The Constant Speedup Mechanism on Adiabatic Quantum Computation

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Copyright © 2016 American Scientific Publishers All rights reserved Printed in the United States of America

Journal of Computational and Theoretical Nanoscience Vol. 13, 7262–7265, 2016

The Constant Speedup Mechanism on Adiabatic Quantum Computation Zhigang Zhang, Songfeng Lu∗ , Jie Sun, and Qing Zhou School of Computer Science and Technology, Huazhong University of Science and Technology, Wuhan 430074, China In the adiabatic quantum computation model, a computational procedure is described by the continuous time evolution of a time dependent Hamiltonian. Classically, the unstructured search problem can be solved only in a running time of order ON. However, by modifying the structure of local Hamiltonian or using specific interpolating functions, it is possible to do the calculation in constant time for a quantum computer. This paper reveals the cause that lead to the speedup. We analyze two kinds of specific adiabatic quantum models, and conclude that the value of relevant elements in back-diagonal of the local Hamiltonian is the main factors affecting the time complexity of adiabatic quantum algorithms. According to the speedup mechanism, we have proposed two kinds of adiabatic quantum algorithms to make a constant time complexity.

Keywords: Quantum Computing, Adiabatic Quantum Computing, Structure of Local Hamiltonian, Interpolating Functions, Speedup Mechanism.

1. INTRODUCTION Quantum adiabatic evolution1 has been developed as an alternative to traditional circuit model of quantum computation, which have some advantages especially like robustness against different kinds of perturbations, such as decoherence,2 and unitary control errors,3 and it is believed to have the power to solving various kinds of NP problems.4–7 In addition, other researchers8 9 also have shown that AQC is equivalent to standard quantum computing. The essence of an adiabatic quantum algorithm is to prepare a quantum system in the ground state of a simple initial Hamiltonian, then slowly transform this Hamiltonian to a final Hamiltonian whose ground state encodes the solution to the problem which will be solved. Suppose the state of a quantum system is 0, which evolves according to the Schrödinger equation. i

d t = Htt dt

(1)

For the sake of convenience, the time-dependent Hamiltonian Ht can be reparametrized as Hs = Ht/T

for 0 ≤ s ≤ 1

Author to whom correspondence should be addressed.

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Hs = 1 − sHi + sHf

for 0 ≤ s ≤ 1

(3)

More specifically, if E0 s and E1 s are the ground and first excited states of the Hamiltonian Hs with energies E0 s and E1 s, we define the minimum gap between these eigenvalues gmin = minE1 s − E0 s

for 0 ≤ s ≤ 1

(4)

and the maximum value of the matrix element of dHs/ds between the eigenstates as dH E0 s for 0 ≤ s ≤ 1 (5) D = max E1 s ds In some typical problems of interest, D will scale polynomially with the problem size, so the efficiency of the algorithm hinges on whether gmin is exponentially small or not. It is not difficult to know: −2 T ∈ gmin

(2)

Usually, the instantaneous Hamiltonian Hs which connects the initial Hamiltonian Hi and final Hamiltonian ∗

Hf of the quantum system is a linear adiabatic evolution path as:

(6)

In terms of an unstructured search problem, where N is the number of items in the√database, it can be solved in a running time of order O N in the adiabatic evolution model by using a local formulation of the adiabatic theorem, which was proposed by Roland et al.10 The same 1546-1955/2016/13/7262/004

doi:10.1166/jctn.2016.5997

Zhang et al.

The Constant Speedup Mechanism on Adiabatic Quantum Computation

As the author of Ref. [15] stated, the adiabatic algorithm will work taking any path, as long as the adiabaticity condition is satisfied. In this section, we will introduce two specific adiabatic quantum models by considering nonlinear interpolation, and discuss why the models owns a constant time complexity in adiabatic evolution. Before we start, it is assumed that throughout the whole paper, the initial Hamiltonian and the final Hamiltonian take the following form as: Hf = I − 1 1

(7)

in which 0 and 1 are the initial state and the final state of quantum system. To facilitate the calculation, by using Gram-Schmidt transformation, the initial state and final state can be rewritten as: 0 = 0

(8)

1 = a0 + b1

(9)

where a = 0 1

b=

√ 1 − a2

(10)

and 0, 1 are orthogonal basic in a two-dimensional Hibernate space. J. Comput. Theor. Nanosci. 13, 7262–7265, 2016

= f sH + gsH + hsH Hs i f e

(11)

where the interpolating functions are set as: f s = 1 − s

gs = s

hs = s1 − s

(12)

and the extra driving Hamiltonian of the quantum system can be simply encoded as: He = 0 1 + 10

(13)

The form of the Hamiltonian as Eq. (11) can be recast as: 2 b gs + 2ahs bhs − abgs Hs = (14) bhs − abgs f s + a2 gs and the eigenvalues of it are easy to get:

√ f s + gs + 2ahs + E0 1 s = 2

(15)

where = f s − gs2 + 4hs2 + 4a2 f sgs − 4ahsf s + gs

2. THE SPECIFIC ADIABATIC QUANTUM MODELS

Hi = I − 0 0

2.1. The O(1) Adiabatic Algorithm with Extra Driving Hamiltonian As the author of Ref. [13] stated, by adding an extra driving Hamiltonian into the adiabatic model as Eq. (19), the adiabatic algorithm can be speeded to a constant time complexity. The following Hamiltonian has been considered as:

(16)

With Eq. (15), we can easily obtain the minimal gap of the system as: 1/4+aa−1∈ O1 (17) gmin0≤s≤1 = E 0 s− E1 s = Why the model as Eq. (11) can achieve a constant time complexity by adding an extra piece of Hamiltonian as Eq. (13)? Because the overlap a = 0 1 of initial Hamiltonian and final Hamiltonian tends to zero, with Eq. (15) we can get: gmin = 4hs2 + 4a2 f sgs − 4ahsf s + gs ∈ O1 + Oa2 − Oa (18) We thus arrive at a constant time complexity for this model. 2.2. The O(1) Adiabatic Algorithm with Specific Interpolating Functions In order to get a constant running time complexity, Das et al.12 had shown a specific model by modifying its interpolating functions to increase the lowest eigenvalue of the system exponentially. The following Hamiltonian has been considered as: Hs = f sHi + gsHf

(19) 7263

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√ speed up, of order O N , has been obtained in another adiabatic evolution model, which was provided by Avatar Tulsi.11 However, it has been shown that the problem can be solved in a constant time by using specific interpolating functions during the computation.12 Andrecut and Ali13 also have proposed another specific adiabatic model by adding an extra driving Hamiltonian to speed the computation in a constant time complexity. In order to further understand the reason that lead to a constant running time for adiabatic quantum computation, we analyze the two specific adiabatic models12 13 as mentioned above, and find that both two models have increased the value of the relevant back-diagonal elements of time-dependent Hamiltonian, which causes the minimal gap increasing to a nonzero constant, so that the time complexity of these models can achieve O1. The result reveal the fact that the relevant back-diagonal elements of time-dependent Hamiltonian is the key factor to affect the time complexity of adiabatic computation, which is just the same as what the Ref. [14] stated, the entanglement of initial state and final state has affected the time complexity of adiabatic computation. The rest of the paper is organized as follows. In Section 2, we analyze two kinds of specific adiabatic quantum models, and compute the running time complexity of these models. In Section 3, we explore the cause that lead to the speed of these models. In the last section, we summarize and draw some conclusions.


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where the interpolating functions are set as: f s = 1 − s + Qs1 − s

gs = s + Qs1 − s (20)

in which Q is a constant and st is a function of time that varies monotonically between 0 and 1. The form of the Hamiltonian as Eq. (19) can be recast as: 2 b gs −abgs Hs = (21) −abgs f s + a2 gs and its two eigenvalues are given by f s + gs ± f s − gs2 + 4f sgsa2 E0 1 s = 2 (22) Then by using Eq. (20), we can easily work out the minimal gap of the system as: gmin0≤s≤1 = E0 s − E1 s = 4a2 f sgs = a +

Qa 2

(23)

It is easy to see that if we choose Q = 1/a, the running time of the model as Eq. (19) tends to a constant. The reason leading to a constant time complexity of the model as Eq. (19) is obvious. The general interpolating functions f s = 1 − s and gs = s belong to O1. However, if we modify them to f s = 1 − s + s1 − s/a and gs = s + s1 − s/a, which contain the factor 1/a, the minimal gap as Eq. (23) can achieve O1.

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The value of the minimal gap is based the equation as: (28) gmin = A − C2 + 4B 2 By using the Eq. (31), we can easily obtain that: A − C ∈ O1

B ∈ Oa

So, we should easily get that the minimal gap as Eq. (28) belongs to O1. However, in terms of a general adiabatic model as Eq. (24) with O1 interpolating functions, it is well known that the time complexity belongs to O0 1 −2 by using Eq. (22). We can determine that the factor A − C can equal zero during the adiabatic evolution. In order to attain an adiabatic algorithm with a constant time complexity, we should increase the value of B to a constant. There are two ways to achieve our goal. The first way is that we assume an element , which is an arbitrary constant. Then we add to matrix element B, and the Hamiltonian matrix will be transformed as: A B+ = Hs (30) B+ C The value of the minimal gap is easily obtained as: (31) gmin = A − C2 + 4B + 2 Now, A − C and B + belong to O1, the minimal gap must belong to O1. More specially, we consider the following Hamiltonian as: = f sH + gsH + hsH

e Hs i f

3. THE SPEED MECHANISM OF ADIABATIC QUANTUM COMPUTATION In the previous section, we have exposed two specific adiabatic algorithms which can even achieve the O1 time complexity by considering nonlinear interpolation. What is the basic reason for the speedup of an adiabatic algorithm? In order to answer the questions, we consider the general adiabatic model as: Hs = f sHi + gsHf

(24)

where f s and gs are arbitrary interpolating functions of time, subjecting to the boundary conditions f 0 = g1 = 1

f 1 = g0 = 0

whose Hamiltonian matrix can be recast as: 2 b gs = A −abgs = B Hs = −abgs = B f s + a2 gs = C And the eigenvalues E can be easily obtained as: A + C ± A − C2 + 4B 2 E0 1 = 2 7264

(25)

(26)

(27)

(29)

(32)

where the range of functions f s, gs and hs is a constant, and

e = 0 0 ⊥ + 0 ⊥0 H (33) in which 0 ⊥ is the orthogonal state to the initial state 0 in the two-dimensional Hibernate space with 0 , 1 basic. By using Eq. (8)–(10), the Hamiltonian can be recast as: hs + −abg b 2 gs (34) Hs = hs + −abg f s + a2 gs The value of the minimal gap is easily obtained as: (35) g˜min = hs2 + a2 gsf s − abhsgs When the overlap a = 0 1 of initial Hamiltonian and final Hamiltonian tends to zero, it is easily to obtain the Eq. (35) belongs to a constant. This kind of mechanism which adds a constant element to the original Hamiltonian needs an extra proper Hamiltonian to achieve an O1 time complexity, and Eq. (11) is a specific realization of the kind of mechanism. The other way is that we can multiple a specific constant factor by the matrix element B of Eq. (26) to make the J. Comput. Theor. Nanosci. 13, 7262–7265, 2016

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The value of the minimal gap is easily obtained as: (37) gmin = A − C2 + 4wB2 Now, A−C and wB belong to O1, the minimal gap must belong to O1. However, according to Eq. (26), if we multiple a factor by the element B, which will change into · A. In order to keep the balance of Hamiltonian matrix, we should consider all elements by using this method, so we choose = 1/a. More generally, the following Hamiltonian can be considered as: Hs = f sHi + gsHf

(38)

where f s = 1 − s/a, and gs = s/a. By using Eqs. (8)–(10), the Hamiltonian as can be recast as: ⎛ ⎞ b 2 gs 1 · −abgs ⎟ ⎜ a a ⎟ (39) Hs = ⎜ ⎝1 f s + a2 gs ⎠ · −abgs a a We can easily get the minimal gap with Eqs. (21) and (22): gmin = 4f sgsmin 0≤s≤1 ∈ O1 (40) It is easy to see that the model as Eq. (19) is a realization of this kind of mechanism, which multiples a factor as 1/a to the original Hamiltonian. In this section, we have proposed the speedup mechanism to a general quantum adiabatic algorithms. We can easily conclude that the time complexity of adiabatic quantum algorithms is dependent on the relevant backdiagonal elements of the structure as 0 1 + 1 0 of local Hamiltonian. When we add an extra suitable efficient Hamiltonian or multiple a suitable factor to the general adiabatic model as Eq. (24), we increase the value of relative back-diagonal elements which is beneficial for the evolution in nature, so the time complexity can be reduced to O1. This conclusion is conforms to what the author of Ref. [14] stated, as the entanglement of initial

state and final state affects the speed of adiabatic quantum computation.

4. CONCLUSION In this paper, we conclude that the value of relevant elements in back-diagonal of the local Hamiltonian is the main factors affecting speedup of adiabatic quantum algorithms. We also have proposed two kinds of adiabatic models by adding an extra suitable efficient Hamiltonian or multiplexing a suitable factor to the general adiabatic model as Eq. (24), which can be generalized two kinds of common models to the Refs. [12] and [13] stated. This may help us better understand the adiabatic analogue of resources required in adiabatic evolution model, just like the number of operations does for conventional quantum computing. Acknowledgments: This work is supported by National Natural Science Foundation of China (Grant Nos. 61173050 and 61402188).

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Received: 25 February 2016. Accepted: 26 March 2016.

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value of B to be a constant. The Hamiltonian matrix will be transformed as: A wB Hs = (36) wB C