Solving a hidden subgroup problem using the adiabatic quantum-computing paradigm
M. V. Panduranga Rao*
Department of Computer Science and Automation, Indian Institute of Science, Bangalore 560 012, India
We present and solve a restricted Abelian hidden subgroup problem using the adiabatic quantum-computing
paradigm. The time step complexity is shown to be a polynomial in the number of input qubits. This paper is
a step towards looking at the Abelian hidden subgroup problem from a quantum adiabatic standpoint.
I. INTRODUCTION
Models of quantum computing in the continuous time domain have evoked a lot of interest in recent times. Adiabatic
quantum computing is one such model. It has been applied to
problems such as restricted versions of Boolean formula satisfiability 关1兴 and the Deutsch-Jozsa problem 关2兴. This model
has also been shown to be ‘‘truly quantum in nature’’ in Ref.
关3兴, where a quadratic speedup in searching an unordered list
is achieved. It may be recalled that a classic result in ‘‘discrete’’ quantum computing, Grover’s algorithm 关4兴, achieves
the same. Another interesting problem that has been closely
examined in the discrete setting is the following 关5兴.
The hidden subgroup problem (HSP). Given an efficiently
computable function ␾ :G→S from a finite group G to a set
S, which is constant on the 共left兲 cosets of some hidden subgroup H and takes distinct values for distinct cosets, find the
generating set for H.
Polynomial time solutions have been proposed for Abelian groups, including those with immense practical applications such as Shor’s algorithm 关6兴 for prime factorization.
In this paper, we take a step towards studying the HSP
‘‘quantum adiabatically.’’ Simon’s problem 关7兴 seems to be
an appropriate starting point in that direction, as it was for
the discrete paradigm. This problem is defined over the
group Z n2 with ‘‘bitwise exclusive OR’’ ( 丢 ) as the group
operation. The problem considered in this paper is a restriction of Simon’s problem in that we disallow the trivial hidden subgroup 共consisting of only the identity element 0 n ).
Modified Simon’s problem (MSP). Given a function
␾ : 兵 0,1其 n → 兵 0,1其 n⫺1 such that ␾ (x)⫽ ␾ (x ⬘ ) if and only if
x ⬘ ⫽x 丣 h, for h苸 兵 0 n ,c 其 , where c苸 兵 0,1其 n \ 0 n .
Problem. Find c.
Note that 兵 0 n ,c 其 is the hidden subgroup in this case. The
original problem of Simon has the range 兵 0,1其 m , m⭓n. Reducing this range to 兵 0,1其 n⫺1 does not significantly reduce
the toughness of the problem.
For a more comprehensive treatment, the reader is referred to
Ref. 关8兴 and Refs. 关1–3兴, respectively.
Let H(s) (0⭐s⫽t/T⭐1) be a time-dependent Hamiltonian Ĥ(t) acting on a system having an N-dimensional
Hilbert space, scaled linearly in time by a delay factor T. At
any instant s, let the eigenstates of H(s) be given by 兩 l;s 典
and the eigenvalues by ␭ l (s), with ␭ 0 (s)⭐␭ 1 (s)⭐•••
⭐␭ N⫺1 (s). Suppose we start with the initial state of the
system 兩 ␺ (0) 典 as 兩 0;0 典 关the ground state of H(0)] and apply
the Hamiltonian H(s), 0⭐s⭐1, to evolve it to 兩 ␺ (1) 典 at s
⫽1. Then the quantum adiabatic theorem states that for a
‘‘large enough’’ delay factor T, the final state of the system
兩 ␺ (1) 典 will be arbitrarily close to the ground state 兩 0;1 典 of
H(1). In addition, for the purpose of efficient computation,
we also require T to be small. Let g min ⫽min0⭐s⭐1关␭1(s)
⫺␭0(s)兴. Then the delay required for such an evolution to
take place is given by
max
TⰇ
0⭐s⭐1
The adiabatic quantum-computing paradigm is based on
the adiabatic theorem of quantum mechanics. The rest of this
section describes the theorem and the computing technique.
*Electronic address: pandurang@csa.iisc.ernet.in
d
H共 s 兲
ds
2
g min
冏冏冏
2
.
共1兲
The interpolating Hamiltonian
In this section, we present an overview of the general
technique. Adiabatic quantum computing is useful when a
problem can be reformulated as a minimization problem—
given a function f : 兵 0,1其 n →R 共computable in time polynok
for which f is minimum.
mial in n), find those inputs 兵 x i 其 i⫽1
We construct the final Hamiltonian H(1) in such a way
that its ground state encodes the required solution set
k
. An obvious way is to associate the eigenvalues of
兵 x i 其 i⫽1
H(1) with the function values so that
H共 1 兲⫽
II. THE ADIABATIC QUANTUM-COMPUTING
PARADIGM
冏冏冏
兺
z苸 兵 0,1其 n
f 共 z 兲 兩 z 典具 z 兩 ,
where 兵 兩 z 典 其 form the ‘‘computational basis’’ of the Hilbert
space of the system.
The initial Hamiltonian is independent of the problem,
with the restriction that it should not be diagonal in the computational basis 关1兴. It is easy to see that the following
Hamiltonian satisfies the above condition:
H共 0 兲⫽
兺
z苸 兵 0,1其
n\0n
兩 ẑ 典具 ẑ 兩 ,
where each 兩 ẑ 典 is a basis vector in the ‘‘Hadamard’’ basis
given by
兩 ẑ 典 ⫽
1
冑2 n
冉
1
1
1
⫺1
III. SOLUTION TO MSP
共2兲
冊
Choose x苸 兵 0,1其 n \ 0 n uniformly at random. With a very
high probability 关 1⫺1/(2 n ⫺1) 兴 , x⫽c. Let the final Hamiltonian be
丢n
兩z典
H共 1 兲⫽
and the ground state is (1/冑2 n ) 兺 z苸 兵 0,1其 n 兩 z 典 , which is easy to
construct. Given the initial and final Hamiltonians, define the
interpolating Hamiltonian as
H 共 s 兲 ⫽ 共 1⫺s 兲 H 共 0 兲 ⫹sH 共 1 兲 .
冋
⫹
冋
N/2⫺2
兿
k⫽1
冋冉 冊 册 冎
1⫺s⫹
N
1
1⫺s
⫺1
⫹n s⫺␭ ⫺
2
N
N
N/2⫺1
⫹ 共 1⫺s⫺␭ 兲
兿
k⫽1
冋
再
冉 冊 册再
n
兩 ␾共 x 兲⫺ ␾共 x 丣 z 兲兩
N
冊
⫹ 共 1⫺ ␦ ␾ (x), ␾ (x 丣 z) 兲 n 兩 z 典具 z 兩 ,
共4兲
N/2⫺1
兿
k⫽1
冋
1⫺s⫹
冉 冊 册
冋冉 冊 册
2
k
⫹n s⫺␭
N
N
1
⫺1
⫹n s⫺␭
2
N
where ␦ i, j is the delta function 共having value 1 if i⫽ j and 0
otherwise兲 and N⫽2 n . Clearly, f, the function to be minimized, is bounded by a polynomial in n. Therefore, the crucial factor in determining the time delay is g min . We take the
initial Hamiltonian as given by Eq. 共2兲 and use Eq. 共3兲 to
obtain H(s).
Lemma 1. The characteristic equation of H(s) is
2
k
⫹n s⫺␭
N
1⫺s⫹
1⫺s⫹
冉
共3兲
We start in the ground state of H(0) and evolve to H(1)
slowly enough and end up in its ground state. However, since
the eigenvalues have been associated with function values,
the ground state is a superposition of all solutions. If f (x) is
bounded by a polynomial in n then so is the numerator of Eq.
共1兲, 兩兩兩 (d/ds)H(s) 兩兩兩 2 . The deciding factor then is the de2
.
nominator g min
c 共 ␭ 兲 ⫽ 共 1⫺s⫺␭ 兲 共 1⫺s⫺␭ 兲
兺
z苸 兵 0,1其
1⫺s⫹
冋 冉 冊 册 冎再
N
1
⫺1
⫹n s⫺␭
2
N
冉 冊 册
2
k
⫹n s⫺␭ ⫹2 共 1⫺s⫺␭ 兲
N
冎册
⫺
1⫺s
共 1⫺s⫺␭ 兲
N
N/2⫺1
兿
k⫽1
冋
共 N⫺1 兲
共 1⫺s 兲
N
N/2⫺2 N/2⫺1
兺 k⫽1
兿
j⫽1
1⫺s⫹
冋
冉 冊 册
冉 冊
k
⫹n s⫺␭
N
j
1⫺s⫹ ⫹n s⫺␭
N
1⫺s⫹
2
冉 冊 册
2
k
⫹n s⫺␭ ⫽0.
N
Proof. To evaluate the eigenvalues ␭(s)’s1 of H(s), we evaluate the determinant
兩 H 共 s 兲 ⫺␭I 兩 ⫽0.
冏
冏
Subtracting the last column of this determinant from all other columns and using x 0 for 1⫺s⫺␭, x 1 for 1⫺s⫹(1/N⫹n)s
⫺␭ and so on up to x N/2⫺1 for 1⫺s⫹ 关 (N/2⫺1)(1/N)⫹n 兴 s⫺␭, we have
1
x0
0
0
•••
•••
⫺
共 1⫺s 兲
N
0
x0
0
•••
•••
⫺
共 1⫺s 兲
N
⯗
0
⯗
x N/2⫺1
⫺x N/2⫺1
⫺x N/2⫺1
•••
We use ␭ and ␭(s) interchangeably.
⫺x N/2⫺1
冉
⯗
x N/2⫺1 ⫺
共 1⫺s 兲
N
冊
⫽0.
N⫻N
FIG. 2. N⫽8.
FIG. 1. N⫽4.
Expanding this determinant gives the required characteristic
equation.
Remark. In the above characteristic equation, N/2 of the N
eigenvalue curves are immediately identifiable 共as factors兲.2
共See Figs. 1–3.兲 They are 共in increasing order at s⫽1)
␭ 1 (s)⫽1⫺s, ␭ 3 (s)⫽1⫺s⫹(1/N⫹n)s, . . . ,␭ N⫺1 ⫽1⫺s
⫹ 关 (N/2⫺1)(1/N)⫹n 兴 s.
We now state a classic result from the theory of equations.
Theorem 1. If a real polynomial f (x) for x⫽a and x⫽b
takes values f (a) and f (b) of opposite signs, so that, for
instance, f (a)⬍0 and f (b)⬎0, then there is a root of the
equation f (x)⫽0 in the interval (a,b).
The above theorem will be used extensively to prove our
result.
Lemma 2. There exists exactly one root 共i.e., an eigenvalue curve兲 of the characteristic equation 共Lemma 1兲 in the
intervals (0,1⫺s), „1⫺s,1⫺s⫹(1/N⫹n)s…, . . . ,„1⫺s
⫹ 关 (N/2⫺2)(1/N)⫹n 兴 s,1⫺s⫹ 关 (N/2⫺1)(1/N)⫹n 兴 s…; s
苸(0,1).
Proof. Consider the interval „0,␭ 1 (s)⫽1⫺s…. Let ␭(s)
⫽ ␦ and ␭(s)⫽1⫺(1⫹ ⑀ )s be two lines, where ␦ and ⑀ are
positive reals tending to 0. Substituting for ␭ in c(␭), we
see that c( ␦ )⬎0 and c 关 1⫺(1⫹ ⑀ )s 兴 ⬍0 for s苸(0,1).
Therefore, by Theorem 1, there exists a root of c(␭)⫽0 in
the interval (0,1⫺s) for s苸(0,1). Similarly, in the interval
„1⫺s,1⫺s⫹(1/N⫹n)s…, c(1⫺s⫹ ␦ s)⬎0 and c 关 1⫺s
⫹(1/N⫹n⫺ ⑀ )s 兴 ⬍0. For „1⫺s⫹(k/N⫹n)s,1⫺s⫹ 关 (k
⫹1)/N⫹n 兴 s…, 1⭐k⭐N/2⫺2, we have c 关 1⫺s⫹(k/N⫹n
⫹ ␦ )s 兴 ⬎0 and c„1⫺s⫹ 关 (k⫹1)/N⫹n⫺ ⑀ 兴 s…⬍0 for s
苸(0,1).
Therefore there is a root in each interval. Given that 共i兲
there are N/2 open intervals bounded by N/2 straight line
eigenvalue curves; 共ii兲 there is at least one root in each interval; and 共iii兲 there are N roots in all, the lemma follows.
The eigenvalue curves ␭ 0 (s) and ␭ 1 (s) intersect at s⫽1
because there is more than one solution (0 and c). Note that
H(s) commutes with the operations 兩 0 n 典 ↔ 兩 c 典 , which rules
out transitions between the eigenstates (1/冑2)( 兩 0 n 典 ⫹ 兩 c 典 )
and (1/冑2)( 兩 0 n 典 ⫺ 兩 c 典 ) corresponding to the eigenvalues
␭ 0 (1) and ␭ 1 (1), respectively. By the same argument as
2
See, example, eigenvalue curves in the Appendix.
given in Sec. 3.2 of Ref. 关1兴, we consider the gap between
␭ 0 (s) and ␭ 2 (s).
We will try to estimate the minimum gap between ␭ 0 (s)
and ␭ 2 (s) by studying their slopes. The Wielandt-Hoffman
theorem 关9兴 is a useful tool for obtaining an upper bound on
their instantaneous slopes.
Theorem 2. Wielandt-Hoffman Theorem 共WHT兲: Let A
and E be real symmetric matrices and let Â⫽A⫹E. Let the
eigenvalues of A be ␭ 0 ⭐␭ 1 ⭐•••⭐␭ N⫺1 and those of  be
␭ˆ 0 ⭐␭ˆ 1 ⭐•••⭐␭ˆ N⫺1 . Then
冋兺
N⫺1
j⫽0
册 冋兺 兺 册
1/2
共 ␭ j ⫺␭ˆ j 兲 2
⭐
N⫺1 N⫺1
i⫽0 j⫽0
兩 E i j兩 2
1/2
.
共5兲
Let E of the WHT be the ‘‘perturbation matrix’’ H(s⫹ds)
⫺H(s). This completely specifies the right-hand side 共rhs兲
of Eq. 共5兲. In the collection of N/2 straight-line eigenvalues,
consider
the
following
set:
兵 ␭ 3 (s)⫽1⫺s⫹(1/N
⫹n)s, . . . ,␭ N⫺1 (s)⫽1⫺s⫹ 关 (N/2⫺1)(1/N)⫹n 兴 s 其 . Their
slopes d␭/ds are known. These lines are closely spaced in
the sense that they start at ␭(0)⫽1 and the gap between
‘‘consecutive’’ lines is just 1/N at s⫽1. Consider any inter1⭐k
val
„1⫺s⫹(k/N⫹n)s,1⫺s⫹ 关 (k⫹1)/N⫹n 兴 s…,
⭐N/2⫺2. There lies an eigenvalue curve ␭ k (s) in it with
␭ k (0)⫽1 and ␭ k (1)⫽n⫹(k⫹1)/N, so that its average
slope is n⫹(k⫹1)/N⫺1. Since the WHT deals with only
squared terms, both positive and negative slopes of equal
magnitude can be treated at par. Due to such a small gap
between 1⫺s⫹(k/N⫹n)s and 1⫺s⫹ 关 (k⫹1)/N⫹n 兴 s, any
significant deviation 共in magnitude兲 of the slope from the
FIG. 3. N⫽16.
冋兺 冉
冋兺 冉
N/2⫺1
average is short lived. Moreover, it has to be compensated
later to maintain the average. For example, ␭ k (s) having a
slope zero at a stretch is possible for a duration of only
2
k N⫺1
n⫹ ⫺
N
N
k⫽2
N/2⫺1
p
,
k⫹1
⫺1
N n⫹
N
冉
冊
共 by simple trigonometry兲 .
0⭐ p⭐1
2
冉
兺
k⫽2
n⫹
k
⫺1
N
冊
冉
2
共 ds 兲 2 ⫹ n⫹
1
⫺1
N
冊
冉 冊
N⫺1
N
N/2⫺1
⫹2
兺
k⫽1
2
共 ds 兲 2 ⫹N
冉
共 N⫺1 兲
N2
k N⫺1
n⫹ ⫺
N
N
冊
共 ds 兲 2 ⫹ 共 ds 兲 2
共 ds 兲 2
2
共6兲
共 ds 兲 2 .
Rearranging some terms, we have
冉
冊 冉 冊
冋冉 冊
册 冋冉
冉 冊册 冋冉兺 冉
冊 冊册
兺 冉
d␭ 0 共 s 兲
ds
d␭ 2 共 s 兲
ds
2
⫹
N⫺1
⭐ 2
N
2
N/2⫺1
⫺
n⫹
k⫽2
N/2⫺1
2
⫹ 2
k
⫺1
N
k⫽2
冉 冊
N⫺1
N
冉
2
⫹
冉
k N⫺1
n⫹ ⫺
N
N
冊
冊
2
.
5
2
N⫺1
⫺1⫽3⫺ ⫹ 2 ⫺1⬇2,
N
N N
冉
冊 冉
冊
1 N⫺1
⫺
N
N
⬇ n⫹
1
⫺1
N
⫺ n⫹
1
⫺1
N
2
⬇ 共 n⫺1 兲 2 ,
2n⫹
冊
冊册
冊
冊册
2
2k 2N⫺1
3
⫺
⫽2n⫺ .
N
N
2
d␭ 0 共 s 兲
ds
冊 冉
2
⫹
d␭ 2 共 s 兲
ds
冊
2
3
3
⭐ 共 n⫺1 兲 2 ⫹2n⫺ ⫹2⫽n 2 ⫹ .
2
2
Corollary 3.1. From the above lemma, it is easy to see
that 冑n 2 ⫹ 32 is an upper bound for both 兩 d␭ 0 /ds 兩 and
兩 d␭ 2 /ds 兩 .
We are now in a position to state the main results of the
paper.
Lemma 4. The minimum time s 1 required for the ␭ 0 (s)
eigenvalue curve to rise to 关within O(1/N) distance of兴 the
␭ 1 (s) curve is 1/(1⫹ 冑n 2 ⫹ 32 ).
Proof. The minimum time will be attained when the ␭ 0 (s)
rises with the maximum available slope. Therefore, (1
⫺s 1 )/s 1 ⫽ 冑n 2 ⫹ 32 , from which we have s 1 ⫽1/(1
冊
2
1
1⫹n⫹
3
4n
.
共7兲
Theorem 3. The minimum gap between ␭ 0 (s) and ␭ 2 (0)
for s苸 关 0,1兴 is O„1/poly(n)….
Proof. Consider the interval „␭ 1 (s)⫽1⫺s,␭ 3 (s)⫽1⫺s
⫹(1/N⫹n)s…. There exists exactly one eigenvalue curve
␭ 2 (s) in this interval. Further, c 关 1⫺s⫹(1/N⫹n⫺ ␦ )s 兴 ⬍0
for a positive ␦ tending to 0. Consider a line ␭ 1⬘ (s)⫽1
⫺(1⫺1/m)s where m⫽poly(n) or a polynomial in n. Since
it lies above3 ␭ 1 (s)⫽1⫺s, the ␭ 2 (s) eigenvalue curve may
not lie between ␭ 3 (s) and ␭ ⬘1 (s) for the entire interval s
苸 关 0,1兴 . To find the interval in which it does lie, we solve4
c 关 1⫺(1⫺1/m)s 兴 ⬎0 for s and invoke Theorem 1.
By above we mean ␭ 1⬘ (s)⬎␭ 1 (s) for s苸(0,1兴 共or any other interval, if specified兲.
4
Note that „1⫺s⫹ 关 (N/2⫺1)(1/N)⫹n 兴 s⫺␭…„关 (N⫺1)/N 兴 (1
⫺s)⫹ 关 (N/2⫺1)(1/N)⫹n 兴 s⫺␭…, the first term in the square
bracket of c(␭), has been approximated by „1⫺s⫹ 关 (N/2
⫺1)(1/N)⫹n]s⫺␭…2 . The error introduced because of this can be
ignored in Eq. 共8兲 and later.
3
2 n⫹
k
⫺1
N
We have ignored small terms 关of O(n/N)] because whatever
little error that occurs turns out to be on the conservative
side. Reassembling the three expressions, we have
2
2
2
冉
n⫹
k
k N⫺1
⫺
⫹n⫹ ⫺1
N
N
N
s 1⬇
Let us evaluate the expressions in square brackets of the rhs
individually for the sake of clarity.
2
冉
k
k N⫺1
⫺
⫺n⫺ ⫹1
N
N
N
N/2⫺1
兺
k⫽2
兺
k⫽2
⫹ 冑n 2 ⫹ 32 ). Using binomial expansion,
2
N⫺1
1 N⫺1
⫺1 ⫹ 2 n⫹ ⫺
⫹
N
N
N
1
⫺ n⫹ ⫺1
N
2
N
⫺
2
⫹ 共 d␭ 0 兲 2 ⫹ 共 d␭ 2 兲 2
⭐2
冉
⫽
n⫹
k⫽2
⫻ n⫹
The maximum can be attained only towards the end (p
⫽1). Therefore, we can approximate the slope of ␭ k (s) by
n⫹(k⫹1)/N⫺1, 1⭐k⭐N/2⫺2 for use in the WHT with
negligible error. But the same cannot be said about the slopes
d␭ 0 (s)/ds and d␭ 2 (s)/ds as ␭ 0 (s) and ␭ 2 (s) are not
bounded by closely spaced lines.
Lemma 3. 关 d␭ 0 (s)/ds 兴 2 ⫹ 关 d␭ 2 (s)/ds 兴 2 ⭐n 2 ⫹ 23 .
Proof. Substitute the values of the d␭’s and the matrix
elements of the perturbation matrix E in the WHT to obtain
N/2⫺1
⫽2
冊
N/2⫺1
2
Theorem 1.
⫺
s
m
再
⫺
⫺
s N⫺2
s
m
s N⫺3
s
m
冉
N/2⫺1
冊
⫺
1⫺s
N
k
1 2
⫹n⫺
N
m
N
1
1
⫺1
⫹n⫺
2
N
m
册冎
兿
k⫽1
冉
N/2⫺1
兿
k⫽1
k
1
⫹n⫺
N
m
2
冊
冉 冊
冋
N/2⫺1
2s N⫺2
兿
k⫽1
冉
k
1
⫹n⫺
N
m
冊
2
⫺
2s N⫺3
s
m
N/2⫺2 N/2⫺1
兺 k⫽1
兿
j⫽1
冉
k
1
⫹n⫺
N
m
j
1
⫹n⫺
N
m
冊
2
⭓0,
which gives
1
s⭓
1⫹
m
冉
N
2
m
N/2⫺2
兺
j⫽1
1
j
1
⫹n⫺
N
m
⫹
1
1
⫺2
m N
1
1
⫺1
⫹n⫺
2
N
m
冉 冊
冊
共8兲
.
But,
N/2⫺2
兺
j⫽1
N/2⫺2
1
j
1
⫹n⫺
N
m
⫽N
兺
1
冉 冊
1
j⫹N n⫺
m
j⫽1
⫽N
1
.
j
兺
N(n⫺1/m)⫹1⭐ j⭐N/2⫺2⫹N(n⫺1/m)
Separating into two summations, we have the sum as
N
冉
兺
1⭐ j⭐N/2⫺2⫹N(n⫺1/m)
冊
1
1
.
⫺
j 1⭐ j⭐N(n⫺1/m) j
兺
The two summations above are in the form of harmonic numbers. The sum to p terms of a harmonic series is bounded as
p
1/k⬍ln(p)⫹1. Therefore,
ln(p)⬍兺k⫽1
N/2⫺2
N
兺
j⫽1
冉 冊
冉 冊
冉
1
N
⫺2⫹N n⫺
1
2
m
1
⬇N ln
⬇N ln 1⫹
1
1
1
j⫹N n⫺
N n⫺
2 n⫺
m
m
m
冉 冊
Expanding the logarithm,
⬇N
冉冉
1
1
2 n⫺
m
冊 冉
⫺
1
1
8 n⫺
m
冊冊
2
冉
1
1⫹
1
1
n⫺
1
m
⫺
1
1
m
1
⫺2
N N
1
N
1
⫺1
⫹n⫺
2
N
m
冉 冊 冉 冊
4 n⫺
1
m
2⫹
.
共9兲
.
Substituting in Eq. 共8兲, we get
s⭓
冊冊
.
关and therefore ␭ 2 (s)⬎␭ 1⬘ (s) for s 2 ⬍s⭐1]. This happens at
Ignoring smaller terms,
s⭓
1
冉 冊
冉 冊
1 2
4 n⫺
m
1⫹
1
4 n⫺
⫺1
m
⫽
1
1
1
m
1⫹n⫺ ⫹
m
1
4 n⫺
⫺1
m
␭ 1⬘ 共 s 2 兲 ⫽␭ 2 共 s 2 兲 ⫽1⫺ 1⫺
冉 冊
which gives
s⭓
冉 冊
,
n⫺
1
1 1
1⫹n⫺ ⫹
m 4
.
共10兲
This implies that the line ␭ 1⬘ (s)⫽1⫺(1⫺1/m)s cuts ␭ 2 (s)
at
s 2 ⫽1/共 1⫹n⫺1/m⫹1/4兲
关1兴 E. Farhi, J. Goldstone, S. Gutmann, and M. Sipser, e-print
quant-ph/0001106.
关2兴 Saurya Das, Randy Kobes, and Gabor Kunstatter, Phys. Rev. A
65, 062310 共2002兲; e-print quant-ph/0111032.
关3兴 W. van Dam, M. Mosca, and U. Vazirani, Proceedings of the
42nd Annual Symposium on Foundations of Computer Science,
共IEEE Computer Society Press, Los Alamitos, CA, 2001兲, pp.
279–287; e-print quant-ph/0206003; Jeremie Roland and
Nicolas J. Cerf, Phys. Rev. A 65, 042308 共2002兲; e-print
quant-ph/0107015.
关4兴 Lov Grover, Proceedings of the 28th Annual Symposium on
1
m
1
,
1 1
1⫹n⫺ ⫹
m 4
which is above the ␭ 1 (s) curve by an inverse polynomial
distance. Clearly, the minimum time required by the ␭ 0 (s)
eigenvalue curve to reach ␭ 0 (s)⫽1⫺s 关given by Eq. 共7兲兴,
s 1 , is greater than s 2 . Therefore, by the time ␭ 0 (s) nears
␭ 1 (s)⫽1⫺s, there exists an inverse polynomial gap between ␭ 2 (s) curve and ␭ 0 (s 1 ) 关 ⬇␭ 1 (s 1 ) 兴 which only increases for s 1 ⬍s⭐1. Since the numerator of Eq. 共1兲 is a
polynomial in n and the denominator is an inverse polynomial in n, the total time delay for evolution to the solution
state is upper bounded by a polynomial.
关5兴
关6兴
关7兴
关8兴
关9兴
Theory of Computing, 共ACM Press, New York, 1996兲, pp.
212–219.
Sean Hallgren, Ph.D. thesis, University of California, Berkeley, CA, 2000 共unpublished兲.
P.W. Shor, SIAM J. Comput. 26, 1484 共1997兲; e-print
quant-ph/9508027.
Daniel R. Simon, SIAM J. Comput. 26, 1474 共1997兲.
Albert Messiah, Quantum Mechanics 共Wiley, New York,
1958兲.
A.J. Hoffman and H.W. Wielandt, Duke Math. J. 20, 37
共1953兲.