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quantum

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startingfromapedagogicalintroductiontoquantumme- d
chanics and machine learning.
a quantum state in the d-dimensional system can be de-
Therestofthispaperisorganizedasfollows. InSecII,
scribed simply by
we will provide a pedagogical introduction to quantum
mechanicsforthosewhoarenotfamiliartoitandfixour
d
notation. In Sec III, we will briefly mention to several (cid:88)
ψ = c i . (4)
i
machinelearningtechniqueslike,linearandnonlinearre- | (cid:105) | (cid:105)
i=1
gressions, temporal machine learning tasks and reservoir
computing. In Sec IV, we will explain QRC and related Thestateissaidtobeasuperpositionstateof i . Theco-
| (cid:105)
approaches, quantum extreme learning machine [39] and efficients c i arecomplex,andcalledcomplexprobability
{ }
quantum circuit learning [43]. The former is a frame- amplitudes. If we measure the system in the basis i ,
{| (cid:105)}
work to use quantum reservoir for non temporal tasks, weobtainthemeasurementoutcomeiwithaprobability
thatis, theinput isfedinto aquantum system, andgen-
p = iψ 2 = c 2, (5)
eralization or classification tasks are performed by a lin- i i
|(cid:104) | (cid:105)| | |
ear regression on a quantum enhanced feature space. In
andhencethecomplexprobabilityamplitudeshavetobe
the latter, the parameters of the quantum system is fur-
normalized as follows
therfine-tunedviathegradientdescentbymeasuringan
analytically obtained gradient, just like the back propa- d
(cid:88)
gationforfeedforwardneuralnetworks. RegardingQRC, ψ ψ 2 = c i 2 =1. (6)
|(cid:104) | (cid:105)| | |
wewillalsoseechaotictimeseriespredictionsasdemon- i=1
strations. Sec.Visdevotedtoconclusionanddiscussion.
In other words, a quantum state is represented as a nor-
malized vector on a complex vector space.
Suppose the measurement outcome i corresponds to a
II. PEDAGOGICAL INTRODUCTION TO certain physical value a , like energy, magnetization and
i
QUANTUM MECHANICS
soon,thentheexpectationvalueofthephysicalvaluable
is given by
In this section, we would like to provide a pedagogical
(cid:88)
introduction to how quantum mechanical systems work a ip i = ψ Aψ A , (7)
(cid:104) | | (cid:105)≡(cid:104) (cid:105)
for those who are not familiar to quantum mechanics. i
If you already familiar to quantum mechanics and its
where we define an hermitian operator
notations, please skip to Sec. III.
(cid:88)
A= a i i, (8)
i
| (cid:105)(cid:104) |
i
A. Quantum state
whichiscalledobservable,andhastheinformationofthe
measurement basis and physical valuable.
A state of a quantum system is described by a state
The state vector in quantum mechanics is similar to
vector,
a probability distribution, but essentially different form
  it, since it is much more primitive; it can take complex
c
1 valueandismorelikeasquarerootofaprobability. The
|ψ (cid:105)=  . . .   (1) unique features of the quantum systems come from this
c property.
d
3
B. Time evolution D. Density operator
Thetimeevolutionofaquantumsystemisdetermined Next, I would like to introduce operator formalism of
byaHamiltonianH,whichisahermitianoperatoracting theabovequantummechanics. Thisdescribesanexactly
on the system. Let us denote a quantum state at time the same thing but sometimes the operator formalism
t = 0 by ψ(0) . The equation of motion for quantum would be convenient. Let us consider an operator ρ con-
| (cid:105)
mechanics, so-called Schr¨odinger equation, is given by structed from the state vector ψ :
| (cid:105)
∂ ρ= ψ ψ . (16)
i ψ(t) =H ψ(t) . (9) | (cid:105)(cid:104) |
∂t| (cid:105) | (cid:105)
If you chose the basis of the system i for the matrix
{| (cid:105)}
This equation can be formally solved by representation, then the diagonal elements of ρ corre-
sponds the probability distribution p = c 2 when the
i i
ψ(t) =e−iHt ψ(0) . (10) system is measured in the basis i . The| re| fore the op-
| (cid:105) | (cid:105) {| (cid:105)}
erator ρ is called a density operator. The probability
Therefore the time evolution is given by an operator distribution can also be given in terms of ρ by
e−iHt, which is a unitary operator and hence the norm
of the state vector is preserved, meaning the probability p =Tr[i iρ], (17)
i
| (cid:105)(cid:104) |
conservation. In general, the Hamiltonian can be time
whereTristhematrixtrace. Anexpectationvalueofan
dependent. Regarding the time evolution, if you are not