Complexity Bounds of Constant-Space Quantum Computation

Abstract

We realize constant-space quantum computation by measure-many two-way quantum finite automata and evaluate their language recognition power by analyzing patterns of their exotic behaviors and by exploring their structural properties. In particular, we show that, when the automata halt "in finite steps" along all computation paths, they must terminate in worstcase liner time. In the bounded-error probability case, the acceptance of the automata depends only on the computation paths that terminate within exponentially many steps even if not all computation paths may terminate. We also present a classical simulation of those automata on two-way multi-head probabilistic finite automata with cut points. Moreover, we discuss how the recognition power of the automata varies as the automata's acceptance criteria change to error free, one-sided error, bounded error, and unbounded error by comparing the complexity of their computational powers. We further note that, with the use of arbitrary complex transition amplitudes, two-way unbounded-error quantum finite automata and two-way bounded-error 2head quantum finite automata can recognize certain non-recursive languages, whereas two-way error-free quantum finite automata recognize only recursive languages.Keywords: constant space, quantum finite automata, cut point, error free, one-sided error, bounded error, unbounded error, absolutely halt, completely halt, determinant 1 over 2-way deterministic finite automata, whereas one-way qfa's fail to capture even regular languages [17]. Despite our efforts over the past 20 years, the behaviors of 2qfa's have remained largely enigmatic to us and the 2qfa's seem to be still awaiting for full investigation of their functionalities.There are four important issues that we wish to address in depth.(1) Acceptance criteria issue. The first issue to contemplate is that, in traditional automata theory, recognizing languages by probabilistic finite automata (or pfa's) has been subject to a threshold of the acceptance probability of the automata under the term of "cut point" and "isolated cut point." In quantum automata theory, on the contrary, the recognition of languages is originally defined in terms of "boundederror probability" of qfa's [17,21] although the "isolated cut point" criterion has been occasionally used in certain literature (e.g., [5]). What is a precise relationship between those two criteria? When automata are particularly limited to one-way head moves, as noted in Lemma 2.4, the cut-point criterion of pfa's coincides with the unbounded-error criterion of qfa's; however, the same equivalence does not hold in a general 2-way case. In this paper, we shorthandedly denote by 2BQFA the family of languages recognized by bounded-error 2qfa's. When we modify this bounded-error criterion of 2qfa's to error free (or exact), one-sided error, and unbounded error probabilities, we further obtain crucial language families ‡ 2EQFA, 2RQFA, and 2PQFA, respectively. With the use of 'cut point," in contrast, two families § 2NQFA and 2C = QFA ...

Nonuniform Families of Polynomial-Size Quantum Finite Automata and Quantum Logarithmic-Space Computation with Polynomial-Size Advice

“…Let L = {(L n , L n )} n∈N be any family in ptime-2BQ and take a family {M n } n∈N of n O(1) -state 2qfa's recognizing L in expected-polynomial-time with bounded-error probability. It is possible to assume that M n uses only real amplitudes (see, e.g., [45,arXiv version]). In a natural way, we express each computation path of M n on input x as strings over an appropriate alphabet, say, Θ.…”

Automata and Quantum Computing

“…We note that the best known upper bound (in terms of complexity classes) for unbounded-error 2QFAs (with algebraicvalued transitions) is P ∩ L 2 [100]. (Also see [117] for certain relations and upper bounds were defined on the running time of 2KWQFAs under different recognition modes.) 5.2.3 Undecidability of emptiness problem.…”