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arXiv:quant-ph/0604085

A Polynomial-Time Algorithm for the Equivalence between Quantum Sequential Machines

Lvzhou Li, Daowen Qiu

Published Apr 12, 2006• 2 citations
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Abstract

Quantum sequential machines (QSMs) are a quantum version of stochastic sequential machines (SSMs). Recently, we showed that two QSMs M_1 and M_2 with n_1 and n_2 states, respectively, are equivalent iff they are (n_1+n_2)^2--equivalent (Theoretical Computer Science 358 (2006) 65-74). However, using this result to check the equivalence likely needs exponential expected time. In this paper, we consider the time complexity of deciding the equivalence between QSMs and related problems. The main results are as follows: (1) We present a polynomial-time algorithm for deciding the equivalence between QSMs, and, if two QSMs are not equivalent, this algorithm will produce an input-output pair with length not more than (n_1+n_2)^2. (2) We improve the bound for the equivalence between QSMs from (n_1+n_2)^2 to n_1^2+n_2^2-1, by employing Moore and Crutchfield's method (Theoretical Computer Science 237 (2000) 275-306). (3) We give that two MO-1QFAs with n_1 and n_2 states, respectively, are equivalent iff they are (n_1+n_2)^2--equivalent, and further obtain a polynomial-time algorithm for deciding the equivalence between two MO-1QFAs. (4) We provide a counterexample showing that Koshiba's method to solve the problem of deciding the equivalence between MM-1QFAs may be not valid, and thus the problem is left open again.

DOI
10.48550/arxiv.quant-ph/0604085

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In this paper, we explore the power of quantum computers with restricted transition amplitudes. In 1997 Adleman, DeMarrais, and Huang showed that quantum Turing machines (QTMs) with the amplitudes from [Formula: see text] are computationally equivalent to ones with the polynomial-time computable amplitudes as machines implementing bounded-error polynomial-time algorithms. We show that QTMs with the amplitudes from [Formula: see text] is polynomial-time equivalent to deterministic Turing machines as machines implementing exact algorithms, i.e., algorithms that output correct answers with certainty. By extending this result, it is shown that exact quantum computers with rational biased coins are equivalent to classical computers. Moreover, we discuss the computational power of exact quantum computers with multiple types of coins. We also show that, from the viewpoint of zero-error polynomial-time algorithms, [Formula: see text] is not more powerful than [Formula: see text] as the set of amplitudes taken by QTMs; however, it is sufficient to solve the factoring problem.

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Quantum sequential machines (QSMs) are a quantum version of stochastic sequential machines (SSMs). Recently, we showed that two QSMs M_1 and M_2 with n_1 and n_2 states, respectively, are equivalent iff they are (n_1+n_2)^2--equivalent (Theoretical Computer Science 358 (2006) 65-74). However, using this result to check the equivalence likely needs exponential expected time. In this paper, we consider the time complexity of deciding the equivalence between QSMs and related problems. The main results are as follows: (1) We present a polynomial-time algorithm for deciding the equivalence between QSMs, and, if two QSMs are not equivalent, this algorithm will produce an input-output pair with length not more than (n_1+n_2)^2. (2) We improve the bound for the equivalence between QSMs from (n_1+n_2)^2 to n_1^2+n_2^2-1, by employing Moore and Crutchfield's method (Theoretical Computer Science 237 (2000) 275-306). (3) We give that two MO-1QFAs with n_1 and n_2 states, respectively, are equivalent iff they are (n_1+n_2)^2--equivalent, and further obtain a polynomial-time algorithm for deciding the equivalence between two MO-1QFAs. (4) We provide a counterexample showing that Koshiba's method to solve the problem of deciding the equivalence between MM-1QFAs may be not valid, and thus the problem is left open again.
We define a formal framework for equivalence checking of sequential quantum circuits. The model we adopt is a quantum state machine, which is a natural quantum generalization of Mealy machines. A major difficulty in checking quantum circuits (but not present in checking classical circuits) is that the state spaces of quantum circuits are continuums. This difficulty is resolved by our main theorem showing that equivalence checking of two quantum Mealy machines can be done with input sequences that are taken from some chosen basis (which are finite) and have a length quadratic in the dimensions of the state Hilbert spaces of the machines. Based on this theoretical result, we develop an (and to the best of our knowledge, the first) algorithm for checking equivalence of sequential quantum circuits with running timeO(23m+5l(23m + 23l))\mathcal {O}(2^{3m+5l}(2^{3m}{\,+\,}2^{3l}))O(23m+5l(23m+23l)), wheremmmandllldenote the numbers of input and internal qubits, respectively. The complexity of our algorithm is comparable with that of the known algorithms for checking classical sequential circuits in the sense that both are exponential in the number of (qu)bits. Several case studies and experiments are presented.
In this paper, we focus on determining the equivalence for {\it 1-way quantum finite automata with control language} (CL-1QFAs) defined by Bertoni et al and {\it measure-many 1-way quantum finite automata} (MM-1QFAs) introduced by Kondacs and Watrous. More specifically, we obtain that: \begin{enumerate} \item[(i)] Two CL-1QFAs A1{\cal A}_1A1​ and A2{\cal A}_2A2​ with control languages (regular languages) L1{\cal L}_1L1​ and L2{\cal L}_2L2​, respectively, are equivalent if and only if they are (c1n12+c2n22−1)(c_1n_1^2+c_2n_2^2-1)(c1​n12​+c2​n22​−1)-equivalent. Furthermore, if L1{\cal L}_1L1​ and L2{\cal L}_2L2​ are given in the form of DFAs, with m1m_1m1​ and m2m_2m2​ states, respectively, then there exists a polynomial-time algorithm running in time O((m1n12+m2n22)4)O ((m_1n_1^2+m_2n_2^2)^4)O((m1​n12​+m2​n22​)4) that takes as input A1{\cal A}_1A1​ and A2{\cal A}_2A2​ and determines whether they are equivalent. \item[(ii)] Two MM-1QFAs A1{\cal A}_1A1​ and A2{\cal A}_2A2​ with n1n_1n1​ and n2n_2n2​ states, respectively, are equivalent if and only if they are (3n12+3n22−1)(3n_1^2+3n_2^2-1)(3n12​+3n22​−1)-equivalent. Furthermore, there is a polynomial-time algorithm running in time O((3n12+3n22)4)O ((3n_1^2+3n_2^2)^4)O((3n12​+3n22​)4) that takes as input A1{\cal A}_1A1​ and A2{\cal A}_2A2​ and determines whether A1{\cal A}_1A1​ and A2{\cal A}_2A2​ are equivalent.
Multi-letter {\it quantum finite automata} (QFAs) were a quantum variant of classical {\it one-way multi-head finite automata} (J. Hromkovič, Acta Informatica 19 (1983) 377-384), and it has been shown that this new one-way QFAs (multi-letter QFAs) can accept with no error some regular languages (a+b)∗b(a+b)^{*}b(a+b)∗b that are unacceptable by the previous one-way QFAs. In this paper, we study the decidability of the equivalence of multi-letter QFAs, and the main technical contributions are as follows: (1) We show that any two automata, a k1k_{1}k1​-letter QFA A1{\cal A}_1A1​ and a k2k_{2}k2​-letter QFA A2{\cal A}_2A2​, over the same input alphabet ΣΣΣ are equivalent if and only if they are (n2mk−1−mk−1+k)(n^2m^{k-1}-m^{k-1}+k)(n2mk−1−mk−1+k)-equivalent, where m=∣Σ∣m=|Σ|m=∣Σ∣ is the cardinality of ΣΣΣ, k=max⁡(k1,k2)k=\max(k_{1},k_{2})k=max(k1​,k2​), and n=n1+n2n=n_{1}+n_{2}n=n1​+n2​, with n1n_{1}n1​ and n2n_{2}n2​ being the numbers of states of A1{\cal A}_{1}A1​ and A2{\cal A}_{2}A2​, respectively. When k=1k=1k=1, we obtain the decidability of equivalence of measure-once QFAs in the literature. It is worth mentioning that our technical method is essentially different from that for the decidability of the case of single input alphabet (i.e., m=1m=1m=1). (2) However, if we determine the equivalence of multi-letter QFAs by checking all strings of length not more than $ n^2m^{k-1}-m^{k-1}+k$, then the worst time complexity is exponential, i.e., O(n6mn2mk−1−mk−1+2k−1)O(n^6m^{n^2m^{k-1}-m^{k-1}+2k-1})O(n6mn2mk−1−mk−1+2k−1). Therefore, we design a polynomial-time O(m2k−1n8+kmkn6)O(m^{2k-1}n^{8}+km^kn^{6})O(m2k−1n8+kmkn6) algorithm for determining the equivalence of any two multi-letter QFAs. Here, the time complexity is concerning the number of states in the multi-letter QFAs, and kkk is thought of as a constant.
One-way quantum finite automata (1QFA) proposed by Moore and Crutchfield and by Kondacs and Watrous accept only subsets of regular languages with bounded error. In this paper, we develop a new computing model of 1QFA, namely, one-way quantum finite automata together with classical states (1QFAC for short). In this model, a component of classical states is added, and the choice of unitary evolution of quantum states at each step is closely related to the current classical state. 1QFAC can accept all regular languages with no error, and in particular, 1QFAC can accept some languages with essentially less number of states than deterministic finite automata (DFA). The main technical results are as follows. (1) We prove that the set of languages accepted by 1QFAC with bounded error consists precisely of all regular languages. (2) We show that, for any prime number m ≥ 2, there exists a regular language L0(m) whose minimal DFA needs O(m) states and that can not be accepted by the 1QFA proposed by Moore and Crutchfield and by Kondacs and Watrous, but there exists 1QFAC accepting L0(m) with only constant classical states and O(log(m)) quantum basis states. Analogous results for multi-letter automata are also established. (3) By a bilinearization technique we prove that any two 1QFAC A1 and A2 are equivalent if and only if they are (k1n1) 2 + (k2n2) 2 − 1-equivalent, and there exists a polynomial-time O([(k1n1) 2 + (k2n2) 2]4) algorithm for determining their equivalence, where k1 and k2 are the numbers of classical states of A1 and A2, as well as n1 and n2 are the numbers of quantum basis states of A1 andA2, respectively. (4) We show that the minimization problem of 1QFAC is in EXPSPACE. As a corollary of this result, we obtain that the minimization problem of MO-1QFA is also in EXPSPACE. Finally, we draw some conclusions and point out open ∗Corresponding author. E-mail address: issqdw@mail.sysu.edu.cn (D. Qiu). †E-mail address: {paulo.mateus,amilcar.sernadas}@math.ist.utl.pt.
While two hidden Markov process (HMP) resp. quantum random walk (QRW) parametrizations can differ from one another, the stochastic processes arising from them can be equivalent. Here a polynomial-time algorithm is presented which can determine equivalence of two HMP parametrizations M1, M2resp. two QRW parametrizations Q1, Q2in time O(|Σ| max(N1, N2)4), where N1,N2are the number of hidden states in M1, M2resp. the dimension of the state spaces associated with Q1, Q2, and Σ is the set of output symbols. Previously avail able algorithms for testing equivalence of HMPs were exponential in the number of hidden states. In case of QRWs, algorithms for testing equivalence had not yet been presented. The core subrou tines of this algorithm can also be used to efficiently test hidden Markov processes and quantum random walks for ergodicity.
In this paper, we explore the power of quantum computers with restricted transition amplitudes. In 1997 Adleman, DeMarrais, and Huang showed that quantum Turing machines (QTMs) with the amplitudes from [Formula: see text] are computationally equivalent to ones with the polynomial-time computable amplitudes as machines implementing bounded-error polynomial-time algorithms. We show that QTMs with the amplitudes from [Formula: see text] is polynomial-time equivalent to deterministic Turing machines as machines implementing exact algorithms, i.e., algorithms that output correct answers with certainty. By extending this result, it is shown that exact quantum computers with rational biased coins are equivalent to classical computers. Moreover, we discuss the computational power of exact quantum computers with multiple types of coins. We also show that, from the viewpoint of zero-error polynomial-time algorithms, [Formula: see text] is not more powerful than [Formula: see text] as the set of amplitudes taken by QTMs; however, it is sufficient to solve the factoring problem.