1. Seminar onQUANTUM AUTOMATA and LANGUAGES PRESENTED BY: AbhijitDoley. RanjanPhukan. Rekhamoni Morang. Roll No-0928027.Roll No-0928026. Roll No-0928017. SEMESTER: 7th. DEPARTMENT OF INFORMATION TECHNOLOGY. 1 31-Jan-11

2. 31-Jan-11 Quantum Automata and Languages 2 Contents Introduction. Bits and Qubits. Brief Introduction to Classical Automata. Probabilistic Automata and Stochastic Languages. Quantum Automata and Quantum Languages. Quantum finite-state automata (QFA). QRL and Pumping lemma for QRL. One-way quantum finite automata (1QFA). Measure Once 1-way QFA. Measure Many 1-way QFA. Multi-letter 1QFA. One-way quantum finite automata together with classical states (1QFAC). Two-way quantum finite automata (2QFA). Two-way finite automata with quantum and classical states (2QCFA). 1.5-way Quantum Finite Automata. Quantum Push-down Automaton (QPDA). Quantum context-free grammars. Sequential Quantum Machines (SQM). Quantum Sequential Machines (QSM) Decidability and Undecidability of Quantum Automata. Conclusion.

3. Introduction Quantum computing is a promising research field, which touches on computer science, quantum physics and mathematics . Quantum computation has received a great deal of interest in both physics and computer science in recent years. Driven by the recent discovery of quantum algorithms for factoring that operate in polynomial time. 31-Jan-11 3 Quantum Automata and Languages

4. Introduction 4 A quantum computer is a device for computation that makes direct use of quantum mechanical phenomena, such as superposition, to perform operations on data. Quantum computers are different from traditional computers based on transistors. To understand computation in a quantum context, it might be useful to translate as many concepts as possible from classical computation theory into the quantum case. Simplest language classes — regular languages. context-free languages. 31-Jan-11 Quantum Automata and Languages

5. Introduction 5 To do this, we define quantum finite-state and push-down automata as two special cases of Quantum Automata. In this setting a formal language becomes a function that assigns quantum probabilities to words. In quantum grammars, we sum over all derivations to find the amplitude of a word. The corresponding languages generated by quantum grammars and recognized by quantum automata have their own properties. 31-Jan-11 Quantum Automata and Languages

6. Evolution of Quantum Automata Quantum events cannot be simulated in classical computers in feasible time. So it was needed to formalize the quantum computers. Quantum automata are the basic model for the quantum computers. Quantum automata are built due to the problems of classical computers with certain mathematical problems. 31-Jan-11 6 Quantum Automata and Languages

8. Quantum Computational Unit (Qubits) The basic unit of information in quantum computing is called the qubit. Two states are labeled as |0> and |1>. An object enclosed using the notation |> can be called a state, a vector or a ket. 8 31-Jan-11 Quantum Automata and Languages

9. Qubits (contd…) A qubit can exist in the state |0> or the state |1>. Can also exist in a state that is a linear combination of the states|0> and |1> Superposition State. A superposition state is written as |ψ> = α|0> + β|1 > Here α, β are complex numbers. 9 31-Jan-11 Quantum Automata and Languages

10. Qubits (contd…) When a qubit is measured, it is only found to be in the state |0> or the state |1>. |α|²: probability of finding |ψ> in state |0>. |β|²: probability of finding |ψ> in state |1>. Example: |ψ >=1/√3 |0> +√(2/3) |1> probability of finding |ψ> in state |0> = | 1/√3 |²=1/3 probability of finding |ψ> in state |1> = | √2/√3 |²=2/3 10 31-Jan-11 Quantum Automata and Languages

11. Qubits (contd…) 31-Jan-11 11 Figure 1: Qubit System Quantum Automata and Languages

12. Brief Introduction to Classical Automata 31-Jan-11 Quantum Automata and Languages 12

13. Alphabet, Strings & Languages Alphabet(∑): Finite non-empty set of symbols. Example:{0,1} is the binary alphabet. String: Finite sequence of symbols chosen from some alphabet. Example: 1011 is string from the alphabet {0,1}. ∑* denotes the set of all strings over alphabet ∑. Language: A set of strings all of which are chosen from some ∑*. Example: The set of even numbers. 13 31-Jan-11 Quantum Automata and Languages

15. Deterministic Finite Automata(DFA) DFA is a 5-tuple (K, , , q0, F) where K is a finite set of states,  is a finite set of input symbols, q0 is the initial state, F is the set of final states,  is the transition function mapping from K *   K, (q1,a)= q2 means when we are in state q1 and read ‘a’ , we move to state q2. 31-Jan-11 15 Quantum Automata and Languages

16. Deterministic Finite Automata(DFA) 31-Jan-11 16 Figure 2: Deterministic Finite Automata Quantum Automata and Languages

17. Non-deterministic Finite Automata(NFA) NFA is a 5-tuple (Q, , , q0, F) where Q is a finite set of states,  is a finite set of input symbols, q0 is the initial state, F is the set of final states,  is the transition function mapping from Q *   2Q. 31-Jan-11 17 Quantum Automata and Languages

18. Non-deterministic Finite Automata(NFA) 31-Jan-11 18 Figure 3: Non-deterministic Finite Automata Quantum Automata and Languages

19. Transition Matrix A Transition Matrix M of an alphabet in  accepted by a DFA with Q states is a |Q| *|Q| matrix with entries 0 or 1. Ma(i,j) = 1, if (qj, a)  qi = 0, otherwise; a is an element of . 31-Jan-11 19 Quantum Automata and Languages

20. Transition Matrix (Example) 31-Jan-11 20 Quantum Automata and Languages

21. Probabilistic Automata (PA) We obtain probabilistic automata if we allow fractional values in transition matrix. Probabilistic Automata accepts regular language. Example: 31-Jan-11 21 Quantum Automata and Languages

22. Quantum Automata 31-Jan-11 Quantum Automata and Languages 22

23. Probabilistic Automata 31-Jan-11 23 A probabilistic automaton is a tuple A = (Q, q0, qf ,Σ, (Xa)a∈Σ) Q = {1, . . . , q} is a finite set of states, q0 ∈ Q is the initial state, qf ⊆ Q is the set of final states, and Σ is a finite alphabet. Each matrix Xa is a q × q stochastic matrix: (Xa)i j is the probability of going from state i to state j when a is the input letter. Quantum Automata and Languages

24. Fundamental properties of Probabilistic Automata 31-Jan-11 24 Each columns adds up to 1. If the rows of all Xa contain exactly one 1 we obtain the model of deterministic finite automata. Quantum Automata and Languages

25. Language Accepted by Probabilistic Automata 31-Jan-11 25 To define the language accepted by a probabilistic automaton, we need to fix a threshold η ∈ [0, 1]. A word w = w1 . . .wn ∈ Σ∗ is accepted if the probability of ending up in qf upon reading w is at least η. A probabilistic automaton A accepts a language L with certainty if Quantum Automata and Languages

27. The language depends on the value of the cut-point η, normally taken to be in the range 0≤ η <1.Quantum Automata and Languages

28. Stochastic Languages 31-Jan-11 27 A language is called η-stochastic if and only if there exists some PA that recognizes the language, for fixed η. A language is called stochastic if and only if there is some 0≤ η <1 for which Lη is η-stochastic. A cut-point is said to be an isolated cut-point if and only if there exists a δ > 0 such that, for all s ∈ Σ∗, Quantum Automata and Languages

29. Properties of Stochastic Languages 31-Jan-11 28 Every regular language is stochastic. More strongly, every regular language is η-stochastic. The general converse does not hold: there are stochastic languages that are not regular. Every η-stochastic language is stochastic, for some 0 < η < 1. If η is an isolated cut-point, then Lη is a regular language. Quantum Automata and Languages

30. Quantum Automata (QA) 29 Quantum automata are obtained by letting the transition matrices have complex entries. We also require each of the matrices to be unitary. Example: Transition Matrix 31-Jan-11 Quantum Automata and Languages

31. Definition of Quantum Automata 30 A Quantum Automaton (QA) Q consists of a Hilbert space H, an initial state vector sinit ∈ H with |sinit|2 = 1, a subspace Haccept ⊂ H and an operator Pacceptthat projects onto it, an input alphabet A, and a unitary transition matrix Ua for each symbol a ∈ A. 31-Jan-11 Quantum Automata and Languages

32. Quantum Language 31 We define the quantum language recognized by the Quantum Automata Q as the function fQ(w) = |sinitUwPaccept|2 from words in A∗ to probabilities in [0, 1]. We start with ‹sinit|, apply the unitary matrices Uwifor the symbols of w in order, Measure the probability that the resulting state is in Haccept by applying the projection operator Paccept. This is a real-time automaton since it takes exactly one step per input symbol, with no additional computation time after the word is input. 31-Jan-11 Quantum Automata and Languages

33. Acceptance Probabilities Let q1 is the starting state of the automaton, Mw|q> is a vector describing a superposition of states. If the jth entry in the vector is αj then αj is the probability that the automaton reaches state qj. | αj |2 is the probability that a measurement will end in state qj . | ∑ qjєF αj |2 gives the probability that the automaton accepts the string w. 32 31-Jan-11 Quantum Automata and Languages

34. Different Classes Of Quantum Automata 33 We can then define different classes of quantum automata by restricting the Hilbert space H and the transition matrices Ua in various ways: to the finite-dimensional case. to an infinite memory in the form of a stack. 31-Jan-11 Quantum Automata and Languages

35. Quantum finite-state automata 34 A quantum finite-state automaton (QFA) is a real-time quantum automaton where H, sinit, and the Ua all have a finite dimensionality n. They are related to quantum computers in a similar fashion as finite automata are related to classical computers. 31-Jan-11 Quantum Automata and Languages

36. Quantum finite-state automata A QFA is a 6-tuple M =(Q, ∑, V, q0,Qacc,Qrej) where Q is a finite set of states. ∑ is an input alphabet. V is a transition function. q0∈Q is a starting state. Qacc⊆Q are accepting states. Qrej⊆Q are sets of and rejecting states (Qacc∩Qrej=∅). Qaccand Qrej, are called halting states. Qnon=Q−(Qacc∪Qrej) are called non-halting states. 35 31-Jan-11 Quantum Automata and Languages

37. Endmarkers We use κ and $ as the left and the right endmarker respectively. They do not belong to ∑. We call Γ= ∑ ∪ {κ; $} the working alphabet of M. 36 31-Jan-11 Quantum Automata and Languages

38. Computation 37 The computation of a QFA starts in the superposition |q›. Thentransformations corresponding to the left endmarkerκ, the letters of the input word x and the right endmarker $ are applied. The transformation corresponding to a∈Γ consists of two steps. First, Vais applied. The new superposition Ψ' is Va(Ψ) where Ψ is the superposition before this step. Then, Ψ' is observed with respect to Eacc; Erej; Enonwhere Eacc=span{|q›:q∈Qacc}, Erej=span{|q›: q∈Qrej}, Enon=span{|q›: q∈Qnon}. 31-Jan-11 Quantum Automata and Languages

40. rejects with probability pr= ∑βj2

41. continues the computation with probability pc= ∑γk2 i.e. applies transformations corresponding to next letters. 31-Jan-11 Quantum Automata and Languages

42. Recognition of languages and QRL 39 We will say that an automaton recognizes a language L with probability p (p>½) if it accepts any word x ∈ L with probability ≥ p and rejects any word x ∈ L with probability ≥ p. A quantum regular language (QRL) is a quantum language recognized by a QFA. 31-Jan-11 Quantum Automata and Languages

43. The pumping lemma for QRLs Theorem: If f is a QRL, then for any word w and any Є> 0, there is a k such that |f(uwkv) − f(uv)| < Є for any words u, v. Moreover, if f’s automaton is n-dimensional, there is a constant c such that k < (cЄ)−n. 40 31-Jan-11 Quantum Automata and Languages

44. Types Of QFA One-way quantum finite automata (1QFA) tape heads move one cell only to right at each evolution. Two-way quantum finite automata (2QFA) tape heads are allowed to move towards right or left, or to be stationary. 41 31-Jan-11 Quantum Automata and Languages

45. One-way quantum finite automata (1QFA) 42 Proposed by Moore and Crutchfield. Represent a theoretical model for a quantum computer with finite memory. Does not allow intermediate measurements, except to decide whether to accept or reject the input. Allows the full range of operations permitted by the laws of quantum physics, subject to a space constraint. 31-Jan-11 Quantum Automata and Languages

46. Definition of One-way quantum finite automata 1-way QFA is a 6-tuple M = (Q,∑, δ, q0,Qacc,Qrej) where Q is a finite set of states ∑ is an input alphabet δ is a transition function q0 ∈ Q is a starting state Qacc ⊂ Q are accepting states Qrej⊂ Q are rejecting states 43 31-Jan-11 Quantum Automata and Languages

47. One-way quantum finite automata The states in Qacc and Qrejare called halting states. The states in Qnon = Q − (Qacc ∪ Qrej) are called non-halting states. ¢ and $ are used as the left and the right endmarker respectively. The working alphabet of M is Γ = ∑ ∪ {¢, $}. δ: Q×Γ×Q×{0,1}C is the transition function. 44 31-Jan-11 Quantum Automata and Languages

48. Example (1QFA) 45 We use a one letter alphabet ∑ = {a}. The state space is Q = {q0, q1, qacc, qrej} with the set of accepting states Qacc = {qacc} and the set of rejecting states Qrej = {qrej}. the starting state is q0. The transition function can be specified in two ways: by specifying δ or by specifying Vxfor all letters x ∈ Γ. Both methods are equivalent: all Vx are determined by δ. 31-Jan-11 Quantum Automata and Languages

50. Defining by δ :δ(q0, a, q0) =½ δ(q0, a, q1) =½ δ (q0, a, qacc) = 0 δ (q0, a, qrej) =1/√2 31-Jan-11 Quantum Automata and Languages

51. Example (contd…)Working steps of the automaton: 47 The automaton starts in |q0›. Then, Va is applied, giving ½ |q0›+ ½ |q1›+ 1/√2 |qrej›. Two outcomes are possible. With probability (1/√2)2 = ½, a rejecting state is observed, the word is rejected and the computation terminates. Otherwise with probability ½ , a non-halting state is observed and the superposition collapses to ½ |q0›+ ½ |q1›.In this case, the computation continues. The word ends and the transformation V$ corresponding to the right endmarker $ is done. It maps the superposition to ½ |qrej› + ½ |qacc›. With probability (½)2 = ¼, the rejecting state qrej is observed. With probability ¼, the accepting state qacc is observed. 31-Jan-11 Quantum Automata and Languages

52. Example (contd…) 48 Probability of accepting and rejecting: The total probability of accepting is ¼. The total probability of rejecting is ½ + ¼ = ¾. 31-Jan-11 Quantum Automata and Languages

53. Languages Accepted by 1-way QFA All languages recognized by 1-way QFAs are regular. There is a regular language that cannot be recognized by a 1-way QFA with probability ½+є for any є > 0. It was generalized by Brodsky and Pippenger. 49 31-Jan-11 Quantum Automata and Languages

54. Advantages & Disadvantages of 1QFA 50 Advantages: Quantum superposition offers some computational advantages on probabilistic superposition. Quantum automata can be exponentially more space efficient than deterministic or probabilistic automata. Disadvantages: Due to limitation of memory, it is sometimes impossible to simulate deterministic automata by quantum automata. Since it is reversible, so it is unable to recognize some regular languages. 31-Jan-11 Quantum Automata and Languages

55. Types of 1QFA 51 The acceptance capability of a 1-way QFA depends on the measurements that the QFA performs during the computation. Two models of 1-way QFAs that differ in the type of measurement that they perform during the computation: Measure Once 1-way QFA Measure Many 1-way QFA 31-Jan-11 Quantum Automata and Languages

56. Measure Once 1-way QFA Introduced by Moore and Crutchfield. It is a 5-tuple (Q, , , q0, Qacc) where Qaccis the set of accepting states. The transition function is defined as  : Q xx Q  C[0,1] that represents the probability that flows from state q to state q′ upon reading symbol σє∑. Measurement is performed after the whole input string is read. The language accepted by MO-1QFA is regular language. 52 31-Jan-11 Quantum Automata and Languages

57. Measure Many 1-way QFA Introduced by Kondacs and Watrous. It is a 7-tuple (Q, , , q0, Qacc, Qrej, Qnh) where Qrej is the set of rejecting states and Qnh = Q – Qacc - Qrej The transition function is defined as  : Q xx Q  C[0,1] Measurement is performed after each input symbol is read. More complex than Measure Once 1-way QFA. The language accepted by MM-1QFA is regular language. 53 31-Jan-11 Quantum Automata and Languages

58. Operation of MM 1QFA 54 After every transition M measures its configuration with respect to the three subspaces that corresponding to the three subsets Qnon, Qacc, and Qrej: Enon = Span( { |q› | q ∈ Qnon} ), Eacc = Span( { |q› | q ∈ Qacc} ), Erej = Span( { |q› | q ∈ Qrej} ). If the configuration of M is in Enon then the computation continues, If the configuration is in Eacc then M accepts, Otherwise it rejects. 31-Jan-11 Quantum Automata and Languages

59. Language Accepted Measure-many model is more powerful than the measure-once model, where the power of a model refers to the acceptance capability of the corresponding automata. MM-1QFA can accept more languages than MO-1QFA. Both of them accept proper subsets of regular languages. 55 31-Jan-11 Quantum Automata and Languages

60. Comparison of MO-1QFA and MM-1QFA MO-1QFA MM-1QFA Initiated by Moore and Crutchfield. There is only one measurement for computing each input string, performing after reading the last symbol. Two results: acceptance and rejection. Initiated by Kondacs and Watrous. Measurement is performed after reading each symbol, instead of only the last symbol. Three results: acceptance, rejection and continuation. 56 31-Jan-11 Quantum Automata and Languages

61. Multi-letter 1QFA Proposed by A. Belovs, A. Rosmanis, J. Smotrovs. Multiple reading heads are present. A k-letter 1QFA is not limited to see the just-incoming input letter, but can see several earlier received letters as well. Quantum state transition which the automaton performs at each step depends on the last k letters received. In the simplest form k =1, it reduces to an MO-1QFA. Any given k-letter QFA can be simulated by some (k + 1)-letter QFA, but the contrary does not hold. 57 31-Jan-11 Quantum Automata and Languages

62. Definition of k-letter 1QFA 58 A k-letter QFA A is defined as a 5-tuple A = (Q,Qacc, |ψ0›,∑, μ), where Q is a set of states, Qacc ⊆ Q is the set of accepting states, |ψ0› is the initial unit state that is a superposition of the states in Q, ∑ is a finite input alphabet, and μ is a function that assigns a unitary transition matrix Uw on C|Q| for each string w ∈ ({Λ} ∪ ∑)k, where |Q| is the cardinality of Q. 31-Jan-11 Quantum Automata and Languages

63. Equivalence of Multi-letter 1QFA 59 Let us consider, a k1-letter QFA A1 and a k2-letter QFA A2. A1 and A2 are equivalent if and only if they are (n1+n2)4+k−1-equivalent, where n1 and n2 are the numbers of states of A1 and A2, respectively. k = max(k1, k2). Two multi-letter QFAs over the same input alphabet are n-equivalent if and only if the accepting probabilities of A1 and A2 are equal for the input strings of length not more than n. 31-Jan-11 Quantum Automata and Languages

64. Language accepted by Multi-letter 1QFA 60 Can accept some regular languages not acceptable by MO-1QFA and MM-1QFA. Accept a proper subset of regular languages. 31-Jan-11 Quantum Automata and Languages

65. Hierarchy of multi-letter QFAs and some relations 61 j-letter QFA are strictly more powerful than i-letter QFAs for 1 ≤ i < j. Let us denote the languages accepted by MO-1QFAs, MM-1QFAs, and multi-letter QFAs, denoted by L(MO), L(MM), and L(QFA*), respectively, then L(MO) ⊆ L(MM) ∩ L(QFA*), where ⊆ may be proper. L(MM) ∪ L(QFA*) is a proper subset of all regular languages. 31-Jan-11 Quantum Automata and Languages

66. One-way quantum finite automata together with classical states (1QFAC) 62 1QFA accepts only subsets of regular languages with bounded error. In 1QFAC the component of classical states together with their transformations is added the choice of unitary evolution of quantum states at each step is closely related to the current classical state. So the classical element is preserved in this quantum device. As MO-1QFA , 1QFAC performs only one measurement for computing each input string, doing so after reading the last symbol. 31-Jan-11 Quantum Automata and Languages

67. One-way quantum finite automata together with classical states (1QFAC) A 1QFAC A is defined by a 9-tuple A= (S,Q,∑, Γ, s0, q0,δ ,U,M) where: ∑ is a finite set of input alphabet. Γ is a finite set of output alphabet. S is a finite set of classical states. Q is a finite set of quantum states. s0 is an element of S (the initial classical state). q0the initial quantum state. δ : S × ∑ S is the classical transition function. U = {Usσ}sЄS,σЄ∑where Usσ: H(Q)  H(Q) is a unitary operator for each s and σ (the quantum transition operator at s and σ). M= {Ms}sЄS where each Ms is a projective measurement over H(Q) with outcomes in Γ (the measurement operator at s). 63 31-Jan-11 Quantum Automata and Languages

68. Computation in 1QFAC 64 At start up, automaton is in an initial classical state and in an initial quantum state. By reading the first input symbol, the classical transformation results in a new classical state as current state. the initial classical state together with current input symbol assigns a unitary transformation to process the initial quantum state, leading to a new quantum state as current state. Similar process for next input symbols read. Continues to operate until the last input symbol has been scanned. According to the last classical state, a measurement is assigned to perform on the final quantum state, producing a result of accepting or rejecting the input string. 31-Jan-11 Quantum Automata and Languages

69. Diagrammatic Representation 65 Figure 4: 1QFAC dynamics as an acceptor of language 31-Jan-11 Quantum Automata and Languages

70. Language Accepted by 1QFAC 1QFAC accepts only regular languages. Can accept same language with essentially less number of states than DFA. It accepts some languages that cannot be accepted by any MO-1QFA and MM-1QFA as well as multi-letter 1QFA. For any prime number m ≥ 2, there exists a regular language whose minimal DFA needs O(m) states, that can not be accepted by the 1QFA, but there exists 1QFAC accepting it with only constant classical states and O(log(m)) quantum basis states. 66 31-Jan-11 Quantum Automata and Languages

71. Equivalence of 1QFAC 67 Any two 1QFAC A1 and A2 over the same input alphabet ∑ are equivalent iff their probabilities for accepting any input string are equal. Two 1QFAC over the same input alphabet ∑ are k-equivalent iff their probabilities for accepting any input string do not differ more than k at each string. 31-Jan-11 Quantum Automata and Languages

72. Two-way quantum finite automata (2QFA) 2-way QFA is a 6-tuple M = (Q,∑, δ, q0,Qacc,Qrej) where Q is a finite set of states ∑ is an input alphabet δ is a transition function q0 ∈ Q is a starting state Qacc ⊂ Q are accepting states Qrej⊂ Q are rejecting states 68 31-Jan-11 Quantum Automata and Languages

73. Two-way quantum finite automata The states in Qacc and Qrejare called halting states. The states in Qnon = Q − (Qacc ∪ Qrej) are called non-halting states. ¢ and $ are used as the left and the right endmarker respectively. The working alphabet of M is Γ = ∑ ∪ {¢, $}. δ: Q×Γ×Q×{-1,0,1}C is the transition function. Tape head can move towards right, left or remain stationary. 69 31-Jan-11 Quantum Automata and Languages

74. Language Accepted by 2-way QFA Can accept all regular languages with certainty. Also accepts some non-regular languages within linear time. 70 31-Jan-11 Quantum Automata and Languages

75. Disadvantage of 2QFA 71 It allows superposition where the head can be in multiple positions simultaneously. To implement such a machine, we need at least O(log n) qubits to store the position of the head (where n is the length of the input). 31-Jan-11 Quantum Automata and Languages

76. Two-way finite automata with quantum and classical states (2QCFA) Proposed by Ambainis and Watrous. It has both quantum states and classical states. 2QCFA is simpler to implement than 2QFA, since the moves of tape heads are classical. Solves the problem of 2QFA, by having the size of the quantum part does not depend on the length of the input. 72 31-Jan-11 Quantum Automata and Languages

77. Two-way finite automata with quantum and classical states (2QCFA) 73 We may describe a 2qcfa as a classical 2-way finite automaton that has access to a fixed size quantum register, upon which it may perform quantum transformations and measurements. It has two transfer functions: One specifies unitary operator or measurement for the evolution of quantum states. The other describes the evolution of classical part of the machine, including the classical internal states and the tape head. 31-Jan-11 Quantum Automata and Languages

78. Formal Definition of 2QCFA A 2QCFA is specified by a 9-tuple M = (Q, S, ∑, θ, δ, q0, s0, Sacc, Srej), where Q and S are finite state sets (quantum states and classical states, respectively). ∑ is a finite alphabet. θ and δ are functions that specify the behavior of M. q0 ∈ Q is the initial quantum state. s0 ∈ S is the initial classical state. Sacc, Srej⊆ S are the sets of (classical) accepting states and rejecting states, respectively. Γ=∑ ∪ {¢, $} are the tape alphabet of M, where ¢ and $ are the left end-marker and right end-marker, respectively. 74 31-Jan-11 Quantum Automata and Languages

79. Transition Functions Function θ specifies the evolution of the quantum portion of the internal state, for each pair (s, σ) ∈ S. Function δ specifies the evolution of the classical part of M and the tape head. δ is defined so that the tape head never moves left when scanning the left end-marker ¢ and never moves right when scanning the right end-marker $. 75 31-Jan-11 Quantum Automata and Languages

80. Languages Recognized By 2QCFA A 2QCFA recognizes all regular languages. Hence it is more powerful than 1QFA. A 2QCFA recognizes some context free languages also. Hence it is more powerful than a DFA. 76 31-Jan-11 Quantum Automata and Languages

81. Example 77 Let us consider the two languages: Lpal= {x ∈ {a, b}∗ | x = xR} (the language consisting of all palindromes over the alphabet {a, b}) and Leq= {anbn | n ∈ N}. No probabilistic 2-way finite automaton can recognize Lpalin any amount of time. No classical 2-way finite automaton can recognize Leq in polynomial time. But there exists an exponential time 2qcfa recognizing Lpal, a polynomial time 2qcfa recognizing Leq. Thereby giving two examples where 2qcfa’s are more powerful than classical 2pfa’s. 31-Jan-11 Quantum Automata and Languages

82. 1.5-way Quantum Finite Automata 78 An intermediate form of QFA. Developed by Amano and Iwama. Tape heads are allowed to move right or to be stationary. 31-Jan-11 Quantum Automata and Languages

84. It was proved earlier by A. Ambainis and R. Freivalds that quantum finite automata with pure states can have an exponentially smaller number of states than deterministic finite automata recognizing the same language.

85. Quantum finite automata with mixed states are no more super-exponentially more concise than deterministic finite automata. It was not known whether the super-exponential advantage of quantum automata is really achievable.79 31-Jan-11 Quantum Automata and Languages

86. Quantum Push-down Automaton (QPDA) 80 A quantum push-down automaton (QPDA) is a real-time quantum automaton where H is the tensor product of a finite-dimensional space Q, which is called the control state, an infinite-dimensional stack space Σ, It is also required that sinit is infinite-dimensional and superposition of a finite number of different initial control and stack states. 31-Jan-11 Quantum Automata and Languages

87. Formal definition of QPDA A quantum pushdown automaton (QPDA) is a 7-tuple A = (Q,∑, T, q0,Qa,Qr, δ) where Q is a finite set of states ∑ is a finite input alphabet T is a stack alphabet. q0 ∈ Q an initial state. Qa ⊂ Q, Qr ⊂ Q of accepting and rejecting states respectively, with Qa∩Qr= ∅ δ : Q × Γ × ∆ × Q × {↓,->} × ∆∗ C[0,1], where Γ = ∑ ∪ {#, $} is the input tape alphabet of A and #, $ are end-markers not in ∑, ∆= T ∪ {Z0} is the working stack alphabet of A . Z0is the stack base symbol {↓,->} is the set of directions of input tape head. 81 31-Jan-11 Quantum Automata and Languages

88. Quantum Push-down Automaton (QPDA) 82 Let q1, q2 ∈ Q are control states and σ1, σ2 ∈ T∗ are stack states. The transition amplitude ‹(q1,σ1)| Ua |(q2, σ2)› can be nonzero only if tσ1 = σ2, σ1 = tσ2, or σ1 = σ2 for some t ∈ T. So, transitions can only push or pop single symbols on or off the stack or leave the stack unchanged. For acceptance the QPDA end in both an accepting control state and with an empty stack. i.e. Haccept = Qaccept ⊗ {∈} for some subspace Qaccept⊂ Q. 31-Jan-11 Quantum Automata and Languages

89. Example of QPDA 83 Figure 5: Quantum Pushdown Automata 31-Jan-11 Quantum Automata and Languages

90. Language Accepted by QPDA Every regular language is recognizable by some QPDA. Can also recognize some languages that are not recognizable by QFA. Languages accepted by QPDA are called Quantum Context free languages(QCFL). 84 31-Jan-11 Quantum Automata and Languages

91. Quantum grammars A quantum grammar G consists of two alphabets V and T , the variables and terminals, an initial variable I ∈ V , and a finite set P of productions α -> β, where α ∈ V∗and β ∈ (V ∪ T )∗. Each production in P has a set of complex amplitudes ck(α -> β) for 1 ≤ k ≤ n, where n is the dimensionality of the grammar. 85 31-Jan-11 Quantum Automata and Languages

92. Quantum context-free grammars A quantum grammar is context-free if only productions where α is a single variable v have nonzero amplitudes. A quantum context-free language (QCFL) is one generated by some quantum context-free grammar. A quantum language is context-free if and only if it is recognized by a generalized QPDA. 86 31-Jan-11 Quantum Automata and Languages

93. Quantum context-free grammars Two quantum grammars G1 and G2 are equivalent if they generate the same quantum language, f1(w) = f2(w) for all w. A quantum context-free grammar is in Greibach normal form if only productions of the form v -> aγ where a ∈ T and γ ∈ V∗can have nonzero amplitudes, i.e. every product β consists of a terminal followed by a (possibly empty) string of variables. 87 31-Jan-11 Quantum Automata and Languages

94. Closure properties of QCFLs 88 Lemma 1: If f is a QCFL and g is a QRL, then fg is a QCFL. Proof: We simply form the tensor product of the two automata. If f and g have finite-dimensional state spaces Q and R, construct a new QPDA with control states Q⊗R, transition matrices U′a = Ufa ⊗Uga and accepting subspace H′ accept = Qaccept ⊗ Raccept ⊗ {∈}. 31-Jan-11 Quantum Automata and Languages

95. Closure properties of QCFLs Lemma 2: If f and g are QCFLs, then f + g is a QCFL. Proof: Suppose the grammars generating f and g have m and n dimensions, variables V and W, and initial variables I and J. We will denote their amplitudes by cfk and cgk. Then create a new grammar with m+ n dimensions, variables V ∪ W ∪ {K}, and initial variable K, with the productions K -> I and K ->J allowed with amplitudes ck = 1. Other productions are allowed with ck = cfk for 1 ≤ k ≤ m and ck = cgk−m for m + 1 ≤ k ≤ m + n. This grammar generates f + g. 89 31-Jan-11 Quantum Automata and Languages

96. Sequential Quantum Machines (SQM) A SQM is a 5-tuple M=(S, s0, I, O, ∂), where S is a finite set of internal states, s0∈S is the start state, I and O are finite input and output alphabets, respectively, and ∂ : I× S × O × S  C is a transition amplitude function, satisfying ∑y, t∂(x,s,y,t) ∂(x,s',y,t)* = ∂s,s' for every x∈I; s,s‘ ∈ S. The symbol * stands for complex conjugation and ∂( x, s, y, t) is interpreted as the transition amplitude that SQM M prints y and enters state t after scanning x in the current state s. 90 31-Jan-11 Quantum Automata and Languages

97. Sequential Quantum Machines Sequential quantum machines (SQMs)was considered by Gudder (2000). Factorizable and strongly factorizable SQMs were also proposed. 91 31-Jan-11 Quantum Automata and Languages

98. Factorizable SQMs An SQM M = (S, s0, I,O, ) is factorizable if there exist some functions ∂1 : I × S × O -> C and ∂2 : I × S × S -> C such that for any (x, s, y, t) ∈ I × S × O × S, ∂(x, s, y, t) = ∂1 (x, s, y) ∂2(x, s, t). 92 31-Jan-11 Quantum Automata and Languages

99. Strongly Factorizable SQMs An SQM M is strongly factorizable if ∂(x, s, y, t) = ∂1 (x, s, y) ∂2(x, s, t). ∑y | ∂1 (x, s, y) |2 = 1, ∑t ∂2(x, s, t) ∂2(x, s', t)∗ = ∂s,s‘ for every x ∈ I , and any s, s'∈ S. 93 31-Jan-11 Quantum Automata and Languages

100. Quantum Sequential Machines (QSM) A QSM is 5-tuple M=(S, ηi0 , I, O, {A(y | x) : y ∈ O, x ∈ I}), where S={s1, s2,……., sn}is a finite set of internal states; ηi0 =(0…1…0)T is a degenerate stochastic column vector of n dimension, that is, the i0th entry is 1; I and O are input and output alphabets, respectively; A(y|x) is an n × n matrix satisfying ∑y∈OA(y|x)A(y|x)T=I for any x ∈ I, where the symbol T denotes Hermitian conjugate operation and I is unit matrix.

101. Stochastic Sequential Machines (SSM) A SSM is a 4 tupleM= (S, I,O, {A(y|x)}) where S, I and O are finite sets (the internal states, inputs, and outputs,respectively), and {A(y|x)} is a finite set containing |I| × |O| square matrices of order |S| such that aij (y|x)≥0 for all i and j , and ∑y∈O∑|S|j=1 aij (y|x) = 1, where A(y|x) = [aij (y|x)], and |I |, |O|, and |S| mean the cardinality of set I , O, and S, respectively.

102. Decidability and Undecidability of Quantum Automata. 96 A language is said to be decidable if there exists a quantum automaton that halts on all the input words of that language. A language is said to be undecidableifthere exists no algorithm by which any quantum automaton fails to halt on some input words of that languages. 31-Jan-11 Quantum Automata and Languages

103. Example of Undecidable Problem About Quantum Automata For a quantum automaton A, ValA(w) is the probability that on any given run of A on the input word w, w is accepted by A. The languages recognized by the automata A with non-strict threshold λ are L≥ = {w : ValA(w) ≥ λ} There is no algorithm that can decide for a given automaton A whether if L≥ is empty. 97 31-Jan-11 Quantum Automata and Languages

104. Conclusion A quantum finite automaton is a theoretical model for a quantum computer with a finite memory. QFA can recognize all regular languages if arbitrary intermediate measurements are allowed. Quantum automata can recognize several languages not recognizable by the corresponding classical model. 1QFAC can accept some languages with essentially less number of states than DFA, but no MO-1QFA or MM-1QFA or multi-letter 1QFA can accept these languages. 2QFA is more powerful than 1QFA. QPDA can accept all regular languages and some non-regular languages. 98 31-Jan-11 Quantum Automata and Languages

105. References Cristopher Moore and James P. Crutchfield “Quantum Automata and Quantum Grammars” (1997) [4-17]. AndrisAmbainisandArnolds Kikusts “Quantum Finite Automata” (2000) [1-5]. Alex Brodsky and Nicholas Pippenger “Characterizations of 1-Way Quantum Finite Automata” (2008) [1-4]. MaratsGolovkins “Quantum Pushdown Automata” (2001) [1-9]. AndrisAmbainis and John Watrous “Two-way finite automata with quantum and classical states” (2008) [1-3]. DaowenQiu “Characterization of Sequential Quantum Machines” (2001) [1-4]. Vincent D. Blondel, Emmanuel Jeandel, Pascal Koiran, NatachaPortier“Decidable And Undecidable Problems About Quantum Automata” (2003) [1-2]. 99 31-Jan-11 Quantum Automata and Languages

106. THANK YOU 100 31-Jan-11 Quantum Automata and Languages