Another approach to the equivalence of measure-many one-way quantum finite automata and its application

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by  Tianrong Lin

Another approach to the equivalence of
measure-many one-way quantum finite
✩
automata and its application
Tianrong Lina,1,∗
2
1 aDepartment of Computer Science, Fujian Normal University,Fuzhou 350007, China
0
2 In memory of my grandparents

n
a
J

6
Abstract

]
C
In this paper, we present a much simpler, direct and elegant approach to the
C
. equivalence problem of measure many one-way quantum finite automata (MM-
s
c
1QFAs). The approach is essentially generalized from the work of Carlyle
[

[J. Math. Anal. Appl. 7 (1963) 167-175]. Namely, we reduce the equivalence
0
1
problem of MM-1QFAs to that of two (initial) vectors.
v
1 As an application of the approach, we utilize it to address the equivalence
8
4 problem of Enhanced one-way quantum finite automata (E-1QFAs) introduced
2
. byNayak[Proceedingsofthe40thAnnualIEEESymposiumonFoundationsof
6
0 Computer Science, 1999,pp. 369-376]. We provethat two E-1QFAsA andA
1 2
1
1 over Σ are equivalence if and only if they are n21+n22−1-equivalent where n1
:
v and n are the numbers of states in A and A , respectively.
2 1 2
i
X
Keywords: quantum finite automata, measure-many one-way quantum finite
r
a automata, enhanced one-way quantum finite automata, equivalence
✩ Most part of this paper was done while the author was a graduate student at Fujian
NormalUniversity
∗Correspondingauthor. Tel.:[phone]
Email address: [email protected] (TianrongLin)
1Present address: Dept. of Computer Science and Communication Engineering for GM
InformationCollege,FujianUniversityofTechnology, Fuzhou350014, China
January 9, 2012
1. Introduction
The theory of quantum computing is unquestionably one of the hottest and
frontresearchfieldsinthetheoryofcomputing[1–3]. Thereexistafewworksde-
veloped quantum computation model, such as quantum Turing machines [5, 6],
Quantumcircuits[7,8],andthequantumgeneralizationsoffiniteautomata,i.e.,
Quantum finite automata (QFAs) [9–16, 22]. In particular, the study of QFAs
provides a good insight into the nature of quantum computation, since QFAs
canbe viewedas the simplest theoreticalmodel basedon quantum mechanism.
Theso-calledmeasure-manyone-wayquantumfiniteautomata(MM-1QFAs),
introducedin[10],isakindofQFAmodelwhosetapeheadissubjectedtomov-
ingonecelltotherightateachcomputationstep,andmeasurementisperformed
after every computation step. There exist a few works dealt with the language
recognized ability of MM-1QFAs, such as [10, 11, 14, 17–21]. Incidentally, the
so-called enhanced one-way quantum finite automata (E-1QFAs) introduced by
Nayak [22] can be viewed as a generalization of MM-1QFAs.
Just as the equivalence problem of the classical finite automata [23–25, 34,
35], the concept of “equivalence” gives us a classificationof the elements of the
set of MM-1QFAs over the same alphabet. On the equivalence issue of MM-
1QFAs, Li and Qiu [26] have shown, with the help of the so-called 1qfa with
control language [11], that two MM-1QFAs A and A over the same alphabet
1 2
are equivalent if and only if they are 3n2 +3n2 −1-equivalent where n and
1 2 1
n are the numbers of states in A and A , respectively, and factor 3 is the
2 1 2
numbers of states in the minimal DFA [23–25] recognized the regular language
g∗a{a,g,r}∗. Incidentally, there exist some works dealt with the equivalence
issue with respect to other quantum finite automata [27–30]. However, the
equivalence problem of E-1QFAs is still open thus far. A more comprehensive
survey on this subject is [31] by Gruska.
We note that the method to the equivalence problem of MM-1QFAs, at-
tributed to Li and Qiu [26], is roundabout and somewhat complicated. There-
fore, the first aim of this paper is to present a much simpler, direct and elegant
2
approach to the equivalence problem of MM-1QFAs. We summarize our moti-
vations as follows. (1) As we know, the mathematical method is the essence of
mathematics. The mathematician usually investigates the same problem with
different mathematical methods and different concepts to fully understand it.
This method can be followed; (2) It is an interesting work of its own to find
a more general method to address the equivalence problem for MM-1QFAs;
(3) We want to know whether the upper-bound 3n2+3n2−1 can be further
1 2
improved. Such considerations lead us to transform the word function of MM-
1QFAs defined in a “cumulation” manner (described in the sequel) to another
versionwhichisina“non-cumulation”manner. Then,weimprovetheprevious
upper-bound to n2+n2−1 by showing the following
1 2
Theorem 1. Let A = (Q ,{U (σ)} ,|π i,O ), i = 1,2, be two MM-
i i i σ∈Σ∪{$} i i
1QFAs over Σ. Then A and A are equivalent iff they are (n2 + n2 − 1)-
1 2 1 2
equivalent,wheren andn arethenumbersofstatesinA andA ,respectively.
1 2 1 2
Asmentionedearlier,theE-1QFAmodel[22]canbeseenasafinitememory
version of the mixed state MM-1QFA. Thus, the approach to the equivalence
problem of MM-1QFAs also can be applied to that of E-1QFAs. Therefore, as
oursecondaim, we utilize the aboveapproachto solve the equivalence problem
of E-1QFAs, which remains open so far, by showing the following
Theorem 2. Let A = (Q ,Q ,Q ,{U(i)} ,ρ ,O ),i = 1,2, be
i i acc,i rej,i σ σ∈Σ∪{#,$} i i
two E-1QFAs over Σ. Then A and A are equivalent iff they are (n2+n2−1)-
1 2 1 2
equivalentwheren andn arethenumbersofstatesin A andA ,respectively.
1 2 1 2
The remainder of the paper is organized in the following way: Section 2 is
the preliminary part where basic concepts and notations used in the sequel are
reviewed. Section 3 and Section 4 are devoted to the proofs of Theorem 1 and
Theorem 2, respectively. Section 5 is the concluding section.
3
2. Preliminaries
For convenience, we briefly review some basic notions needed in the sequel.
To a more exhaustive illustration about linear algebra, we refer to [32]. Also,
we refer to [1–3] for a through treatment on the quantum theory.
2.1. Some notation on Linear algebra
a ··· a
11 1n
LetCdenotethefieldofcomplexnumber,M acomplexmatrix,i.e., ··· ··· ··· 
 
 am1 ··· amn 
with aij ∈ C for all 1 ≤ i ≤ m and 1 ≤ j ≤ n. Some times, we use (aij)m×n 
to denote M. In particular, 1×n (resp. n×1) complex matrices are called n
dimensional row vectors (resp. column vectors). If m = n, then M is called a
complex square matrix of order n (or m), and sometimes M is called a n-order
(or m-order) complex matrix. Let M = (a ) be a m ×n complex ma-
ij m×n
trix, then the transpose of M is denoted as M′, i.e., M′ = (a ) , and the
ji n×m
conjugate-transposeofM isdenotedasM†. Inthispaper,thesetofalln-order
complex matrices will be denoted as M (C). For any H ∈ M (C), H is said
n n
to be Hermitian if H† = H, and is said to be Unitary if H†H = HH† = I
n
whereI denotesthen-orderidentitymatrix. SupposethatAandB aremand
n
n-ordercomplex matrix,respectively,we define the “diagonal sum” ofA andB
to be
A 0
A⊕B , .
 
0 B
 
Therefore, A⊕B is a (m+n)-order complex matrix.
Let A = (a ) be an n×n matrix over C, let Tr(A) denote the trace of A,
ij
n
i.e., Tr(A)= a . It is well known that
ii
i=1
P
Tr(AB)=Tr(BA), and Tr(λ A+λ B)=λ Tr(A)+λ Tr(B)
1 2 1 2
where λ ∈C.
i
4
Let V be a finite dimensional vector space over C, and B ={η ,η ,···,η }
1 2 n
a basis for V over C. This means that for any vector α ∈ V, it has a unique
expression as a linear combination
α = c η +c η +···+c η
1 1 2 2 n n
where c ∈ C. The dimension of V, denoted by dimV, is defined to be the
i
cardinal number of B. Let span{B} denote the vector space generated by the
vectorsin B. Then, as a matter of fact, V =span{B}. Furthermore, M (C) is a
n
vector space over C with the dimension n2.
2.2. Some notation on Quantum mechanics
In quantum theory, for any isolated physical system, it is associated with a
(finite-dimensional) Hilbert space, denoted as H, which is called the state space
of the system. In Dirac notation, the row vector (resp. column vector) ϕ is
denoted as hϕ| (resp. |ϕi). Incidentally, hϕ| is the conjugate-transpose of |ϕi,
i.e., hϕ| = |ϕi†. The inner product of two vectors |ϕi and |ψi is denoted as
hϕ|ψi. The norm (or length) of the vector |ϕi, denoted by k|ϕik, is defined
as k|ϕik = hϕ|ϕi. A vector |ϕi (resp. hϕ|) is said to be unit if k|ϕik = 1
(resp. khϕ|kp=1).
SupposethatQ={q ,q ,···,q }isthebasicstatesetofaquantumsystem.
1 2 m
Then the corresponding Hilbert space is H = span{|q i|q ∈ Q,1 ≤ i ≤ m}
m i i
where |q i = (0,···,0,1,0,···,0)′ is a m dimensional column vector having
i
only 1 at the (i,1) entry, together with the inner product h·|·i, defined to be
hα|βi = m x∗y where λ∗ stands for the conjugate of λ for each complex
i=1 i i
number λP∈ C, |αi = (x1,x2,···,xm)′ and |βi = (y1,y2,···,ym)′ are two
vectors in H . At any time, the state of this system is a superposition of |q i,
m i
m
1 ≤ i ≤ m, and can be represented by a unit vector |φi = c |q i with
i=1 i i
ci ∈C such that mi=1|ci|2 =1. One can perform a measure oPn Hm to extract
some informationPabout the system. A measurement can be described by an
observable, i.e., a Hermitian matrix O = λ P + ··· + λ P where λ is its
1 1 s s i
eigenvalue and P is the projector onto the eigenspace corresponding to λ .
i i
5
Theabovemathematicaldescriptionsofquantumsystemarebasedon“pure
state”. We need some descriptions based on “mixed states”. In mixed states
picture, the states of quantum device are represented by density operator ρ ∈
L(H), i.e., ρ is self-adjoint, ρ ≥ 0 (semi-positive definite) and Tr(ρ) = 1. The
evolutionofa closedquantumsystemis characterizedby aunitary operationU
which maps ρ to UρU†. However, a general quantum operation U from L(H )
1
to L(H ) is a trace-preservingcompletely positive mapping [1–3] with the form
2
U(ρ) = M ρM† for any ρ ∈ L(H ), where {M } are Kraus operators of U
i i 1 i
i
satisfyinPg M†M =I . LetH=P ⊕P ⊕···⊕P be a decomposition.
i i i dimH1 1 2 k
Then, forPany ρ ∈ L(H), Tr(Pjρ) (equivalent to Tr(PjρPj†)) is the probability
that the property P is observed.
j
2.3. On relevant definitions of MM-1QFAs
ForanyfinitesetS,|S|denotesthecardinalityofS. Throughoutthispaper,
Σdenotesthenon-emptyfinitealphabet. Awordoverthe alphabetΣisafinite
sequence of symbols chosen from Σ. Let Σ∗ denote the set of all words over Σ.
For any word ω ∈Σ∗, |ω| denotes the length of ω. Let Σn denote the set of all
words of length n over Σ where n is a non-negative integer. Then Σ∗ can be
represented as Σ∗ =ǫ∪Σ∪Σ2∪··· where ǫ denotes the empty word.
ForafixedalphabetΣ,letM(x ),wherex ∈Σ,becomplexsquarematrices
i i
1
indexed by x . For convenience, we define the formal product M(x ) by
i i
i=n
Q
1
M(x ) , M(x )M(x )···M(x ).
i n n−1 1
i=n
Y
Now, we state the definition of MM-1QFA as follows.
Definition 1. Formally, an MM-1QFA with m states on the alphabet Σ is a
quadruple tuple
A = (Q,{U(σ)} ,|πi,O)
σ∈Σ∪{$}
where Q = {q ,q ,···,q } is the basic state set, |πi is the initial state vector
1 2 m
with k|πik = 1, $ ∈/ Σ is an end-mark, for each σ ∈Σ∪{$}, U(σ) ∈ M (C) is
m
6
an unitary matrix, and O is an observable with results in {a,r,g}, completely
described by the projectors P(a), P(r) and P(g).
The projectors P(a), P(g) and P(r) are given by
P(a)= |qihq|, P(g)= |qihq|, P(r)= |qihq|
q∈XQacc q∈XQnon q∈XQrej
where Q = Q\(Q ∪Q ) is the set of non-halting states, Q ⊆ Q and
non acc rej acc
Q ⊆Q (with Q ∩Q = ∅) are the sets of accepting states and rejecting
rej acc rej
states, respectively, and |qihq| denotes the matrix product of column vector |qi
and row vector hq|.
Fed with x x ···x $ where x x ···x ∈Σ∗, A computes as follows: start-
1 2 n 1 2 n
ing from |πi, U(x ) is applied and a measurement of O is performed reaching
1
a new current state. If the measurement result is ‘g’, then U(x ) is applied
2
and a new measurement of O is performed. This process continues as far as
measurements yields the result ‘g’. As far as the result of measurement is ‘a’,
the computation stops and the word is accepted. If the measurement result is
‘r’, then the computation stops and the word is rejected. Therefore, A induces
a word function p :Σ∗$→[0,1] in a “cumulation” manner, i.e.,
A
n+1 1 2
p (x x ···x $) = P(a)U(x ) P(g)U(x ) |πi (1)
A 1 2 n k i
(cid:13) ! (cid:13)
kX=1(cid:13)(cid:13) i=Yk−1(cid:0) (cid:1) (cid:13)(cid:13)
(cid:13) (cid:13)
1 (cid:13) (cid:13)
where x denotes $. By P(g)U(x ) we mean that
n+1 i
i=0
Q (cid:0) (cid:1)
1
P(g)U(x ) = I
i m
i=0
Y(cid:0) (cid:1)
i.e., the m-order (m = |Q|) identity matrix. Further, the probability of A
accepting the word x x ···x is defined as
1 2 n
P (x x ···x ) = p (x x ···x $). (2)
A 1 2 n A 1 2 n
Definition 2. Two MM-1QFAs A and A over Σ are said to be equivalent
1 2
(resp.t-equivalent)ifP (ω)=P (ω)forallω ∈Σ∗ (resp.forallω ∈Σ∗ with
A1 A2
|ω|≤t).
7
The probability P (ω) of A accepting the word ω given in terms of Eq. (2)
A
is somewhat complicated. Now, we define another “probability function” of A
‘accepting’ the word ω as follows.
P (x x ···x )−P (x x ···x ), ω =x x ···x ;
A 1 2 n A 1 2 n−1 1 2 n
F (ω) = (3)
A 
P (ǫ), ω =ǫ.
 A

Remark 1. Note that, if n=1 in Eq. (3), then x x ···x denotes the empty
1 2 0
word ǫ. More specifically, we define F (x) to be the value: P (x)−P (ǫ) for
A A A
any x∈Σ.
Forreadability,weintroducethe conceptof“β-equivalence”forMM-1QFAs
in terms of Eq. (3) as follows.
Definition 3. TwoMM-1QFAsA andA overthesameinputalphabetΣare
1 2
saidtobeβ-equivalent(resp.t-β-equivalent)ifF (ω)=F (ω)forallω ∈Σ∗
A1 A2
(resp. for all ω ∈Σ∗ with |ω|≤t).
ThefollowingTheoremisthebasisthatallowedustopresentamuchsimpler
approachto the equivalence problem of MM-1QFAs.
Theorem 3. Let A and A be two MM-1QFAs over Σ. Then A and A are
1 2 1 2
equivalent iff they are β-equivalent.
Proof. Weshowfirstthe“onlyif”part. AssumethatA andA areequivalent,
1 2
then we have
P (ω) = P (ω) (∀ω ∈Σ∗). (4)
A1 A2
We assert that F (ω) = F (ω) for all ω ∈ Σ∗. By Eq. (3) and Eq. (4), the
A1 A2
assertionis obvious when ω =ǫ; For the case when ω =x x ···x with n≥1,
1 2 n
by Eq. (4) we have
P (x ···x )−P (x ···x ) = P (x ···x )−P (x ···x )
A1 1 n A1 1 n−1 A2 1 n A2 1 n−1
i.e., F (x ···x )=F (x ···x ). Thus the assertion holds for all ω ∈Σ∗.
A1 1 n A2 1 n
8
We show next the “if” part of the Theorem. By hypothesis
F (ω) = F (ω) (∀ω ∈Σ∗) (5)
A1 A2
Also,itisclearthatP (ω)=P (ω)whenω =ǫ. Assumethatω =x x ···x
A1 A2 1 2 n
with n≥1. For simplicity, denote
a = P (x ···x )
n A1 1 n
and
b = P (x ···x )
n A2 1 n
for all n ≥ 1. Setting a = P (ǫ) and b = P (ǫ), then by Eq. (3), we find
0 A1 0 A2
that
F (x ···x )=a −a and F (x ···x )=b −b .
A1 1 n n n−1 A2 1 n n n−1
Thus,
n
P (x ···x ) = a + (a −a )
A1 1 n 0 k k−1
k=1
X
n
= F (ǫ)+ F (x ···x )
A1 A1 1 k
k=1
X
n
= F (ǫ)+ F (x ···x ) (by Eq. (5))
A2 A2 1 k
k=1
X
n
= b + (b −b )=P (x ···x ).
0 k k−1 A2 1 n
k=1
X
Theorem 3 follows. (cid:4)
Remark 2. In fact, it is clear that the proof of Theorem 3 can be extended to
prove that two MM-1QFAs A and A are t-equivalent if and only if they are
1 2
t-β-equivalent.
Forconvenience,weexpandEq.(3)asfollows. Notethat,ifω =x x ···x ,
1 2 n
then we have
F (ω) = P (x x ···x )−P (x x ···x )
A A 1 2 n A 1 2 n−1
9
†
1 1
= hπ| (P(g)U(x )) U(x )†P(a)†P(a)U(x ) (P(g)U(x )) |πi
i n n i
! !
i=n−1 i=n−1
Y Y
1 † 1
+ hπ| (P(g)U(x )) U($)†P(a)†P(a)U($) (P(g)U(x )) |πi
i i
! !
i=n i=n
Y Y
1 † 1
− hπ| (P(g)U(x )) U($)†P(a)†P(a)U($) (P(g)U(x )) |πi
i i
! !
i=n−1 i=n−1
Y Y
Setting A(σ) = P(g)U(σ) for each σ ∈ Σ and noting that P(a)2 = P(a),
P(a)† =P(a), we find that
F (ω) = hπ|η (ω)|πi (6)
A A
where
1 † 1
A(x ) δ (x ) A(x ) , ω =x x ···x ∈Σn;
i A n i 1 2 n
ηA(ω) =  (cid:16)i=Qn−1 (cid:17) (cid:16)i=Qn−1 (cid:17)
U($)†P(a)U($), ω =ǫ.
and δ (x ) is given by
A n
δ (x ) = U(x )†P(a)U(x )+A(x )†U($)†P(a)U($)A(x )−U($)†P(a)U($).
A n n n n n
Wefurtherintroducethefollowingauxiliarydefinitionsneededinthesequel.
Definition 4. Let A = (Q ,{U (σ)} ,|π i,O ), i = 1,2, be two MM-
i i i σ∈Σ∪{$} i i
1QFAsoverthealphabetΣ,whereO ={P (a),P (g),P (r)}andO ={P (a),P (g),P (r)}.
1 1 1 1 2 2 2 2
The diagonal sum ofA andA , denotedby A ⊕A , is anMM-1QFA,defined
1 2 1 2
to be
A=A ⊕A =(Q,{U(σ)} ,|ϑi,O)
1 2 σ∈Σ∪{$}
whereQ=Q ∪Q withQ ∩Q =∅,U(σ)=U (σ)⊕U (σ)foreachσ ∈Σ∪{$},
1 2 1 2 1 2
|ϑi ∈ H is an arbitrary unit vector and O = {P (a)⊕P (a),P (g)⊕
|Q1|+|Q2| 1 2 1
P (g),P (r)⊕P (r)}.
2 1 2
10