19

sively used to study v arious physical systems. The repre-

sentational power and the en tanglement properties of the

RBM states have been inv estigated and the RBM repre-

sentation of diﬀerent systems suc h as the Ising Mo del,

T oric co de, graph states and stabiliser codes have been

constructed 117 . Also, the representational power of the

other neural netowork arc hitectures such as the Deep

Boltzmann Machine (DBM) 120 are under active in ves-

tigation.

A. Variational representation of man y b ody

systems in RBM networks

A neural network can represen t a quantum state of a

physical system in terms of its netw ork parameters 105 .

The Restricted Boltzmann Machine architecture consists

of a visible layer of N neurons and a hidden la yer of M

neurons. The neurons of the visible layer and hidden

layers are connected but there are no in tra-layer connec-

tions. As the spin of the neurons in the RBM network can

have the v alues ± 1, the spins of the neurons of the visible

layer can be mapped to the spins of the physical system

they represent. Moreover a set of weights is assigned to

the visible ( a i for the i th visible neuron), hidden ( b i for

the i th hidden neuron) and to the couplers connecting

them( W ij for the coupler connecting the i th visible neu-

ron with the j th hidden neuron) 121 . Then, wave function

ansatz for the N-dimensional quantum state of spin v ari-

able conﬁguration S = { s i } N

i =1 would be given by 105 , 121 .

ψ M ( S , W ) = X

{ h i }

e P i a i s i + P j b j h j + P ij W ij s i h j (21)

where s i and h i denote the spins of the visible and

hidden neurons respectively and the whole state is given

by the superposition of all the spin conﬁguration states

with ψ ( S ) as the amplitude of the |S i state 121 , 122 .

| ψ i = X

S

ψ ( S ) |Si (22)

XIII. CLASSICAL SIMULATION OF

QUANTUM COMPUT ATION USING NEURAL

NETWORKS

Since the neural networks are able to represen t vari-

ous quantum states eﬃciently , a natural question to be

posed is whether they can also simulate various quan-

tum algorithms. Interestingly , the netw orks are also

able to simulate the action of v arious quantum gates.

This has been investigated in the DBM and the RBM

architectures 120 , 123 . As mentioned before, the represen-

tation of a quantum state by a neural net work depends on

its network parameters. Thus, the action of various gates

FIG. 13. The structure of a Restricted Boltzmann

Machine 121 . The spin conﬁguration of the N visible neu-

rons is represented by { s i } N

i =1 ( s i is the spin value of the i t h

neuron). Also, there are M hidden neurons in one more layer

(called the hidden layer). The coupler connecting the i th vis-

ble neuron with the j th hidden neuron has the weight W ij .

Howev er, there are no intra-lay er connections. The wav efun-

tion ansatz for the system represented by this net work is given

by Eqn. 21 .

can be simulated by appropriately changing the net work

parameters in a way that the new quan tum state repre-

sented by the net work with the new parameters is the

same as would hav e b een obtained by applying the quan-

tum gate to the initial quantum state. Also in a recen t

work, the methods to prepare speciﬁc initial states initial

states in RBM analogous to those used as initial states

while implementing a quantum algorithms in a quan tum

circuit model has been discussed 121 . The prepared states

were shown to eﬃcien tly simulate the action of the Pauli

X gate. These results have opened up a great possibility

of solving various quan tum mechanical problems using

neural networks. F uture investigations in this direction

may include the implementations of quan tum algorithms

in various neural net work architectures and the exploita-

tion of the machine learning techniques to ac hieve higher

accuracy in solving the quantum mechanical problems 121 .

XIV. IMPLEMENTA TION OF QUANTUM

MACHINE LEARNING ALGORITHMS ON

QUANTUM COMPUTERS

In this section we discuss the implementation of some

quantum machine learning algorithms with the help of

quantum logic and quantum gates.

Earlier this year H. Liu’s 124 paper ﬁrst proposed a

quantum algorithm to obtain the classical gradients.

can be regarded as the inner product of two vec-

tors ( p ( x 1 ; w ) − y 1 , ..., p ( x N ; w ) − y N ) and ( x j

1 , ..., x j

N ).

T o achieve this, their quantum algorithm consists of

two steps: generate an intermediate quantum state

1

√ N Σ N

i =1 | i i| p ( x i ; w ) i mainly based on amplitude estima-

tion; (2)perform swap test to obtain O w j in the classical

form. Then the parameters w is updated according to

the iterative rules via simple calculations. The entire al-

gorithm process is shown in Fig. 13.