Optimization of the Memory Reset Rate of a Quantum Echo-State Network for Time Sequential Tasks

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Optimization of the Memory Reset Rate of a Quantum

Echo-State Network for Time Sequential Tasks

Riccardo Moltenia,c, Claudio Destric, Enrico Pratia,b,∗

aIstituto di Fotonica e Nanotecnologie, Consiglio Nazionale delle Ricerche, Piazza

Leonardo da Vinci 32, Milan, I-20133, Italy

bDipartimento di Fisica “Aldo Pontremoli”, Università degli Studi di Milano, Milan,

I-20133, Italy

cDipartimento di Fisica ”G. Occhialini”, Universitá degli Studi di Milano Bicocca, Milan,

I-20133, Italy

Abstract

Quantum reservoir computing is a class of quantum machine learning algorithms

involving a reservoir of an echo state network based on a register of qubits,

but the dependence of its memory capacity on the hyperparameters is still

rather unclear. In order to maximize its accuracy in time–series predictive

tasks, we investigate the relation between the memory of the network and

the reset rate of the evolution of the quantum reservoir. We benchmark

the network performance by three non–linear maps with fading memory on

IBM quantum hardware. The memory capacity of the quantum reservoir is

maximized for central values of the memory reset rate in the interval [0,1].

As expected, the memory capacity increases approximately linearly with the

number of qubits. After optimization of the memory reset rate, the mean

squared errors of the predicted outputs in the tasks may decrease by a factor

∼1/5with respect to previous implementations.

Keywords: quantum machine learning, quantum reservoir computing,

quantum echo state network

∗Corresponding author

Email address: enrico.prati@cnr.it (Enrico Prati)

Preprint submitted to Physics Letters A October 4, 2022

arXiv:2210.01052v1 [quant-ph] 3 Oct 2022

1. Introduction

The continuous development of quantum computers [1][2][22][24][23][25]

has found in machine learning one of its natural applications [3][4]. In

particular, the ﬁeld of quantum machine learning (QML) has been widely

explored mainly through hybrid quantum-classical algorithms which take

advantage of both quantum and classical parts during the training. Many of

such applications involve variational algorithms to solve classiﬁcation problems

[12][11], data reconstruction [3] and generative adversarial networks[5]. Diﬀe-

rently, quantum reservoir computing (QRC) deals primarily with sequential

temporal tasks where the fading memory of the system plays a crucial role.

Based on the concept of reservoir computing [7] originally embodied by

echo state networks [21] and liquid state machines [9] respectively, quantum

reservoir computing was ﬁrst proposed in 2017 by Fujii and Nakajima [6].

There, an encoding of a quantum reservoir on a register of qubits is proposed.

The implementation consisted of the simulation of a NMR machine [14, 10,

18]. Other proposals based on gate-model hardware have followed [15, 13].

In QRC, instead of usual rate-neurons, a register of qubits provides the

reservoir of an echo-state network [19, 16, 21, 20], thus realizing a quantum

ESN (qESN). By a suitable quantum evolution of the system, the reservoir

expands non-linearly the input values in a high dimensional system so that

it is suﬃcient to add a ﬁnal linear readout to make the network capable of

emulating any nonlinear map. Furthermore, the reservoir retains information

about past inputs in its quantum state enabling the network to tackle temporal

tasks for which memory is essential.

Quantum reservoir computing, and in particular quantum echo-state net-

works, appear as promising implementations of quantum machine learning

for temporal tasks, but a detailed analysis of the dependence of the memory

properties of a qESN on its hyperparameters is still largely unadressed. With

the spirit of ﬁlling such gap, we study the memory dependence on the reset

rate which regulates the probability with which the quantum reservoir resets

its internal state during the evolution.

Previously, some of the authors explored QML through quantum annealers

for unsupervised learning [3], quantum generative adversarial networks [5, 8],

quantum feed-forward neural networks [11] and quantum variational tensor

networks for multiclass supervised learning [12]. Here we explore supervised

learning of time-series by a qESN through an embodiment on gate model

quantum computers. The implementation of the qESN used in our work is

particularly suited for noisy intermediate-scale quantum (NISQ) computers.

In particular we take in consideration the subclass of universal QRs which

evolves with a resettable convex linear combination of complete trace preserving

(CPTP) maps, introduced in Ref. [15].

We investigate the memory of the reservoir by employing the memory

capacity (MC) metric [6]. The memory capacity quantiﬁes the short-term

memory of the reservoir by testing its ability to reproduce previously received

input values by using its current internal state. In particular, in Ref. [6] an

empirical evaluation suggested that a qESN of 5 qubits could exhibit a higher

memory capacity than a classical ESN of 500 nodes.

In our experiments we measured the memory capacity for diﬀerent values

of the reset rate , a fundamental hyperparameter of the QR subclass studied

in this work, which regulates the rate at which the quantum state of the

reservoir is reset during the time evolution, thus driving its fading memory

behaviour. We compare the memory capacity of diﬀerent reservoirs based on

3, 5 and 7 qubits, respectively.

Furthermore, the value of inﬂuences also the overall performance of the

network. We tested our qESN over three diﬀerent tasks, all of them requiring

the emulation of a nonlinear map with fading memory, that is with outputs

which depend increasingly less on previous inputs as time ﬂows. We employed

two, relatively simple, second-order maps and a more involved realistic map

consisting in a pair of diﬀerential equations related to the simpliﬁed dynamics

of an aircraft. For all of the tasks we injected into the system random input

variables and trained the network to predict the exact output values for input

data not used during the training, according to the speciﬁc map. For each

task we tested the implementation on a simulator with diﬀerent values of

. Next we selected the optimal one to run the experiments on quantum

hardware.

The qESN is realized via quantum circuits on both the Qiskit simulator

and the IBM ibmq_casablanca quantum hardware.

The results of the memory investigation show that the MC is maximized

for the central values of ∈[0,1], while it sensibly decreases for  > 0.5.

Furthermore for the optimal = 0.5the MC increases approximately linearly

with the number of qubits of the reservoir, conﬁrming the result obtained in

Ref. [6] with a diﬀerent evolution for the reservoir by using a simulated NMR

machine.

The qESN is able to predict outputs which match those expected with

low errors, measured as normalized mean squared errors (NMSE), for all the

three maps on which it was tested. In particular, by choosing the optimal 

of the network for each task, lower errors are obtained in comparison with

previous implementations in Ref. [15].

The lowest error in the tasks is obtained in correspondence of the values

for which the MC is maximum.

The paper is organized as follows. In Section 2 we summarize the theory

behind QRC and in particular quantum echo-state network while in Section 3

we describe the actual implementation as a circuit on a gate model quantum

computer. The results are shown in Section 4 along with the detailed descrip-

tion of the tasks and the comparison between diﬀerent settings for the reservoir

with various values of the reset rate and numbers of qubits. The conclusions

are drawn in the last Section 5.

2. Method

The key idea of QRC resides in using a N−qubits quantum register

as reservoir for an ESN. In such a way it is possible to exploit the high

dimensionality of the Hilbert space associated to Nqubits, which eﬀectively

acts as a reservoir with a large number of nodes. Speciﬁcally, the nodes of

the reservoir can be associated to the basis elements of the operator space,

which for a Nqubits system amount to 4Nelements. We consider as basis

elements the N-qubit Pauli operators deﬁned as:

Pi≡Pi1i2...iN=

k=1

σik, ik∈ {00,01,10,11}(1)

where {σ01, σ10, σ11}={X, Y, Z}are the three Pauli matrices while σ00 =I,

the 2×2identity matrix, and iis a natural number described by a binary

string ranging from 000...0to 111...1, build from the indexes of the Pauli

matrices in the tensor product.

For a Nqubit reservoir, a single operator Piis the tensor product between

Nmatrices σ∈ {σ01, σ10, σ11, σ00}. There are 4Npossible total combinations

of such matrices, so that i= 1,2, ...4N. Thanks to the orthogonality property

Tr(PiPj)=2Nδij (2)

a generic density matrix ρdescribing the quantum state of the system can

be written as

ρ=

i=1

xiPi(3)

where the real numbers

xi= 2−NTr(Piρ)(4)

can be regarded as classical coordinates, or nodes, of our qESN. Assuming

P4N=NN

k=1 Ik, one observes that the last element of Eq. (4) is constrained

by normalization, i.e. x4N= 1. Therefore, for a system of Nqubits there are

4N−1internal nodes of the reservoir.

We are interested in the learning of a temporal tasks where the input

signal is an ordered array {u(k)}L

k=1 of Lreal values with u(k)∈[0,1] for

every k. The input can be injected into the quantum reservoir with a pure

states encoding, for example by initializing at each time step the ﬁrst qubit

of the reservoir in the state |ψ1i=pu(k)|0i+p1−u(k)|1i[6]. Instead, we

follow the approach of Ref.[15] so to encode the input as a classical mixture.

Such an approach corresponds to initializing an ancilla qubit in the diagonally

mixed state:

ρa(u(k)) = u(k)|0ih0|+ (1 −u(k))|1ih1|(5)

The ancilla qubit is then coupled with the N−qubits register of the reservoir

in order to evolve the system with an input dependent map Tacting on

the density matrix ρRof the reservoir, initialized at t= 0 in the pure state

ρR(0) = |0ih0|⊗N. In particular, at each time step t=kthe density matrix

ρRevolves as:

ρR(k) = T(u(k))ρR(k−1) (6)

where T(u(k)) is a map which acts within the space of density matrices (i.e.

T:CN×CN→CN×CN) whose structure is described in Section 3. The

time evolution of the reservoir serves the purpose of nonlinearly expanding

the input signal, which is a sequence of scalar values, in a higher dimensional

space, i.e. the reservoir degree. Indeed, by choosing a suitable map T(u(k)),

the output signals extracted from the reservoir are nonlinear functions of the

input values [6], [14]. In this way the network is able to emulate a nonlinear

map by just adding a linear readout at its output.

After each injection of an input value and one–step evolution of the

reservoir state, Nsignals are extracted from the system and passed to the

linear readout layer. The expectation values of the Zoperators i.e. {hZii}N

i=1

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OptimizationoftheMemoryResetRateofaQuantumEcho-StateNetworkforTimeSequentialTasksRiccardoMoltenia,c,ClaudioDestric,EnricoPratia,b,aIstitutodiFotonicaeNanotecnologie,ConsiglioNazionaledelleRicerche,PiazzaLeonardodaVinci32,Milan,I-20133,ItalybDipartimentodiFisicaAldoPontremoli,UniversitàdegliStudid...

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Optimization of the Memory Reset Rate of a Quantum Echo-State Network for Time Sequential Tasks

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