arXiv subjects

Hirotada Kobayashi

Publications and source records attributed to Hirotada Kobayashi.

25 records · Page 2Linked to original sources

arXiv2007

General Properties of Quantum Zero-Knowledge Proofs

This paper studies the complexity classes QZK and HVQZK of problems having a quantum computational zero-knowledge proof system and an honest-verifier quantum computational zero-knowledge proof system, respectively. The results proved in this paper include: (a) HVQZK = QZK, (b) any problem in QZK has a public-coin quantum computational zero-knowledge proof system, (c) any problem in QZK has a quantum computational zero-knowledge proof system of perfect completeness, and (d) any problem in QZK has a three-message public-coin quantum computational zero-knowledge proof system of perfect completeness with arbitrarily small constant error in soundness. All the results above are unconditional and do not rely any computational assumptions. For the classes QPZK, HVQPZK, and QSZK of problems having a quantum perfect zero-knowledge proof system, an honest-verifier quantum perfect zero-knowledge proof system, and a quantum statistical zero-knowledge proof system, respectively, the following new properties are proved: (e) HVQPZK = QPZK, (f) any problem in QPZK has a public-coin quantum perfect zero-knowledge proof system, (g) any problem in QSZK has a quantum statistical zero-knowledge proof system of perfect completeness, and (h) any problem in QSZK has a three-message public-coin quantum statistical zero-knowledge proof system of perfect completeness with arbitrarily small constant error in soundness. It is stressed that our proofs are direct and do not use complete promise problems or those equivalents. This gives a unified framework that works well for all of quantum perfect, statistical, and computational zero-knowledge proofs, and enables us to prove properties even on the computational and perfect zero-knowledge proofs for which no complete promise problems are known.

quant-ph↗

arXiv2004

Universal Test for Quantum One-Way Permutations

The next bit test was introduced by Blum and Micali and proved by Yao to be a universal test for cryptographic pseudorandom generators. On the other hand, no universal test for the cryptographic one-wayness of functions (or permutations) is known, though the existence of cryptographic pseudorandom generators is equivalent to that of cryptographic one-way functions. In the quantum computation model, Kashefi, Nishimura and Vedral gave a sufficient condition of (cryptographic) quantum one-way permutations and conjectured that the condition would be necessary. In this paper, we affirmatively settle their conjecture and complete a necessary and sufficient for quantum one-way permutations. The necessary and sufficient condition can be regarded as a universal test for quantum one-way permutations, since the condition is described as a collection of stepwise tests similar to the next bit test for pseudorandom generators.

quant-ph↗

arXiv2003

Quantum Multi-Prover Interactive Proof Systems with Limited Prior Entanglement

This paper gives the first formal treatment of a quantum analogue of multi-prover interactive proof systems. It is proved that the class of languages having quantum multi-prover interactive proof systems is necessarily contained in NEXP, under the assumption that provers are allowed to share at most polynomially many prior-entangled qubits. This implies that, in particular, if provers do not share any prior entanglement with each other, the class of languages having quantum multi-prover interactive proof systems is equal to NEXP. Related to these, it is shown that, in the case a prover does not have his private qubits, the class of languages having quantum single-prover interactive proof systems is also equal to NEXP.

cs.CC↗

arXiv2003

Analysis of Absorbing Times of Quantum Walks

Quantum walks are expected to provide useful algorithmic tools for quantum computation. This paper introduces absorbing probability and time of quantum walks and gives both numerical simulation results and theoretical analyses on Hadamard walks on the line and symmetric walks on the hypercube from the viewpoint of absorbing probability and time.

quant-ph↗

arXiv2002

Non-Interactive Quantum Statistical and Perfect Zero-Knowledge

This paper introduces quantum analogues of non-interactive perfect and statistical zero-knowledge proof systems. Similar to the classical cases, it is shown that sharing randomness or entanglement is necessary for non-trivial protocols of non-interactive quantum perfect and statistical zero-knowledge. It is also shown that, with sharing EPR pairs a priori, the class of languages having one-sided bounded error non-interactive quantum perfect zero-knowledge proof systems has a natural complete problem. Non-triviality of such a proof system is based on the fact proved in this paper that the Graph Non-Automorphism problem, which is not known in BQP, can be reduced to our complete problem. Our results may be the first non-trivial quantum zero-knowledge proofs secure even against dishonest quantum verifiers, since our protocols are non-interactive, and thus the zero-knowledge property does not depend on whether the verifier in the protocol is honest or not. A restricted version of our complete problem derives a natural complete problem for BQP.

quant-ph↗

arXiv2001

Two-way Quantum One-counter Automata

After the first treatments of quantum finite state automata by Moore and Crutchfield and by Kondacs and Watrous, a number of papers study the power of quantum finite state automata and their variants. This paper introduces a model of two-way quantum one-counter automata (2Q1CAs), combining the model of two-way quantum finite state automata (2QFAs) by Kondacs and Watrous and the model of one-way quantum one-counter automata (1Q1CAs) by Kravtsev. We give the definition of 2Q1CAs with well-formedness conditions. It is proved that 2Q1CAs are at least as powerful as classical two-way deterministic one-counter automata (2D1CAs), that is, every language L recognizable by 2D1CAs is recognized by 2Q1CAs with no error. It is also shown that several non-context-free languages including {a^n b^{n^2}} and {a^n b^{2^n}} are recognizable by 2Q1CAs with bounded error.

cs.CC↗

arXiv2001

Quantum Certificate Verification: Single versus Multiple Quantum Certificates

The class MA consists of languages that can be efficiently verified by classical probabilistic verifiers using a single classical certificate, and the class QMA consists of languages that can be efficiently verified by quantum verifiers using a single quantum certificate. Suppose that a verifier receives not only one but multiple certificates. In the classical setting, it is obvious that a classical verifier with multiple classical certificates is essentially the same with the one with a single classical certificate. However, in the quantum setting where a quantum verifier is given a set of quantum certificates in tensor product form (i.e. each quantum certificate is not entangled with others), the situation is different, because the quantum verifier might utilize the structure of the tensor product form. This suggests a possibility of another hierarchy of complexity classes, namely the QMA hierarchy. From this point of view, we extend the definition of QMA to QMA(k) for the case quantum verifiers use k quantum certificates, and analyze the properties of QMA(k). To compare the power of QMA(2) with that of QMA(1) = QMA, we show one interesting property of ``quantum indistinguishability''. This gives a strong evidence that QMA(2) is more powerful than QMA(1). Furthermore, we show that, for any fixed positive integer $k \geq 2$, if a language L has a one-sided bounded error QMA(k) protocol with a quantum verifier using k quantum certificates, L necessarily has a one-sided bounded error QMA(2) protocol with a quantum verifier using only two quantum certificates.

quant-ph↗