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

Abstract

The state complexity of a finite(-state) automaton intuitively measures the size of the description of the automaton. Sakoda and Sipser [STOC 1972, pp. 275-286] were concerned with nonuniform families of finite automata and they discussed the behaviors of nonuniform complexity classes defined by families of such finite automata having polynomial-size state complexity. In a similar fashion, we introduce nonuniform state complexity classes using families of quantum finite automata. Our primarily concern is one-way quantum finite automata empowered by garbage tapes. We show inclusion and separation relationships among nonuniform state complexity classes of various one-way finite automata, including deterministic, nondeterministic, probabilistic, and quantum finite automata of polynomial size. For two-way quantum finite automata equipped with garbage tapes, we discover a close relationship between the nonuniform state complexity of such a polynomial-size quantum finite automata family and the parameterized complexity class induced by quantum logarithmic-space computation assisted by polynomial-size advice.

How Does Adiabatic Quantum Computation Fit into Quantum Automata Theory?

“…As their variants and extensions, numerous models have been proposed for one-way and two-way quantum finite automata. To empower early models of quantum finite automata, a more general model, called two-way quantum finite automata with mixed states and quantum operations, was studied in [2,14,31] under various names and is computationally equivalent to a garbage-tape model of two-way quantum finite automata [37]. A nonuniform analogue of a quantum automata family was lately discussed in, e.g., [30,37].…”

“…Given a language L over an alphabet Σ and a constant ε ∈ [0, 1], we say that M recognizes with error probability at most ε if (i) for any x ∈ L, tr(Π acc A | cx$ (ρ 0 )) ≥ 1 − ε and (ii) for any x ∈ Σ * − L, tr(Π rej A | cx$ (ρ 0 )) ≥ 1 − ε, where ρ 0 = |q 0 q 0 |. This model is computationally equivalent to a garbage-tape model of 1qfa's used in [37] (implicitly in [34]). In the case of k = 1, we identify A σ,1 with A σ and then obtain A σ (H) = A σ HA † σ .…”

“…As shown in [37], regular languages are recognized by 1qfa's with garbage tapes. The proof of Theorem 5.2 hinges at a critical simulation of garbage-tape 1qfa's by appropriate AEQSs under the desired conditions.…”

“…In Lemma 3.1, we have already shown how to approximate each AEQS and this approximation can be carried out by certain forms of resource-bounded QTMs with advice of polynomial size. Such advised QTMs induce the nonuniform analogue of ptime-BQL, denoted ptime-BQL/poly [37], which is composed of all languages recognized by QTMs with polynomial-size advice strings (one per every input size) using logarithmic space in expected polynomial runtime. Unlike the previous sections, we wish to limit the use of transition amplitudes of 1moqqaf's within polynomial-time approximable complex numbers.…”

A Quantum Finite Automata Approach to Modeling the Chemical Reactions

“…Initially, Kondacs and Watrous [8], and Moore and Crutchfield [9] proposed the concept of quantum automata separately. Since then, a variety of quantum automata models have been studied and demonstrated in various directions, such as QFAs, Latvian QFA, 1.5-way QFA, two-way QFA (2QFA), quantum sequential machine, quantum pushdown automata, quantum Turing machine, quantum multicounter machines, quantum queue automata [10], quantum multihead finite automata, QFAs with classical states (2QCFA) [11,12], state succinctness of two-way probabilistic finite automata (2PFA), QFA, 2QFA, and 2QCFA [13][14][15], interactive proof systems with QFAs [16,17], quantum finite state machines of matrix product state [18], promise problems recognition by QFA [19][20][21][22], quantum-omega automata [23] and semi-quantum two-way finite automata [24][25][26], time complexity advantages of QFA [27], nonuniform classes of polynomial size QFA [28,29], QFA and linear temporal logic relationship [30], and many more since the past 2 decades [31][32][33][34]. These models are effective in determining the boundaries of various computational features and expressive power [35][36][37].…”