Seminar on quantum automata (complete)

Seminar on
QUANTUM AUTOMATA and
LANGUAGES

PRESENTED BY:
Abhijit Doley. Ranjan Phukan. Rekhamoni Morang.

SEMESTER: 8th.

DEPARTMENT OF INFORMATION TECHNOLOGY.

1 7-Mar-12

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) and its types.
 Two-way quantum finite automata (2QFA) and its types.
 1.5-way Quantum Finite Automata.
 Quantum Push-down Automaton (QPDA) and Quantum context-free grammars.
 One-way General Quantum Finite Automata (1gQFA).
 Quantum One-Counter Automata (Q1CA).
 Sequential Quantum Machine (SQM) & Quantum Sequential Machines (QSM).
 Two-way Quantum Finite Automaton With Reset.
 Minimization of a quantum automaton: The transducer.
 Decidability and Undecidability of Quantum Automata.
 Conclusion.
 Future Work.

2 Quantum Automata and Languages 7-Mar-12

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.


Introduction
 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.
4  context-free languages. and Languages
Quantum Automata 7-Mar-12

Introduction
 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.


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.


Classical Computational Unit (Bits)
 A building block of classical computational
devices is a two-state system.
0 and 1
 Indeed, any system with a finite set of
discrete, stable states, with controlled
transitions between them will do.


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.


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


Qubits (contd…)

Figure 1: Qubit System

Brief Introduction to
Classical Automata


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.


Finite Automata
• Collection of three things:
 A finite set of states
 One of them is the start state and
 Some (or none) are final states.
 An alphabet set (∑) containing symbols to
construct input strings .
 A finite set of transitions denoting the states
it goes next on accepting each letter.
• Languages accepted by FA are called regular
languages.


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.


Deterministic Finite Automata(DFA)

Figure 2: Deterministic Finite Automata


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.


Non-deterministic Finite Automata(NFA)

Figure 3: Non-deterministic Finite Automata


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 .


Transition Matrix (Example)


Probabilistic Automata (PA)

 We obtain probabilistic automata if we
allow fractional values in transition matrix.
 Probabilistic Automata accepts regular
language.
 Example:


Probabilistic Automata
 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.


Fundamental properties of
Probabilistic Automata
 Each columns adds up to 1.
 If the rows of all Xa contain exactly
one 1 we obtain the model of
deterministic finite automata.


Language Accepted by Probabilistic
Automata

 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


Stochastic Languages
 The set of languages recognized by probabilistic
automata are called stochastic languages.
 Let Qaccept be the set of "accepting" or "final" states of
the automaton.
 It has a 1 at the places corresponding to elements in
Qaccept, and a 0 otherwise.
 The language recognized by a specific automaton is then
defined as

• Σ * is the set of all strings in the alphabet Σ.
• The language depends on the value of the cut-point η, normally taken to be in
the range 0≤ η <1.


Stochastic Languages
 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∗,


Properties of Stochastic Languages
 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


Quantum Automata (QA)

 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


Definition of Quantum Automata
 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 Paccept that
projects onto it,
 an input alphabet A, and
 a unitary transition matrix Ua for each symbol a ∈ A.


Quantum Language
 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 Uwi for
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.


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 .
 | ∑ qj єF αj |2 gives the probability that the
automaton accepts the string w.

Different Classes Of Quantum Automata
 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.


Quantum finite-state automata
 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.


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=∅).
 Qacc and Qrej, are called halting states.
 Qnon=Q−(Qacc∪Qrej) are called non-halting states.


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.


Computation
 The computation of a QFA starts in the superposition
|q›.
 Then transformations corresponding to the left
endmarker κ, the letters of the input word x and the
right endmarker $ are applied.
 The transformation corresponding to a∈Γ consists of
two steps.
 First, Va is 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; Enon

›
where Eacc=span{|q :q∈Qacc}, Erej=span{|q›:
q∈Qrej}, Enon=span{|q›: q∈Qnon}.

Computation
 If the system‟s state before the measurement was

then the measurement,
 accepts Ψ' with probability pa=∑αi2 ,
 rejects with probability pr= ∑βj2
 continues the computation with probability pc= ∑γk2
i.e. applies transformations corresponding to next
letters.


Recognition of languages and QRL
 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.


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.


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.


One-way quantum finite automata (1QFA)
 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.


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


One-way quantum finite automata
 The states in Qacc and Qrej are 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.


Example (1QFA)
 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 Vx for all letters x ∈ Γ.

 Both methods are equivalent: all Vx are determined
by δ.


Example (contd…)
 Defining by Vx :

 Defining by δ :
δ(q0, a, q0) =½
δ(q0, a, q1) =½
δ (q0, a, qacc) = 0
δ (q0, a, qrej) =1/√2


Example (contd…)
Working steps of the automaton:
 ›
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.


Example (contd…)
 Probability of accepting and rejecting:
 The total probability of accepting is ¼.
 The total probability of rejecting is ½ + ¼ = ¾.


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.


Advantages & Disadvantages of 1QFA
 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.

Types of 1QFA

 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

Measure Once 1-way QFA
 Introduced by Moore and Crutchfield.
 It is a 5-tuple (Q, , , q0, Qacc) where Qacc is the set
of accepting states.
 The transition function is defined as
:Qx xQ C[0,1]
that represents the probability that flows from state q
to state q′ upon reading symbol σ є ∑.
 Measurement is performed after the whole input
string is read.
 The language accepted by MO-1QFA is regular
language.

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
:Qx xQ 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.


Operation of MM 1QFA
 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.

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.


Comparison of MO-1QFA and MM-1QFA

MO-1QFA MM-1QFA

 Initiated by Moore and  Initiated by Kondacs and
Crutchfield. Watrous.
 There is only one  Measurement is
measurement for performed after reading
computing each input each symbol, instead of
string, performing after only the last symbol.
reading the last symbol.  Three results:
 Two results: acceptance acceptance, rejection and
and rejection. continuation.


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.

Definition of k-letter 1QFA
 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.


Equivalence of Multi-letter 1QFA
 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.


Language accepted by Multi-letter 1QFA
 Can accept some regular languages not acceptable
by MO-1QFA and MM-1QFA.
 Accept a proper subset of regular languages.


Hierarchy of multi-letter QFAs and some
relations
 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.


One-way Quantum Automata with Control
Language (CL-1QFAs)
 Computation is performed after each input
symbol is read.
 An observable O is considered with a fixed, but
arbitrary, set of possible results C = c1,…, cn.
 On any given input word x, the computation
displays a sequence y C* of results of O with a
certain probability p(y|x).
 The computation is accepted if and only if y
belongs to a fixed regular control language L
C*.


One-way quantum finite automata together
with classical states (1QFAC)
 1QFA accepts only subsets of regular languages with
bounded error.
 1QFAC is the combination of the concepts of both
quantum and classical finite automata.
 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.

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).

Computation in 1QFAC
 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.

Diagrammatic Representation

Figure 4: 1QFAC dynamics as an acceptor of language


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.

Equivalence of 1QFAC
 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.


Determining the equivalence for 1QFA
 Two QFA are equivalent if for any input string
x, the two automata accept x with equal
probability.
 Two QFA are n-equivalent if and only if the
acceptance probabilities of the two QFAs are
equal for the input strings of length not more than
n.


Two-way quantum finite automata (2QFA)
 2-way QFA is a 6-tuple M = (Q,∑, δ, q0,Qacc,Qrej)
where
 ∑ 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


Two-way quantum finite automata
 The states in Qacc and Qrej are 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.


Language Accepted by 2-way QFA

 Can accept all regular languages with certainty.
 Also accepts some non-regular languages within
linear time.


Disadvantage of 2QFA
 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).


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.


Two-way finite automata with quantum and
classical states (2QCFA)
 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.


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.


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 $.


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.


Example
 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 Lpal in 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.


1.5-way Quantum Finite Automata
 An intermediate form of QFA.
 Developed by Amano and Iwama.
 Tape heads are allowed to move right or to be
stationary.


Improved Constructions Of Mixed State
Quantum Automata
• Quantum finite automata with mixed states are proved to
be super-exponentially more concise rather than
quantum finite automata with pure states.
• 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.
• 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.

Quantum Push-down Automaton (QPDA)
 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.


Formal definition of QPDA
 A quantum pushdown automaton (QPDA) is a 7-tuple A =
(Q,∑, T, q0,Qa,Qr, δ) where
 ∑ 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 .
 Z0 is the stack base symbol
 {↓,→} is the set of directions of input tape head.


Quantum Push-down Automaton (QPDA)
 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.


Example of QPDA

Figure 5: Quantum Pushdown Automata

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).


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.


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.


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.


Closure properties of QCFLs
 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 ⊗
{∈}.


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.
91
 This grammar generatesAutomata and Languages 7-Mar-12
Quantum
f + g.

Quantum Pushdown Automaton with a
Classical Stack (QCPA)
 A Quantum Pushdown Automaton with a Classical Stack
(QCPA) has two state controls,
 one is a quantum state control for moving its tape head,
 and the other is a classical state control for dealing with the
stack.
 It has an input tape to which a quantum head is attached
and a classical stack to which a classical stack top
pointer is attached.
 The classical state control reads the stack top symbol
and the result of an observation of the quantum part.
 The quantum state control reads the stack top symbol
pointed by the classical stack top pointer, and the input
symbol pointed by the quantum head.
 The outputs are determined based on a state of the
classical state control.

Quantum Pushdown Automaton with a
Classical Stack (QCPA) (contd…)
 Transition Function: (q, a, b, q',D) = means
that the amplitude of the transition from q to q'
moving its head to D (1 means right and 0
means stay) and reading an input symbol a and
a stack symbol b is .
 Language Accepted: QCPAs can recognize
every deterministic context-free language and
some non-context-free languages.


One-way General Quantum
Finite Automata (1gQFA)
 Generalized version of 1QFA.
 The unitarity puts limit on the computational power of
quantum finite automata (QFA).
 In 1gQFA each symbol in the input alphabet induces
a trace-preserving quantum operation.
 It is of two types:
 measure-once one-way general quantum finite automata
(MO-1gQFA).
 measure-many one-way general quantum finite automata
(MM-1gQFA).


Measure-once One-way General
Quantum Finite Automata (MO-1gQFA)
 Generalized version of MO-1QFA.
 Can simulate any probabilistic
automaton.
 Thus it can recognize any regular
language.
 Cannot accept non-regular languages.
 Studied from three aspects:
the closure property,

 the computational power,
 the equivalence problem.

Closure properties of MO-1gQFA
 If f is a function induced by an MO-1gQFA, then (1-f)
is also induced by an MO-1gQFA.

 If f1, f2,…,fk are functions induced by MO-
1gQFA, then ∑ikcifi is also induced by an MO-1gQFA
for any real constants ci > 0 such that ∑ikci=1.

 If f1, f2,…,fk are functions induced by MO-
1gQFA, then f1f2…..fk is also induced by an MO-
1gQFA.

The computational power of MO-1gQFA
 Theorem 1: The languages recognized by
MO-1gQFA with bounded error are regular.

 Theorem 2: MO-1gQFA recognize all
regular languages with certainty.


The equivalence problem of MO-1gQFA
 Two MO-1gQFA M1 and M2 on the same
input alphabet ∑ are equivalent if and only
if they are (n1 + n2)2-equivalent.
 where ni = dimHi for i = 1,2.
 Hi is the finite-dimensional Hilbert space.


Measure-many One-way General
Quantum Finite Automata (MM-1gQFA)
 Generalized version of MM-1QFA.
 Studied from two aspects:
 The computational power,
 The equivalence problem.


The computational power of MM-1gQFA
 Theorem : The languages recognized by MM-
1gQFA with bounded error are exactly regular
languages.

 Thus, MM-1gQFA and MO-1gQFA have the
same computational power.
 unlike MO-1QFA and MM-1QFA.


The equivalence problem of MM-1gQFA
 Two MM-1gQFA M1 and M2 on the same
input alphabet ∑ are equivalent if and only
if they are (n1 + n2)2-equivalent.
 where ni = dimHi for i = 1,2.
 Hi is the finite-dimensional Hilbert space.


Quantum One-Counter Automata
 In 1QFA and 2QFA, as the tape head is
allowed in quantum superposition,
logarithmically many qubits are necessary
to store the position of the tape head.
 Due to this problem, quantum one-counter
automata was developed that has a
counter which can store arbitrarily large
integer value and can track the position of
the tape head.


Functioning of Quantum Finite One-
counter Automata
 There is a counter that contains an arbitrary large
integer value.
 It is 0 at the beginning of computation.
 ←, ↓, → respectively, decreases by one, retains the
same and increases by one the value of the counter.
 The automaton reads a letter of the word written on
the tape and checks the value of the counter.
 According to the transition function, it changes to a
new state and updates the value of the counter.


Types of Quantum One-Counter Automata
 It is of two types:
 One-way Quantum One-Counter Automata (1Q1CA),
where tape head can move towards one direction.
 Two-way Quantum One-Counter Automata (2Q1CA),
where tape head can move towards both directions.
 1Q1CAs can recognize several non-context-free
languages while there are some regular languages
that cannot be recognized by 1Q1CAs.
 2Q1CA can recognize some non-context-free
languages in addition with context-free languages.


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.

Sequential Quantum Machines
 Sequential quantum machines (SQMs)was
considered by Gudder (2000).
 Two types of SQMs:
 Factorizable SQMs and
 Strongly factorizable SQMs


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).


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.


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.

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.


Equivalence of QSM and SQM
 Theorem 1: For any SQM M there is a
corresponding QSM M′ with the same input
and output alphabets, such that M and M′ are
equivalent, and vice versa.
 Theorem 2: Two machines (SQMs or QSMs)
M1 and M2 with n1 and n2
states, respectively, and the same input and
output alphabets, are equivalent iff they are
(n1 + n2)2-equivalent.


Two-way Quantum Finite Automaton With
Reset
 Introduced due to the weakness of QFA in
language recognition power.
 It is an enhancement to the 2QFA.
 It has capability of resetting the position of
the tape head to the left end of the tape in
a single move during the computation.


2-way Quantum Finite Automaton With
Reset
 The configurations of a 2QFA with reset are pairs of
the form (state, head position).
 Initially, the head is on the left end-marker and the
machine starts computation in the superposition
|q0, 0>.
 Transition function is given by:
δ(q, σ,q′, dq‟) = <q′ | Uσ|q> where
o the machine currently in state q and scanning symbol σ
will jump to state q‟ and move the head in direction dq‟.
o Uσ is the unitary operator.


Language Accepted
 A 2-way Quantum Finite Automaton With Reset M is
said to recognize a language L with error bounded
by ε if M‟s computation results in “accept” being
measured for all members of L with probability at
least 1− ε, and “reject” being measured for all other
inputs with probability at least 1 − ε .


Other Variants Of Two-way Automata With
Reset
 A two-way quantum finite automaton with
restart is a restricted 2QFA with reset, in
which the reset moves can target only the
original start state of the machine.
 A one-way quantum finite automaton with
reset is a restricted 2QFA with reset which
uses neither “move one square to the left” nor
“stay put” transitions, and whose tape head is
classical.
 A one-way quantum finite automaton with
restart is a variant where the reset moves
can target only the original start state.

Minimization of a Quantum Automaton:
The Transducer
 Quantum transducer can be used for control, identification and
simulation of quantum systems.
 Quantum transducer is formed by two parts:
 The quantum memory composed by a finite number of two-
level quantum particles, where the information is encoded in
the form of qubits.
 The classical device composed by:
 a tape, where the input symbols, the program and the
output symbols are written;
 the tape head that scans the input symbols;
 a finite set of quantum gates corresponding to a finite set
of input symbols and
 a measurement device.


Functioning of Transducer
 The quantum particles are prepared in an initial
quantum state.
 Then the tape head reads each input symbol and
apply the corresponding gate to the quantum part.
 After all the letters of the input string are read, the
QA is observed.
 Finally, the output of the quantum measurement is
written in the tape.
 It is similar to the measure-once quantum finite
automata.


Functioning of Transducer
 The flexibility of the model resides in the
quantum memory and in the classical program.
 The set of gates and the measurement device
are always the same as well as the number of
qubits.
 The string of input symbols can be changed
freely by an exterior user, without changing the
QA.
 Running each string of input symbols in a QA is
equivalent to simulate a quantum circuit.


Advantage of the Transducer
 Transducer is able to simulate a great
variety of quantum circuits with the same
finite set of quantum gates and the same
set of quantum particles.
 Therefore, it is possible to minimize the
number of qubits in a given quantum circuit
and thus minimizing the quantum
automata.


Decidability and Undecidability of Quantum
Automata.
 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 undecidable if there
exists no algorithm by which any quantum
automaton fails to halt on some input words of
that languages.


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.


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.
 We have two major types of QFA: 1QFA and 2QFA.
 2QFA is more powerful than 1QFA.
 QPDA can accept all regular languages and some
non-regular languages.
 Various new models of quantum automata are
developed namely 1gQFA, Q1CA, SQM,QSM, 2QFA
122 with reset, Transducer etc. and Languages 7-Mar-12
Quantum Automata

Future Work
 There are many problems related to this topic which
are still unsolved.
 When we remove the real-time restriction, allowing
the machine to choose when to read an input symbol
in classical DFAs and PDAs, it adds no power. But
its affect on quantum automata is still uncovered.
 It is future work to solve whether each QRL is
recognized by a unique QFA with the minimal
number of dimensions or not.
 It is also a future work to study on the quantum
pushdown automata whose tape head is
implemented as a classical part as well as the stack.


References
 Cristopher Moore and James P. Crutchfield “Quantum
Automata and Quantum Grammars” [4-17] (1997).
 Andris Ambainis and Arnolds Kikusts “Quantum Finite
Automata” [1-5] (2000).
 Alex Brodsky and Nicholas Pippenger “Characterizations
of 1-Way Quantum Finite Automata” [1-4](2008).
 Marats Golovkins “Quantum Pushdown Automata” [1-9]
(2001) .
 Andris Ambainis and John Watrous “Two-way finite
automata with quantum and classical states” (2008) [1-3].
 Daowen Qiu “Characterization of Sequential Quantum
Machines” [1-4](2001).
 Vincent D. Blondel, Emmanuel Jeandel, Pascal
Koiran, Natacha Portier “Decidable And Undecidable
Problems About Quantum Automata” [1-2] (2003) .

References
 Maksim Kravtsev “Quantum Finite One-Counter
Automata” , [1-6], (1999).
 Lvzhou Li, Daowen Qiu, Xiangfu Zou, Lvjun Li, Lihua
Wu “Characterizations of one-way general quantum
finite automata” , [1-28], (2009).
 Abuzer Yakaryilmaz and A.C. Cem Say
“Succinctness of two-way probabilistic and quantum
finite automata”, [1-5], (2009).
 Lvzhou Li, Daowen Qiu “Determining the
equivalence for 1-way quantum finite automata” ,[1-
14], (2007).
 A. M. Martins “Minimization of a quantum automaton:
The transducer”, [1-2], (2008).

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