Quantum Büchi Automata

04/24/2018 ∙ by Qisheng Wang, et al. ∙ University of Technology Sydney Tsinghua University 0

This paper defines a notion of quantum Büchi automaton (QBA for short) with two different acceptance conditions for ω-words: non-disturbing and disturbing. Several pumping lemmas are established for QBAs. The relationship between the ω-languages accepted by QBAs and those accepted by classical Büchi automata are clarified with the help of the pumping lemmas. The closure properties of the languages accepted by QBAs are studied in the probable, almost sure and threshold semantics. The decidability of the emptiness problem for the languages accepted by QBAs is proved using the Tarski-Seidenberg elimination.

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1 Introduction

As acceptors for infinite words (i.e. -words), Büchi automata Büchi (1962) are widely applied in model-checking, program analysis and verification, reasoning about infinite games and decision problems for various logics. Many variants of Büchi automata have been defined in the literature, with acceptance conditions different from the original one in Büchi (1962) (e.g. Muller, Rabin and Street conditions Thomas (1990)). More recently, probabilistic generalisations of Büchi automata and other -automata have been systematically studied in Baier and Größer (2005); Baier et al. (2012).

In quantum computing, quantum automata over finite words were introduced almost 20 years ago and have been extensively studied since then; see for example Kondacs and Watrous (1997); Ambainis and Freivalds (1998); Moore and Crutchfield (2000); Brodsky and Pippenger (2002). To the best of our knowledge, however, quantum automata over infinite words were only very briefly considered in Ruks̆āne et al. (2009); Dzelme-Bērziņa. (2010) where Büchi, Street and Rabin acceptance conditions were defined. The only result obtained in Ruks̆āne et al. (2009) is an example -language accepted by a -way quantum automaton but not by any -way quantum automaton, and in our opinion, the definition of quantum Büchi automata given in Ruks̆āne et al. (2009); Dzelme-Bērziņa. (2010) is problematic (see Subsection 4.1).

The overall aim of this paper is to properly define the notion of quantum Büchi automata and systematically study their fundamental properties, with the expectation that the results obtained here can serve as the mathematical tools needed in the areas like model-checking quantum systems Gay et al. (2006, 2008); Feng et al. (2013); Ying et al. (2014); Feng et al. (2015), semantics and verification of quantum programs and quantum cryptographic protocols Brunet and Jorrand (2004); Chadha et al. (2006); Lago and Zorzi (2014); Feng et al. (2007); Hasuo and Hoshino (2011); Pagani et al. (2014); Selinger (2004); Staton (2015); Ying (2009); Ying et al. (2013); Ying (2016); Anticoli et al. (2016) and analysis of quantum games Eisert et al. (1998); Meyer (1999); Gutoski and Watrous (2007); Zhang (2012).

One of the major differences between classical and quantum automata is: in a classical automaton, checking the acceptance condition does not disturb the state of the system. In contrast, checking the acceptance condition in a quantum automaton may require to perform a quantum measurement which can disturb the state of the system. Thus, different decisions about where a quantum measurement be introduced lead to different languages accepted by the same quantum automaton. In the case of quantum automata over finite words, two modes of acceptance have been defined in the literature:

  • Measure-once (MO): The measurement for checking acceptance condition is performed only at the end of the system’s unitary evolution Moore and Crutchfield (2000).

  • Measure-many (MM): The measurement for checking acceptance condition is performed after each of the system’s unitary transformations Kondacs and Watrous (1997).

To define a quantum Büchi automaton, we consider an infinite sequence of check points in the evolution of the system over an infinite word, and then two scenarios naturally arise:

  • Non-disturbing acceptance (ND): For a given state in the accepting space, we perform a measurement at a check point to see whether the system’s state coincides with it, and then the system’s post-measurement state is discarded (for a more detailed description, see Definition 3.1 and Remark 3.1).

  • Disturbing acceptance (D): For a given state in the accepting space, we perform a measurement at a check point to see whether the system’s state coincides with it, the system evolves from the post-measurement state and we perform the same measurement at the next check point. Thus, the system’s state is disturbed by the measurements (see Definition 3.2 and Remark 3.2).

1.1 Contributions of this paper

Under both the non-disturbing and disturbing acceptance conditions, we define the probable, almost sure and threshold semantics for quantum Büchi automata, which generalise the corresponding semantics defined in Baier and Größer (2005); Baier et al. (2012) for probabilistic Büchi automata to the quantum case. Within these three semantics, we study the closure properties of the -languages accepted by quantum Büchi automata under boolean operations and the emptiness decision problems for these languages. The main technical contributions include:

  1. Pumping lemmas for both QBA|NDs (QBAs with non-disturbing acceptance) [Theorems 5.1 and 5.2] and QBA|Ds (QBAs with disturbing acceptance) [Theorems 5.3 and 5.4].

  2. The precise threshold of QBA|NDs doesn’t matter for the threshold semantics [Theorem 6.2].

  3. There is a QBA|ND (resp. QBA|D) under the probable semantics that cannot be simulated by any QBA|ND (resp. QBA|D) under the threshold semantics [Theorem 6.4].

  4. There is a language accepted by a QBA|NDs under the threshold semantics which is not -regular [Theorem 7.1].

  5. There is a language accepted by a QBA|ND (resp. QBA|D) under the almost sure semantics that is not -context-free (resp. -regular) [Theorem 7.1 and Theorem 7.2].

  6. The QBA|NDs under the probable semantics is closed under union but not under intersection and complement [Theorem 8.1].

  7. The emptiness problem for the languages accepted by QBA|NDs or QBA|Ds is decidable [Theorem 9.2, Theorem 9.4 and Theorem 9.6].

1.2 Organisation of the paper

In Section 2, we review the notion of quantum finite automata from the previous literature. In Section 3, we define the acceptance conditions and various semantics for quantum Büchi automata. Section 4 establishes several basic properties of the accepting probabilities by QBAs. The pumping lemmas for both QBA|NDs and QBA|Ds are presented in Section 5. Sections 6 and 7 are devoted to examine the relationship between different semantics of QBAs, and the relationship between QBAs and classical Büchi automata. The closure properties of QBAs are given in Section 8. Several decision problems about emptiness of the languages accepted by QBAs are discussed in Section 9. A brief conclusion is drawn and in particular, several open problems are raised in Section 10. For readability, the proofs of all results are put into the Appendices (some of them are quite lengthy and tedious).

2 Quantum Finite Automata

For convenience of the reader, in this preliminary section we recall some basic notions of quantum finite automata from Kondacs and Watrous (1997); Moore and Crutchfield (2000).

2.1 Definition of Quantum Automata

Definition 2.1.

A quantum (finite) automaton (QFA) is a 5-tuple

where

  • is a finite-dimensional Hilbert space;

  • is a pure state in , called the initial state;

  • is a finite alphabet;

  • For each , is a unitary operator on ;

  • is a subspace of , called the space of accepting states.

For a finite word , we write:

Let and be the projections onto and (the ortho-complement of ), respectively. We further set and

A major difference between classical or probabilistic automata and quantum automata is that the latter check the acceptance condition by a quantum measurement, which can disturb the state of the system. Therefore, acceptance of by quantum automaton can be defined in the following two different ways:

  • Measure-once (MO): The probability that is accepted by in the MO scenario is:

  • Measure-many (MM): The probability that is accepted by in the MM scenario is:

Intuitively, in the MO scenario, automaton starts in state and executes unitary transformations . At the end, is in state . Then we perform a yes/no measurement with and to see whether the state of is in accepting space , and is the probability that the measurement outcome is “yes”. In the MM scenario, starts in state and at each step, say step , it executes unitary transformation and then perform measurement : if the outcome is “yes”, it terminates; otherwise it enters step . Then is the probability that is accepted within steps.

2.2 Semantics of Quantum Automata

Similar to the case of probabilistic automata, we can define probable, almost sure and threshold semantics for quantum automata.

Definition 2.2.

Let be a quantum automaton, stand for the measure-once or measure-many scenario, and . The language accepted by in the scenario is:

  • Probable semantics:

  • Almost sure semantics:

  • Threshold semantics:

Definition 2.3.

The class of languages accepted by s in the scenario (s for short) is:

  • Probable semantics:

  • Almost sure semantics:

  • Threshold semantics:

2.3 Operations of Quantum Automata

As in the case of classical and probabilistic automata, several operations can be defined for quantum automata.

Definition 2.4.

Suppose that

are quantum automata, and are two complex numbers with .

  1. The weighted direct sum of and is:

  2. The tensor product of

    and is:

  3. The ortho-complement of is:

3 Basic Definitions of Quantum Büchi Automata

Now we start to define quantum Büchi automata. Recall that in a classical Büchi automaton, a run is Büchi accepted if there is a state in the accepting subset which appears infinitely many times in the run. In the quantum case, each state defines a yes/no measurement:

where This measurement is used to check whether the system is in the state . However, such a measurement can disturb the state of the automaton whenever it is performed.

3.1 Non-disturbing Büchi Acceptance

Let us first consider the non-disturbing scenario. Assume that is a quantum automaton and an infinite word. Then the non-disturbing run of over is the infinite sequence of states with for all .

Definition 3.1.

The probability that is non-disturbingly Büchi accepted by is:

(1)

where is the non-disturbing run of over , ranges over all infinite sequences with , and each is called a checkpoint.

Intuitively, in Eq. (1) can be understood as the similarity degree between states and .

Remark 3.1.

The physical interpretation of the sequence is given by the following experiment. Take a system and perform measurement on it after it runs steps, is the probability that the measurement outcome is “yes”, then discard the system. Take a second, identically prepared system, let it run steps, perform measurement on it, is the probability that the outcome is “yes”, then discard the system. We can continue similarly for an arbitrary number of steps. In fact, this procedure was often adopted by physicists in studying recurrence behaviour of quantum systems (see tefaák et al. (2008)

for example of quantum Markov chains).

The following simple example can help us to see how an infinite word is non-disturbingly accepted by a quantum automaton.

Example 1.

Consider quantum automaton , where

  1. ,

  2. ,

  3. ,

  4. , and

  5. and .

Here, stands for the rotation about the axe of the Bloch sphere; that is,

for any real number , and

The non-disturbing run of over is and

for all . Since there is a sequence , such that , we have

3.2 Disturbing Büchi Acceptance

Now we turn to consider a different scenario. For every infinite sequence such that , if we actually perform measurement at each checkpoint , then the disturbing run of over an infinite word under the measurement with checkpoints is the infinite sequence of states where for any :

  • ,

  • for all , .

Definition 3.2.

The probability that is disturbingly Büchi accepted by is:

where is the disturbing run of over under with , and ranges over all infinite sequences with .

Remark 3.2.

The disturbing scenario is widely adopted in defining quantum walks with absorbing boundary Ambainis et al. (2001) and in studying the recurrence of quantum Markov chains, see for example Grünbaum et al. (2013).

The following example is a modification of Example 1, from which we can see the difference between non-disturbing and disturbing acceptance.

Example 2.

We consider a quantum automaton similar to the one in Example 1. Let be a quantum automaton, where

  1. ,

  2. ,

  3. ,

  4. , and

  5. and .

Put word . Then we only need to consider measurement with and . One should notice that whichever the checkpoints is chosen, is either (when is even) or (when

is odd), where

is the disturbing run of over under with the checkpoints . Once is fixed, and if it is even, we can choose for all and then for all . If we choose some odd , and the remaining can be set similarly, then the disturbing run is , which leads to

3.3 Semantics of Quantum Büchi Automata

The probable, almost sure and threshold semantics for quantum finite automata given in Definitions 2.2 and 2.3 can be easily generalised into the quantum case.

Definition 3.3.

Let be a quantum automata, stand for non-disturbing or disturbing acceptance condition, and . Then the language accepted by with the acceptance is defined as follows:

  • Probable semantics:

  • Almost sure semantics:

  • Threshold semantics:

Definition 3.4.

The class of languages accepted by quantum Büchi automata with the acceptance is defined as follows:

  • Probable semantics:

  • Almost sure semantics:

  • Threshold semantics:

Here, we use to indicate that is a quantum Büchi automaton.

The above two definitions are quantum generalisations of the corresponding semantics defined in Baier and Größer (2005); Baier et al. (2012) for probabilistic -automata.

To conclude this section, let us see a simple example showing how the semantics define above can be actually used to describe certain behaviours of quantum systems.

Example 3.

Consider a quantum system with Hilbert space and initial state . It behaves as follows: repeatedly choose one of the two unitary operators:

and apply it. Our question is whether the system’s state can be arbitrarily close to infinitely often. Formally, we construct quantum automaton

and then the question is: whether ?

4 Properties of Accepting Probabilities by Quantum Büchi Automata

In this section, we examine some basic properties of the accepting probabilities by quantum Büchi automata. These properties will serve as a step stone for studying the languages accepted by quantum Büchi automata.

4.1 An Alternative Definition of QBANDs

A Büchi non-disturbing acceptance condition for quantum automata was introduced in Ruks̆āne et al. (2009). Using the notations introduced in this paper, it can be rephrased as the following:

Definition 4.1.

The probability that is non-disturbingly Büchi accepted by is:

where is the non-disturbing run of over , is the projector onto , ranges over all infinite sequences with , and each is called a checkpoint.

Intuitively, for , a state in is said to be -accepted if An infinite word is Büchi accepted by with probability if there are in infinite sequence such that and is -accepted for every . At the first glance, this acceptance looks very different from Definition 3.1, and it is hard to be regarded as a quantum counterpart of Büchi acceptance because it only guarantees that (as a subspace) is hit infinitely often, but Büchi acceptance requires that some (single) state in is visited infinitely often. Actually, it is a quantum generalisation of reachability of (see Ying et al. (2013); Barry et al. (2014); Ying and Ying (2014)

for the definition of reachability in quantum Markov chains and Markov decision processes). But surprisingly, Definitions

3.1 and 4.1 are equivalent; more precisely, we have:

Proposition 4.1.

For any quantum automaton and ,

4.2 Accepting Probability by a QBA as a Limit of Accepting Probabilities by QFAs

We recall that for any classical nondeterministic finite automaton , and for any , we have:

where ,

are the characteristic functions of the languages accepted by (nondeterministic) Büchi automata

and finite automata , respectively, and stands for the prefix of of length for all . This conclusion can be generalised into the quantum case.

Proposition 4.2.

Suppose that is a quantum automaton. Then for any :

The above proposition establishes a connection between quantum finite automata in the measure-once (MO) scenario and quantum Büchi automata with the non-disturbing acceptance. It will be extensively used to prove closure properties of quantum Büchi automata.

Corollary 4.3.

For any two quantum automata and , we have:

This corollary shows that the equivalence of two quantum automata in the measure-once (MO) scenario implies the equivalence of them as quantum Büchi automata with the non-disturbing acceptance. But the following example shows that its inverse is not true.

Example 4.

Let be a quantum automaton, where

  1. ,

  2. ,

  3. ,

  4. , and

  5. ,

and is the same as except for . It is obvious that for any , but

4.3 Accepting Probabilities by Operations of QBANDs

Several operations of quantum automata were reviewed in Subsection 2.3. Now we consider how do they accept infinite words. The following proposition establishes a relationship between the acceptance probabilities by two QBAs and the acceptance probability by their weighted direct sum and by their tensor product as well as a relationship between the acceptance probability by a QBA and that by its ortho-complement.

Proposition 4.4.

Let and be two quantum automata and two complex numbers with . Then for any , we have:

  1. and the equality holds if and only if exists.

Corollary 4.5.
  1. For any two complex number with , and for any , we have:

  2. for any and any positive integer , we have:

The next proposition shows the existence of QBA’s with their acceptance probabilities being that of a given QBA modified by a constant.

Proposition 4.6.
  1. For any , there is a quantum automaton such that for all .

  2. For any quantum automaton and , there is a quantum automaton such that for all .

  3. For any quantum automaton and , there is a quantum automaton such that for all .

4.4 Accepting Probabilities by Operations of QBADs

The behaviours of QBA|Ds are much more complicated than that of QBA|NDs. At this moment, we don’t know as much about the accepting probabilities by the former as that by the latter. But we are able to prove the following:

Proposition 4.7.
  1. For any , there is a quantum automaton such that for all .

  2. Let and be two quantum automata and two complex numbers with . Then for any , we have:

5 Pumping Lemmas

In this section, we prove several pumping lemmas for quantum Büchi automata with non-disturbing acceptance (QBA|NDs for short) or disturbing acceptance (QBA|Ds for short). They will be used in Section 7 to prove non-inclusion between the -languages accepted by classical and quantum automata.

5.1 Pumping Lemmas for QBANDs

We first establish a pumping lemma for QBANDs in terms of their acceptance probabilities.

Theorem 5.1.

Let be a quantum automaton. For any and any , there is a positive integer such that

(2)

for any and . Moreover, if is -dimensional, there is a constant such that .

The above theorem is an -generalisation of the pumping lemma for quantum automata over finite words given in Moore and Crutchfield (2000). The following pumping lemma is a corollary of the above theorem.

We can establish a pumping lemma for QBANDs in terms of their accepted words rather than acceptance probabilities. Occasionally, it is more convenient to use than Theorem 5.1 (see the proof of Theorem 6.2, Theorem 6.4, Theorem 7.1 and Theorem 8.1).

Theorem 5.2.

Let for some .

  1. For any , and , implies there are infinitely many positive integers such that .

  2. For any , there are infinitely many prefixes of such that for all .

5.2 Pumping Lemmas for QBADs

We are also able to prove several pumping lemmas for QBADs.

Theorem 5.3.

Let be a quantum automaton. For any and any , there is a positive integer such that

(3)

for any and . Moreover, if is -dimensional, there is a constant such that .

The above theorem is a counterpart of Theorem 5.1 in the case of disturbing acceptance. It is worthy noting the difference between Theorem 5.1 and Theorem 5.3. The inequality (2) in Theorem 5.1 implies both inequality (3) in Theorem 5.3 and

(4)

However, inequality (4) does not hold for disturbing acceptance, and the following example presents a quantum automaton such that there are , , and with for all .

Example 5.

Consider quantum automaton , where

  1. ,

  2. ,

  3. ,

  4. , and

  5. , and .

Let , and . It can be verified that We consider the word for all positive integer . Let us construct a sequence as follows: , and for all . We write for the disturbing run of over . Then

and . We always have

Theorem 5.4.

Let for some .

  1. For any , and , implies there are infinitely many positive integers such that .

  2. For any , there are infinitely many prefixes of such that for all .

  3. For any , each can be written as , where , and

    1. ;

    2. ;

    3. for all .

Obviously, Clauses 1 and 2 of the above theorem is a counterpart of Theorem 5.2 in the case of disturbing acceptance.

6 Relationship between Different Semantics

In this section, we clarify the relationships between: (1) the languages accepted by QBAs and QFAs; (2) two different acceptance conditions QBANDs and QBADs; and (3) the probable, almost sure and threshold semantics of QBANDs and QBADs.

6.1 Relationship between QBAs and QFAs

First of all, we noticed that a simple relationship between the accepted languages by QBANDs and QFAs can be derived from Proposition 4.2. In this subsection, we establishes a connection between quantum finite automata in the measure-once (MO) scenario and quantum Büchi automata with the disturbing acceptance. Recall that for any class of finite languages, its -Kleene closure is defined as follows:

Especially, and are the classes of -regular languages and -context-free languages, respectively. If is an -regular language, then

for some positive integer , where and are regular languages. This conclusion can also be generalised into the quantum case. Let be a quantum automaton. We define its modification: where .

Proposition 6.1.

For any quantum automaton , and , we have: if and only if , where:

  1. ;

for some real number and state .

6.2 Relationship between Different Semantics of QBANDs

In this subsection, we examine the relationship between the probable, almost sure and threshold semantics of QBA|NDs.

Theorem 6.2.
  1. For any , we have:

  2. For any , we have:

  3. For any , it holds that

Parts 1) and 2) of the above theorem indicates that in the threshold semantics, the concrete threshold value is not essential. Moreover, it is interesting to see from parts 2) and 3) that is included in the non-strict threshold semantics but not in the strict threshold semantics . But it is still not aware whether the inclusion with given in part 1) is proper or not.

6.3 Relationship between Different Semantics of QBADs

The following theorem clarifies the relationship between the almost sure and threshold semantics of QBA|Ds.

Theorem 6.3.

For any , we have:

The above theorem is the disturbing acceptance counterpart of Theorem 6.2 3). But at this moment we don’t know whether the conclusions given Theorem 6.2 1) and 2) hold for disturbing acceptance or not.

6.4 Relationship between QBANDs and QBADs

In this subsection, we consider the relationship between various semantics of QBA|NDs and that of QBA|Ds.

Theorem 6.4.
  1. For any and , it holds that

  2. For any , we have:

  3. For any , we have:

Part 1) of the above theorem shows that the threshold semantics of quantum Büchi automata with non-disturbing acceptance is not included in that with disturbing acceptance. But it is still an unsolved problem whether the reverse inclusion is true. Furthermore, parts 2) and 3) shows that the almost sure semantics of quantum Büchi automata with non-disturbing (resp. disturbing) acceptance is not included in the threshold semantics with disturbing (resp. non-disturbing) acceptance.

7 Relationship between Quantum Büchi Automata and Classical -Automata

The aim of this section is to clarify the relationship between classical -automata and quantum Büchi automata with the two different acceptance conditions QBANDs and QBADs.

7.1 Relationship between QBANDs and Classical -Automata

The following theorem shows the relationship between the languages accepted by QBA|NDs and classical -regular languages and -context free languages.

Theorem 7.1.
  1. For any , we have:

  2. For any , we have:

  3. For any , it holds that

  4. .

  5. .

Parts 1) and 2) of the above theorem indicates that both -regular and context-free languages are not included in the threshold semantics of quantum Büchi automata with non-disturbing acceptance. Parts 3) asserts that the threshold semantics of quantum Büchi automata with non-disturbing acceptance is not included in -regular languages, and the almost sure semantics is not included in either -regular or context-free languages.

7.2 Relationship between QBADs and Classical -Automata

The following theorem describes the relationship between the languages accepted by QBA|Ds and classical -regular languages and -context free languages.

Theorem 7.2.
  1. For any , we have:

  2. For any , we have:

  3. .

The above theorem is the disturbing acceptance counterpart of Theorem 7.1 1), 2) and 4), but it is still unknown whether the conclusions given in Theorem 7.1 3) and 5) are valid for disturbing acceptance.

8 Closure Properties

The aim of this section is to investigate the closure properties of the languages accepted by quantum Büchi automata under the Boolean operations.

8.1 Closure Properties of QBANDs

In this section, we consider the closure properties of the languages accepted by QBA|NDs under the threshold semantics with respect to Boolean operations: union, intersection and complement.

Theorem 8.1.
  1. is closed under union:

    • if , then .

  2. For , is closed under union in the limit:

    • if , then there is a sequence of -languages such that

  3. is not closed in the limit for .

  4. is not closed under complementation for