Computational complexity of uniform quantum circuit families and quantum Turing machines Communicated by O. Watanabe

Abstract

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 simulating multi-tape QTMs efficiently. 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 one another as computing models implementing not only Monte Carlo algorithms but exact (or errorfree) ones.

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

“…In our previous investigation [19], we instituted the complexity theory of uniform QCFs . For this purpose, we rigorously introduced the notion of uniformity of QCFs and based on that we defined the complexity classes, what we called BUPQC, ZUPQC, and EUPQC, which correspond to BQP, ZQP, and EQP, respectively.…”

“…Using this formulation and Yao's quantum circuit construction for simulation of QTMs, we showed the following results on the computational equivalence between QTMs and uniform QCFs: (i) BUPQC = BQP [19] (QTMs and uniform QCFs are computationally equivalent in the bounded-error setting). (ii) ZQP ⊆ ZUPQC and EQP ⊆ EUPQC while the simulation does not work to show the converse [19,20] (the computational equivalence between the two models are open in the zero-error and exact setting). Thus, the following question still remained: In the zero-error and exact setting, what restriction for the uniform QCF guarantees the computational equivalence between the two models?…”

“…To this end, we introduce the notion of finitely generated uniform QCFs based on finite sets of elementary gates, which is a subclass of the uniform QCFs. In [19], we briefly mentioned the notion of semi-uniform QCFs, which is also based on finite sets of elementary gates. Finitely generated uniform QCFs are regarded as those semi-uniform QCFs whose amplitudes are taken from polynomial-time computable numbers.…”

“…The proof requires more minute arguments than that of the computational equivalence up to bounded error simulation. For the proof, we revisit properties of polynomial-time computable numbers, which was implicitly used in [19], and Yao's quantum circuit construction. We combine them with the exact decomposition of unitary matrices.…”

Universality of quantum Turing machines with deterministic control

“…The s-m-n result allowed the definition of a robust notion of quantum Kolmogorov complexity. Notwithstanding the well known equivalence between quantum circuits and Deutsch machines [14,15], the results we established in this paper show that a quantum transition table is not required for achieving similar equivalence results, at least in the Monte Carlo scenario.…”