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wesley7137
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quantum

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dimensional input 0 x ,x 1. Class 0 and 1 are The gradient of the loss function can be obtained as fol-
0 1
≀ ≀
defined to be those being subject to (x 0.5)2+(x lows:
0 1
βˆ’ βˆ’
0.5)2 0.15 and > 0.15, respectively. The linear read-
out we≀ ights w are learned with 1000 randomly chosen βˆ‚ L( Ο† )= βˆ‚ (cid:88) ( A( Ο† ,x(j)) y(j))2
i k k
{ } βˆ‚Ο† { } βˆ‚Ο† (cid:104) { } (cid:105)βˆ’
trainingdataandpredictionisperformedwith1000ran- l l j
domly chosen inputs. The class 0 and 1 are determined (cid:88) βˆ‚
whether or not the output y is larger than 0.5. Quan- = 2( A( Ο† k ,x(j)) y(j)) A( Ο† k ,x(j)) .
(cid:104) { } (cid:105)βˆ’ βˆ‚Ο† (cid:104) { } (cid:105)
l
tum extreme learning machine with an 8-qubit quantum j
circuit shown in Fig. 2 (a) succeeds to predict the class
Therefore if we can measure the gradient of the observ-
with 95% accuracy. On the other hand, a simple linear
able A( Ο† ,x(j)) , the loss function can be minimized
regression for (x ,x ) results in 39%. Moreover, quan- (cid:104) { k } (cid:105)
0 1 according to the gradient descent.
tum extreme learning machine with U = I, meaning no
If the unitary operation u(Ο† ) is given by
entangling gate, also results in poor, 42%. In this way, k
the feature space enhanced by quantum entangling op- u(Ο† )=W eβˆ’i(Ο†k/2)Pk, (71)
k k
erations is important to obtain a good performance in
quantum extreme learning machine. whereW isanarbitraryunitary,andP isaPaulioper-
k k
ator. Then the partial derivative with respect to the lth
parameter can be analytically calculated from the out-
B. Quantum circuit learning puts A( Ο† ,x(j)) with shifting the lth parameter by
k
(cid:104) { } (cid:105)
(cid:15) [43, 61]:
Β±
Inthesplitofreservoircomputing,dynamicsofaphys-
βˆ‚
icalsystemisnotfine-tunedbutnaturaldynamicsofthe A( Ο† ,x(j))
k
systemisharnessedformachinelearningtasks. However, βˆ‚Ο† l(cid:104) { } (cid:105)
ifweseethe-state-of-the-artquantumcomputingdevices, = 1 ( A( Ο† ,...,Ο† +(cid:15),Ο† ,... ,x(j))
1 l l+1
theparameterofquantumoperationscanbefinelytuned 2sin(cid:15) (cid:104) { } (cid:105)
as done for universal quantum computing. Therefore it A( Ο† ,...,Ο† (cid:15),Ο† ,... ,x(j)) ).
1 l l+1
is natural to extend quantum extreme learning machine βˆ’(cid:104) { βˆ’ } (cid:105)
bytuningtheparametersinthequantumcircuitjustlike
feedfoward neural networks with back propagation. By considering the statistical error to measure the ob-
Using parameterized quantum circuits for supervised servable A , (cid:15) should be chosen to be (cid:15) = Ο€/2 so as to
machineleaningtaskssuchasgeneralizationofnonlinear make the(cid:104) d(cid:105) enominator maximum. After measuring the
functions and pattern recognitions have been proposed partial derivatives for all parameters Ο† and calculating
k
in Refs. [43, 47], which we call quantum circuit learning. thegradientofthelossfunctionL( Ο† ),theparameters
k
Letusconsiderthesamesituationwithquantumextreme are now updated by the gradient d{ esce} nt:
learning machine. The state before the measurement is
βˆ‚
given by ΞΈ(m+1) =ΞΈ(m) Ξ± L( Ο† ). (72)
UV(x)0 βŠ—n. (68) l l βˆ’ βˆ‚Ο† l { k }
| (cid:105)
In the case of quantum extreme learning machine the The idea of using the parameterized quantum cir-
unitary operation for a nonlinear transformation with cuits for machine learning is now widespread. After
respect to the input parameter x is randomly chosen. the proposal of quantum circuit learning based on the
However, the unitary operation U may also be parame- analytical gradient estimation above [43] and a similar
terized: idea [47], several researches have been performed with
(cid:89) various types of parameterized quantum circuits [48–52]
U( Ο† )= u(Ο† ). (69)
{ k } k andvariousmodelsandtypesofmachinelearninginclud-
k ing generative models [54, 55] and generative adversarial
Thereby, the output from the quantum circuit with re- models[56–58]. Moreover,anexpressionpowerofthepa-
spect to an observable A rameterized quantum circuits and its advantage against
A( Ο† ,x) = 0βŠ—nV†(x)U( Ο† )†Z U( Ο† )V(x)0 βŠ—n classicalprobabilisticmodelshavebeeninvestigated[59].
k k i k
(cid:104) { } (cid:105) (cid:104) | { } { } | (cid:105) Experimentally feasible ways to measure an analytical
becomes a function of the circuit parameters Ο† in ad-
{ k } gradientoftheparameterizedquantumcircuitshavebeen
ditiontotheinputx. Thentheparameters Ο† istuned
{ k } investigated[60–62]. Anadvantageofusingsuchagradi-
soastominimizetheerrorbetweenteacherdataandthe
ent for the parameter optimization has been also argued
output, for example, by using the gradient just like the
in a simple setting [63], while the parameter tuning be-
output of the feedforward neural network.
comes difficult because of the vanishing gradient by an
Let us define a teacher dataset x(j),y(j) and a
{ } exponentially large Hilbert space [64]. Software libraries
quadratic loss function
for optimizing parameterized quantum circuits are now
(cid:88)
L( Ο† )= ( A( Ο† ,x(j)) y(j))2. (70) developing [65,66]. Quantummachinelearningonnear-
k k
{ } (cid:104) { } (cid:105)βˆ’
term devices, especially for quantum optical systems, is
j
9
(a) Quantum Reservoir Computing Now we see, from Eq. (74), a time evolution similar