabilities of the prover. Our technique exploits an extremality-based fidelity bound whose estimation relies on non-Gaussian state nullifiers, which we introduce on the way as a byproduct result. The certification of many-mode photonic networks, as those used for photonic quantum simulations, boson samplers, and quantum metrology, is now within reach.Comment: 8 pages + 20 pages appendix, 2 figures, results generalized to scenarios with post-selection, presentation improve