Tomoyuki Yamakami

Quantum Merlin-Arthur Proof Systems: Are Multiple Merlins More Helpful to Arthur?

This paper introduces quantum “multiple-Merlin”-Arthur proof systems in which Arthur uses multiple quantum proofs unentangled with each other for his verification. Although classical multi-proof systems are obviously equivalent to classical single-proof systems, it is unclear whether quantum multi-proof systems collapse to quantum single-proof systems. This paper presents a necessary and sufficient condition under which the number of quantum proofs is reducible to two. It is also proved that using multiple quantum proofs does not increase the power of quantum Merlin-Arthur proof systems in the case of perfect soundness, and that there is a relativized world in which co-NP (actually co-UP) does not have quantum Merlin-Arthur proof systems even with multiple quantum proofs.

Polynomial time quantum computation with advice

Advice is supplementary information that enhances the computational power of an underlying computation. This paper focuses on advice that is given in the form of a pure quantum state and examines the influence of such advice on the behaviors of an underlying polynomial-time quantum computation with bounded-error probability.

Immunity and pseudorandomness of context-free languages

We discuss the computational complexity of context-free languages, concentrating on two well-known structural properties?immunity and pseudorandomness. An infinite language is REG-immune (resp., CFL-immune) if it contains no infinite subset that is a regular (resp., context-free) language. We prove that (i) there is a context-free REG-immune language outside REG/n and (ii) there is a REG-bi-immune language that can be computed deterministically using logarithmic space. We also show that (iii) there is a CFL-simple set, where a CFL-simple language is an infinite context-free language whose complement is CFL-immune. Similar to the REG-immunity, a REG-primeimmune language has no polynomially dense subsets that are also regular. We further prove that (iv) there is a context-free language that is REG/n-bi-primeimmune. Concerning pseudorandomness of context-free languages, we show that (v) CFL contains REG/n-pseudorandom languages. Finally, we prove that (vi) against REG/n, there exists an almost 1?1 pseudorandom generator computable in nondeterministic pushdown automata equipped with a write-only output tape and (vii) against REG, there is no almost 1?1 weakly pseudorandom generator computable deterministically in linear time by a single-tape Turing machine.

An application of quantum finite automata to interactive proof systems

Quantum finite automata have been studied intensively since their introduction in late 1990s as a natural model of a quantum computer working with finite-dimensional quantum memory space. This paper seeks their direct application to interactive proof systems in which a mighty quantum prover communicates with a quantum-automaton verifier through a common communication cell. Our quantum interactive proof systems are juxtaposed to Dwork-Stockmeyers classical interactive proof systems whose verifiers are two-way probabilistic finite automata. We demonstrate strengths and weaknesses of our systems by studying how various restrictions on the behaviors of quantum-automaton verifiers affect the power of quantum interactive proof systems.

ANALYSIS OF QUANTUM FUNCTIONS

This paper initiates a systematic study of quantum functions, which are (partial) functions defined in terms of quantum mechanical computations. Of all quantum functions, we focus on resource-bounded quantum functions whose inputs are classical bit strings. We prove complexity-theoretical properties and unique characteristics of these quantum functions by recent techniques developed for the analysis of quantum computations. We also discuss relativized quantum functions that make adaptive and nonadaptive oracle queries.

Resource bounded immunity and simplicity

Revisiting the 30-years-old notions of resource-bounded immunity and simplicity, we investigate the structural characteristics of various immunity notions: strong immunity, almost immunity, and hyperimmunity as well as their corresponding simplicity notions. We also study limited immunity and simplicity, called k-immunity and feasible k-immunity, and their simplicity notions. Finally, we propose the k-immune hypothesis as a working hypothesis that guarantees the existence of simple sets in NP.

The efficiency of quantum identity testing of multiple states

We examine two quantum operations, the permutation test and the circle test, which test the identity of n quantum states. These operations naturally extend the well-studied swap test on two quantum states. We first show the optimality of the permutation test for any input size n as well as the optimality of the circle test for three input states. In particular, when n = 3, we present a semi-classical protocol, incorporated with the swap test, which approximates the circle test efficiently. Furthermore, we show that, with the help of classical preprocessing, a single use of the circle test can approximate the permutation test efficiently for an arbitrary input size n.

Theory of One Tape Linear Time Turing Machines

A theory of one-tape linear-time Turing machines is quite different from its polynomial-time counterpart. This paper discusses the computational complexity of one-tape Turing machines of various machine types (deterministic, nondeterministic, reversible, alternating, probabilistic, counting, and quantum Turing machines) that halt in time O(n), where the running time of a machine is defined as the height of its computation tree. We also address a close connection between one-tape linear-time Turing machines and finite state automata.

An application of quantum finite automata to interactive proof systems (extended abstract)

Quantum finite automata have been studied intensively since their introduction in late 1990s. This paper seeks their direct application to interactive proof systems in which a mighty quantum prover communicates with a quantum-automaton verifier through a common communication cell.

Pseudorandom generators against advised context-free languages

Pseudorandomness has played a central role in modern cryptography, finding theoretical and practical applications to various fields of computer science. A function that generates pseudorandom strings from shorter but truly random seeds is known as a pseudorandom generator. Our generators are designed to fool languages (or equivalently, Boolean-valued functions). In particular, our generator fools advised context-free languages, namely, context-free languages assisted by external information known as advice, and moreover our generator is made almost one-to-one, stretching