Open Systems & Information Dynamics Vol. 22, No. 3 (2015) 1550018 (6 pages) DOI:10.1142/S1230161215500183 c World Scientific Publishing Company
A Modified Adiabatic Quantum Algorithm for Evaluation of Boolean Functions Jie Sun1,2 , Songfeng Lu1,∗ and Fang Liu1 1
School of Computer Science and Technology Huazhong University of Science and Technology Wuhan 430074, China 2
College of Educational Information and Technology Hubei Normal University Huangshi 435002, China (Received: March 26, 2015; Accepted: June 24, 2015; Published: September 30, 2015) Abstract. In this paper, we propose a modified construction of the quantum adiabatic algorithm for Boolean functions studied by M. Andrecut et al. [13, 14]. Our algorithm has the time complexity O(1) for the evaluation of Boolean functions, without additional computational cost of implementing the driving Hamiltonian, which is required by the adiabatic evolution described in [13, 14].
1.
Introduction
There has been a growing interest in quantum computation especially after the famous quantum algorithm for factoring [1], unstructured database quantum search [2] were presented. The two algorithms above were based on the quantum circuit model. Quantum adiabatic evolution [3] proposed by Farhi et al., is considered as a promising and alternative approach to quantum computing. The equivalence between these two models has already been shown [4, 5], and the adiabatic versions of factoring [6, 7] and quantum search [8] have also been studied. Adiabatic quantum computation is believed to possess some special features, such as the robustness against perturbations, e.g. unitary control errors [9] and decoherence [10]. The basic process of adiabatic quantum computing can be summarised as follows. Suppose that the evolution of a quantum system can be fully characterised by a Hamiltonian H(t), and it is driven by the Schr¨ odinger equation d (1.1) i |ψ(t)i = H(t)|ψ(t)i , dt ∗
Corresponding author (
[email protected]) 1550018-1
J. Sun, S. F. Lu, and F. Liu
where |ψ(t)i is the instantaneous state of the system. Denote |E0 (t)i and |E1 (t)i, respectively, E0 (t) and E1 (t) as the eigenstates and the corresponding eigenvalues of the quantum Hamiltonian H(t). The quantum adiabatic theorem [11] tells that if the ground state of an initial Hamiltonian Hs evolves sufficiently slowly to the ground state of a final Hamiltonian He (the latter ground state is also the solution to the problem under consideration), then we know that 2 (1.2) hE0 (T )|ψ(T )i ≥ 1 − ε2 , provided that
Dmax ≤ ε, 2 gmin
(1.3)
with Dmax gmin
D dH(t) E = max E1 (t) E0 (t) , 0≤t≤T dt = min [E1 (t) − E0 (t)] , 0≤t≤T
(1.4) (1.5)
and 0 < ε ≪ 1 in (1.2). Generally speaking, the required evolution time T for finding out the solution will be predominantly determined by gmin , as long as Dmax is polynomially bounded. For convenience, the time-dependent Hamiltonian H(t) can be rewritten as b H(s) = H(t/T ) ,
for
0 ≤ s ≤ 1.
(1.6)
b The Hamiltonian H(s) which interpolates the initial Hamiltonian Hs and the final one can simply take a linear form b H(s) = (1 − s)Hs + sHe .
(1.7)
In this paper, we will construct a quantum adiabatic algorithm using this kind of system Hamiltonian for the evaluation of an n-bit Boolean function. 2.
The Original Adiabatic Algorithm for the Evaluation of Boolean Functions
Suppose we are given a Boolean function such as f : {0, 1}n −→ {0, 1} .
(2.1)
Such a function is considered a “black box”, and its implementation is often neglected for quantum computational purposes. In the circuit model 1550018-2
A Modified Adiabatic Quantum Algorithm for Evaluation of Boolean Functions
of quantum computing, a way of computing this function is to consider a two-qubit computer which starts in the state |x, yi. With an appropriate sequence of logic gates, it is easy to show that this state can be transformed into |x, y ⊕ f (x)i, where ⊕ indicates addition modulo 2. Obviously, if y = 0, then the final state of the second qubit is just the value of f (x). More generally, it can be shown that if there exists a classical circuit which computes the function f , there is a quantum circuit of better performance which computes the following unitary transformation Uf
|x, yi −→ |x, y ⊕ f (x)i
(2.2)
on a quantum computer. One of the distinguished features that makes quantum computing really appealing is its action on a superposition of different inputs. Taking the Boolean function above as an example, a quantum computer can determine the values of f in parallel, and produce a superposition of f (x) for allPx in a single run. It has to be noted that doing a measurement of the state x |x, f (x)i would give only f (x) for a single value of x. That is, quantum computing is inherently probabilistic. In [13, 14], the authors studied the Boolean function above in the framework of adiabatic quantum computing model. Specifically, they gave an adiabatic quantum algorithm which can evaluate the Boolean function in (2.1) with a minimum delay schedule of O(1) time complexity. For completeness, we will rewrite the adiabatic Hamiltonian shown in [13, 14], which takes the following nonlinear interpolating form: b H(s) = (1 − s)Hs + sHe + s(1 − s)Hs,e ,
in which Hs = He =
n −1 2X h
i Ex0 |x, 0ihx, 0| + Ex1 |x, 1ihx, 1| ,
x=0 n −1 2X n x=0
and Hs,e =
n −1 2X
x=0
By assuming that
(2.3)
(2.4)
[(1 − f (x))Ex0 + f (x)Ex1 ]|x, 0ihx, 0|
o + [(1 − f (x))Ex1 + f (x)Ex0 ]|x, 1ihx, 1| ,
f (x)(Ex1 − Ex0 )[|x, 0ihx, 1| + |x, 1ihx, 0|] .
(2.5)
(2.6)
E00 = 0 , E01 = 1 , E10 = 2 , . . . , E20n −1 = 2n+1 −2 , E21n −1 = 2n+1 −1 , (2.7) 1550018-3
J. Sun, S. F. Lu, and F. Liu
it can be shown that the minimum delay schedule for the adiabatic computation of an n-bit Boolean function is in O(1). Comparing (2.3) and (1.7), we see that there is an extra driven Hamiltonian Hs,e in (2.3) for the adiabatic evolution of a Boolean function. This driven Hamiltonian is necessary to make sure that the switching between the eigenstates |x, 0i and |x, 1i corresponding to the eigenvalues Ex0 and Ex1 of the adiabatic computation is done without crossing the eigenvalues during the adiabatic evolution. It is easy to verify that the adiabatic evolution would take infinite running time to accomplish without this driven Hamiltonian. Here we show how to reduce the implementation effort, by using the linear interpolating scheme [15], without resorting to a driven Hamiltonian. 3.
The Modified Adiabatic Algorithm for the Evaluation of Boolean Functions
We can overcome the difficulty mentioned above simply by designing a global adiabatic algorithm [12] for Boolean functions without resorting to adding a driven Hamiltonian. Specifically, we adopt the linear interpolating Hamiltonian form (1.7) for the adiabatic evolution, in which Hs = I − |αihα| , where
(3.1)
P2n −1 0 [Ex |x, 0i + Ex1 |x, 1i] |αi = qx=0 P2n −1 0 2 1 2 x=0 [(Ex ) + (Ex ) ]
and
He = I − |βihβ| ,
(3.2)
(3.3)
where
|βi =
n −1 2X n
x=0
[(1 − f (x))Ex0 + f (x)Ex1 ]|x, 0i + [(1 − f (x))Ex1 + f (x)Ex0 ]|x, 1i v u2n −1 h i uX t (E 0 )2 + (E 1 )2 x
o
.
x
x=0
(3.4) In (3.2) and (3.4), the values for Exi , x = 0, 1, . . . , 2n − 1, i = 0, 1, remain the same as those given in (2.7). Now we turn to estimate the time complexity of this new adiabatic quantum algorithm for Boolean functions. For this, firstly we assume that b H(s)|E k (s)i = Ek (s)|Ek (s)i , 1550018-4
k = 0, 1 .
(3.5)
A Modified Adiabatic Quantum Algorithm for Evaluation of Boolean Functions
By multiplying hα| to the above equation and combining with (1.7), we can get s hα|Ek (s)i = hβ|Ek (s)i|hα|βi| . (3.6) s − Ek (s) In the meantime, multiplying hβ| to (3.5) and again with (1.7), we can obtain hβ|Ek (s)i =
1−s hα|Ek (s)i|hα|βi| . 1 − s − Ek (s)
(3.7)
By substituting (3.6) into (3.7), we can arrive at the following equation: Ek2 (s) − Ek (s) + s(1 − s)(1 − |hα|βi|2 ) = 0 ,
(3.8)
b from which we can get the two lowest eigenvalues for the Hamiltonian H(s) as p 1 ∓ 1 − 4s(1 − s)(1 − |hα|βi|2 ) Ek (s) = , k = 0, 1 . (3.9) 2 So the minimum eigenvalue gap between these eigenvalues is given by
gmin = min [E1 (s)−E0 (s)] = |hα|βi| = 1− 2n −1 0≤s≤1 X
n −1 2X
f (x)
x=0
[(Ex0 )2
. (3.10) +
(Ex1 )2 ]
x=0
By the fact that 2n+1 X−1
i2 =
i=1
2n (2n+1 − 1)(2n+2 − 1) , 3
(3.11)
we get P2n −1
x=0 f (x) 0 2 1 2 x=0 [(Ex ) + (Ex ) ]
1 − P2n −1
≥ 1−
3 , (2n+1 − 1)(2n+2 − 1)
(3.12)
which implies that the time complexity for our construction of the adiabatic evolution for Boolean functions is in O(1) by (1.3). 4.
Conclusion
In conclusion, we have proposed a modification of the adiabatic quantum algorithm for Boolean functions, studied in [13, 14]. The complexity of the adiabatic algorithm presented here is equivalent to that given in [13, 14]. Also, 1550018-5
J. Sun, S. F. Lu, and F. Liu
our construction simplifies the implementation by using a linear interpolating scheme, without the need of an extra driven Hamiltonian to guarantee that the computation is done without crossing the eigenvalues during the adiabatic evolution. Although the linear interpolating scheme in adiabatic algorithms has its intrinsic limitations for some special problems, it is sufficient for using in most occasions. The problem studied here is just such a good example. Acknowledgments J. Sun gratefully acknowledges the support from the China Postdoctoral Science Foundation under Grant No. 2014M552041. This work is also supported by the National Natural Science Foundation of China under Grant Nos. 61402188, 61173050 and U1233119. Bibliography [1] P. W. Shor, SIAM Rev. 41(2), 303 (1999). [2] L. K. Grover, Phys. Rev. Lett. 79, 325 (1997). [3] E. Farhi, J. Goldstone, S. Gutmann, J. Lapan, A. Lundgren, and D. Preda, Science 292, 472 (2001). [4] A. Mizel, D. A. Lidar, and M. Mitchell, Phys. Rev. Lett. 99, 070502 (2007). [5] D. Aharonov, W. v. Dam, J. Kempe, Z. Landau, S. Lloyd, and O. Regev, SIAM J. Comput. 37 (1), 166 (2007). [6] X. H. Peng, Z. Y. Liao, N. Y. Xu, G. Qin, X. Y. Zhou, D. Suter, and J. F. Du, Phys. Rev. Lett. 101, 220405 (2008). [7] N. Y. Xu, J. Zhu, D. W. Lu, X. Y. Zhou, X. H. Peng, and J. F. Du, Phys. Rev. Lett. 108, 130501 (2012). [8] J. Roland and N. J. Cerf, Phys. Rev. A 65, 042308 (2002). [9] A. M. Childs, E. Farhi, and J. Preskill, Phys. Rev. A 63, 012322 (2001). [10] J. ˚ Aberg, D. Kult, and E. Sj¨ oqvist, Phys. Rev. A 71, 060312 (2005). [11] A. Messiah, Quantum Mechanics, Chap. XVII, Dover, New York, 1999. [12] E. Farhi, J. Goldstone, S. Gutmann, and M. Sipser, arXiv:quant-ph/0001106 (2000). [13] M. Andrecut and M. K. Ali, J. Phys. A: Math. Gen. 37, L267-L273 (2004). [14] M. Andrecut and M. K. Ali, J. Phys. A: Math. Gen. 37, L421-L427 (2004). [15] M. Andrecut and M. K. Ali, Int. J. Theor. Phys. 43, 925 (2004).
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