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.Comment: 11page

Local Transition Functions of Quantum Turing Machines

Foundations of the notion of quantum Turing machines are investigated. According to Deutsch's 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.Comment: 22 pages, LaTeX, revised and extended, to appear in RAIRO Theor. Inform. App

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