Generalized Quantum Arthur-Merlin Games

This paper investigates the role of interaction and coins in public-coin quantum interactive proof systems (also called quantum Arthur-Merlin games). While prior works focused on classical public coins even in the quantum setting, the present work introduces a generalized version of quantum Arthur-Merlin games where the public coins can be quantum as well: the verifier can send not only random bits, but also halves of EPR pairs. First, it is proved that the class of two-turn quantum Arthur-Merlin games with quantum public coins, denoted qq-QAM in this paper, does not change by adding a constant number of turns of classical interactions prior to the communications of the qq-QAM proof systems. This can be viewed as a quantum analogue of the celebrated collapse theorem for AM due to Babai. To prove this collapse theorem, this paper provides a natural complete problem for qq-QAM: deciding whether the output of a given quantum circuit is close to a totally mixed state. This complete problem is on the very line of the previous studies investigating the hardness of checking the properties related to quantum circuits, and is of independent interest. It is further proved that the class qq-QAM_1 of two-turn quantum-public-coin quantum Arthur-Merlin proof systems with perfect completeness gives new bounds for standard well-studied classes of two-turn interactive proof systems. Finally, the collapse theorem above is extended to comprehensively classify the role of interaction and public coins in quantum Arthur-Merlin games: it is proved that, for any constant m>1, the class of problems having an m-turn quantum Arthur-Merlin proof system is either equal to PSPACE or equal to the class of problems having a two-turn quantum Arthur-Merlin game of a specific type, which provides a complete set of quantum analogues of Babai's collapse theorem.Comment: 31 pages + cover page, the proof of Lemma 27 (Lemma 24 in v1) is corrected, and a new completeness result is adde

Quantum Arthur-Merlin Games

This paper studies quantum Arthur-Merlin games, which are Arthur-Merlin games in which Arthur and Merlin can perform quantum computations and Merlin can send Arthur quantum information. As in the classical case, messages from Arthur to Merlin are restricted to be strings of uniformly generated random bits. It is proved that for one-message quantum Arthur-Merlin games, which correspond to the complexity class QMA, completeness and soundness errors can be reduced exponentially without increasing the length of Merlin's message. Previous constructions for reducing error required a polynomial increase in the length of Merlin's message. Applications of this fact include a proof that logarithmic length quantum certificates yield no increase in power over BQP and a simple proof that QMA is contained in PP. Other facts that are proved include the equivalence of three (or more) message quantum Arthur-Merlin games with ordinary quantum interactive proof systems and some basic properties concerning two-message quantum Arthur-Merlin games.Comment: 22 page

Non-locality and Communication Complexity

Quantum information processing is the emerging field that defines and realizes computing devices that make use of quantum mechanical principles, like the superposition principle, entanglement, and interference. In this review we study the information counterpart of computing. The abstract form of the distributed computing setting is called communication complexity. It studies the amount of information, in terms of bits or in our case qubits, that two spatially separated computing devices need to exchange in order to perform some computational task. Surprisingly, quantum mechanics can be used to obtain dramatic advantages for such tasks. We review the area of quantum communication complexity, and show how it connects the foundational physics questions regarding non-locality with those of communication complexity studied in theoretical computer science. The first examples exhibiting the advantage of the use of qubits in distributed information-processing tasks were based on non-locality tests. However, by now the field has produced strong and interesting quantum protocols and algorithms of its own that demonstrate that entanglement, although it cannot be used to replace communication, can be used to reduce the communication exponentially. In turn, these new advances yield a new outlook on the foundations of physics, and could even yield new proposals for experiments that test the foundations of physics.Comment: Survey paper, 63 pages LaTeX. A reformatted version will appear in Reviews of Modern Physic

Quantum interactive proofs with short messages

This paper considers three variants of quantum interactive proof systems in which short (meaning logarithmic-length) messages are exchanged between the prover and verifier. The first variant is one in which the verifier sends a short message to the prover, and the prover responds with an ordinary, or polynomial-length, message; the second variant is one in which any number of messages can be exchanged, but where the combined length of all the messages is logarithmic; and the third variant is one in which the verifier sends polynomially many random bits to the prover, who responds with a short quantum message. We prove that in all of these cases the short messages can be eliminated without changing the power of the model, so the first variant has the expressive power of QMA and the second and third variants have the expressive power of BQP. These facts are proved through the use of quantum state tomography, along with the finite quantum de Finetti theorem for the first variant.Comment: 15 pages, published versio