Quantum finite automata: Advances on Bertoni's ideas

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Quantum finite automata: Advances on Bertoni’s ideas ✩ Maria Paola Bianchi a , Carlo Mereghetti b,∗ , Beatrice Palano b a b

Department of Computer Science, ETH-Zurich, Universitätstrasse 6, CH-8092 Zürich, Switzerland Dipartimento di Informatica, Università degli Studi di Milano, via Comelico 39, 20135 Milano, Italy

a r t i c l e

i n f o

Article history: Received 15 April 2015 Received in revised form 19 December 2015 Accepted 31 January 2016 Available online xxxx To our friend and mentor

Alberto Bertoni (1946–2014†) Keywords: Quantum finite automata Descriptional complexity

a b s t r a c t We first outline main steps and achievements along Bertoni’s research path in quantum finite automata theory – from the very basic definitions of the models of quantum finite automata throughout the investigation of their computational and descriptional power. Next, we choose to focus on Bertoni’s studies on quantum finite automata descriptional complexity. In particular, we expand on a statistical framework for the synthesis of succinct quantum finite automata, discussing its adaptation to the case of multiperiodic events and languages. We then improve such a framework to obtain even more succinct quantum finite automata for some multiperiodic languages. Finally, we introduce some promise problems for multiperiodic inputs, showing that even on this class of problems the descriptional power of quantum finite automata greatly outperforms that of equivalent classical finite automata. © 2016 Elsevier B.V. All rights reserved.

1. Some aspects of Alberto Bertoni’s explorations in quantum finite automata theory Quantum computing is a prolific research area, halfway between physics and computer science [39,43,65]. Most likely, its origins may be dated back to 70’s, when some works on quantum information began to appear (see, e.g., [47,50]). In early 80’s, R.P. Feynman suggested that the computational power of quantum mechanical processes might be beyond that of traditional computation models [35]. Almost at the same time, P. Benioff already proved that such processes are at least as powerful as Turing machines [7]. In 1985, D. Deutsch [33] proposed the notion of a quantum Turing machine as a physically realizable model for a quantum computer. From the point of view of structural complexity, E. Bernstein and U. Vazirani introduced in [9] the class BQP of problems solvable in polynomial time on quantum Turing machines, focusing attention on relations with the corresponding deterministic and probabilistic classes P and BPP, respectively. Further works in the literature explored classical issues in complexity theory from the quantum paradigm perspective (see, e.g., [8,72,73]). The first impressive result witnessing quantum power was P. Shor’s algorithm for integer factorization, which could run in polynomial time on a quantum computer [70]. (It should be stressed that no classical polynomial time factoring algorithm is currently known. On this fact, the security of many nowadays cryptographic protocols actually relies.) Another relevant progress was made by L. Grover [38], who√proposed a quantum algorithm for searching an item in an unsorted database containing n items, which runs in time O ( n). Being both a physicist and a computer scientist, Alberto Bertoni naturally approached the study of quantum computing at the beginning of 90’s. His first deep investigations in the field are most likely to be singled out in his collaboration with

✩

*

Partially supported by the Italian MIUR under the project PRIN-2010LYA9RH_005 “PRIN: Automi e Linguaggi Formali: Aspetti Matematici e Applicativi.” Corresponding author. E-mail addresses: [email protected] (M.P. Bianchi), [email protected] (C. Mereghetti), [email protected] (B. Palano).

http://dx.doi.org/10.1016/j.tcs.2016.01.045 0304-3975/© 2016 Elsevier B.V. All rights reserved.

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M. Carpentieri, a PhD student at the Department of Computer Science – University of Milano during 1995–99. Carpentieri’s PhD activity, supervised by Bertoni, almost entirely dealt with quantum computing, and his doctoral dissertation [30] covered several aspects of the discipline. Of particular interest here is Bertoni and Carpentieri’s contribution to the novel (at that time) theory of quantum finite automata. In this regard, we feel it noteworthy to emphasize that the main body of their research on quantum finite automata was already presented in a version of Carpentieri’s doctoral dissertation dating 1996 (the second author of the present paper was an internal reviewer of the dissertation). If we add to the fact that the foundational works on quantum finite automata theory are unanimously considered to be the papers by A. Kondacs and J. Watrous [51] and by C. Moore and J. Crutchfield [63] both issued in 1997, then one may truly grasp the importance of Bertoni and Carpentieri’s work within quantum finite automata theory. Informally, a quantum finite automaton can be obtained by imposing the quantum paradigm – complex state superposition, unitary evolution, quantum measurement – on classical finite automata, e.g., deterministic or probabilistic. Thus, quantum finite automata may represent a theoretical model for a quantum computational device with finite memory. Several observations motivate the introduction and study of quantum finite automata, both theoretical and applied. From a theoretical viewpoint, quantum finite automata computations exhibit all the relevant ingredients of general quantum computing in a slightly more simplified form. So, tackling problems on such “simple” devices may be more manageable and shed some light on questions pertaining to general quantum computers. Yet, it is natural to seek for the simplest model of computation where the quantum paradigm may possibly outperform the classical one. From application perspective, while we can hardly expect to see a full-featured quantum computer in the near future, it is reasonable to envision classical computing devices incorporating small quantum components, i.e., with memory consisting of few quantum bits only. Thus, it is well worth modeling such small components by quantum finite automata, as a tool to explore their computational capabilities. Bertoni and Carpentieri formally settled the most basic and widely studied model of a quantum finite automaton, named measure-once quantum finite automaton later on in the literature [29,40]. Such a model served as a basis for several variants of quantum finite automata introduced and studied in a plenty of contributions (see, e.g., [2,5,17,76]). Being the only model to be considered in the present paper, from now on for the sake of brevity we will simply write “quantum finite automaton” instead of “measure-once quantum finite automaton”. The “hardware” of a quantum finite automaton is that of a classical finite automaton. Thus, we have an input tape scanned by a one-way input head moving one position forward at each move,1 plus a finite state control. At any given time during the computation, the state of the quantum finite automaton is represented by a complex linear combination of classical states, called a superposition. At each step, a unitary transformation associated with the currently scanned input symbol makes the automaton evolve to the next superposition. Superposition dynamics can transfer the complexity of the problem from a large number of sequential steps to a large number of coherently superposed quantum states. At the end of the input processing, the automaton is observed in its final superposition. This operation makes the superposition collapse to a particular classical state with a certain probability. The probability that the automaton accepts the input word is given by the probability of observing (collapsing into) an accepting state. Quantum finite automata exhibit both advantages and disadvantages with respect to their classical counterparts. Basically, quantum superposition offers some computational advantages on probabilistic superposition. On the other hand, quantum dynamics must be reversible, and this requirement may impose severe computational limitations to finite memory devices. As a matter of fact, as we will see later on, it is sometimes impossible to simulate classical finite automata by quantum finite automata. Bertoni’s work contributed to explicitly single out both strength and weakness of quantum finite automata. Weakness are pointed out by Bertoni and Carpentieri since the very beginning. In fact, they established the exact computational power of the model, proving that quantum finite automata are strictly less powerful than classical finite automata. Precisely, by using a Rabin-like technique and the compactness of the metric space (unit sphere) containing quantum superpositions, they showed that the class of languages accepted with isolated cut point by quantum finite automata coincides with the class of group languages [66], a proper subclass of regular languages. This fundamental result was published only in 2001 [11], and independently proved in [29] though in a slightly less general form. Further relevant results concerning the power of quantum finite automata (among others: some closure properties, regularity conditions, and a “pumping lemma” for languages accepted by quantum finite automata) may be found in [12], again published only in 2001. From the scenario depicted so far, it was clear that the strength of quantum finite automata has not to be found in “what” they do, but possibly in “how” they work. Following this guideline, Bertoni, together with the authors of the present paper, switched the research focus from the computational to the descriptional power of quantum finite automata, thus settling investigations on the area of descriptional complexity. In this discipline, roughly speaking, the models of computation are studied on the basis of their size. Typical questions under examination are, e.g., size upper and lower limits for accomplishing certain tasks, or comparing the size of different models to single out their descriptional power, i.e., the ability to operate succinctly. For finite automata, a natural and widely used size measure is the number of control states. In this regard, probably the first well known result in the descriptional complexity is the optimal exponential gain on the descrip-

1

This type of automaton is sometimes referred to as real time automaton [40,63], stressing the fact that it can never perform stationary moves.

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tional power allowed by nondeterminism [62,67]. Since the very beginning of automata theory, a well consolidated trend in the literature has been exploring the descriptional power of finite automata, with impacts in several areas of theoretical computer science. The reader is referred to, e.g., [48,74] for introductory surveys. Most likely, the first contribution explicitly studying the descriptional power of quantum finite automata in comparison with classical finite automata models is [3]. There, the descriptional superiority of quantum finite automata is emphasized by focusing on the unary language L p = {ak | k ≡ 0 mod p } for a prime p. Such a language is shown to be accepted with isolated cut point on a quantum finite automaton with only O (log p ) states, whereas p states are necessary and sufficient on deterministic, nondeterministic, probabilistic, and even two-way classical finite automata [60]. Since [3], many contributions investigated the descriptional power of quantum finite automata witnessing the extreme succinctness of the quantum paradigm (see, e.g., [3,4,20,26,40,71]). Almost all of them are obtained by constructing quantum finite automata accepting ad hoc languages or solving suitably defined problems. As an opposite to this attitude, Bertoni’s first results in descriptional complexity aimed to provide general methodologies for the synthesis of small size quantum finite automata. In [54,55,59], the problem was tackled of building concise quantum finite automata inducing linear approximations of given stochastic events p : a∗ → [0, 1] of period n, i.e., satisfying the rela√ tion p (ak ) = p (ak+n ) for any k ≥ 0. A time efficient algorithm was proposed, returning quantum finite automata with O ( n) states. More specifically, upper and lower bounds on the size of resulting quantum finite automata inducing periodic events were related to some interesting combinatorial objects, namely, Golomb rulers and difference covers [16,57], connected to the harmonic structure (i.e., the discrete Fourier transform) of events. By these relations, an efficient algorithm using finite projective geometry arguments was designed in [16], for building quantum finite automata whose size matches the theoretical lower bound. As a general consequence, these results highlighted an entire meaningful class of languages, namely, unary periodic languages [61], where the adoption of the quantum paradigm leads to quadratically more succinct accepting automata. Always in the spirit of providing general tools for building small quantum finite automata, Bertoni and his collaborators defined in [18,20] a statistical framework within which to attack the following problem: given a family { p α :  ∗ → [0, 1] | α ∈ I } of stochastic events, each induced by an m-state quantum finite automaton, find a succinct quantum finite automaton A inducing a stochastic event p which is a δ -approximation of a convex linear combination q of p α ’s, i.e., satisfying the inequality supω∈ ∗ {| p (ω) − q(ω)|} ≤ δ . By viewing this question as a problem of uniform convergence of empirical averages to their expectations and using tools from machine learning, a bound of O ((m · d/δ 3 ) log2 (d/δ 2 )) states for A was obtained, where d is the Vapnik dimension of the class { p α (ω) | ω ∈  ∗ } of random variables. This result was then specialized and improved for interesting families of stochastic events with various types of periodicity (e.g., simple, commutative, multi) [18,20,21,58]. As a consequence, small quantum finite automata were obtained, approximating periodic events whose discrete Fourier transform fulfills certain norm conditions. This in turn leaded to single out classes of periodic languages (for which the language L p above addressed is a special case) accepted by quantum finite automata which are exponentially more succinct than equivalent classical finite automata. To conclude the overview of Bertoni’s investigations in quantum descriptional complexity, it is worth mentioning his contribution in establishing size lower bounds for quantum finite automata. By refining the Rabin-like analysis above recalled and used in [11] to characterize the computational power of quantum finite automata as the class of group languages, a quasi-optimal lower bound on the size of quantum finite automata was proved in [21], which is logarithmic in the size of equivalent minimal deterministic finite automata. This result improves a lower bound obtained in [1] by a similar approach, and inspired the investigation of size lower bounds for other more general variants of quantum finite automata (see, e.g., [23,25]). Other lower bounds on the size of quantum finite automata accepting finite languages were obtained in [19], by using quantum information arguments. More generally, an (exponential time) algorithm was proposed in [21], for determining the size of minimal accepting quantum finite automata for any given unary periodic language. So far, we aimed to quickly outline the research path in quantum finite automata theory coherently followed by Alberto Bertoni, from the computational to descriptional power. Nevertheless, he contributed to several other aspects of quantum finite automata, some of which will be briefly addressed in the concluding section. Among many topics and possible investigations suggested by Bertoni’s research, we here chose to expand and build on his results and techniques for designing small quantum devices. As above recalled, by specializing a general result on the synthesis of small quantum finite automata inducing approximation of stochastic events, Bertoni and his collaborators proposed in [18,20,21] a statistical framework within which to prove the existence of small quantum finite automata approximating periodic events and accepting periodic languages. This framework was then generalized in [58] to deal with multiperiodicity. In short, given an alphabet  = {σ1 , . . . , σ H }, an event p :  ∗ → [0, 1] (a language L ⊆  ∗ ) is said to be (n1 , . . . , n H )-periodic whenever, for any ω ∈  ∗ , the value of p (ω) (the membership of ω in L) depends only on the number modulo nh of occurrences of σh in ω , for every 1 ≤ h ≤ H. We here restate the statistical framework in [58] in a more accurate form, emphasizing the modular architecture of quantum finite automata resulting from this framework. We prove that any convex linear combination of (n1 , . . . , n H )-periodic events, each induced by a quantum finite automaton (a module), can be δ -approximated by the event induced by a quanH tum finite automaton built by suitably composing O (( t =1 log nt )/δ 2 ) modules. We then apply this design pattern to the construction of succinct quantum finite automata for multiperiodic languages. In particular, we focus on the language L ∧(n1 ,...,n H ) ⊆ {σ1 , . . . , σ H }∗ consisting of those words in which the number of occurrences of each symbol σi is a multiple of ni . We obtain a quantum finite automaton which accepts L ∧(n1 ,...,n H ) in Monte Carlo mode with the error probability  ,

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whose inner architecture consists of the composition of O ((

H H

H

H +2

t =1 log nt )/

 2 ) modules each with O ((4/ ) H ) states. Globally,

this automaton has O ((4 ) states, and turns out to be exponentially smaller than any deterministic finite t =1 log nt )/ automaton for L ∧(n1 ,...,n H ) . Next, we improve the above design pattern by reducing both the number and size of the modules involved in quantum finite automata construction. This is achieved by considering a statistical tool from [4]. This way, we obtain a Monte Carlo quantum finite automaton with the error probability  for L ∧(n1 ,...,n H ) , in which the number of modules is reduced to

H

O (( t =1 log nt )/ ) and where each module features 2 H states, this number of states no longer depending on the error probability  . In conclusion, the final number of states of the quantum finite automaton for L ∧(n1 ,...,n H ) turns out to be

H

H

O (2 H ( t =1 log nt )/ ), rather closely approaching the theoretical lower bound ( t =1 log nt )/2 log(1 + 2/ ) on the number of states of any quantum finite automaton accepting L ∧(n1 ,...,n H ) with an isolated cut point. Finally, again inspired by Bertoni’s approach to the synthesis of succinct quantum finite automata, we tackle the sizeefficient solution of promise problems within the quantum paradigm. For recent results on this topic, we refer the reader to, e.g., [6,24,36,41,42]. Promise problems [34] may be regarded as a generalization of language recognition, and have wide application to several areas of theoretical computer science [37]. A promise problem on an alphabet  is specified by two nonempty disjoint subsets of  ∗ called yes-instances and no-instances. Unlike language recognition, the union of the yes-instances and no-instances may be a proper subset of  ∗ . A device which solves the promise problem accepts yes-instances, rejects no-instances, and is allowed arbitrary behavior on the remaining strings. Intuitively, this device is “promised” that the input is either a yes-instance or a no-instance, and is only required to distinguish between these two cases. In the spirit of our results on multiperiodic language recognition, we define a promise problem on multiperiodic inputs which generalizes some promise problems proposed in the literature [6,24,41]. We prove that such a promise problem can be solved exactly (i.e., without any error probability) by a quantum finite automaton with a constant number of states, while any solution by deterministic finite automata must employ a number of states which depends on the values of the considered periods. This result, once again, shows the descriptional superiority, in some cases, of the quantum vs. classical paradigm even on the class of promise problems. The paper is organized as follows. In Section 2, we collect basics and main results on quantum finite automata. In Section 3, we tackle the synthesis of small quantum finite automata on multiperiodic inputs. First, we describe the statistical framework within which to prove the existence of small modular quantum finite automata inducing multiperiodic events and accepting multiperiodic languages. Then, we improve the design pattern of such quantum finite automata by reducing the number and size of modules they consist of. In Section 4, we define some promise problems on multiperiodic inputs and study their size-efficient solutions by quantum and classical finite automata. Finally in Section 5, for the sake of completeness, we quickly overview other aspects of Alberto Bertoni’s explorations in quantum finite automata theory. 2. Preliminaries 2.1. Linear algebra We quickly recall some notions of linear algebra, useful to describe the quantum world. For more details, we refer the reader to, e.g., [53,69]. The field of complex numbers is denoted by√C. Given a complex number z = a + ib, for reals a and b, we denote its conjugate by z∗ = a − ib and its modulus by | z| = zz∗ . We let Cn×m and Cn (shorthand for C1×n ) denote, respectively, the set of n × m matrices and n-dimensional row vectors with entries in C. Given a matrix M ∈ Cn×m (a vector ϕ ∈ Cn ), we let M i j ((ϕ ) j ) denote its (i , j )th entry ( jth component). We denote by [0]n×m the zero matrix in Cn×m . The identity matrix in Cn×n is denoted by I n . Given a vector ϕ ∈ Cn , we denote by diag (ϕ ) ∈ Cn×n the diagonal matrix having ϕ on its main diagonal and 0 elsewhere, i.e., (diag (ϕ ))i j = (ϕ )i if i = j, 0 otherwise. The transpose of M ∈ Cn×m is the matrix M T ∈ Cm×n satisfying M T i j = M ji , while we let M ∗ be the matrix satisfying M ∗ i j = ( M i j )∗ . The adjoint of M is the matrix ∗ M † = ( M T ) . The sum of the matrices A , B ∈ Cn×m  is the n × m matrix ( A + B )i j = A i j + B i j . The product of the matrices m C ∈ Cn×m and D ∈ Cm×r is the n × r matrix (C D )i j = k=1 C ik D kj . For matrices A ∈ Cn×m and B ∈ C p ×q , their direct sum and Kronecker (or tensor or direct) product are the (n + p ) × (m + q) and np × mq matrices defined, respectively, as

 A⊕B=

A

[0] p ×m

[0]n×q B

⎛

 ,

··· ⎜ .. .. A⊗B =⎝ . . A n1 B · · · A 11 B

A 1m B

.. .

⎞

⎟ ⎠.

A nm B

Whenever such operations can be performed, we have the identities ( A ⊗ B ) · (C ⊗ D ) = AC ⊗ B D and ( A ⊕ B ) · (C ⊕ D ) = AC ⊕ B D. For vectors ϕ ∈ Cn and ψ ∈ Cm , their direct sum is the vector ϕ ⊕ ψ = ((ϕ )1 , . . . , (ϕ )n , (ψ)1 , . . . , (ψ)m ) ∈ Cn+m , and their Kronecker (or tensor or direct) product is the vector ϕ ⊗ ψ = ⊕nj=1 (ϕ ) j · ψ ∈ Cnm . A Hilbert space of dimension n is the linear space Cn of n-dimensional complex row vectors equipped with sum and † n n product by elements √ in C, in which the inner product ϕ , ψ = ϕ ψ is defined, for ϕ , ψ ∈ C . The norm of a vector ϕ ∈ C is given by ϕ  = ϕ , ϕ . If ϕ , ψ = 0 (and ϕ  = 1 = ψ), than ϕ and ψ are orthogonal (orthonormal). Two subspaces X , Y ⊆ Cn are orthogonal if any vector in X is orthogonal to any vector in Y . In this case, we denote by X  Y the linear space generated by X ∪ Y . For vectors ϕ and ψ , we have that ϕ ⊗ ψ = ϕ  · ψ.

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A matrix M ∈ Cn×n is unitary whenever M M † = I = M † M. Equivalently, M is unitary if and only if it preserves the norm, i.e., ϕ M  = ϕ  for any ϕ ∈ Cn . Another characterization states that M is unitary if and only if its rows or columns are mutually orthonormal vectors. It is easy to see that, given two unitary matrices A and B, the matrices A B, A ⊕ B, and A ⊗ B are unitary as well. In what follows, a particular role will be played by the 2 × 2 unitary matrix



Rθ =

cos θ i sin θ

i sin θ cos θ



.

The reader may easily verify that, for any θ and θ  , we have

R θ · R θ  = R θ +θ  = R θ  · R θ .

(1)

A matrix H ∈ Cn×n is said to be Hermitian (or self-adjoint) whenever H = H † . A matrix P ∈ Cn×n is a projector if and only if P is Hermitian and idempotent, i.e., P = P † and P 2 = P . Given the Hermitian matrix H , let c 1 , . . . , c s be its eigenvalues and E 1 , . . . , E s the corresponding eigenspaces. It is well known that each eigenvalue ck is real, that E i is orthogonal to E j for any i = j, and that E 1  · · ·  E s = Cn . Thus, every vector ϕ ∈ Cn can be decomposed as ϕ = ϕ1 + · · · + ϕs for unique ϕ j ∈ E j . The linear transformation ϕ → ϕ j is the projector P j onto s the subspace E j . Actually, the Hermitian matrix H is biunivocally determined by its eigenvalues and projectors as H = i =1 c i P i . We note that { P 1 , . . . , P s } is a complete set of

s

†

mutually orthogonal projectors, i.e., i =1 P i = I n and P i P j = [0]n×n for i = j. Evolutions in quantum systems can be described by unitary matrices, while observables to be measured can be described by Hermitian matrices. 2.2. Formal languages, classical and quantum finite automata

We assume familiarity with basics in formal language and classical finite automata theory (see, e.g., [46]). The set of all words (including the empty word) on a finite alphabet  is denoted by  ∗ . For a word ω ∈  ∗ , we let: |ω| denote its length, ωi its ith symbol, and |ω|σ the number of occurrences of σ ∈  . A language on  is any subset of  ∗ . A deterministic finite automaton (dfa) is formally defined as a 5-tuple D = ( S , , τ , s0 , F ), where S is the finite set of states,  the finite input alphabet, s0 ∈ S the initial state, F ⊆ S the set of accepting states, and τ : S ×  → S the transition function. Denoting by τ ∗ the canonical extension of τ to  ∗ , the language accepted by D is the set L D = {ω ∈  ∗ | τ ∗ (s0 , ω) ∈ F }. Let us now introduce the model of a measure-once one-way quantum finite automaton [11,12,29,63]. From now on, for the sake of brevity and since we shall be dealing with this model only, we will simply abbreviate as qfa understanding the designation “measure-once one-way”. A qfa on an input alphabet  with m basis states (also an m-state qfa) is formally defined as a 3-tuple A = (ϕ , {U (σ )}σ ∈ , η) where:

• ϕ ∈ Cm , with ϕ  = 1, is the initial superposition of the basis states; (ϕ )i , with |(ϕ )i | ≤ 1, is the amplitude of the ith basis state, while |(ϕ )i |2 is the probability of observing the qfa being in the ith basis state. • U (σ ) ∈ Cm×m is the unitary evolution matrix on σ ∈  ; (U (σ ))i j , with |(U (σ ))i j | ≤ 1, is the amplitude of the transition from the ith to jth basis state upon reading σ , while |(U (σ ))i j |2 is the related probability. • η ∈ {0, 1}m is the characteristic vector of the accepting states, i.e., (η)i = 1 if and only if the ith basis state is accepting. The computation of A on an input word ω ∈  ∗ starts in the initial superposition ϕ . After reading the first k input symbols, k the state of A is the superposition ϕ i =1 U (ωi ) whose norm is 1 since ϕ  = 1 and U (ωi )’s are unitary. After entering the

| ω |

final superposition ξ = ϕ i =1 U (ωi ), we measure the probability of A being in an accepting state, namely, the probability  that A accepts ω . Such a probability is easily seen to be p acc (ω) = {i | (η)i =1} |(ξ )i |2 and can be defined according to the axioms of quantum mechanics (see, e.g., [49]) as follows. We use the standard observable completely described by an Hermitian matrix with diag (η) and I m − diag (η) being the complete set of mutually orthogonal projectors projecting onto the two orthogonal subspaces of Cm spanned by the accepting and nonaccepting states, respectively. So, the probability of accepting the input word ω is given by the square norm of the projection of ξ onto the subspace spanned by the accepting states. Formally:

2 ⎛ ⎞ |ω|

 ⎝ ⎠ p acc (ω) = ϕ U (ωi ) diag (η) |(ξ )i |2 . = { i | ( η ) i =1 } i =1 The stochastic event induced by A is the function p A :  ∗ → [0, 1] defined, for any ω ∈  ∗ , as p A (ω) = p acc (ω). In what follows, we will exploit the possibility of inducing stochastic events by suitably composing qfas as in Proposition 1. Let A= (ϕ A , {U A (σ )}σ ∈ , η A ) be an m-state qfa. nAn m-state qfa B can be built, satisfying p B = 1 − pn A . For 1 ≤ i ≤ n, let A i = (ϕ A i , U A i (σ ) σ ∈ , η A i ) be an mi -state qfa. A ( i =1 mi )-state qfa C can be built, satisfying p C = i =1 p A i . For

nonnegative reals α1 , . . . , αn with

n

i =1

n

αi = 1, a (

i =1 mi )-state

qfa D can be built, satisfying p D =

n

i =1

αi p A i .

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Proof. Define B as A except for the accepting states whose characteristic vector is now given by the bitwise negation of η A . √ Define C = (⊗ni=1 ϕ A i , {⊗ni=1 U A i (σ )}σ ∈ , ⊗ni=1 η A i ). Define D = (⊕ni=1 αi ϕ A i , {⊕ni=1 U A i (σ )}σ ∈ , ⊕ni=1 η A i ). It is not hard to verify that B, C , and D are well formed qfas with the claimed number of states and inducing the desired events. 2 Concerning the qfas built in Proposition 1, with a slight abuse of terminology we say that the qfa C is the Kronecker product of the qfas A i ’s, and that D is a direct sum. In particular, for this latter case, we say that D is the uniform direct sum of A i ’s whenever all αi ’s are equal. A language L ⊆  ∗ is said to be accepted by a qfa A with the cut point λ ∈ [0, 1] if and only if L = {ω ∈  ∗ | p A (ω) > λ}. In addition, if there exists an  ∈ (0, 1/2] such that | p A (ω) −λ| ≥  holds true for every ω ∈  ∗ , then we say that λ is isolated by  and that A is an isolated cut point qfa for L. The relevance of isolated cut point acceptance on finite automata is due to the fact that, in this case, we can arbitrarily reduce the classification error probability of an input word ω by repeating a constant number of times (not depending on the length of ω ) its parsing and taking the majority of the answers. Notice that the proof of Proposition 1 explicitly displays the isolated cut point qfas for the complementation, intersection and union of languages accepted by isolated cut point qfas. An acceptance mode that is more reliable than isolated cut point acceptance is represented by Monte Carlo acceptance. A language L ⊆  ∗ is said to be accepted by a qfa A in Monte Carlo mode if and only if there exists an  ∈ (0, 1/2] such that, for any ω ∈  ∗ , ω ∈ L implies p A (ω) = 1, while ω ∈ / L implies p A (ω) ≤  . In this case, we say that A is a Monte Carlo qfa for L with the error probability  . The construction for the product of events given in the proof of Proposition 1 directly yields a Monte Carlo qfa for the intersection of languages accepted by Monte Carlo qfas. The same does not hold for the other two constructions. We recall that another acceptance mode, called with bounded error, is often considered in the literature on qfas. Precisely, a qfa A is said to accept a language L with a bounded error  ∈ [0, 1/2) if and only if p A (ω) ≥ 1 −  for every ω ∈ L, and p A (ω) ≤  for every ω ∈ / L. The reader may easily verify that the acceptance with a bounded error  coincides with the acceptance with the cut point 1/2 isolated by 1/2 −  . Vice versa, the acceptance with the cut point 1/2 isolated by  is easily seen to coincide with the acceptance with a bounded error 1/2 −  . Moreover, according to [60], paying by one new basis state, the acceptance with a cut point λ = 1/2 isolated by  can be turned into an acceptance with a bounded error 1/2 −  /(2(1 − α )) with α = λ if λ < 1/2, and α = 1 − λ if λ > 1/2. This direct connection between isolated cut point and bounded error policies enables to straightforwardly migrate state complexity results for qfas working by an acceptance mode to qfas adopting the other mode and vice versa. The computational capabilities of qfas have been widely investigated by several authors. Among others (see, e.g., [29]), Bertoni and Carpentieri independently established the exact computational power of isolated cut point qfas by using a Rabin-like approach: Theorem 2. (See [11,29].) The class of languages accepted by qfas with isolated cut point coincides with the class of group languages. We recall that a language L ⊆  ∗ is a group language if and only if it is accepted by a dfa D = ( S , , τ , s0 , F ) where, for every σ ∈  , the function τ (−, σ ) : S → S is a permutation. Equivalently, L is a group language if and only if its syntactic semigroup is a finite group [66]. Group languages form a proper subclass of regular languages, thus implying that isolated cut point qfas are strictly less powerful than classical finite automata. Although weaker from a computational power viewpoint, qfas may greatly outperform classical finite automata when descriptional power is at stake. In this realm, models of computation are compared on the basis of their size and, in case of finite state machines, a commonly assumed size measure is the number of finite control states. Bertoni, in collaboration with the authors of this paper, stated the maximal theoretical distance between the descriptional power of quantum and classical finite automata: Theorem 3. (See [21].) For a group language L, let D ( L ) ( Q ( L ,  )) denote the size of the minimal dfa (qfa with cut point isolated by  ) accepting L. Then, we have

Q (L,  ) ≥

log( D ( L )) 2 log(1 + 2/ )

.

This result, improving a previous gap in [1] and actually embodied in a more general algebraic framework aiming to tackle some decidability questions on qfas, lays down a potential exponential gap between the succinctness of qfas and dfas. Nevertheless, a subsequent fundamental step to be taken is the study of the optimality of such a gap. As a matter of fact, some results in the literature already certify quasi-optimality, e.g., by the witness unary language L p = {ak | k ≡ 0 mod p } for a prime p. Such a language is shown in [3] to be accepted by a Monte Carlo qfa with the error probability  and with only O ((1/ ) O (1) log p ) states, while p states are necessary and sufficient on deterministic, nondeterministic, probabilistic, and two-way classical finite automata [60]. Aiming to get even closer to the theoretical lower bound in Theorem 3 and, more generally, to provide a uniform framework within which to obtain succinct qfas, Bertoni and his collaborators proposed in [18,20,21] a statistical approach to

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show the existence of small qfas approximating convex linear combinations of stochastic events. By this approach, very small qfas for a great variety of periodic languages were designed. Among others, a Monte Carlo qfa with the error probability  for the language L p for any p > 1 is proposed, featuring O ((1/ 3 ) log p ) states. Indeed, in the spirit of Bertoni’s approach and using different statistical tools, the number of states is further reduced to O ((1/ ) log p ) in [4]. In the next section, we start giving details on Bertoni’s framework for the synthesis of small qfas. In particular, we survey the adaptation of this framework to qfas exhibiting multiperiodic behaviors [58]. Next, using Bertoni’s ideas and tools from [4], we propose improved constructions of concise qfas for families of multiperiodic languages. 3. Small size QFAs for multiperiodic events and languages The ability to exhibit periodic behaviors is a key feature of qfas, a benchmark on which the descriptional power of the quantum paradigm is proved to greatly outperform that of classical devices. The simplest notion of periodicity is represented by a periodic event. Given a unary alphabet {a}, a stochastic event p : a∗ → [0, 1] is said to be n-periodic whenever p (ak ) = p (ak+n ) holds true for any k ≥ 0. A first generalization to alphabets with more than one symbol was addressed in [20] with the notion of commutative periodicity. For  the sake of conciseness and readability, from now on we will use the notation xn = x mod n. In addition, we let Zn = xn | x ∈ Z and we will use on Zn operations modulo n. For a general alphabet  = {σ1 , . . . , σ H }, a stochastic event p :  ∗ → [0, 1] is said to be commutative n-periodic if there exists a function pˆ : Zn H → [0, 1] satisfying the condition p (ω) = pˆ ( |ω|σ1 n , . . . , |ω|σ H |n ) for any ω ∈  ∗ . Roughly speaking, the value of a commutative n-periodic event on any word depends only on the number, modulo n, of occurrences of every symbol of the alphabet in the word. Bertoni and his collaborators proposed in [20] a general statistical framework for the synthesis of small qfas inducing approximations of convex linear combinations of commutative periodic events. We recall that a δ -approximation of a given stochastic event q :  ∗ → [0, 1] is any stochastic event p :  ∗ → [0, 1] satisfying the inequality supω∈ ∗ {| p (ω) − q(ω)|} ≤ δ . ∗ Moreover, a convex linear combination of the stochastic events in a family  = { p α :    → [0, 1] | α ∈ I } is any event q :  ∗ → [0, 1] defined as q(ω) = α ∈ I bα p α (ω), for real bα ≥ 0 satisfying the condition α ∈ I bα = 1. We are now going to present the main ingredients of Bertoni’s statistical framework for the synthesis of small qfas, adapted to the case of multiperiodic events. In particular, we give a more accurate version of the results in [58], aiming to precisely point out the modular architecture of the resulting succinct Monte Carlo qfas for some multiperiodic languages. Finally, by considering some statistical tools from [4], we will further improve on the size of such qfas by reducing both the number and size of modules. 3.1. Approximating multiperiodic events by qfas A multiperiodic event generalizes the notion of a commutative periodic event in that a different period ni is associated with every symbol σi of the alphabet. More precisely: Definition 4. Given an alphabet  = {σ1 , . . . , σ H }, a stochastic event p :  ∗ → [0, 1] is said to be (n1 , . . . , n H )-periodic if there exists a function pˆ : Zn1 × · · · × Zn H → [0, 1] satisfying the relation p (ω) = pˆ ( |ω|σ1 n1 , . . . , |ω|σ H n H ) for any ω ∈  ∗ . We are going to rephrase the result from [20] for the case of multiperiodic events. We need to recall the well known Höffding’s inequality [45]. If X i ’s are independent identically distributed random variables with values in [0, 1] and expectation E [ X i ] = μ, then for any S ≥ 1

   S  1  2   prob  X i − μ ≥ δ ≤ 2e−2δ S .  S i =1

Theorem 5. Let  be a family of (n1 , . . . , n H )-periodic events each  induced by an m-state  qfa on an alphabet with H symbols. For H 2 any convex linear combination q of the events in  , there exists an O log n -tuple of these m-state qfas whose uniform /δ t t =1 direct sum induces a δ -approximation of q.

, 1] | α ∈ I } be the family of (n1 , . . . , n H )-periodic events Proof. Let the alphabet  = {σ1 , . . . , σ H } and  = { p α :  ∗ → [0 – the event p α being induced by the m-state qfa A α . Let q = α ∈ I bα p α be a convex linear combination of the events in  . Now, choose independently S qfas A α1 , . . . , A α S with probabilities bα1 , . . . , bα S (αi ∈ I ), respectively, and construct their uniform direct sum as addressed in Proposition 1. The resulting qfa B induces the (n1 , . . . , n H )-periodic event p B =  (1/ S ) iS=1 p αi for which we get:



prob



sup {| p B (ω) − q(ω)|} ≥ δ = prob

ω∈ ∗



max

k∈Zn1 ×···×Zn H

  | pˆ B (k) − qˆ (k)| ≥ δ



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≤

H

 nt ·

max

k∈Zn1 ×···×Zn H

t =1

≤

H



   prob | pˆ B (k) − qˆ (k)| ≥ δ

(by union bound)

nt · 2e−2δ S . 2

(by Höffding’s inequality)

t =1

By requiring that

H

t =1 nt

· 2e−2δ

2

S

< 1, we get the result. 2

3.2. Succinct Monte Carlo qfas for multiperiodic languages We now turn to the notion of a multiperiodic language. For an alphabet  = {σ1 , . . . , σ H }, a language L ⊆  ∗ is said to be (n1 , . . . , n H )-periodic if there exists a set S ⊆ Zn1 × · · · × Zn H such that

L = {ω ∈  ∗ | ( |ω|σ1 n1 , . . . , |ω|σ H n H ) ∈ S }.

(2)

H

It is not hard to verify that L can be accepted by a (

t =1 nt )-state

dfa D on  with:

• Zn1 × · · · × Zn H as the set of states, with (0, . . . , 0) as the initial state, • τ : Zn1 × · · · × Zn H ×  → Zn1 × · · · × Zn H as a transition function defined, for any state (x1 , . . . , x H ) and symbol σt , by τ ((x1 , . . . , xt , . . . , x H ), σt ) = (x1 , . . . , xt + 1nt , . . . , x H ), • S as a set of final states. Here, we focus on the multiperiodic language

L ∧(n1 ,...,n H ) = {ω ∈  ∗ | |ω|σ1 n1 = 0 ∧ · · · ∧ |ω|σ H n H = 0} which, obviously, can be defined as in (2) with S = {(0, . . . , 0)}. For L ∧(n1 ,...,n H ) , the dfa D above described may be shown to

H

be minimal. In fact, assume by contradiction that there exists a dfa A for L ∧(n1 ,...,n H ) with less than (

t =1 nt )

states. Simple

ν = σ1k1 · · · σ Hk H and ω = σ1s1 · · · σ Hs H , with kt , st ∈ Znt (n −k ) (n −k ) for 1 ≤ t ≤ H , taking A to the same state s. Let us add to ν and ω the suffix γ = σ1 1 1 · · · σ H H H . Clearly, νγ ∈ L ∧(n1 ,...,n H ) since each σt occurs nt times, and hence A reaches a final state from s upon consuming γ . Moreover, the fact that A is deterministic implies that ωγ is accepted as well. Thus, ωγ ∈ L ∧(n1 ,...,n H ) which, by definition, implies that counting arguments show the existence of two distinct input words

st − kt + nt nt = 0 for 1 ≤ t ≤ H . This, together with the condition kt , st ∈ Znt , yields kt = st for 1 ≤ t ≤ H , against the hypothesis ν = ω . H

Hence, t =1 nt states are necessary and sufficient for accepting L ∧(n1 ,...,n H ) by dfas. In what follows, by using the approximation technique addressed in Theorem 5, we can design an exponentially smaller Monte Carlo qfa for L ∧(n1 ,...,n H ) . Let us start by emphasizing some properties of a matrix family that will turn out to be useful in the construction of qfas exhibiting multiperiodic behaviors. 1 H −t a family U 1 , . . . , U H ∈ Cn×n of matrices, for 1 ≤ t ≤ H define the n H × n H matrix M t = (⊗ts− Lemma 6. Given =1 I n ) ⊗ U t ⊗ (⊗s=1 I n ). m For any product j =1 M i j , denote by kt the number of occurrences of M t within the product, i.e., kt = |{i j | 1 ≤ j ≤ m and i j = t }| for

1 ≤ t ≤ H . Then, we have

m

j =1

M i j = ⊗tH=1 U t kt .

Proof. Notice that the matrices M 1 , . . . , M H commute. So, the product

m

j =1

by the properties of the Kronecker product and matrix multiplication (see Section 2.1), one gets M t kt =

(⊗sH=−1t I n ), whence the result follows. 2

H

kt t =1 M t . Now, t −1 (⊗s=1 I n ) ⊗ U t kt ⊗

M i j can be rearranged as

Lemma 6 provides a basic machinery to build qfas inducing particular multiperiodic events: Lemma 7. Let the alphabet  = {σ1 , . . . , σ H }. For any vector (n1 , . . . , n H ) of periods and any vector v ∈ Zn1 × · · · × Zn H of parameters, there exists a 2 H -state qfa A inducing the (n1 , . . . , n H )-periodic event p A :  ∗ → [0, 1] defined as

p A (ω) =

H

t =1

 2

cos

π ( v )t |ω|σt nt

 .

Proof. For any 1 ≤ t ≤ H , set the matrix U t of Lemma 6 as the matrix R π ( v )t pointed out in Section 2.1 and obtain the nt

2 H × 2 H matrices M t accordingly. Then, define the desired 2 H -state qfa A = (⊗iH=1 (1, 0), {U (σt ) = M t }σt ∈ , ⊗iH=1 (1, 0)). For

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any ω ∈  ∗ , let us evaluate p A (ω) by letting, for the sake of readability, kt = |ω|σt for 1 ≤ t ≤ H . By Lemma 6 and the properties of matrices R displayed in Equation (1), Section 2.1, we have

⎛ ⎞   H  2  H   H  2 |ω| H

    H k p A (ω) = (1, 0) ⎝ U (ωi )⎠ diag (1, 0) (1, 0) R π ( v )t t diag (1, 0) = n t i =1 t =1 i =1 i =1 i =1 i =1 H 2   H 

π ( v )t kt . 2 = (1, 0) R π (v )t kt diag ((1, 0)) = cos2 nt nt t =1

t =1

It may be interesting to notice that the qfa A in the previous lemma, upon fixing v = (1, . . . , 1), accepts with certainty (with probability strictly less than 1) the words in (not in) L ∧(n1 ,...,n H ) . Thus, A turns out to be an isolated cut point qfa for L ∧(n1 ,...,n H ) with a constant (not depending on the values of the periods n1 , . . . , n H ) number of basis states. This is in sharp

H

contrast with the deterministic case where, as above observed, we cannot employ less than i =1 ni states. Nevertheless, it should be stressed that the parsing precision of A tends to diminish in the sense that the isolation around the cut point tends to zero for increasing values of the periods n1 , . . . , n H . We are now going to improve parsing reliability still keeping an extremely compact size, by showing the existence of a Monte Carlo qfa for L ∧(n1 ,...,n H ) with the error probability  , which is exponentially smaller than equivalent dfas. Such a Monte Carlo qfa has the following modular architecture: (i) It consists of the uniform direct sum of simple qfas (see Proposition 1). (ii) Both the number and size of the qfas in the direct sum depend on the error probability  . The following theorem, which restates [58] in a more accurate form, precisely accounts for the aspects (i) and (ii). Theorem 8. Let  be an alphabet with || = H . For any probability  , consisting of the uniform direct sum of O (( total O ((4 H

H

t =1 log nt )/

 H +2 ) basis states.



 > 0, there exists a Monte Carlo qfa B for L ∧(n1 ,...,n H ) with the error 2 H t =1 log nt )/ ) qfas having O ((4/ ) ) basis states each. Thus, B has in

H



Proof. We let S = log2 2 , and define a suitable family  of (n1 , . . . , n H )-periodic events on the alphabet  = {σ1 , . . . , σ H }. The events in  are indexed by H × S matrices M with M t j ∈ Znt , for 1 ≤ t ≤ H and 1 ≤ j ≤ S. We call M the set of such

H

matrices. Clearly, |M| = ( define  as

=

⎧ ⎨ ⎩

p M (ω) =

t =1 nt )

H S

S

= ||. Again for the sake of readability, for any ω ∈  ∗ , we let kt = |ω|σt . Thus, we

 cos2

π Mt j kt nt

j =1 t =1

By Lemma 7, for 1 ≤ j ≤ S, the event

⎫   ⎬   M ∈M .  ⎭ 

H

t =1 cos

2



π M t j kt nt HS



is induced by a 2 H -state qfa A j . Furthermore, by Proposition 1,

for any M ∈ M the event p M is induced by the 2 -state qfa A M obtained as the Kronecker product of A j ’s. Notice that A M accepts with certainty the words in L ∧(n1 ,...,n H ) . Let us now consider the following convex linear combination of the events in  :

q(ω) =  H

1

t =1 nt

We observe that

(e + iθ

S

nt −1 m=0

 M ∈M

p M (ω) =  H

t =1 nt



cos2 π nm kt t



n −1 H t



1 S

t =1

m =0

 2

cos

π m kt nt

 S .

yields nt whenever kt is a multiple of nt . Otherwise, by using the identity cos θ =

e −i θ )/2, the sum is easily seen to equal nt /2. So, we get q(ω) S

= 1 if ω ∈ L ∧(n1 ,...,n H ) , otherwise q(ω) ≤ 1/2 S . By

Theorem  H5, we can (1/2 )-approximate the convex linear combination q by a qfa B obtained as a uniform direct sum of O (22S t =1 log nt ) many A M ’s, each with 2 H S basis states. For properties of A M ’s, we have that B induces the event

 p B (ω)

=1 ≤ 22S

if ω ∈ L ∧(n1 ,...,n H ) otherwise.





By recalling that S = log2 2 , we get the promised Monte Carlo qfa for L ∧(n1 ,...,n H ) .

2

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3.3. Even more succinct Monte Carlo qfas for multiperiodic languages The Monte Carlo qfa for L ∧(n1 ,...,n H ) exhibited in the previous section consists of a proper composition of a certain number of modules. According to Theorem 8, both the number and size of modules depend on the desired precision  : the former as 1/ 2 , the latter as (4/ ) H . Now, we are going to improve this design pattern by keeping the same modular architecture of the resulting qfa, but reducing both the number and size of modules. Precisely, the former will depend on 1/ , while the latter turn out to be a constant no longer depending on  . To achieve this enhancement, we consider a statistical tool adopted in [4], where a framework similar to ours is proposed for building succinct qfas on unary alphabets. We shall make use of a variant of Azuma’s theorem [64], hereafter called Azuma’s inequality. If X i ’s are independent random variables with values in [−1, 1] and expectation E [ X i ] = 0, then for any S ≥ 1

   S   δ2   prob  X i  ≥ δ ≤ 2e− 2S .   i =1

To avoid some tedious technicalities, as in [4], we assume that input words for qfas are terminated by a right endmarker symbol ‘’ not contained in the input alphabet. As shown in [24], the presence of endmarkers on qfas may be assumed without loss of generality. Theorem 9. Let  be an alphabet with || = H . For any ror probability  , consisting of a uniform direct sum of O (( O ((2 H

H

t =1 log nt )/

 ) basis states.

Proof. Let  = {σ1 , . . . , σ H }, and set S = A = (ϕ , {U (σ )}σ ∈∪{} , η) where:

• ϕ=

√1

#S

S

t =1

 > 0, there exists a Monte Carlo qfa A for L ∧(n1 ,...,n H ) with the erH H t =1 log nt )/ ) qfas having 2 basis states each. Thus, A has in total

  H 2 

h=1 log(

√ H



2 · nh ) . We directly display the final form of the desired qfa

$H

h=1 (1, 0),

• for σh ∈  , we let U (σh ) =

#S

t =1

 $h−1



j =1 I 2 ⊗ R 2π Mht ⊗ n h

$ H −h H

j =1 I 2



with M being an H × S matrix with M ht ∈ Znh ,

for 1 ≤ h ≤ H and 1 ≤ t ≤ S; we call M the set of all the ( h=1 nh ) S matrices of this form, • U () is a unitary matrix having ϕ T as its first column; the remaining 2 H S − 1 columns may be filled by any set of vectors forming with

• η ∈ {0, 1}

ϕ T an orthonormal basis,

2H S

is the vector having 1 in the first component and 0 elsewhere. Therefore, the matrix U () diag (η) has as its first column and 0 elsewhere.

ϕT

So, A has the same modular architecture as the Monte Carlo qfa B designed in Theorem 8. That is, A comes as a uniform direct sum of S qfas, each featuring 2 H basis states. Such qfas are basically the modules A M ’s used in the construction of B (with a slight modification in the angles involved in the evolution matrices), but now their size does not depend on  . Let us now inspect the event p A :  ∗ → [0, 1] induced by A. First of all, by Lemma 6 and properties of matrices R displayed in Equation (1), Section 2.1, it is not hard to verify that, for any ω ∈  ∗ : |ω|

U (ω j ) =

j =1

H S  % t =1 h =1

R 2π Mht |ω|σh . nh

Thus, we get

2  ⎛ ⎞ 2  |ω| H S  1 %

⎝ U (ω j )⎠ U () diag (η) = √ p A (ω) = (1, 0) R 2π Mht |ω|σh U () diag (η) ϕ S nh t =1 h =1 j =1   S H  2  2 H S 1  2π M ht |ω|σh  2π M ht |ω|σh 1   = cos cos .  = 2 S  nh nh S t =1 h =1

(3)

t =1 h =1

If ω ∈ L ∧(n1 ,...,n H ) , then it is easy to see that p A (ω) = 1. On the other hand, to get the desired Monte Carlo behavior, we must ensure the property “p A (ω) <  for every ω ∈ / L ∧(n1 ,...,n H ) ”. To this aim, for each 1 ≤ t ≤ S, we use the random variable X t (ω) =

H

h=1 cos

 2π M |ω |  σh ht nh

, where M is picked uniformly at random from the set M of H × S matrices above

introduced. So, (3) can be rewritten as p A (ω) = L ∧(n1 ,...,n H ) ” is ensured by requiring that

S

1 ( S2

t =1

X t (ω))2 , and hence that the property “p A (ω) <  for every

ω ∈/

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  S  √    X t (ω) <  S for every ω ∈ / L ∧ (n1 , . . . , n H ).   

(4)

t =1

Clearly, the random variable X t (ω) has values in [−1, 1]. So, to use Azuma’s inequality on (4), we must show that the expectation of X t (ω) is 0. For the sake of readability, we let |ω|σh = kh for 1 ≤ h ≤ H . Then H 

1 E [ X t (ω)] =  H

h=1 nh M ∈M h=1

1

= H

H n h −1

h=1 nh h=1 mh =0

Since

 cos

2π M h,t kh



nh



cos

2π m h k h nh

n 1 −1



1

= H

h=1 nh m1 =0



···

n H H −1 m H =0 h =1

 cos

2π m h k h



nh

.

iθ −i θ ω ∈/ L ∧(n1 ,...,n H ) , there exists  1 ≤ z ≤ H satisfying k z nz = 0. For such z, by the identity cos θ = (e + e )/2, one may

easily get

nz −1

m z =0 cos

2π m z k z nz

= 0, and the claim is settled. So, by Azuma’s inequality, the probability that Property (4) S

does not hold is bounded above by 2e− 2 . Since the behavior of the qfa A is (n1 , . . . , n H )-periodic, we can clearly restrict our analysis of the property “p A (ω) <  for every ω ∈ / L ∧(n1 ,...,n H ) ” to words ω coming only from the set C = {σ1 k1 · · · σ H k H | (k1 , . . . , k H ) ∈ Zn1 × · · · × Zn H }. We let N = C \ L ∧(n1 ,...,n H ) be the set of words in C not belonging to L ∧(n1 ,...,n H ) . It is easy to see that | N | =  

recalling that S =

H 2



h=1 log(

√ H

H

h=1 nh

− 1. So, by

2 · nh ) , for the probability that the desired property does not hold, i.e., that p A (ω) ≥  for

ω ∈ N, we have     S   S   √  √  &      prob X t (ω) ≥  S prob  X t (ω) ≥  S (by union bound) ≤      ω∈ N ω∈ N t =1 t =1  H  H   H √

H − 2S ≤ nh − 1 · 2e ≤ nh − 1 · 2 e− log( 2·nh ) (by Azuma’s inequality)

a word

h =1

 =

H

h =1

 nh − 1 · 2

h =1 H

h =1

√ H

1

2 · nh

h =1

1

= 1 − H

h=1 nh

.

Since this probability is smaller than 1, there must exist at least an S-tuple of random variables X t (ω) satisfying Property (4). In turn, this clearly implies the existence of the claimed Monte Carlo qfa for L ∧(n1 ,...,n H ) . 2 3.4. Some remarks and open problems As shown in Section 3.2, the minimal dfa for L ∧(n1 ,...,n H ) has

H

t =1 nt

states. So, by Theorem 3, the minimal number of

H

basis states for a qfa accepting L ∧(n1 ,...,n H ) with a cut point isolated by  is ( t =1 log nt )/2 log(1 + 2/ ). The size of the Monte Carlo qfa obtained in Theorem 9 gets rather close to such a theoretical lower bound, but still does not completely witness its optimality. This suggests two possible research lines: (i) To investigate the possibility of rising the lower bound. (ii) To devise new constructions leading to even smaller isolated cut point qfas for L ∧(n1 ,...,n H ) . Beside L ∧(n1 ,...,n H ) , several other families of multiperiodic languages are considered in [58], for which succinct qfas are built. It would be interesting to investigate the possibility of extending the analysis proposed in Theorem 9 even to such language families. 4. Succinct QFAs for promise problems on multiperiodic inputs The notion of a promise problem was introduced by S. Even, A.L. Selman, and Y. Yacobi [34] in the realm of cryptography. Nevertheless, promise problems show up in many areas of theoretical computer science. We refer the reader to [37] for a thoughtful survey on this subject. Formally, a promise problem on an alphabet  is defined as a pair  = (yes , no ), where yes , no ⊆  ∗ are nonempty disjoint sets. An automaton A solves the promise problem  with an isolated cut point λ whenever there exists  ∈ (0, 1/2] such that

• for any ω ∈ yes , we have p A (ω) ≥ λ +  , and • for any ω ∈ no , we have p A (ω) ≤ λ −  . If λ = 1/2 =  , then we say that A solves the promise problem  exactly. Notice that A is not required to exhibit a particular behavior on words in  ∗ \ (yes ∪ no ). Intuitively, A is “promised” that the input is either a yes-instance or a no-instance,

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and it is only required to distinguish between these two cases. It is easy to see that the classical membership problem for a nonempty language L ⊆  ∗ may be regarded as the particular promise problem ( L ,  ∗ \ L ). Recently, several contributions tackled the solution of promise problems on different types of quantum finite automata (see, e.g., [6,24,36,41,42,68,75]), in all cases emphasizing a drastic reduction in the number of states with respect to solutions by classical finite automata. We are now going to settle a promise problem on multiperiodicity, which actually generalizes several promise problems proposed in the literature [6,24,41]. By using the design tools from the previous section, we build a qfa that solves this promise problem and compare its state complexity with that of dfas. We get that, beside language recognition, even on promise problems the quantum paradigm greatly outperforms the classical one from a descriptional power viewpoint. Let  = {σ1 , . . . , σ H } be an alphabet with associated periods n1 , . . . , n H . Moreover, we let a sequence of remain(1) (2) (1) (2) (1) (2) (1) (2) ders r1 , r1 , . . . , r H , r H satisfy rh , rh ∈ Znh and rh = rh , for all 1 ≤ h ≤ H . We define the promise problem P ∧∨ = ∧∨ ∧∨ ( P yes , P no ) where

'H (1) ∧∨ = {ω ∈  ∗ | • P yes ( |ω|σh nh = rh )}, and (hH=1 (2) ∧∨ ∗ • P no = {ω ∈  | h=1 ( |ω|σh nh = rh )}. By adapting the technique from [41] to multiperiodic qfas obtained by the design patterns from the previous section, we (1) (2) get a constant (not depending on the values of nh ’s, rh ’s, and rh ’s) upper bound on the size of a qfa solving P ∧∨ exactly. Theorem 10. The promise problem P ∧∨ can be solved exactly by a qfa with 3 H basis states. (2)

(1)

Proof. For the sake of readability, for 1 ≤ h ≤ H , we let h = rh − rh nh and θh =

√

in [41] such that the numbers α = − cos(h θh )/(1 − cos(h θh )) and β = [0, 1]. Then, for 1 ≤ h ≤ H , we introduce the unitary matrices

⎛

α

T h = ⎝ −β 0

β 0

⎞

α 0 ⎠, 0

1

⎛

1 0 U h = ⎝ 0 cos θh 0 − sin θh

√

2π t h nh

, where th is an integer chosen as

1/(1 − cos(h θh )) are both from the interval

⎞

0 sin θh ⎠ , cos θh

and the 3-state qfa A h = (ϕh , { M h (σ )}σ ∈∪{} , ηh ) with2 : −r

(1 )

• ϕh = (1, 0, 0) T h U h h , • M h (σh ) = U h , M h () = T h−1 , and M h (σk ) = I 3 for all 1 ≤ k ≤ H with k = h, • ηh = (1, 0, 0). The qfa A solving P ∧∨ is now the Kronecker product (see Proposition 1) of qfas A 1 , . . . , A H . The reader may easily check that A is a 3 H -state qfa inducing the event p A :  ∗ → [0, 1] defined, for any ω ∈  ∗ , as

⎞ 2 ⎛ ⎛ ⎞ H |ω|

 . ⎠ ⎝ϕh ⎝ ⎠ p A (ω) = M ( ω ) M () diag η ( ) j h h h h =1 j =1 Moreover, being M h (σk ) = I 3 for all k = h, this event can be rewritten as

H H 2    2 |ω|σh p A (ω) = ϕh ( M h (σh )) M h () diag (ηh ) = ϕh ( M h (σh ))|ω|σh M h () diag (ηh ) h =1

=

h =1

  H 2 (1) (1) |ω|σ −r |ω|σh −rh −1 T h diag (ηh ) = T h U h h h T h−1 (1, 0, 0) T h U

h =1

Let us now inspect the possible values of p A (ω), for

h =1

.

(5)

1,1

∧∨ ∪ P ∧∨ : ω ∈ P yes no (1)

(1 )

|ω|σh −rh

∧∨ , then for all 1 ≤ h ≤ H we have |ω|  • If ω ∈ P yes σh nh = rh , implying U h H H −1 2 2 h=1 (( T h T h )1,1 ) = h=1 (( I 3 )1,1 ) = 1.

2

2

H

Again, without loss of generality, we assume inputs for qfas are terminated by a right endmarker symbol ‘’.

= I 3 in (5). Hence, p A (ω) =

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∧∨ , then by definition there exists an integer 1 ≤ k ≤ H such that |ω|  = r . Therefore, |ω| − r  =  . • If ω ∈ P no σk nk σk nh k k k n Let us now focus on the kth factor of the product in (5). Since U k k = I 3 , for suitable γ , γ  , γ  we have (2)



(1)

|ω|σk −rk

Tk Uk

−1

Tk



2

= 1,1

(1)

  2   2  (α , β, 0) U k k T k−1 = (α , cos(k θk ) β, γ ) T k−1 1

1



2

= (α 2 + cos(k θk ) β 2 , γ  , γ  )1 2  cos(k θk ) − cos(k θk ) = =0 + 1 − cos(k θk ) 1 − cos(k θk ) Hence, p A (ω) = 0. Thus, we can conclude that the 3 H -state qfa A solves P ∧∨ exactly.

2

Let us now investigate the solution of P ∧∨ by classical finite automata. The following theorem proposes a dfa whose number of states depends on the values of the parameters of the promise problem. In what follows, we let a | b (a  b) denote the fact that a divides (does not divide) b. Theorem 11. The promise problem P ∧∨ can be solved (exactly) by a dfa with (2)

(1)

smallest integer satisfying dh | nh and dh  rh − rh nh .

H

h=1 dh

states where, for 1 ≤ h ≤ H , we let dh be the

Proof. The architecture and dynamics of a dfa D solving P ∧∨ resemble those of a dfa for a multiperiodic language presented in Section 3.2. Informally, D consists of the Cartesian product of H cycles of lengths d1 , . . . , d H . Each state of D is an H -tuple that can be used to store the number modulo dh of occurrences of each symbol σh within the input word ω ∈  ∗ . This is achieved by setting (0, . . . , 0) as the initial state and, after reading σh , incrementing by one modulo dh the hth component of the current state. Thus, after processing ω , we find D in the state ( |ω|σ1 d1 , . . . , |ω|σ H d H ). We let

∧∨ given ( r1(1) d1 , . . . , r (H1) d H ) be the only accepting state, which is clearly reached by D after processing any word in P yes that dh | nh for any 1 ≤ h ≤ H . ∧∨ we have |ω| n = r (2) for some 1 ≤ k ≤ H . Moreover, since On the other hand, by definition, for any word ω ∈ P no σk k k (2) (1) (1) dk  rk − rk nk , we get |ω|σk dk = rk dk implying that D cannot reach the accepting state after consuming ω . This proves the correctness of D. 2

We conjecture that the dfa D proposed in Theorem 11 for the promise problem P ∧∨ on an alphabet with H symbols is minimal. This is actually the case when H = 1, as shown in [41]. In addition to the unary case, it has been proved in [24] that D is a minimal nondeterministic, probabilistic, and two-way deterministic finite automaton for solving P ∧∨ . It should be stressed that the proofs of all these minimality conditions crucially rely on the existence of normal forms for finite automata working on unary alphabets [27,32,52]. Thus, as a per se interesting intermediate step toward proving minimality of D for general alphabets, it may be worth investigating normal forms for finite automata accepting multiperiodic languages. Nevertheless, by the results for the unary case, what can be safely affirmed is that a minimal dfa for P ∧∨ on a H -symbol alphabet cannot employ less than min {d1 , . . . , d H } states. Moreover, e.g., by assuming prime periods n1 , . . . , n H , one may easily see that dh = nh for every 1 ≤ h ≤ H . These simple observations show that the size of a minimal dfa for P ∧∨ on H symbols must depend on the values of periods, and cannot be a constant as proved in Theorem 10 for exact qfas. This fact, once again, witnesses the superiority of quantum vs. classical devices from a descriptional point of view, and highlights promise problems as another interesting research field, beside language membership, in which one can explore the quantum paradigm full power. 5. Final remarks on Alberto Bertoni’s explorations in quantum finite automata theory In this contribution, we focused on Alberto Bertoni’s outcomes on computational and descriptional power of quantum finite automata. Nevertheless, his activity touched many other relevant topics in quantum finite automata theory. In this concluding section, we would like to only add few pointers to other aspects of Bertoni’s work in quantum finite automata theory, of course, without claiming to be exhaustive. In [17,56], Bertoni and his collaborators introduced the model of a quantum finite automaton with control language (qcf), one among the first “hybrid” models of computation (see, e.g. [5,44,76]) featuring cooperating quantum and classical components. Several motivations lead to propose and study hybrid automata. First of all, the need to enhance the computational power of “purely quantum” finite automata which, as noted in the introductory sections, are strictly less powerful than classical finite automata. Then, the need to provide theoretical models for a physically plausible quantum computer

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consisting of a classical device incorporating small “costly” quantum components. Finally, the need to provide a general unifying framework within which to represent and study a great variety of quantum finite automata models proposed in the literature. By introducing the notion of linear representation of a qcf and studying boundedness properties of formal power series generated by qcfs, it was proved that the class of languages accepted with an isolated cut point by qcfs coincides with the class of regular languages. It is worth noticing that the model of qcfs and tools used in its analysis have been used, e.g., in [23,25] to provide the size lower bounds for several models of quantum finite automata. In [22], the model of a measure only quantum finite automaton (mon-qfa) is introduced, where the computational steps are observations (measurements) only. Namely, with each input symbol an observable (hence, an Hermitian instead of unitary matrix) is associated, so that processing a word reduces to repeatedly observe the system according to symbols the word consists of. To analyze the computational power of mon-qfas, the theory of free partially commutative monoids is brought into play. Hence, an interesting connection is established between quantum finite automata and trace language theory, a field where Bertoni also greatly contributed (see, e.g., [10,15]). An algebraic characterization of the class of languages accepted with isolated cut point by mon-qfas is provided in [31]. Finally, we point out Bertoni’s work on decidability problems for quantum finite automata. This topic was probably approached thanks to his previous explorations on analogous problems for probabilistic finite automata [14]. In [21], some elements of the theory of compact groups [28] were used to investigate the solution of classification problems by quantum finite automata. A k-quantum classifier is a k-tuple A 1 , . . . , A k of quantum finite automata on input alphabet  satisfying the following conditions:

• each automaton accepts at least one word from  ∗ , and • every word from  ∗ is accepted by at most one automaton. A k-quantum classifier is complete whenever, in addition, every word in  ∗ is accepted by some automaton. For any k ≥ 2, the problem of deciding whether a k-tuple of quantum finite automata is a (complete) k-quantum classifier was shown to be (un)decidable. With similar tools, it is proved in [13] that, given a language accepted by a quantum finite automaton and a linear context-free language, it is decidable whether or not they have a nonempty intersection. Acknowledgements The authors wish to thank the anonymous referees for careful reading and valuable suggestions. References [1] F. Ablayev, A. Gainutdinova, On the lower bounds for one-way quantum automata, in: Proc. 25th Int. Symp. on Mathematical Foundations of Computer Science, MFCS, in: LNCS, vol. 1893, Springer, 2000, pp. 132–140. [2] A. Ambainis, M. Beaudry, M. Golovkins, A. Kikusts, M. Mercer, D. Thérien, Algebraic results on quantum automata, Theory Comput. Syst. 39 (2006) 165–188. [3] A. Ambainis, R. Freivalds, 1-way quantum finite automata: strengths, weaknesses and generalizations, in: Proc. 39th Symp. on Foundations of Computer Science, FOCS, 1998, pp. 332–342. [4] A. Ambainis, N. Nahimovs, Improved constructions of quantum automata, Theoret. Comput. Sci. 410 (2009) 1916–1922. [5] A. Ambainis, J. Watrous, Two-way finite automata with quantum and classical states, Theoret. Comput. Sci. 287 (2002) 299–311. [6] A. Ambainis, A. Yakaryilmaz, Superiority of exact quantum automata for promise problems, Inform. Process. Lett. 112 (2012) 289–291. [7] P. Benioff, Quantum mechanical Hamiltonian models of Turing machines, J. Stat. Phys. 29 (1982) 515–546. [8] C.H. Bennett, E. Bernstein, G. Brassard, U. Vazirani, Strength and weakness of quantum computing, SIAM J. Comput. 26 (1997) 1510–1523. [9] E. Bernstein, U. Vazirani, Quantum complexity theory, SIAM J. Comput. 26 (1997) 1411–1473, A preliminary version appeared in: Proc. 25th ACM Symp. on Theory of Computing, STOC, 1993, pp. 11–20. [10] A. Bertoni, M. Brambilla, G. Mauri, N. Sabadini, An application of the theory of free partially commutative monoids: asymptotic densities of trace languages, in: Proc. 10th Int. Symp. on Mathematical Foundations of Computer Science, MFCS, in: LNCS, vol. 118, Springer, 1981, pp. 369–376. [11] A. Bertoni, M. Carpentieri, Regular languages accepted by quantum automata, Inform. and Comput. 165 (2001) 174–182. [12] A. Bertoni, M. Carpentieri, Analogies and differences between quantum and stochastic automata, Theoret. Comput. Sci. 262 (2001) 69–81. [13] A. Bertoni, C. Choffrut, F. D’Alessandro, On the decidability of the intersection problem for quantum automata and context-free languages, Internat. J. Found. Comput. Sci. 25 (2014) 1065–1082. [14] A. Bertoni, G. Mauri, M. Torelli, Some recursive unsolvable problems relating to isolated cutpoints in probabilistic automata, in: Proc. 4th Int. Coll. on Automata, Languages, and Programming, ICALP, in: LNCS, vol. 52, 1977, pp. 87–94. [15] A. Bertoni, G. Mauri, N. Sabadini, Membership problems for regular and context-free trace languages, Inform. and Comput. 82 (1989) 135–150. [16] A. Bertoni, C. Mereghetti, B. Palano, Golomb rulers and difference sets for succinct quantum automata, Internat. J. Found. Comput. Sci. 14 (2003) 871–888. [17] A. Bertoni, C. Mereghetti, B. Palano, Quantum computing: 1-way quantum automata, in: Proc. 7th Conf. on Developments in Language Theory, DLT, in: LNCS, vol. 2710, Springer, 2003, pp. 1–20. [18] A. Bertoni, C. Mereghetti, B. Palano, Approximating stochastic events by quantum automata, in: Proc. ERATO Conf. on Quantum Information Science, 2003, pp. 43–44. [19] A. Bertoni, C. Mereghetti, B. Palano, Lower bounds on the size of quantum automata accepting unary languages, in: Proc. 8th Italian Conf. on Theoretical Computer Science, ICTCS, in: LNCS, vol. 2841, Springer, 2003, pp. 86–96. [20] A. Bertoni, C. Mereghetti, B. Palano, Small size quantum automata recognizing some regular languages, Theoret. Comput. Sci. 340 (2005) 394–407. [21] A. Bertoni, C. Mereghetti, B. Palano, Some formal tools for analyzing quantum automata, Theoret. Comput. Sci. 356 (2006) 14–25. [22] A. Bertoni, C. Mereghetti, B. Palano, Trace monoids with idempotent generators and measure-only quantum automata, Nat. Comput. 9 (2010) 383–395. [23] M.P. Bianchi, C. Mereghetti, B. Palano, Size lower bounds for quantum automata, Theoret. Comput. Sci. 551 (2014) 102–115.

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