Harumichi Nishimura

Computational complexity of uniform quantum circuit families and quantum Turing machines

Deutsch proposed two sorts of models of quantum computers, quantum Turing machines (QTMs) and quantum circuit families (QCFs). In this paper we explore the computational powers of these models and re-examine the claim of the computational equivalence of these models often made in the literature without detailed investigations. For this purpose, we formulate the notion of the codes of QCFs and the uniformity of QCFs by the computability of the codes. Various complexity classes are introduced for QTMs and QCFs according to constraints on the error probability of algorithms or transition amplitudes. Their interrelations are examined in detail. For Monte Carlo algorithms, it is proved that the complexity classes based on uniform QCFs are identical with the corresponding classes based on QTMs. However, for Las Vegas algorithms, it is still open whether the two models are equivalent. We indicate the possibility that they are not equivalent. In addition, we give a complete proof of the existence of a universal QTM efficiently simulating multi-tape QTMs. We also examine the simulation of various types of QTMs such as multi-tape QTMs, single tape QTMs, stationary, normal form QTMs (SNQTMs), and QTMs with the binary tapes. As a result, we show that these QTMs are computationally equivalent to one another as computing models implementing not only Monte Carlo algorithms but also exact (or error-free) ones

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.

Constructing quantum network coding schemes from classical nonlinear protocols

The k-pair problem in network coding theory asks to send k messages simultaneously between k source-target pairs over a directed acyclic graph. In a previous paper [ICALP 2009, Part I, pages 622–633] the present authors showed that if a classical k-pair problem is solvable by means of a linear coding scheme, then the quantum k-pair problem over the same graph is also solvable, provided that classical communication can be sent for free between any pair of nodes of the graph. Here we address the main case that remained open in our previous work, namely whether nonlinear classical network coding schemes can also give rise to quantum network coding schemes. This question is motivated by the fact that there are networks for which no linear solutions exist to the k-pair problem, whereas nonlinear solutions exist. In the present paper we overcome the limitation to linear protocols and describe a new communication protocol for perfect quantum network coding that improves over the previous one as follows: (i) the new protocol does not put any condition on the underlying classical coding scheme, that is, it can simulate nonlinear communication protocols as well, and (ii) the amount of classical communication sent in the protocol is significantly reduced.

Perfect quantum network communication protocol based on classical network coding

This paper considers a problem of quantum communication between parties that are connected through a network of quantum channels. The model in this paper assumes that there is no prior entanglement shared among any of the parties, but that classical communication is free. The task is to perfectly transfer an unknown quantum state from a source subsystem to a target subsystem, where both source and target are formed by ordered sets of some of the nodes. It is proved that a lower bound of the rate at which this quantum communication task is possible is given by the classical min-cut max-flow theorem of network coding, where the capacities in question are the quantum capacities of the edges of the network.

General Scheme for Perfect Quantum Network Coding with Free Classical Communication

This paper considers the problem of efficiently transmitting quantum states through a network. It has been known for some time that without additional assumptions it is impossible to achieve this task perfectly in general -- indeed, it is impossible even for the simple butterfly network. As additional resource we allow free classical communication between any pair of network nodes. It is shown that perfect quantum network coding is achievable in this model whenever classical network coding is possible over the same network when replacing all quantum capacities by classical capacities. More precisely, it is proved that perfect quantum network coding using free classical communication is possible over a network with k source-target pairs if there exists a classical linear (or even vector-linear) coding scheme over a finite ring. Our proof is constructive in that we give explicit quantum coding operations for each network node. This paper also gives an upper bound on the number of classical communication required in terms of k , the maximal fan-in of any network node, and the size of the network.

Local Transition Functions of Quantum Turing Machines

Foundations of the notion of quantum Turing machines are investigated. According to Deutschs formulation, the time evolution of a quantum Turing machine is to be determined by the local transition function. In this paper, the local transition functions are characterized for fully general quantum Turing machines, including multi-tape quantum Turing machines, extending the results due to Bernstein and Vazirani.

Perfect computational equivalence between quantum Turing machines and finitely generated uniform quantum circuit families

In order to establish the computational equivalence between quantum Turing machines (QTMs) and quantum circuit families (QCFs) using Yao’s quantum circuit simulation of QTMs, we previously introduced the class of uniform QCFs based on an infinite set of elementary gates, which has been shown to be computationally equivalent to the polynomial-time QTMs (with appropriate restriction of amplitudes) up to bounded error simulation. This result implies that the complexity class BQP introduced by Bernstein and Vazirani for QTMs equals its counterpart for uniform QCFs. However, the complexity classes ZQP and EQP for QTMs do not appear to equal their counterparts for uniform QCFs. In this paper, we introduce a subclass of uniform QCFs, the finitely generated uniform QCFs, based on finite number of elementary gates and show that the class of finitely generated uniform QCFs is perfectly equivalent to the class of polynomial-time QTMs; they can exactly simulate each other. This naturally implies that BQP as well as ZQP and EQP equal the corresponding complexity classes of the finitely generated uniform QCFs.

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.

(4,1)-Quantum random access coding does not exist—one qubit is not enough to recover one of four bits

An (n,1,p)-quantum random access (QRA) coding, introduced by Ambainis et al (1999 ACM Symp. Theory of Computing p 376), is the following communication system: the sender which has n-bit information encodes his/her information into one qubit, which is sent to the receiver. The receiver can recover any one bit of the original n bits correctly with probability at least p, through a certain decoding process based on positive operator-valued measures. Actually, Ambainis et al shows the existence of a (2,1,0.85)-QRA coding and also proves the impossibility of its classical counterpart. Chuang immediately extends it to a (3,1,0.79)-QRA coding and whether or not a (4,1,p)-QRA coding such that p > 1/2 exists has been open since then. This paper gives a negative answer to this open question. Moreover, we generalize its negative answer for one-qubit encoding to the case of multiple-qubit encoding

Quantum Query Complexity of Boolean Functions with Small On-Sets

The main objective of this paper is to show that the quantum query complexity Q(f) of an N-bit Boolean function f is bounded by a function of a simple and natural parameter, i.e., M = |{x|f(x) = 1}| or the size of fs on-set. We prove that: (i) For