Seminar on quantum automata (complete)

Apr 7, 20231

Seminar onQUANTUM AUTOMATA and

LANGUAGES

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

SEMESTER: 8th.

DEPARTMENT OF INFORMATION TECHNOLOGY.

Apr 7, 2023Quantum Automata and Languages2

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.

Contents


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. context-free languages.


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…) 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.


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 functionfQ(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 automataA 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=∑αi

2 ,

rejects with probability pr= ∑βj2

continues the computation with probability pc= ∑γk

2 i.e. applies transformations corresponding to next letters.


Recognition of languages and QRLWe 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) =½δ(q0, a, q1) =½δ (q0, a, qacc) = 0δ (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

: Q x x Q 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 : 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.


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 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.


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 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


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 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 . 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 = Uf

a ⊗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 cg

k. 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.


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 QuantumFinite 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> whereo 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 with reset, Transducer etc.


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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