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quantum

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( )= i i( )i i, (27)  . 
M ··· | (cid:105)(cid:104) | ··· | (cid:105)(cid:104) |
r
i 11...1
the above non-selective measurement (forgetting about where r = 1/2n because of Tr[ρ] = 1. Moreover,
00...0
the measurement outcomes) is simply written by because P(i) is hermitian, r is a real vector. The super-
operator is a linear map for the operator, and hence
(ρ). (28) K
M can be represented as a matrix acting on the vector r:
In general, any physically allowed quantum operation
K ρ(cid:48) = (ρ) r(cid:48) =Kr, (37)
that maps a density operator to another can be repre- K ⇔
sented in terms of a set of operators K i being subject where the matrix element is given by
{ }
to K†K =I with an identity operator I:
i i Kij =Tr[P(i) (P(j))]/2n. (38)
(cid:88) K
(ρ)= K ρK†. (29)
K i i In this way, a density operator ρ and a quantum oper-
i ation on it can be represented by a vector r and a
The operators K i are called Kraus operators. matrixK K, respectively.
{ }
E. Vector representation of density operators III. MACHINE LEARNING AND RESERVOIR
APPROACH
Finally, we would like to introduce a vector represen-
tation of the above operator formalism. The operators In this section, we briefly introduce machine learning
themselves satisfy axioms of the linear space. Moreover, and reservoir approaches.
we can also define an inner product for two operators,
so-called Hilbert-Schmidt inner product, by
A. Linear and nonlinear regression
Tr[A†B]. (30)
The operators on the n-qubit system can be spanned by A supervised machine learning is a task to construct a
the tensor product of Pauli operators I,X,Y,Z ⊗n, model f(x) from a given set of teacher data x(j),y(j)
{ } { }
and to predict the output of an unknown input x. Sup-
n
(cid:79) pose x is a d-dimensional data, and f(x) is one dimen-
P(i)= σ . (31)
i2k−1i2k sional,forsimplicity. Thesimplestmodelislinearregres-
k=1
sion,whichmodelsf(x)asalinearfunctionwithrespect
where σ ij is the Pauli operators: to the input:
(cid:18) (cid:19) (cid:18) (cid:19)
1 0 0 1 d
I =σ = ,X =σ = , (cid:88)
00 0 1 10 1 0 f(x)= w ix i+w 0. (39)
i=1
(cid:18) (cid:19) (cid:18) (cid:19)
1 0 0 i Theweights w andbiasw arechosensuchthataner-
Z =σ 01 = 0 1 ,Y =σ 11 = i − 0 . (32) ror between f{ (xi )} and the out0 put of the teacher data, i.e.
−
5
loss, becomes minimum. If we employ a quadratic loss market,whicharecalledtemporaltasks,thenetworkhas
function for given teacher data x(j) ,y(j) , the prob- tohandletheinputdatathatisgiveninasequentialway.
lem we have to solve is as follows{ :{ i } } To do so, the recurrent neural network feeds the previ-
ous states of the nodes back into the states of the nodes
d
min(cid:88) ((cid:88) w x(j) y(j))2, (40) at next step, which allows the network to memorize the
{wi} i i − past input. In contrast, the neural network without any
j i=0
recurrency is called a feedforward neural network.
where we introduced a constant node x = 1. This cor- Letusformalizeatemporalmachinelearningtaskwith
0
responds to solving a superimposing equations: therecurrentneuralnetwork. Forgiveninputtimeseries
x L and target time series y¯ L , a temporal ma-
y=Xw, (41) { k }k=1 { k }k=1
chinelearningisatasktogeneralizeanonlinearfunction,
w soh lve ere dy bj y= usy in(j g), tX heji M= oox r( i ej -) P, ea nn rd osw ei p= seuw di o. T inh vi es rsc ean Xb +e y¯ k =f( {x j }k j=1). (47)
,
which can be defined from the singular value decomposi- For simplicity, we consider one-dimensional input and
tion of X=UDVT to be output time series, but their generalization to a multi-
dimensional case is straightforward. To learn the non-
X+ =VDUT. (42) linear function f( x k ), the recurrent neural network
{ j }j=1
canbeemployedasamodel. Supposetherecurrentneu-
Unfortunately, the linear regression results in a poor
ral network consists of m nodes and is denoted by m-
performance in complicated machine learning tasks, and
dimensional vector
any kind of nonlinearity is essentially required in the
model. A neural network is a way to introduce non-  r 
1
linearity to the model, which is inspired by the human r = . . . (48)
brain. In the neural network, the d-dimensional input  . 
data x is fed into N-dimensional hidden nodes with an r m
N d input matrix Win:
× To process the input time series, the nodes evolve by
Winx. (43)
r(k+1)=σ[Wr(k)+Winx ], (49)
k
Then each element of the hidden nodes is now processed
where W is an m m transition matrix and Win is an
by a nonlinear activation function σ such as tanh, which
×
m 1 input weight matrix. Nonlinearity comes from
is denoted by
×
the nonlinear function σ applied on each element of the
σ(Winx). (44) nodes. Theoutputtimeseriesfromthenetworkisdefined