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wesley7137
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quantum

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in terms of a 1 m readout weights by
FinallytheoutputisextractedbyanoutputweightWout ×
(1 N dimensional matrix): y =Woutr(k). (50)
× k
Woutσ(Winx). (45) Thenthelearningtaskistodeterminetheparametersin
Win,W,andWout byusingtheteacherdata x ,y¯ L
The parameters in Win and Wout are trained such that { k k }k=1
so as to minimize an error between the teacher y¯ and
the error between the output and teacher data becomes { k }
the output y of the network.
minimum. While this optimization problem is highly { k }
nonlinear, a gradient based optimization, so-called back
propagation, can be employed. To improve a representa-
C. Reservoir approach
tion power of the model, we can concatenate the linear
transformation and the activation function as follows:
While the representation power of the recurrent neu-
(cid:16) (cid:16) (cid:17)(cid:17)
Woutσ W(l) σ W(1)σ(Winx) , (46) ral network can be improved by increasing the number
···
of the nodes, it makes the optimization process of the
whichiscalledmulti-layerperceptronordeepneuralnet- weights hard and unstable. Specifically, the back prop-
work. agation based methods always suffer from the vanishing
gradient problem. The idea of reservoir computing is to
resolvethisproblembymappinganinputintoacomplex
B. Temporal task higher dimensional feature space, i.e., reservoir, and by
performing simple linear regression on it.
The above task is not a temporal task, meaning that Let us first see a reservoir approach on a feedforward
theinputdataisnotsequentialbutgivensimultaneously neural network, which is called extreme learning ma-
like the recognition task of images for hand written lan- chine [44]. The input data x is fed into a network like
guage, pictures and so on. However, for a recognition multi-layerperceptron, whereallweightsarechosenran-
of spoken language or prediction of time series like stock domly. The states of the hidden nodes at some layer is
6
now regarded as basis functions of the input x in the A. Quantum extreme learning machine
feature space:
The idea of quantum extreme learning machine lies in
φ 1(x),φ 2(x),...,φ N(x) . (51) using a Hilbert space, where quantum states live, as an
{ }
enhanced feature space of the input data. Let us denote
Nowtheoutputisdefinedasalinearcombinationofthese thesetofinputandteacherdataby x(j),y¯(j) . Suppose
{ }
we have an n-qubit system, which is initialized to
(cid:88)
w iφ i(x)+w 0 (52) 0 ⊗n. (54)
i | (cid:105)
In order to feed the input data into quantum system, a
and hence the coefficients are determined simply by the unitary operation parameterized by x, say V(x), is ap-
linear regression as mentioned before. If the dimen- plied on the initial state:
sion and nonlinearity of the the basis functions are high
V(x)0 ⊗n. (55)
enough, we can model a complex task simply by the lin- | (cid:105)
ear regression. For example, if x is one-dimensional data and normal-
Theechostatenetworkissimilarbutemploysthereser- ized to be 0 x 1, then we may employ the Y-basis
e−iθY≤ wit≤
voir idea for the recurrent neural network [21, 22, 45], rotation h an angle θ =arccos(√x):
which has been proposed before extreme learning ma-
e−iθY 0 =√x0 +√1 x1 . (56)
chine appeared. To be specific, the input weights Win | (cid:105) | (cid:105) − | (cid:105)
and weight matrix W are both chosen randomly up to The expectation value of Z with respect to e−iθY 0 be-
| (cid:105)
an appropriate normalization. Then the learning task is comes
done by finding the readout weights Wout to minimize
Z =2x 1, (57)
the mean square error (cid:104) (cid:105) −
and hence is linearly related to the input x. To enhance
(cid:88)
(y y¯ )2. (53) the power of quantum enhanced feature space, the input
k k
−
could be transformed by using a nonlinear function φ:
k
(cid:112)
θ =arccos( φ(x)). (58)
This problem can be solved stably by using the pseudo
inverse as we mentioned before. The nonlinear function φ could be, for example, hyper-
Forbothfeedforwardandrecurrenttypes,thereservoir bolic tangent, Legendre polynomial, and so on. For
approach does not need to tune the internal parameters simplicity, below we will use the simple linear input
ofthenetworkdependingonthetasksaslongasitposses θ =arccos(√x).
sufficientcomplexity. Therefore,thesystem,towhichthe Ifweapplythesameoperationoneachofthenqubits,
machine learning tasks are outsourced, is not necessarily we have
the neural network anymore, but any nonlinear physical
V(x)0 ⊗n =(√x0 +√1 x1 )⊗n
system of large degree of freedoms can be employed as | (cid:105) | (cid:105) − | (cid:105)
a reservoir for information processing, namely, physical (cid:88) (cid:89)(cid:114) x ik
=(1 x)n/2 i ,...,i .
reservoir computing [23–36]. − 1 x | 1 n (cid:105)
i1,...,in k −
Therefore, we have coefficients that are nonlinear with
respect to the input x because of the tensor product
IV. QUANTUM MACHINE LEARNING ON
structure. Still the expectation value of the single qubit
NEAR-TERM QUANTUM DEVICES
operator Z on the kth qubit is 2x 1. However, if we
k
−
measureacorrelatedoperatorlikeZ Z , wecanobtaina
1 2
In this section, we will see QRC and related frame- second order nonlinear output
works for quantum machine learning. Before going deep
into the temporal tasks done on QRC, we first explain Z 1Z 2 =(2x 1)2 (59)