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Quantum reservoir computing: a reservoir approach toward quantum machine
learning on near-term quantum devices
Keisuke Fujii1,∗ and Kohei Nakajima2,†
1Graduate School of Engineering Science, Osaka University,
1-3 Machikaneyama, Toyonaka, Osaka 560-8531, Japan.
2Graduate School of Information Science and Technology,
The University of Tokyo, Bunkyo-ku, 113-8656 Tokyo, Japan
(Dated: November 11, 2020)
Quantum systems have an exponentially large degree of freedom in the number of particles and
hence provide a rich dynamics that could not be simulated on conventional computers. Quantum
reservoircomputingisanapproachtousesuchacomplexandrichdynamicsonthequantumsystems
asitisfortemporalmachinelearning. Inthischapter,weexplainquantumreservoircomputingand
0202
relatedapproaches,quantumextremelearningmachineandquantumcircuitlearning,startingfrom
apedagogicalintroductiontoquantummechanicsandmachinelearning. Allthesequantummachine
learningapproachesareexperimentallyfeasibleandeffectiveonthestate-of-the-artquantumdevices.
voN
I. INTRODUCTION a useful task like machine learning, now applications of
suchanear-termquantumdeviceforusefultasksinclud-
01 Over the past several decades, we have enjoyed expo- ing machine leanring has been widely explored. On the
nential growth of computational power, namely, Moore’s other hand, quantum simulators are thought to be much
law. Nowadays even smart phone or tablet PC is much easier to implement than a full-fledged universal quan-
]hp-tnauq[ more powerful than super computers in 1980s. Even tum computer. In this regard, existing quantum sim-
though, people are still seeking more computational ulators have already shed new light on the physics of
power,especiallyforartificialintelligence(machinelearn- complex many-body quantum systems [9–11], and a re-
ing), chemical and material simulations, and forecasting stricted class of quantum dynamics, known as adiabatic
complexphenomenalikeeconomics,weatherandclimate. dynamics, has also been applied to combinatorial opti-
Inadditiontoimprovingcomputationalpowerofconven- misation problems [12–15]. However, complex real-time
tional computers, i.e., more Moore’s law, a new genera- quantum dynamics, which is one of the most difficult
tionofcomputingparadigmhasbeenstartedtobeinves- tasksforclassicalcomputerstosimulate [16–18]andhas
1v09840.1102:viXra tigatedtogobeyondMoore’slaw. Amongthem, natural greatpotentialtoperformnontrivialinformationprocess-
computing seeks to exploit natural physical or biological ing,isnowwaitingtobeharnessedasaresourceformore
systems as computational resource. Quantum reservoir general purpose information processing.
computing is an intersection of two different paradigms
Physical reservoir computing, which is the main sub-
of natural computing, namely, quantum computing and
ject throughout this book, is another paradigm for ex-
reservoir computing.
ploiting complex physical systems for information pro-
Regarding quantum computing, the recent rapid ex-
cessing. In this framework, the low-dimensional input is
perimentalprogressincontrollingcomplexquantumsys-
projectedtoahigh-dimensionaldynamicalsystem,which
tems motivates us to use quantum mechanical law as a
is typically referred to as a reservoir, generating tran-
new principle of information processing, namely, quan-
sient dynamics that facilitates the separation of input
tum information processing [2, 3]. For example, certain
states [19]. If the dynamics of the reservoir involve both
mathematical problems, such as integer factorisation,
adequate memory and nonlinearity [20], emulating non-
which are believed to be intractable on a classical com-
linear dynamical systems only requires adding a linear
puter, are known to be efficiently solvable by a sophis-
andstaticreadoutfromthehigh-dimensionalstatespace
ticatedly synthesized quantum algorithm [4]. Therefore,
of the reservoir. A number of different implementations
considerableexperimentalefforthasbeendevotedtoreal-
of reservoirs have been proposed, such as abstract dy-
ising full-fledged universal quantum computers [5, 6]. In
namical systems for echo state networks (ESNs) [21] or
the near feature, quantum computers of size >50 qubits
modelsofneuronsforliquidstatemachines[22]. Theim-
with fidelity > 99% for each elementary gate would ap-
plementationsarenotlimitedtoprogramsrunningonthe
peartoachievequantumcomputationalsupreamcybeat-
PCbutalsoincludephysicalsystems,suchasthesurface
ingsimulationonthe-state-of-the-artclassicalsupercom-
of water in a laminar state [23], analogue circuits and
puters [7, 8]. While this does not directly mean that a
optoelectronic systems [24–29], and neuromorphic chips
quantum computer outperforms classical computers for
[30]. Recently, it has been reported that the mechani-
cal bodies of soft and compliant robots have also been
successfully used as a reservoir [31–36]. In contrast to
∗ fujii@qc.ee.es.osaka-u.ac.jp the refinements required by learning algorithms, such as
† k nakajima@mech.t.u-tokyo.ac.jp in deep learning [37], the approach followed by reservoir
2
computing, especially when applied to real systems, is onacomplexd-dimensionalsystemCd,wherethesymbol
to find an appropriate form of physics that exhibits rich is called ket and indicates a complex column vector.
|·(cid:105)
dynamics, thereby allowing us to outsource a part of the Similarly, is called bra and indicates a complex row
(cid:104)·|
computation. vector, and they are related complex conjugate,
Quantum reservoir computing (QRC) was born in the
ψ = ψ † =(cid:0) c∗ c∗ (cid:1) . (2)
marriage of quantum computing and physical reservoir (cid:104) | | (cid:105) 1 ··· d
computingabovetoharnesscomplexquantumdynamics
With this notation, we can writte an inner product of
as a reservoir for real-time machine learning tasks [38].
two quantum state ψ and φ by ψ φ . Let us define
Since the idea of QRC has been proposed in Ref. [38], | (cid:105) | (cid:105) (cid:104) | (cid:105)
an orthogonal basis
itsproof-of-principleexperimentaldemonstrationfornon
temporal tasks [39] and performance analysis and im-      0 
1 0
provement [40–42] has been explored. The QRC ap- .
p a n pnr e ro ord i va n ic q dgh u eaat atno tt e buq n ru m ota i aon ds nt tu a p[m t 7 ie c1 t–t upa 7 rrs 3 eek ]p .s oa frs Iau Qntc Rih to h Cna iss ah b nq a dosu oa rb kn eet le c au hn tm ea dr pet to ac eem prn , po t rwg l oyr e aa cgp w hah eir ly s- l |1 (cid:105)=      0 . . . . . .       ...
.  .