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quantum

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to the recurrent neural network, r(cid:48) = tanh(Wr). How-
virtual
nodes
ever, there is no nonlinearity such as tanh in each quan-
input } tum operation W. Instead, the time evolution W can
output
be changed according to the external input x , namely
true nodes k
x’i (t ) LR W xk, which contrasts to the conventional recurrent neu-
ralnetworkwheretheinputisfedadditivelyWr+Winx .
k
hidden nodes 1 This allows the quantum reservoir to process the input
0.8
information x nonlinearly, by repetitively feeding the
k
0.6 { }
(b) virtual nodes input.
0.4
Suppose the input x is normalized such that 0
0.2 { k } ≤
0 ……… x k ≤ 1. As an input, we replace a part of the qubits to
0 .1 8 Input virtual node the quantum state. The density operator is given by
000 ... r0246 u0000 .... d01 2468 01 Inp Iu n rt p eInu nt oy vutk
r¯ 1 nl o2 e3 0000 .... 2468 4 0000 .... 01 2468 0000 .... 01 2468 l i p ae Iu ndta puor t {ut Wwei lg ,h s t ρ xk = I+(2x 2k −1)Z . (75)
t e 5 }l,v
virtual node Forsimplicity,belowweconsiderthecasewhereonlyone
qubitisreplacedfortheinput. CorrespondingmatrixS
xk
FIG.3. (a)Quantumreservoircomputing. (b)Virtualnodes is given by
and temporal multiplexing.
(cid:26) (cid:27)
I+(2x 1)Z
(S xk)ji =Tr P(j) 2k − ⊗Tr replace[P(i)] /2N,
proposedinRefs[67,68]. Quantumcircuitlearningwith
parameterized quantum circuits has been already exper- where Tr replace indicates a partial trace with respect to
imentally demonstrated on superconducting qubit sys- the replaced qubit. With this definition, we have
tems [46, 69] and a trapped ion system [70]. ρ(cid:48) =Tr [ρ] ρ r(cid:48) =S r. (76)
replace xk xk
⊗ ⇔
The unitary time evolution, which is necessary to ob-
C. Quantum reservoir computing tain a nonlinear behavior with respect to the input valu-
able x , is taken as a Hamiltonian dynamics e−iHτ for
k
Now we return to the reservoir approach and extend a given time interval τ. Let us denote its representation
quantum extreme learning machine from non temporal on the vector space by U :
τ
tasks to temporal ones, namely, quantum reservoir com-
ρ(cid:48) =e−iHτρeiHτ r(cid:48) =U r. (77)
puting [38]. We consider a temporal task, which we ex- τ
⇔
plained in Sec. IIIB. The input is given by a time series
Then, a unit time step is written as an input-depending
x L and the purpose is to learn a nonlinear temporal
{ k }k linear transformation:
function:
y¯ =f( {x }k j). (73) r((k+1)τ)=U τS xkr(kτ). (78)
k j
To this end, the target time series y¯ L is also pro- where r(kτ) indicates the hidden nodes at time kτ.
{ k }k=1 Since the number of the hidden nodes are exponen-
vided as teacher.
tially large, it is not feasible to observe all nodes from
Contrast to the previous setting with non temporal
experiments. Instead, a set of observed nodes r¯ M ,
tasks, we have to fed input into a quantum system se- { l }l=1
which we call true nodes, is defined by a M 4N matrix
quentially. This requires us to perform an initializa-
×
R,
tion process during computation, and hence the quan-
tumstateofthesystembecomesmixedstate. Therefore, (cid:88)
r¯ l(kτ)= R liri(kτ). (79)
in the formulation of QRC, we will use the vector rep-
i
resentation of density operators, which was explained in
Sec. IIE. The number of true nodes M has to be a polynomial
In the vector representation of density operators, the in the number of qubits N. That is, from exponentially
quantum state of an N-qubit system is given by a vector many hidden nodes, a polynomial number of true nodes
ina4N-dimensionalrealvectorspace,r R4N . InQRC, are obtained to define the output from QR (see Fig. 3
∈
similarly to recurrent neural networks, each element of (a)):
the 4N-dimensional vector is regarded as a hidden node
(cid:88)
of the network. As we seen in Sec. IIE, any physical y = Woutr¯(kτ), (80)
k l l
operation can be written as a linear transformation of l
the real vector by a 4N 4N matrix W:
× where W out is the readout weights, which is obtained
r(cid:48) =Wr. (74) by using the training data. For simplicity, we take the
10
single-qubit Pauli Z operator on each qubit as the true That is, the system learns the input of the next step.
nodes, i.e., Once the system successfully learns y¯ , by feeding the
k
output into the input of the next step of the system, the
r¯ l =Tr[Z lρ], (81) system evolves autonomously.
Here we employ the following target time series from
sothatifthereisnodynamicsthesenodessimplyprovide
chaotic attractors: (i) Lorenz attractor,
a linear output (2x 1) with respect to the input x .
k k
−