Quantum artificial neural networks with applications - Semantic Scholar

Information Sciences 290 (2015) 1–6

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Quantum artificial neural networks with applications q Huaixin Cao a,⇑, Feilong Cao b, Dianhui Wang c a b c

College of Mathematics and Information Science, Shaanxi Normal University, Xi’an 710062, China Institute of Metrology and Computational Science, China Jiliang University, Hangzhou 310018, China Department of Computer Science and Computer Engineering, La Trobe University, Melbourne, VIC 3086, Australia

a r t i c l e

i n f o

Article history: Received 22 March 2014 Received in revised form 4 August 2014 Accepted 13 August 2014 Available online 23 August 2014 Keywords: Quantum system Quantum state Quantum artificial neural network Universal approximation theorem Schrödinger equation

a b s t r a c t Since simulations of classical artificial neural networks (CANNs) run on classical computers, the massive parallel processing speed advantage of a neural network is lost. A quantum computer is a computation device that makes direct use of quantum–mechanical phenomena while large-scale quantum computers will be able to solve certain problems much quicker than any classical computer using the best currently known algorithms. Combining the advantages of quantum computers and the idea of CANNs, we propose in this paper a new type of neural networks, named a quantum artificial neural network (QANN), which is presented as a system of interconnected ‘‘quantum neurons’’ which can compute quantum states from input-quantum states by feeding information through the network and can be simulated on quantum computers. To show the ability of approximation of a QANN, we prove a universal approximation theorem (UAT) which reads every continuous mapping that transforms n quantum states as a non-normalized quantum state can be uniformly approximated by a QANN. The UAT implies that QANNs would suggest a potential computing tool for dealing with quantum information. For instance, we prove that the state of a quantum system driven by a time-dependent Hamiltonian can be approximated uniformly by a QANN. This provides a possible way for finding approximate solution to a Schrödinger equation with a time-dependent Hamiltonian. Ó 2014 Elsevier Inc. All rights reserved.

1. Introduction In computer science and related fields, classical artificial neural networks (CANNs) are computational models and capable of machine learning and pattern recognition. A CANN is usually presented as a system of interconnected ‘‘neurons’’ which can compute values from inputs by feeding information through the network. Since simulations of artificial neural networks run on classical computers, the massive parallel processing speed advantage of a neural network is lost [22]. Clearly, it would be better to utilize the intrinsic physics of a physical system to perform the computation. Many efforts have been expended in this direction, using systems ranging from nonlinear optical materials to proteins [15]. At the same time, many other researchers have been exploring the possibility of building quantum computers [2,1,6]. By using arrays of coupled quantum dot molecules, a quantum cellular automata has been posed in [11], which provides a valuable concrete example of quantum computation in which a number of fundamental issues come to light. An architecture for a quantum neural computer has

q This subject was supported by the NNSF Grants of China (Nos. 11171197, 11371012, 61272023) and the FRF for the Central Universities (No. GK201301007) ⇑ Corresponding author. E-mail addresses: [email protected] (H. Cao), fl[email protected] (F. Cao), [email protected] (D. Wang).

http://dx.doi.org/10.1016/j.ins.2014.08.033 0020-0255/Ó 2014 Elsevier Inc. All rights reserved.

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H. Cao et al. / Information Sciences 290 (2015) 1–6

been proposed in [14] in light of the real time evolution of quantum dot molecules, and simulations have proved that such an architecture can perform any classical logic gate, which can be used to calculate a purely quantum gate (a unitary matrix). In this paper, we propose an analog of a CANN, named a quantum artificial neural network (QANN), and prove that every continuous mapping that maps n quantum states as a non-normalized quantum state can be uniformly approximated by a QANN. As an application, we show that the state of a quantum system driven by a time-dependent Hamiltonian can be approximated uniformly by a QANN. 2. Construction of a quantum artificial neural network Let Cd ¼ fðz1 ; z2 ; . . . ; zd ÞT : zk 2 Cðk ¼ 1; 2; . . . ; dÞg be the d-dimensional complex Hilbert space with the inner product

hxjyi ¼

d X xk yk

ð2:1Þ

k¼1

for all elements jxi ¼ ðx1 ; x2 ; . . . ; xd ÞT and jyi ¼ ðy1 ; y2 ; . . . ; yd ÞT , where xk denotes the conjugate of the complex number xk and

hxj ¼ jxiy ¼ ðx1 ; x2 ; . . . ; xd Þ:

ð2:2Þ

The norm induced by the inner product above reads

kjxik ¼ hxjxi

1=2

¼

d X jxk j2

!1=2 ð2:3Þ

:

k¼1

In quantum mechanics, a d-dimensional quantum system is described by the Hilbert space Cd and quantum states of the system are described by unit vectors in Cd . Let Sd ðCÞ be the set of all quantum states of the quantum system Cd and define

Snd ðCÞ ¼ fðjx1 i; jx2 i; . . . ; jxn iÞT : jxk i 2 Sd ðCÞðk ¼ 1; 2; . . . ; nÞg;

ð2:4Þ

which is clearly a closed bounded subset of the Hilbert space n

ðCd Þ ¼ fðjx1 i; jx2 i; . . . ; jxn iÞT : jxk i 2 Cd ðk ¼ 1; 2; . . . ; nÞg  Cnd : First, we define a mapping T :

Snd ðCÞ

!R

2nd

ð2:5Þ

as

T jxi ¼ ðRejx1 i; Imjx1 i; Rejx2 i; Imjx2 i; . . . ; Rejxn i; Imjxn iÞT T

for all jxi ¼ ðjx1 i; jx2 i; . . . ; jxn iÞ 2

Snd ðCÞ.

ð2:6Þ

pffiffiffi We call T the realization mapping. Clearly, kT jxik ¼ n for all jxi 2 Snd ðCÞ. Put

pffiffiffi  Dn2d ðRÞ ¼ T ðSnd ðCÞÞ  jyi 2 R2nd : kjyik ¼ n : Then Dn2d ðRÞ is a compact subset of R2nd , and T : Snd ðCÞ ! Dn2d ðRÞ becomes a homeomorphism. Next, we let rk : R ! Rðk ¼ 1; 2; . . . ; MÞ be M real-valued functions. For real

numbers

ðiÞ aðiÞ j;k ; hj;k ði ¼ 1; 2;

ðiÞ

j ¼ 1; 2; . . . ; N; k ¼ 1; 2; . . . ; MÞ and vectors jwj;k iði ¼ 1; 2; j ¼ 1; 2; . . . ; N; k ¼ 1; 2; . . . ; MÞ in R2nd , we define a mapping Qk :

Snd ðCÞ

! C as follows:

Qk ðjxiÞ ¼

N  X



ð1Þ ð1Þ ð2Þ ð2Þ ð2Þ að1Þ j;k rk ðhwj;k jT jxi þ hj;k Þ þ iaj;k rk ðhwj;k jT jxi þ hj;k Þ

ð2:7Þ

j¼1

for all jxi 2 Snd ðCÞ where hwj jT jxi denotes the inner product of jwj i and T jxi. Last, we define Q : Snd ðCÞ ! CM as ðiÞ

ðiÞ

QðjxiÞ ¼ ðQ1 ðjxiÞ; Q2 ðjxiÞ; . . . ; QM ðjxiÞÞT

ð2:8Þ

Snd ðCÞ.

for all jxi 2 We call such a mapping Q a quantum artificial neural network (QANN). If we use fje1 i; je2 i; . . . ; jeM ig to denote the canonical basis for CM , then the QANN above can be rewritten as

QðjxiÞ ¼

M X N  X



ð1Þ ð1Þ ð2Þ ð2Þ ð2Þ aj;k rk ðhwð1Þ j;k jT jxi þ hj;k Þ þ iaj;k rk ðhwj;k jT jxi þ hj;k Þ jek i:

k¼1 j¼1

Put ðiÞ yj;k

¼ rk

n X t¼1

! ðiÞ hwj;k ðtÞjT

jxt i þ

ðiÞ hj;k

ðiÞ

; jak i ¼

N X

ðiÞ ðiÞ aj;k yj;k jek i:

j¼1

Then a QANN can be illustrated by Figs. 1 and 2 below.

ð2:9Þ

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H. Cao et al. / Information Sciences 290 (2015) 1–6

ðiÞ

Fig. 1. The output jak i of a QANN, where i ¼ 1; 2; k ¼ 1; 2; . . . ; M.

Fig. 2. The output Q ðjxiÞ of a QANN.

3. Universal approximation theorem and applications Recall that a function

r : R ! R is called a sigmoidal function if it satisfies

lim rðtÞ ¼ 0; lim rðtÞ ¼ 1:

t!1

ð3:1Þ

t!þ1

Let CðSnd ðCÞ; CM Þ be the set of all continuous mappings from Snd ðCÞ into CM . Theorem 3.1. Let rk : R ! Rðk ¼ 1; 2; . . . ; MÞ be continuous sigmoidal functions. Then for every f 2 CðSnd ðCÞ; CM Þ and every positive number e, there exists a QANN QðjxiÞ such that

8jxi 2 Snd ðCÞ:

kf ðjxiÞ  QðjxiÞk < e;

Proof. Let f 2 CðSnd ðCÞ; CM Þ and

ð3:2Þ

e > 0. Then f can be written as T

f ðjxiÞ ¼ ðf 1 ðjxiÞ; f 2 ðjxiÞ; . . . ; f M ðjxiÞÞ ; where f k :

Snd ðCÞ

ð1Þ g k ðjyiÞ

for all jyi 2

Since D :¼

! Cðk ¼ 1; 2; . . . ; MÞ are all continuous. Define functions

¼ Ref k ðT

Dn2d ðRÞ.

f k ðjxiÞ ¼

ð3:3Þ

Then

ð1Þ g k ðT

Dn2d ðRÞ

1

ð2Þ jyiÞ; g k ðjyiÞ

ð1Þ gk

jxiÞ þ

and

ð2Þ ig k ðT

ð2Þ gk

¼ Imf k ðT

1

ð1Þ ð2Þ gk ; gk

:

Dn2d ðRÞ

! R as follows:

jyiÞ

ð3:4Þ

are all continuous and

jxiÞ; 8jxi 2 Snd ðCÞ:

ð3:5Þ

is a compact subset of Rn , there exists a positive number r such that

D  ½r; r2nd ¼ fðx1 ; . . . ; xn ÞT 2 Rn : r 6 xj 6 rðj ¼ 1; 2; . . . ; 2ndÞg: Thus, K :¼ ð2rÞ1 D þ jei  ½0; 12nd , where jei ¼ ð1=2; 1=2; . . . ; 1=2ÞT 2 R2nd . Define uðjxiÞ ¼ ð2rÞ1 jxi þ jei, then we get a ðiÞ

ðiÞ

homeomorphism u : D ! K with the continuous inverse u1 ðjyiÞ ¼ ð2rÞjyi  jei. Since hk :¼ g k  u1 ði ¼ 1; 2; k ¼ 2nd

1; 2; . . . ; MÞ are all continuous real-valued functions on the compact subset K of ½0; 1

, the Tietze extension theorem ðiÞ

[10] implies that there exist real-valued continuous functions F k on ½0; 12nd such that F k ðjyiÞ ¼ hk ðjyiÞ for all jyi 2 K. For each i ¼ 1; 2; k ¼ 1; 2; . . . ; M, we see from Theorem 2 in [8] that there exists a CANN of the form ðiÞ

ðiÞ

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H. Cao et al. / Information Sciences 290 (2015) 1–6

ðiÞ

Lk ðjyiÞ ¼

N X

ðiÞ ðiÞ aðiÞ j;k rk ðhyj;k jyi þ cj;k Þ

ð3:6Þ

j¼1

such that

e

jF k ðjyiÞ  Lk ðjyiÞj < pffiffiffiffiffiffiffiffi ; 8jyi 2 ½0; 12nd ; 2M ðiÞ

ðiÞ

where aj;k ; cj;k 2 R; yj;k 2 Rn . Put Gk ¼ Lk  u. Then for every jxi 2 D, we have jyi ¼ uðjxiÞ 2 K  ½0; 12nd and ðiÞ

ðiÞ

ðiÞ

Gk ðjxiÞ ¼

ðiÞ

ðiÞ

ðiÞ

Nk X

ðiÞ ðiÞ aj;k rk ðhwðiÞ j;k jxi þ hj;k Þði ¼ 1; 2Þ;

ð3:7Þ

j¼1 ðiÞ

ðiÞ

ðiÞ

ðiÞ

ðiÞ

where jwj;k i ¼ ð2rÞ1 jyj;k i 2 Rn ; hj;k ¼ hyj;k jei þ cj;k 2 R. Furthermore, we have ðiÞ

ðiÞ

ðiÞ

e

ðiÞ

jg k ðjxiÞ  Gk ðjxiÞj ¼ jF k ðuðjxiÞÞ  Lk ðuðjxiÞÞj < pffiffiffiffiffiffiffiffi 2M

ð3:8Þ

for all jxi 2 D. Put

Qk ðjxiÞ ¼ Gk ðT jxiÞ þ iGk ðT jxiÞ; 8jxi 2 Snd ðCÞ; ð1Þ

ð2Þ

then

Qk ðjxiÞ ¼

Nk  X



ð1Þ ð1Þ ð2Þ ð2Þ ð2Þ að1Þ j;k rk ðhwj;k jT jxi þ hj;k Þ þ iaj;k rk ðhwj;k jT jxi þ hj;k Þ

j¼1

and for all jxi 2 Snd ðCÞ, Eq. (3.8) yields that

kf ðjxiÞ  QðjxiÞk2 ¼

M X

jf k ðjxiÞ  Qk ðjxiÞj2 ¼

k¼1

M  X

 ð1Þ ð1Þ ð2Þ ð2Þ jg k ðT jxiÞ  Gk ðT jxiÞj2 þ jg k ðT jxiÞ  Gk ðT jxiÞj2 < e2 :

k¼1

For the construction of Q k , please refer to Fig. 3 below. Now, we take a positive integer N > maxfN 1 ; N 2 ; . . . ; N d g and define ðiÞ ðiÞ aðiÞ j;k ¼ hj;k ¼ 0; jwj;k i ¼ 0; 8N k < j 6 N:

Then we get a QANN

QðjxiÞ ¼ ðQ1 ðjxiÞ; Q2 ðjxiÞ; . . . ; Qd ðjxiÞÞT satisfying

Qk ðjxiÞ ¼

N  X



ð1Þ ð1Þ ð2Þ ð2Þ ð2Þ að1Þ j;k rk ðhwj;k jT jxi þ hj;k Þ þ iaj;k rk ðhwj;k jT jxi þ hj;k Þ

j¼1

and

8jxi 2 Snd ðCÞ:

kf ðjxiÞ  QðjxiÞk < e; This completes the proof.

h

Remark 3.1. From the proof of Theorem 3.1, we see that ð2Þ

ð2Þ

ð2Þ

(1) When the function f is real, i.e., f k ’s are all real-valued, we can choose the coefficients aj;k ¼ 0; jwj;k i ¼ 0 and hj;k ¼ 0 for all j; k so that QðjxiÞ is real. (2) When n ¼ 1, the resulted QANN can uniformly approximate any continuous mapping f that maps a quantum state jxi into a non-normalized state f ðjxiÞ. Especially, every quantum gate (unitary matrix) U can be uniformly approximated

ð1Þ

ð2Þ

Fig. 3. The construction of the function Q k , explicitly, Q k ¼ Lk uT þ iLk uT .

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H. Cao et al. / Information Sciences 290 (2015) 1–6

by a QANN. Thus, it is possible to learn an unknown quantum gate from QANNs based on some given inputs and related outputs. (3) When the approximated function f is time-dependent, i.e., a function from Snd ðCÞ  ½0; T into Cn such that f ð; tÞ is conðiÞ

ðiÞ

ðiÞ

tinuous for each t 2 ½0; T, the coefficients aj;k ; jwj;k i and hj;k are all time-dependent for all i; j; k so that QðjxiÞ is timedependent, denoted by Qðjxi; tÞ. More explicitly, we arrive at the following theorem.

Corollary 3.1. Let rk : R ! R be continuous sigmoidal functions ðk ¼ 1; 2; . . . ; MÞ. Then for every mapping f : Snd ðCÞ  ½0; T ! CM such that f ð; tÞ is continuous for each t 2 ½0; T and every positive number e, there exists a time-dependent QANN:

Qðjxi; tÞ ¼

M X N  X

ð1Þ ð1Þ að1Þ j;k ðtÞrk ðhwj;k ðtÞjT jxi þ hj;k ðtÞÞ

k¼1 j¼1

 ð2Þ ð2Þ ð2Þ þiaj;k ðtÞrk ðhwj;k ðtÞjT jxi þ hj;k ðtÞÞ jek i such that

kf ðjxi; tÞ  Qðjxi; tÞk < e;

8jxi 2 Snd ðCÞ; 8t 2 ½0; T;

ðiÞ ðiÞ j;k ðtÞ; jwj;k ðtÞi

ð3:9Þ

ðiÞ hj;k ðtÞ

where the coefficients a and are real-valued functions of t. Next, we discuss an application of Corollary 3.1. Consider the following Schrödinger equation [17]

_ ¼ HðtÞjwðtÞi; t 2 ½0; T; ijwðtÞi

ð3:10Þ

_ is the derivative of where HðtÞ is the Hermitian Hamiltonian of the quantum system C , i.e., a d  d Hermitian matrix, jwðtÞi d

the quantum state jwðtÞi 2 S1d ðCÞ w.r.t. time t. It is well-known that for every initial state jw0 i 2 S1d ðCÞ, Eq. (3.10) has a unique solution jwðtÞi 2 S1d ðCÞ with jwð0Þi ¼ jw0 i. Also, this solution can be written as jwðtÞi ¼ UðtÞjw0 i for all t 2 ½0; T where UðtÞ is a d  d unitary matrix. In the case where the Hamiltonian HðtÞ is independent of t, i.e., HðtÞ  H; UðtÞ ¼ eitH and so jwðtÞi ¼ eitH jw0 i for all t 2 ½0; T. Usually, the quantum gate UðtÞ exists, but it is difficult to be found. In quantum mechanics, the quantum adiabatic theorem [3] enables us to obtain an approximate solution to Schrödinger equation Eq. (3.10) provided that HðtÞ evolves slowly enough. This theorem is one of the most important theories and has been discussed recently [4,5,7,9,12,13,16,18–21,23,24]. As an application of Corollary 3.1, we have the following corollary, which suggests an approximation to solution jwðtÞi to Eq. (3.10) by a time-dependent QANN. Corollary 3.2. Let rk : R ! R be continuous sigmoidal functions ðk ¼ 1; 2; . . . ; dÞ; jw0 i 2 S1d ðCÞ any quantum state of Cd and let jwðtÞi ¼ UðtÞjw0 i be the solution to Eq. (3.10) with jwð0Þi ¼ jw0 i. Then for every positive number e, there exists a time-dependent QANN:

Qðjxi; tÞ ¼

d X N  X



ð1Þ ð1Þ ð2Þ ð2Þ ð2Þ að1Þ j;k ðtÞrk ðhwj;k ðtÞjT jxi þ hj;k ðtÞÞ þ iaj;k ðtÞrk ðhwj;k ðtÞjT jxi þ hj;k ðtÞÞ jek i

k¼1 j¼1

such that

8jxi 2 Snd ðCÞ; 8t 2 ½0; T:

kUðtÞjxi  Qðjxi; tÞk < e;

ð3:11Þ

Especially,

kjwðtÞi  Qðjw0 i; tÞk < e; ðiÞ

8t 2 ½0; T; ðiÞ

ð3:12Þ

ðiÞ

where the parameters aj;k ðtÞ; jwj;k ðtÞi and hj;k ðtÞ are real-valued functions of t. Proof. Use Corollary 3.1 for f ðjxi; tÞ ¼ UðtÞjxi.

h

4. Conclusions As an analog of the classical artificial neural network (CANN), a new type of neural networks, named quantum artificial neural network (QANN), is introduced in this paper. Universal approximation theorem (Theorem 3.1) shows that every continuous mapping that maps continuously n quantum states (d-dimensional complex unit vectors) as a non-normalized quantum state (an M-dimensional complex vector) can be uniformly approximated by the proposed quantum artificial neural networks. This implies that a QANN may be a potentially and powerfully computational tool for dealing with quantum information. Corollary 3.2 tells us that states of a quantum system driven by a time-dependent Hamiltonian can be approximated uniformly by a time-dependent QANN. Moreover, it is capable, at least in principle, of performing computations that cannot be done, classically.

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Acknowledgments The authors would like to thank the editor and the referees of this paper for their kind comments and valuable suggestions. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24]

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