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quantum

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(cid:104) (cid:105) −
how complicated quantum natural dynamics can be ex-
with respect to the input x. To measure a correlated
ploit as generalization and classification tasks. This can
operator, itisenoughtoapplyanentanglingunitaryop-
be viewed as a quantum version of extreme learning ma-
eration like CNOT gate Λ(X)= 0 0 I+ 1 1 X:
chine [39]. While it is an opposite direction to reser- | (cid:105)(cid:104) |⊗ | (cid:105)(cid:104) |⊗
voircomputing,wewillalsoseequantum circuit learning ψ Λ 1,2(X)Z 1Λ 1,2(X)ψ = ψ Z 1Z 2 ψ . (60)
(cid:104) | | (cid:105) (cid:104) | | (cid:105)
(QCL) [43], where the parameters in the complex dy-
Ingeneral,ann-qubitunitaryoperationU transformsthe
namics is further tuned in addition to the linear readout
observable Z under the conjugation into a linear combi-
weights. QCLisaquantumversionofafeedforwardneu-
nation of Pauli operators:
ral network. Finally, we will explain quantum reservoir
(cid:88)
computing by extending quantum extreme learning ma- U†Z 1U = αiP(i). (61)
chine for temporal learning tasks. i
7
(a) x0 x1
0 ✓
| i 0
0 ✓
| i 1
0 ✓
| i 2
0 ✓
| i 3
0 ✓
| i 4
0 ✓
| i 5
0 ✓
| i 6
0 ✓
| i 7
×２ 1.0
(b) training data leaned output
threshold at 0.5
FIG.2. (a)Thequantumcircuitforquantumextremelearn-
FIG.1. Theexpectationvalue(cid:104)Z(cid:105)oftheoutputofaquantum
ing machine. The box with theta indicates Y-rotations by
k
circuit as a function of the input (x ,x ).
0 1 angles θ . The red and blue boxes correspond to X and Z
k
rotations by random angles, Each dotted-line box represent
a two-qubit gate consisting of two controlled-Z gates and 8
Thus if you measure the output of the quantum circuit X-rotations and 4 Z-rotations. As denoted by the dashed-
after applying a unitary operation U, linebox,thesequenceofthe7dottedboxesisrepeatedtwice.
The readout is defined by a linear combination of (cid:104)Z (cid:105) with
i
UV(x)0 ⊗n, (62) constantbiasterm1.0andtheinput(x ,x ). (b)(Left)The
| (cid:105) 0 1
training data for a two-class classification problem. (Middle)
you can get a complex nonlinear output, which could Thereadoutafterlearning. (Right)Predictionfromtheread-
be represented as a linear combination of exponentially out with threshold at 0.5.
many nonlinear functions. U should be chosen to be ap-
propriatelycomplexwithkeepingexperimentalfeasibility
but not necessarily fine-tuned. becomesminimum. Aswementionedpreviously,thiscan
Toseehowtheoutputbehavesinanonlinearwaywith besolvedbyusingthepseudoinverse. Inshort,quantum
respect to the input, in Fig. 1, we will plot the output extreme learning machine is a linear regression on a ran-
Z for the input (x ,x ) and n = 8, where the inputs domlychosennonlinearbasisfunctions,whichcomefrom
0 1
(cid:104) (cid:105)
are fed into the quantum state by the Y-rotation with thequantumstateinaspaceofanexponentiallylargedi-
angles mension, namely quantum enhanced feature space. Fur-
thermore,undersometypicalnonlinearfunctionanduni-
θ 2k =karccos(√x 0) (63) taryoperationssettingstotransformtheobservables,the
θ =karccos(√x ) (64) outputinEq. (66)canapproximateanycontinuousfunc-
2k+1 1
tionoftheinput. Thispropertyisknownastheuniversal
on the 2kth and (2k+1)th qubits, respectively. Regard- approximation property (UAP), which implies that the
ingtheunitaryoperationU, randomtwo-qubitgatesare quantum extreme learning machine can handle a wide
sequentially applied on any pairs of two qubits on the class of machine learning tasks with at least the same
8-qubit system. power as the classical extreme learning machine [75].
Suppose the Pauli Z operator is measured on each Here we should note that a similar approach, quan-
qubit as an observable. Then we have tum kernel estimation, has been taken in Ref. [46]. In
quantum extreme learning machine, a classical feature
z = Z , (65)
i (cid:104) i (cid:105) vector φ i(x) Φ(x)Z i Φ(x) is extracted from observ-
for each qubit. In quantum extreme learning machine, ables on the q≡ ua(cid:104) ntum| fe| ature(cid:105) space Φ(x) V(x)0 ⊗n.
| (cid:105)≡ | (cid:105)
the output is defined by taking linear combination of Then linear regression is taken by using the classical
these n output: feature vector. On the other hand, in quantum kernel
estimation, quantum feature space is fully employed by
(cid:88)n using support vector machine with the kernel functions
y = w iz i. (66) K(x,x(cid:48)) Φ(x)Φ(x(cid:48)) , which can be estimated on a
≡ (cid:104) | (cid:105)
i=1 quantum computer. While classification power would be
Now the linear readout weights w are tuned so that better for quantum kernel estimation, it requires more
i
the quadratic loss function { } quantum computational costs both for learning and pre-
dictionincontrasttoquantumextremelearningmachine.
(cid:88)
L= (y(j) y¯(j))2 (67) In Fig. 2, we demonstrate quantum extreme learn-
−
ing machine for a two-class classification task of a two-