# Entangled Quantum Memristors

Shubham Kumar  International Center of Quantum Artificial Intelligence for
Science and Technology (QuArtist)
and Physics Department, Shanghai University, 200444 Shanghai, China  Francisco
A. Cárdenas-López  International Center of Quantum Artificial Intelligence for
Science and Technology (QuArtist)
and Physics Department, Shanghai University, 200444 Shanghai, China  Narendra.
N. Hegade  International Center of Quantum Artificial Intelligence for Science
and Technology (QuArtist)
and Physics Department, Shanghai University, 200444 Shanghai, China  Xi Chen
Department of Physical Chemistry, University of the Basque Country UPV/EHU,
Apartado 644, E-48080 Bilbao, Spain  Francisco Albarrán-Arriagada
<EMAIL_ADDRESS>International Center of Quantum Artificial
Intelligence for Science and Technology (QuArtist)
and Physics Department, Shanghai University, 200444 Shanghai, China  Enrique
Solano<EMAIL_ADDRESS>International Center of Quantum Artificial
Intelligence for Science and Technology (QuArtist)
and Physics Department, Shanghai University, 200444 Shanghai, China
Department of Physical Chemistry, University of the Basque Country UPV/EHU,
Apartado 644, E-48080 Bilbao, Spain  IKERBASQUE, Basque Foundation for
Science, Plaza Euskadi, 5, 48009 Bilbao, Spain  Kipu Quantum, Kurwenalstrasse
1, 80804 Munich, Germany  Gabriel Alvarado Barrios<EMAIL_ADDRESS>International Center of Quantum Artificial Intelligence for Science and
Technology (QuArtist)
and Physics Department, Shanghai University, 200444 Shanghai, China

###### Abstract

We propose the interaction of two quantum memristors via capacitive and
inductive coupling in feasible superconducting circuit architectures. In this
composed system the input gets correlated in time, which changes the dynamic
response of each quantum memristor in terms of its pinched hysteresis curve
and their nontrivial entanglement. In this sense, the concurrence and
memristive dynamics follow an inverse behavior, showing maximal values of
entanglement when the hysteresis curve is minimal and vice versa. Moreover,
the direction followed in time by the hysteresis curve is reversed whenever
the quantum memristor entanglement is maximal. The study of composed quantum
memristors paves the way for developing neuromorphic quantum computers and
native quantum neural networks, on the path towards quantum advantage with
current NISQ technologies.

##  I Introduction

The memristor was proposed as the fourth basic circuit element with a flux-
charge relation IEEE.1971 ; IEEE.1976 . The first experimental memristor was
claimed by HP labs Nat.2008 , which turned into a growing interest in the
field. A key feature is its nonlinear current/voltage response exhibiting
memory effects, which is used in the area of neuromorphic computing
Schuman2017 . The goal is to develop a computational paradigm based on neuron-
like devices to mimic the human brain capabilities, allowing to simulate from
learning processes Song.2019 ; Lin.2019 ; Gao.2020 ; Xia.2020  and artificial
neural communication Laughlin2003 , to simulate biological processes such as
brain synapses Adv.2020 ; Nat.2020 .

The development of neuromorphic computing has focused into the design of
single memristive systems and structures as the cross-bar arrays, where
memristors work as switches according to its resistance value Borghetti2010 .
These structures have been used in dense arrays of memristors Nat.2015 ;
NatComm.2018  for realizing brain-inspired devices Nat.2015 ; Adv.2020 ;
Davies.2019 . Nevertheless, little attention has been paid into studying the
direct coupling between these system, and only few proposals about
parallel/series, and wireless connections have been considered Budhathoki2013
; Luo2017 . Therefore, characterizing the dynamics of coupled memristors will
advance neuromorphic computing.

The implementation of neural networks and brain-inspired quantum algorithm on
quantum hardware gives rise to the field of neuromorphic quantum computing
(NQC), where quantum features such as superposition and entanglement lead to
calculation speedup Markovica2020 . There exists two NQC approaches, the first
one aims at implementing neural networks on quantum processors through
parametrized quantum circuits using classical/quantum machine learning
techniques Cao.2017 ; Taccinoj.2019 ; Li.2020 ; Kristensen.2021 . The second
approach, closer to the classical neuromorphic computing, attempts to take
advantage of the system dynamics for implementing neuronal and synaptic
processes, specifically a quantum memristor. There are several proposals for
implementing a quantum memristor in quantum platforms such as photonics
sanz2018 ; materials.2020 ; Spagnolo.2021 , quantum dots patent.2017 , and
superconducting circuits SciRep.2016 ; PRapp.2016 ; PRapp.2018 ;
Phys.Rev.Applied.2014 ; Sci.Rep.2016 , as well as memristor-inspired
experiments for quantum advantage Zhou2020arXiv . In the latter, the
memristive behavior appears due to the non-linearity of the Josephson
junctions (JJs), which contains a current term depending on its
superconducting phase Joseph.1962 . It is possible to increase this current
contribution by coupling two Josephson junctions on a closed-loop, forming a
device known as a conductance asymmetric SQUID Phys.Rev.Applied.2014 ;
Sci.Rep.2016 . These works have focused on developing a single quantum
memristor, and not much attention has been paid to develop and characterize
the systems formed by coupled quantum memristors.

In this work, we show how to entangle two quantum memristors (QMs) via
inductive and capacitive couplings within current state of the art
superconducting circuit technologies. We observe that the input signals of the
memristors get correlated in time, which induces the shrinking and expansion
of the hysteresis curves. This behavior is closely related to the rise and
decay of entanglement in the coupled system. We find that the entanglement and
memristive dynamics follow an inverse behavior, where maximal values of
entanglement happen when the form factor of the hysteresis curve is minimal
and vice versa. In addition, we find that the coupling causes the direction of
the hysteresis curve to be reversed as the relative phase between current and
voltage changes in time. Interestingly, this change in direction of the
hysteresis curve happens whenever the entanglement between the memristors is
maximal.

The organization of this article is as follows. In Sec. II we review the
mathematical model of a classical memristor and its generalization to
memristive systems. In Sec. III, we describe the composite system of quantum
memristors. The main part of the manuscript is devoted to the the analysis of
the capacitive interaction between the quantum memristors. In Sec. IV, we
describe the system dynamics via coupled differential equations obtained by
solving the master equation of the coupled open quantum systems. In Sec. V, we
show the dynamical reponses of each quantum memristor in the coupled system.
In Sec. VI, we characterize the perfomance of each quantum memristor using a
quantity called as the Form factor. In Sec. VI, we show the entanglement and
memristive dynamics of the coupled system and investigate the relationship
between them. We devote the Sec. VI to derive and explain the main conclusions
of our work. In the Appendices, we provide the technical details on the
derivation of the Hamiltonian (Appendix A) and the memristive equations
(Appendix B). Furthermore we have included the results of the inductive
interaction (Appendix C) and the interaction caused by using an inductor and a
capacitor simultaneously (Appendix D).

##  II Classical Memristor

The memristor is a two terminal device with resistive memory effects, in which
the resistance depends on the history of the voltage passed through the
device. An ideal memristor is fully described by its state-dependent Ohm’s law
IEEE.1971 . Accordingly, based on the intrinsic variables of the device, the
memristor can be classified as charge-controlled or flux-controlled with the
constitutive relations

$\displaystyle V(t)$ $\displaystyle=$ $\displaystyle
M(q)~{}I(t)\quad\text{with}\quad M(q)=\frac{d\phi}{dq}\,,$ (1a) $\displaystyle
I(t)$ $\displaystyle=$ $\displaystyle G(\phi)~{}V(t)\quad\text{with}\quad
G(\phi)=\frac{dq}{d\phi}\,.$ (1b)

Here, $I(t)$, $V(t)$, $q(t)$, and $\phi(t)$ stand for the current, voltage,
charge, and flux on the device. Furthermore, $M(q)$ and $G(\phi)$ are termed
as memristance and memductance, with units of electrical resistance and
electrical conductance, respectively. The generalization of the device is
termed as memristive systems IEEE.1976 . The main difference between them
corresponds to the explicit dependence of the voltage (current) on the
memductance (conductance). The dynamics of the memristive system is given by
the state dependent Ohm’s law, together with the evolution of the state
variable provided by

$\displaystyle V(t)=M(x,I,t)~{}I(t),\quad f(x,I,t)=\frac{dx}{dt},$ (2a)
$\displaystyle I(t)=G(x,V,t)V(t)\quad g(x,V,t)=\frac{dx}{dt}.$ (2b)

Here, $M(x,I,t)$ and $G(x,V,t)$ are the memristance and memductance, which are
now functions of the internal variable $x$ ruled by $f(x,I,t)$ and $g(x,V,t)$,
respectively.

Figure 1: Schematic illustration of two coupled superconducting quantum
memristors, capacitively and inductively, respectively. (a) Each QM consists
of a CA-SQUID connected to an inductance $L_{\ell}$ in a loop threaded by a
signal $\phi_{d\ell}(t)$. We describe the CA-SQUID as a smaller loop
interrupted by two different junctions and threaded by a static flux
$\phi_{s}$, which cancels the critical current. (b) Effective RLC circuit
describing the system, where the resistance is modelled as a time-dependent
admittance. We depict the current with red arrows.

##  III The model

Let us study the circuit depicted in Fig. 1a, consisting of a pair of quantum
memristors (QMs) coupled with either a capacitor or an inductor. We form each
QM with a conductance asymmetric SQUID (CA-SQUID) connected in parallel to an
inductor $L_{\ell}$ forming a closed-loop threaded by an external signal
$\phi_{d\ell}(t)$. The CA-SQUID consists of a small loop embedded by two
different JJs threaded by a static DC magnetic flux. Each Josephson junction
forming the CA-SQUID has two current contributions. The first is a non-
dissipative current due the tunnelling of Cooper-pairs proportional to the
critical current $I_{C\ell}=2eE_{J}/\hbar$ Golubov2004RMP . The second is a
dissipative contribution due to the tunneling of quasiparticles proportional
to the conductance Catelani2011prl ; Catelani2011prb . The memristive behavior
arises when the latter current dominates over the former. This is achieved
when the junctions satisfy $G_{N1}/G_{N2}=\Delta_{1}/\Delta_{2}$ PRL1963 ;
Phys.Rev.Applied.2014  ($G_{Ni}$ is the junction conductance, $\Delta_{i}$ is
the energy bandgap) and the loop is threaded with a flux $\phi_{s\ell}=\pi$.
In the limit $\Delta_{1}>>\Delta_{2}$, the first junction acts as a shunt.
Then, we model the CA-SQUID as a capacitor coupled to a time-dependent
admittance Sci.Rep.2016 . Therefore, the whole system corresponds to a pair of
coupled RLC resonators with time-dependent resistance (see Fig. 1b) described
by the Hamiltonian (see Appendix A for derivation)

$\displaystyle\hat{\mathcal{H}}=\sum_{\ell=1,2}E_{C_{\ell,\ell}}\hat{n}^{2}_{\ell}+\frac{E_{L_{\ell,\ell}}}{2}\hat{\phi}_{\ell}^{2}+E_{C_{1,2}}\hat{n}_{1}\hat{n}_{2}-E_{L_{1,2}}\hat{\phi}_{1}\hat{\phi}_{2}\,.$

Here, $\hat{n}_{\ell}$ and $\hat{\phi}_{\ell}$ are the dimensionless number
and phase operators respectively, $E_{C}=2e^{2}\hat{C}^{-1}$ is the charge
energy matrix and $\hat{C}^{-1}$ is the inverse of the capacitance matrix.
Moreover, $E_{L}=\varphi_{0}^{2}\hat{L}^{-1}$ is the inductive energy matrix
where $\hat{L}^{-1}$ is the inverse of the inductive matrix, and
$\varphi_{0}=\hbar/2e$ is the reduced quantum flux. Furthermore, $n_{\ell}$
and $\phi_{\ell}$ correspond to the dimensionless charge and phase operators
of the $\ell$th QM, respectively. The first two terms of Eq. (III) are the
free Hamiltonian of each memristive system, whereas the last two terms are the
capacitive and inductive coupling interaction, respectively.

For simplicity we redefine the operators in terms of creation and annihilation
operators,

$\displaystyle\hat{n}_{\ell}$ $\displaystyle=$
$\displaystyle\frac{i}{4g_{\ell}}(\hat{a}_{\ell}^{\dagger}-\hat{a}_{\ell}),$
$\displaystyle\hat{\phi}_{\ell}$ $\displaystyle=$ $\displaystyle
2g_{\ell}(\hat{a}_{\ell}^{\dagger}+\hat{a}_{\ell}),$ (4)

with $g_{\ell}=(E_{C_{\ell,\ell}}/32E_{L_{\ell,\ell}})^{1/4}$. Then, we obtain

$\displaystyle\hat{\mathcal{H}}=\sum_{\ell=1,2}\hbar\omega_{\ell}\hat{a}_{\ell}^{\dagger}\hat{a}_{\ell}-\sqrt{\omega_{1}\omega_{2}}(\alpha-\beta)(\hat{a}_{1}^{\dagger}\hat{a}_{2}+\hat{a}_{1}\hat{a}_{2}^{\dagger}),$
(5)

where $\omega_{\ell}=\sqrt{2E_{C_{\ell,\ell}}E_{L_{\ell,\ell}}}/\hbar$ is the
frequency of the $\ell$th QM. Moreover, parameter
$\alpha=E_{L_{1,2}}/\sqrt{E_{L1}E_{L2}}$ and parameter
$\beta=E_{C_{1,2}}/\sqrt{E_{C1}E_{C2}}$ is the inductive and charge energy
ratios, respectively. We describe the energy loss of the admittance as a
quasiparticle bath at zero temperature whose dynamics follows the time-
dependent master equation ($\hbar=1$)

$\displaystyle\dot{\hat{\rho}}(t)$ $\displaystyle=$ $\displaystyle
i\big{[}\mathcal{\hat{H}},{\hat{\rho}}\big{]}+\sum_{\ell=1,2}\frac{\Gamma_{\ell}(t)}{2}\bigg{[}\mathcal{L}_{\ell}\hat{\rho}\mathcal{L}_{\ell}^{{\dagger}}-\frac{1}{2}\\{\mathcal{L}_{\ell}^{{\dagger}}\mathcal{L}_{\ell},\hat{\rho}\\}\bigg{]},$
(6)

here, $\mathcal{\hat{H}}$ is the Hamiltonian shown in Eq. (5), while
$\mathcal{L}_{\ell}=\sqrt{\Gamma_{\ell}(t)}\hat{a}_{\ell}$ is the collapse
operator describing the quasiparticle tunneling of the $\ell$th QMs at rate
$\Gamma_{\ell}(t)=\lvert\left\langle{0}\right|\sin(\hat{\phi}_{\ell}/2)\left|{1}\right\rangle\lvert^{2}S_{\rm{qp}}(\omega_{\ell})$
Catelani2011prl ; Catelani2011prb . Where,
$S_{\rm{qp}}(\omega_{\ell})\approx\omega_{\ell}$ is the spectral density of
the quasiparticle bath Catelani2011prl ; Catelani2011prb . Due to the fluxoid
quantization rule on the outer loop, it is possible to write
$\Gamma_{\ell}(t)$ in terms of the external magnetic flux $\phi_{d\ell}(t)$.
This means
$\Gamma_{\ell}(t)=g_{\ell}^{2}\omega_{\ell}\exp(-g_{\ell}^{2})[(1+\cos[\phi_{d\ell}(t)])/2]$
Sci.Rep.2016 , we note that $g_{\ell}$ is proportional to the zero-point
fluctuation of the phase operator $\hat{\phi}_{\ell}$. Moreover,
$\phi_{d\ell}(t)=\phi_{0,\ell}+A\sin(\omega_{\ell}t)$ is the external magnetic
flux threading the outer loop, where we consider a sinusoidal modulation for
observing the memristive behavior of the coupled system. The phase dependence
in the decay rate is due to the quasiparticle current term PRB.2011 , which is
intrinsic to JJs. This allows to control the quasiparticle dynamics via the
external flux and thus have a flux controlled memristive device. Since, we are
interested in investigating the memristive behaviour originating from the
phase dependent conductance, we omit the decoherence induced by other sources
such as inductive and radiative losses. Works on superconducting circuits have
studied dominant quasiparticle tunneling PRB.2011 ; Martinis.2009 ; Vool.PRL
and recent works have progressed towards mitigating the radiative decay
Martinis.npj  which justifies our assumption.

Figure 2: Normalized pinched hysteresis curves of identical quantum
memristors. QM1 is initialized in the state
$\left|{\Psi(\pi/4,\pi/2)}\right\rangle$, whereas QM2 is in the state (a)
$\left|{\Psi(\pi/3,\pi/2)}\right\rangle$ and (b)
$\left|{\Psi(\pi/4,0)}\right\rangle$. The system parameters are
$C_{\Sigma_{1}}=C_{\Sigma_{2}}=3.6~{}[\rm{fF}]$, $C_{c}=0.9~{}[\rm{fF}]$,
$L_{1}=L_{2}=6.1~{}[\rm{\mu H}]$, leading to memristor frequency of
$\omega_{1}=\omega_{2}=5.03~{}[\rm{GHz}]$. Figure 3: Form factor of the
hysteresis curves as a function of the period for (a) identical and (b) non-
identical QMs. QM1 and QM2 are shown using diamonds and dots, respectively.
For identical QMs, the parameters are
$\left|{\Psi(\pi/4,\pi/2)}\right\rangle$,
$C_{\Sigma_{1}}=C_{\Sigma_{2}}=3.6~{}[\rm{fF}]$, $L_{1}=L_{2}=6.1~{}[\rm{\mu
H}]$, leading to memristor frequency of
$\omega_{1}=\omega_{2}=5.03~{}[\rm{GHz}]$. For non identical QMs,
$\left|{\Psi(\pi/3,\pi/2)}\right\rangle$, $C_{\Sigma_{1}}=3.6~{}[\rm{fF}]$,
$C_{\Sigma_{2}}=2.6~{}[\rm{fF}]$, $L_{1}=~{}[\rm{\mu H}]$, $L_{2}=~{}[\rm{\mu
H}]$, leading to a memristor frequency of $\omega_{1}=6~{}[\rm{GHz}]$, and
$\omega_{2}=8~{}[\rm{GHz}]$.

##  IV Coupled differential equations

The current-voltage relation of each QM is obtained through the equation of
motion for the operators $\hat{n}_{\ell}$ and $\hat{\phi}_{\ell}$. We can
compute them from the master equation of Eq. (6) (see Appendix B), obtaining

$\displaystyle\frac{d}{dt}\langle\hat{n}_{\ell}\rangle=$ $\displaystyle
E_{L_{\ell,\ell}}\langle\hat{\phi}_{\ell}\rangle-
E_{L_{1,2}}(\delta_{1,\ell}\langle\hat{\phi}_{2}\rangle-\delta_{2,\ell}\langle\hat{\phi}_{1}\rangle)$
(7a) $\displaystyle-\frac{{\Gamma}_{\ell}(t)}{2}\langle\hat{n}_{\ell}\rangle,$
$\displaystyle\frac{d}{dt}\langle{\hat{\phi}}_{\ell}\rangle=$
$\displaystyle-2E_{C_{\ell,\ell}}\langle{\hat{n}}_{\ell}\rangle-2E_{C_{1,2}}(\delta_{1,\ell}\langle\hat{n}_{2}\rangle+\delta_{2,\ell}\langle\hat{n}_{1}\rangle)$
(7b)
$\displaystyle-\frac{\Gamma_{\ell}(t)}{2}\langle{\hat{\phi}}_{\ell}\rangle.$

We can produce analytical solutions for the averaged quantities
$\langle\hat{n}_{\ell}(t)\rangle$ and $\langle\hat{\phi}_{\ell}(t)\rangle$ for
identical QMs. For the initial state
$\langle\hat{n}_{1}(0)\rangle=\langle\hat{n}_{2}(0)\rangle=n_{0}$ and
$\langle\hat{\phi}_{1}(0)\rangle=\langle\hat{\phi}_{2}(0)\rangle=0$, we get

$\displaystyle\langle\hat{n}_{\ell}\rangle$ $\displaystyle=$ $\displaystyle
n_{0}\exp\bigg{[}-\int_{0}^{t}\frac{\Gamma(s)}{2}ds\bigg{]}\cos(\omega^{\prime}t),$
(8) $\displaystyle\langle{\hat{\phi}}_{\ell}\rangle$ $\displaystyle=$
$\displaystyle
n_{0}\sqrt{\frac{E_{L_{\ell,\ell}}-E_{L_{1,2}}}{{2(E_{C_{\ell,\ell}}-E_{C_{1,2}}})}}\exp\bigg{[}-\int_{0}^{t}\frac{\Gamma(s)}{2}ds\bigg{]}\sin(\omega^{\prime}t).$

Here,
$\omega^{\prime}=\sqrt{2(E_{C_{\ell,\ell}}-E_{C_{1,2}})(E_{L_{\ell,\ell}}-E_{L_{1,2}})}=\omega\sqrt{(1-\alpha)(1-\beta)}$
is the effective frequency of the coupled system. From the equation of motion
in Eq. (7a) and Eq. (7b), we obtain that the quasiparticle current of each QM
is $I_{\rm{qp},\ell}(t)=G_{\ell}(t)V_{\rm{cap},\ell}(t)$. Here,
$G_{\ell}=C_{\Sigma\ell}\Gamma(t)/2$ is the conductance of each quantum
memristor Sci.Rep.2016  and $V_{\rm{cap},\ell}(t)=-2e\langle
n_{\ell}(t)\rangle/C_{\Sigma\ell}$ is the voltage across the capacitor in each
CA-SQUID. From these expressions, we will study the response of the composite
system under variation of the initial input voltage of each QM. Furthermore,
we assume that the QMs operate within the two-level approximation satisfying
the adiabatic limit Phys.Rev.A.2005 . In the following analysis, we will focus
on the capacitive coupling. However, we extend the same analysis to the
inductive interaction (see Appendix C) and the interaction using an inductor
and a capacitor simultaneously (see Appendix D).

Figure 4: Entanglement and memristive dynamics for identical a), (d) and non-
identical QMs (b), (c), (e) . We show the memristive dynamics for four
oscillations in each case. The form factor (normalized with respect to the
concurrence) is plotted along with the concurrence, using the same colors for
each oscillation as the hysteresis curves. The form factors for QM1 and QM2
are shown using dots and diamond markers, respectively. Memory revivals can be
observed after three oscillations in (d) and after two oscillations for QM1 in
(e). For QM2, it can be observed at later times since the dynamics is plotted
in the timescale of QM1. The insets show ESD and ESB. We have used the same
initial state and system parameters as those in Fig. 3.

##  V Dynamic response

Let us fix the initial input of the first QM and vary the initial input of the
second one. We choose the input through the initial state
$\left|{\Psi_{\ell}(\theta_{\ell},\varphi_{\ell})}\right\rangle=\cos(\theta_{\ell}/2)\left|{0}\right\rangle+e^{i\varphi_{\ell}}\sin(\theta_{\ell}/2)\left|{1}\right\rangle$.
Depending on the value of $\theta$, it is possible to control the initial
voltage/current on the QM, whereas the relative phase $\varphi$ gives us the
information whether we initialize the system with an input current
($\varphi=0$) or input voltage ($\varphi=\pi/2$). At the starting point, the
composite system is in the product state
$\left|{\Psi_{1}}\right\rangle\otimes\left|{\Psi_{2}}\right\rangle$. Figure 2
shows the hysteresis curves for both QMs, where we have initialized the second
one with an initial voltage and an initial current, respectively. From Fig.
2a, as expected, different voltages do not affect the memristive behavior of
the device and only the shape of the hysteresis curves change. Opposite to the
initial input current case (see Fig. 2b, we observe that the lobes of the
hysteresis curves tends to shrink, with a considerably reduced enclosed area.
Thus, the hysteretic behavior of each memristor is very sensitive to initial
conditions.

##  VI Performance

For comparing the performance of each QM in the uncoupled or composite cases,
we will focus on their memory effects. It has been shown that the area
enclosed by the pinched hysteresis loop is related to the memory effects of
the device Radwan2010 ; Biolek2012 ; Biolek2014 . Here, we will characterize
the memristive features using the form factor given by

$\mathcal{F}=4\pi\frac{A}{P^{2}}\,,$ (10)

where, $A$ is the area enclosed by the pinched hysteresis loop and $P$ is its
perimeter. By using a dimensionless quantity, that is invariant under scaling,
we can compare memory effects and I/V characteristics despite the decay
generated by the dissipative dynamics. We calculate the form factor over
consecutive periods where the Lissajous curves reach zero (origin), thus
forming a closed loop. Figure 3 shows the form factor of the hysteresis curves
as a function of the period for the uncoupled and coupled QMs. We see that for
the uncoupled QMs, the form factor is essentially constant as the area and the
square of the perimeter decay at the same rate. This decay appears since the
system does not receive additional energy and the input voltage decays due to
dissipation, so that the hysteresis follows a constant decay over time. For
the coupled QMs, the form factor shows damped oscillations related to the
correlation of the inputs. In this case, because the hysteresis curves shrink
and expand, it is natural to observe the same pattern in the dynamics of the
form factor. For the identical QMs, we see an improvement in the performance
of the coupled system. In this manner, the peaks of the form factor overpass
the uncoupled case, exhibiting robustness against the loss of its memristive
capacities due to dissipation. For the case of non-identical QMs, we can see
that there is a trade-off in memristive capacities since there is a phase
difference between the oscillations of the form factor of each coupled QMs.
This interesting feature manifests since in the timescale of oscillation of
any one of them, the decrease and increase in the area of one is compensated
by the other. Thus, we observe that the coupling of QMs leads to correlated
inputs as the system evolves, inducing periodic decay and revival of the
memory.

##  VII Entanglement and memristive dynamics

Following the previous discussions, which hints a relationship between the
quantum correlations and the memristive behavior, we investigate the interplay
between the entanglement and the memristive dynamics. We compare the
memristive dynamics with the evolution of the quantum correlations computed
via the concurrence PhysRevLett.80.2245 ; Comp.Phys.2013  which, for a
$2\otimes 2$ bipartite state $\rho$ has a closed form

$\mathcal{C}=\max\\{0,\lambda_{1}-\lambda_{2}-\lambda_{3}-\lambda_{4}\\}$ (11)

where, $\lambda_{i}$ are the eigenvalues of the matrix
$R=\sqrt{\rho^{1/2}\tilde{\rho}\rho^{1/2}}$ ordered in decreasing order, with
$\tilde{\rho}=(\sigma_{y}\otimes\sigma_{y})\rho^{*}(\sigma_{y}\otimes\sigma_{y})$
where $\sigma_{y}$ is the $y$-component Pauli matrix. $\mathcal{C}$ takes
values from zero to one for uncorrelated and maximally correlated states,
respectively. Figure 4 shows the memristive dynamics and the concurrence for
the cases of identical ($\omega_{1}=\omega_{2}$) and non-identical
($\omega_{1}\neq\omega_{2}$) QMs. The selection of system parameters is based
on the adiabatic evolution of open quantum systems Yi.J.Phys  and according to
the current state of the art experimental setups for superconducting circuits.
The hysteresis plots are shown for the first four oscillations and the form
factor is plotted along with the concurrence in the timescale of QM1.
Comparing the timescales of the form factor and the concurrence, Fig. 4d, for
the identical QMs we see that the periodic shrink and expansion of the
hysteresis curves coincide with the increase and decrease of the entanglement
over each period. Accordingly, this shows an inverse relationship between the
memristive dynamics and the quantum correlations. Since current and voltage
are local observables, an increase in the entanglement leads to a loss of
locality, producing hysteresis curves with a smaller area. When the QMs are
detuned, Fig. 4e, the distribution of information exchange between the QMs are
uneven due to which the shrink/expansion of the hysteresis and the
corresponding rise/decay of the quantum correlations do not happen in the same
timescale. This induces a phase shift between the oscillations of the quantum
correlations and the memristive behavior.

Figure 5: Direction of the hysteresis curve(left) and corresponding time
dependence of current and voltage(right) at the (a) first oscillation, (b)
third oscillation and (c) fourth oscillation for the case of identical
memristors. The first(second) row corresponds to $\textrm{QM}_{1(2)}$. We can
see that the relative phase between voltage and current changes in time. In
the fourth oscillation (c), this phase has reversed, which changes the
direction of the hysteresis curve. We have used the same initial state and
system parameters as in the Fig. 3.

It is interesting to notice that the entanglement dynamics are also related to
the direction of the hysteresis curve. For a given memristor, the relative
phase between the voltage and current dictates the direction in which the
hysteresis curve is drawn in time. In our case, the interaction between
memristors makes the relative phase time-dependent. As a consequence, the
direction of the hysteresis curve is periodically reversed during the time
evolution. The first reversal of the direction of the hysteresis curve is
shown in Fig. 5. Specifically, in Fig. 5a (left), we see the pinched
hysteresis loop of both memristors, while the right panels show the
corresponding time dependence of voltage and current. Notice that the
memristors have opposite direction in their hysteresis curves. We can also see
that for each memristor, as time increases, the relative phase between current
and voltage decreases and eventually reverses, which changes the direction of
the hysteresis curve, as can be seen in Fig. 5c. Interestingly, this change in
direction happens when the concurrence between the memristors is maximum, as
can be seen by comparing Fig. 5 with Fig. 4a. There, the colors are chosen to
correspond to the same oscillations. Then, entanglement reaches its maximum
value when the relative phase between current and voltage is minimum.

We also observe that the system exhibits periodic entanglement sudden death
(ESD) and sudden birth (ESB) PRL.2008 , as shown in the insets of Fig. 4 for
identical and non-identical QMs. The entanglement dynamics in the two coupling
schemes differ by timescales and the maximal peak of the entanglement. With
detuning, the time taken to reach the death state and the maximal entanglement
tends to decrease. ESD/ESB appears due to two identified factors. The first
one corresponds to the competition between the two timescales of the system
dynamics. Faster oscillations at time $\tau_{1}=2\pi/\omega_{\ell}$ correspond
to the updated time of the voltage input of the QM and an enveloping at time
$\tau_{2}=2\pi/g$, where $g=\sqrt{\omega_{1}\omega_{2}}\alpha(\beta)$,
coinciding with the entanglement/disentanglement of the system. The second
factor stems from the existence of an effective interaction mediated between
subsystem-subsystem, environment-environment, and subsystem-environment. ESB
implies the existence of correlations between the system and the environment.
The oscillatory behavior of the entanglement and memristive dynamics is a
consequence of them, causing a backflow of information. Moreover, the loss of
of information due to a decay of quantum correlations shows up in the
reservoir, increasing the memory and vice versa. Notice that in our system the
QMs become entangled because of the nature of their interaction. There are
other ways for entangling quantum systems by combination of strong driving and
dissipative processes Ana.PRA . This is an interesting possibility that could
be considered for future works.

##  VIII Conclusions

We have studied the dynamics of two coupled quantum memristors, observing that
the capacitive and inductive interactions correlate the input, producing
shrink and expansion of the hysteresis loop. As a consequence, periodic decays
and revivals of their associated memories emerge. This behavior is inversely
related to the quantum correlations generated in the composite quantum system.
We find that coupling the memristors changes the relative phase between the
current and voltage in time. Therefore, the direction of the hysteresis curve
is periodically reversed during the time evolution. We find that the change in
direction happens whenever the entanglement reaches a maximal value. In
addition, we observe that the composite system exhibits entanglement sudden
death and sudden birth due to a combination of the effect of the periodic
adiabatic driving and the coupling.

Compared to classical coupled memristors Budhathoki2013 ; Luo2017 , we obtain
correlated inputs and nonclassical correlations improving the memristive
features of our quantum device. The enhancement in the memristive properties
could lead to improvements in neuromorphic algorithms if we replace the
classical memristors by their quantum counterparts, a good example is provided
by ref. Spagnolo.2021  which employs quantum memristors for reservoir
computing. Other application may involve analog simulation of differential
equations Zidan.Nat , or analog quantum neural networks composed of quantum
memristors similar to the ones implemented with memristive crossbar latch
architectures Miguel.Nat .

Ordered behavior in nonlinear systems are useful in realizing neuromorphic
architectures. This work advances the understanding of the interplay between
memristivity and entanglement in open quantum systems. Furthermore, it lays
the foundations for developing neuromorphic quantum computing and quantum
neural networks with accessible superconducting technologies, on the way to
reach quantum advantage with NISQ architectures.

##  IX Acknowledgements

The authors acknowledge support from Spanish PGC2018-095113-B-I00
(MCIU/AEI/FEDER), Basque Government IT986-16, projects QMiCS (820505) and
OpenSuperQ (820363) of EU Flagship on Quantum Technologies, NSFC (12075145),
Shanghai STCSM (Grant 2019SHZDZX01-ZX04), and Program for Eastern Scholar. X.
C. acknowledges support from the Spanish Ramón y Cajal program
(RYC-2017-22482).

##  Appendix A Derivation of the circuit Hamiltonian

The Lagrangian of the circuit shown in the Fig. 1 from main text is given by

$\displaystyle\mathcal{L}=$
$\displaystyle\sum_{\ell=1,2}\bigg{[}\frac{C_{\Sigma,\ell}}{2}\dot{\varphi}_{\ell}^{2}+E_{J\ell}\cos\bigg{(}\frac{\varphi_{\ell}+\phi_{d\ell}(t)}{\varphi_{0}}\bigg{)}+E_{J\ell}\cos\bigg{(}\frac{\varphi_{\ell}+\phi_{d\ell}(t)+\phi_{s\ell}}{\varphi_{0}}\bigg{)}-\frac{\varphi_{\ell}^{2}}{2L_{\ell}}\bigg{]}$
(12)
$\displaystyle+\frac{C_{c}(\dot{\varphi}_{2}-\dot{\varphi}_{1})^{2}}{2}-\frac{(\varphi_{2}-\varphi_{1})^{2}}{2L_{c}},$

where, $C_{\Sigma,1}=C_{J1}+C_{J2}$ and $C_{\Sigma,2}=C_{J3}+C_{J4}$ are the
effective capacitance of the CA-SQUIDs, $E_{J\ell}$ is the Josephson energy of
the $\ell$ junction, $\phi_{s\ell}$ is the static magnetic flux threading each
SQUID, $L_{\ell}$ is the inductance of the $\ell$th quantum memristor,
$\phi_{d\ell}(t)$ is the time dependent magnetic flux thought each quantum
memristor, and $\varphi_{0}=\hbar/2e$ is the reduced flux-quanta. Finally,
$C_{c}$ and $L_{c}$ are the coupling capacitance and inductance, respectively.
Notice that for static magnetic flux at $\phi_{s\ell}/\varphi_{0}=\pi$ the
Josephson energy contributions cancel each other, and the system Lagrangian
simplifies to

$\displaystyle\mathcal{L}=\sum_{\ell=1,2}\bigg{[}\frac{(C_{\Sigma,\ell}+C_{c})}{2}\dot{\varphi}_{\ell}^{2}-\bigg{(}\frac{1}{2L_{\ell}}+\frac{1}{2L_{c}}\bigg{)}\varphi_{\ell}^{2}\bigg{]}-C_{c}\dot{\varphi}_{1}\dot{\varphi}_{2}+\frac{\varphi_{1}\varphi_{2}}{L_{c}}.$
(13)

Now, the conjugate momenta
$q_{\ell}=(\partial\mathcal{L}/\partial\dot{\varphi_{\ell}})$ are

$\displaystyle q_{1}$ $\displaystyle=$
$\displaystyle(C_{\Sigma,1}+C_{c})\dot{\varphi}_{1}-C_{c}\dot{\varphi}_{2}\,,$
(14) $\displaystyle q_{2}$ $\displaystyle=$ $\displaystyle-
C_{c}\dot{\varphi}_{1}+(C_{\Sigma,2}+C_{c})\dot{\varphi}_{2}\,.$ (15)

We can write $\vec{q}=\hat{C}\vec{\dot{\varphi}}$, where
$\vec{q}=(q_{1},q_{2})$, and $\vec{\varphi}=(\varphi_{1},\varphi_{2})$ and

$\displaystyle\hat{C}=\left(\begin{array}[]{cc}C_{\Sigma,1}+C_{c}&-C_{c}\\\
-C_{c}&C_{\Sigma,2}+C_{c}\\\ \end{array}\right)\,.$ (18)

The Hamiltonian of the circuit is given by the transformation
$\mathcal{H}=\vec{q}^{~{}~{}T}\vec{\dot{\varphi}}-\mathcal{L}=\vec{q}^{~{}~{}T}\hat{C}^{-1}\vec{q}-\mathcal{L}$,
then

$\displaystyle\mathcal{H}$ $\displaystyle=$
$\displaystyle\sum_{\ell=1,2}\bigg{[}\frac{\hat{C}^{-1}_{\ell,\ell}q_{\ell}^{2}}{2}+\frac{\hat{L}^{-1}_{\ell,\ell}\varphi_{\ell}^{2}}{2}\bigg{]}+\hat{C}^{-1}_{1,2}q_{1}q_{2}-\hat{L}^{-1}_{1,2}\varphi_{1}\varphi_{2}\,.$

Here, $\hat{C}^{-1}_{j,k}$, and $\hat{L}^{-1}_{j,k}$ are the matrix element
$(j,k)$ of the inverse of the capacitance and inductance matrix given by

$\displaystyle\hat{C}^{-1}=\frac{1}{C^{\star}}\left(\begin{array}[]{cc}C_{c}+C_{\text{$\Sigma$2}}&C_{c}\\\
C_{c}&C_{c}+C_{\text{$\Sigma$1}}\\\
\end{array}\right),\qquad\hat{L}^{-1}=\left(\begin{array}[]{cc}\frac{1}{L_{c}}+\frac{1}{L_{1}}&-\frac{1}{L_{c}}\\\
-\frac{1}{L_{c}}&\frac{1}{L_{c}}+\frac{1}{L_{2}}\\\ \end{array}\right),$ (23)

with
$C^{\star}=C_{c}\left(C_{\text{$\Sigma$1}}+C_{\text{$\Sigma$2}}\right)+C_{\text{$\Sigma$1}}C_{\text{$\Sigma$2}}$.
To quantize the circuit Hamiltonian, we promote the variables to operators. It
means, $q_{\ell}\rightarrow\hat{q}_{\ell}=-2e\hat{n}_{\ell}$, and
$\varphi_{\ell}/\varphi_{0}\rightarrow\hat{\phi}_{\ell}$, with
$[\hat{\phi}_{\ell},\hat{n}_{\ell^{\prime}}]=i\delta_{\ell,\ell^{\prime}}$.
Therefore, the quantum Hamiltonian reads

$\displaystyle\mathcal{H}$ $\displaystyle=$
$\displaystyle\sum_{\ell=1,2}\bigg{[}E_{C_{\ell,\ell}}\hat{n}_{\ell}^{2}+\frac{E_{L_{\ell,\ell}}}{2}\hat{\phi}_{\ell}^{2}\bigg{]}+E_{C_{1,2}}\hat{n}_{1}\hat{n}_{2}-E_{L_{1,2}}\hat{\phi}_{1}\hat{\phi}_{2},$
(24)

where $E_{C}=2e^{2}\hat{C}^{-1}$ is the charge energy and
$E_{L}=\varphi_{0}^{2}\hat{L}^{-1}$ is the inductive energy. For simplicity,
we can redefine $\hat{q}_{\ell}$ and $\hat{\phi}_{\ell}$ in terms of creation
and annihilation operators as

$\displaystyle\hat{n}_{\ell}$ $\displaystyle=$
$\displaystyle\frac{i}{4g_{\ell}}(a_{\ell}^{{\dagger}}-a_{\ell}),$ (25)
$\displaystyle\hat{\phi}_{\ell}$ $\displaystyle=$ $\displaystyle
2g_{\ell}(a_{\ell}^{{\dagger}}+a_{\ell}),$ (26)

where $g_{\ell}=(E_{C_{\ell,\ell}}/32E_{L_{\ell,\ell}})^{1/4}$ is proportional
to the zero-point fluctuation of the phase operator. Finally, the quantum
Hamiltonian reads

$\displaystyle\hat{\mathcal{H}}=\sum_{\ell=1,2}^{1}\hbar\omega_{\ell}\hat{a}_{\ell}^{\dagger}\hat{a}_{\ell}-\sqrt{\omega_{1}\omega_{2}}(\alpha-\beta)(\hat{a}_{1}^{\dagger}\hat{a}_{2}+\hat{a}_{1}\hat{a}_{2}^{\dagger}),$
(27)

where, $\omega_{\ell}=\sqrt{2E_{C_{\ell,\ell}}E_{L_{\ell,\ell}}}/\hbar$ is the
frequency of the $\ell$th QMs. Moreover,
$\alpha=E_{L_{1,2}}/\sqrt{E_{L1}E_{L2}}$, and
$\beta=E_{C_{1,2}}/\sqrt{E_{C1}E_{C2}}$ stand for the inductive and charge
energy ratios.

##  Appendix B Memristive equations

We derive the equation of motion for the mean value of the observables
$\hat{n}_{\ell}$, and $\hat{\phi}_{\ell}$, which are related to the current
flowing and the voltage across the memristor. We will consider the dynamics in
the Schrödinger picture, where it is governed by the following master equation

$\displaystyle\dot{\hat{\rho}}(t)$ $\displaystyle=$
$\displaystyle-\frac{i}{\hbar}\big{[}\mathcal{H},\hat{\rho}\big{]}+\sum_{\ell=1,2}\frac{\Gamma_{\ell}(t)}{2}\bigg{[}\mathcal{L}_{\ell}\hat{\rho}\mathcal{L}^{{\dagger}}-\frac{1}{2}\\{\mathcal{L}^{{\dagger}}\mathcal{L},\hat{\rho}\\}\bigg{]}\,.$
(28)

Here, $\hat{\rho}(t)$ is the density matrix describing the system and
$\dot{\hat{\rho}}(t)=d\hat{\rho}(t)/dt$ represents the derivative with respect
to time. Moreover, $\mathcal{H}$ is the system Hamiltonian in Eq. (27), while
$\mathcal{L}_{\ell}=\sqrt{\Gamma_{\ell}(t)}a_{\ell}$ is the collapse operator
describing the quasiparticle tunneling of the $\ell$th QMs at the rate
$\Gamma_{\ell}(t)=\lvert\left\langle{0}\right|\sin(\phi_{\ell}/2)\left|{1}\right\rangle\lvert^{2}S_{\rm{qp}}(\omega_{\ell})$.
We are interested in the time evolution of the expectation value
$\langle\hat{\mathcal{O}}(t)\rangle={\rm{Tr}}[\hat{\mathcal{O}}\hat{\rho}(t)]$
with $\hat{\mathcal{O}}=\\{\hat{n}_{\ell},\hat{\varphi}_{\ell}\\}$. Using Eq.
(28), we obtain the equation of motion for $\hat{\mathcal{O}}$ as

$\displaystyle\frac{d\langle\hat{\mathcal{O}}(t)\rangle}{dt}=-\frac{i}{\hbar}{\rm{Tr}}\bigg{[}[\mathcal{H},\mathcal{O}]\hat{\rho}(t)\bigg{]}+{\rm{Tr}}\bigg{[}\tilde{\mathcal{D}}[\mathcal{O}]\hat{\rho}(t)\bigg{]},$
(29)

where
$\tilde{\mathcal{D}}[\mathcal{O}]=\Gamma(t)(a^{{\dagger}}\mathcal{O}a-\\{a^{{\dagger}}a,\mathcal{O}\\}/2)$
corresponds to the Lindbladian for the operator $\mathcal{O}$. For
$\hat{n}_{k}$, we obtain

$\displaystyle\frac{d\langle\hat{n}_{k}\rangle}{dt}$ $\displaystyle=$
$\displaystyle E_{L_{k,k}}\langle\hat{\phi}_{k}\rangle-
E_{L_{1,2}}(\langle\hat{\phi}_{1}\rangle\delta_{2,k}-\langle\hat{\phi}_{2}\rangle\delta_{1,k})-{{\rm{Tr}}}\bigg{[}\tilde{\mathcal{D}}[\hat{n}]\hat{\rho}(t)\bigg{]}$
(30a)

and

$\displaystyle\tilde{\mathcal{D}}[\hat{n}_{k}]=\frac{4i\Gamma(t)}{g_{0}}\bigg{[}-\frac{a_{k}^{{\dagger}}}{2}+\frac{a_{k}}{2}\bigg{]}=-\frac{\Gamma(t)}{2}\hat{n}_{k}\,.$
(31)

Thus, the equation of motion reads

$\displaystyle\frac{d\langle\hat{n}_{k}\rangle}{dt}$ $\displaystyle=$
$\displaystyle E_{L_{k,k}}\langle\hat{\phi}_{k}\rangle-
E_{L_{1,2}}(\langle\hat{\phi}_{1}\rangle\delta_{2,k}-\langle\hat{\phi}_{2}\rangle\delta_{1,k})-\frac{\Gamma(t)}{2}\langle\hat{n}_{k}\rangle.$
(32)

Analogously for $\hat{\phi}_{k}$, we obtain

$\displaystyle\frac{d\langle\hat{\phi}_{k}\rangle}{dt}$ $\displaystyle=$
$\displaystyle-2E_{C_{k,k}}\langle\hat{n}_{k}\rangle-
E_{C_{1,2}}(\langle\hat{n}_{1}\rangle\delta_{2,k}+\langle\hat{n}_{2}\rangle\delta_{1,k})-\frac{\Gamma(t)}{2}\langle\hat{\phi}_{k}\rangle.$
(33)

##  Appendix C Inductively coupled quantum memristors

Figure 6: Inductively coupled QMs. Entanglement and memristive dynamics for
the case of identical memristors (a), (d) and non-identical memristors (b),
(c), (e). Memristive dynamics are shown for five oscillations using five
different colors. The corresponding form factor (normalized with respect to
the concurrence) is plotted using the same color combination of points for
each oscillation along with the concurrence. The parameter choice and
initialization is the same as in Fig. 3 of the main text.

In this section, we discuss the results of the inductive interaction between
the QMs. We omit the dynamic response and the performance since it is included
in the main text for the capacitively coupled QMs. Here, we show that the
inverse relation between entanglement and memristive dynamics is also
satisfied for the inductive interaction along with the observation of ESD and
ESB.

Fig. 6 shows the entanglement and memristive dynamics for the QMs coupled via
inductive interaction. The form factor is shown along with the concurrence
with the same color points as their corresponding pinched hysteresis loop. We
observe that the coupling leads to the periodic shrink and expansion of the
hysteresis curve along with the periodic rise and decay of the concurrence. To
reflect upon this fact, we can compare the timescales of the form factor and
the concurrence for identical QMs, Fig. 6 (d), where we can find that the loss
of entanglement is accompanied by the increase in the form factor. It hints
that that the entanglement and memristive dynamics may generally be inversely
related to each other. Furthermore, the interaction leads to the competition
between the input voltage update time ($\tau_{1}=2\pi/\omega_{\ell}$) and the
enveloping time ($\tau_{2}=2\pi/g$) of the QM inducing ESD (shown in the
insets) and ESB which can be observed for both the cases. On the other hand,
similar to the capacitive coupling, when we detune the QMs, Fig. 6 (e), the
distribution of information exchange between them are uneven due to which the
shrink/expansion of the hysteresis and the corresponding rise/decay of the
quantum correlations do not happen in the same timescale.

In conclusion, despite the minor differences in the plots of the inductive and
capacitive coupling, which are mainly due to the very nature of the circuit
elements, the interaction produces correlated inputs during the system
evolution leading to periodic shrink/expansion of the hysteresis and the
rise/decay of the entanglement. The inverse behavior of the entanglement and
memristive dynamics can be generally observed when the detuning is low.

Figure 7: QMs coupled via an inductor and a capacitor simultaneously. Form
factor (area/perimeter2) of the hysteresis curves as a function of the period
for (a) identical and (b) non-identical QMs. QM1 and QM2 are shown using
diamonds and dots, respectively. The parameter choice and initialization is
the same as in Fig. 3 of the main text. Figure 8: QMs coupled via an inductor
and a capacitor simultaneously. Entanglement and memristive dynamics for the
cases of identical QMs (a), (d) and non-identical QMs (b), (c), (e) .
Memristive dynamics are shown for four oscillations in the timescale of QM1
using different colors. The corresponding form factor is plotted using the
same color combination of points for each oscillation along with the
concurrence. The parameter choice and initialization is the same as in Fig. 3
of the main text. Figure 9: QMs coupled via an inductor and a capacitor
simultaneously. Direction of the hysteresis curve(left) and corresponding time
dependence of current and voltage(right) at the (a) first oscillation, (b)
14th oscillation and (c) 29th oscillation for the case of identical
memristors. The first(second) row corresponds to $\textrm{QM}_{1(2)}$. We can
see that the relative phase between voltage and current changes in time. In
the 14th oscillation (b), this phase has reversed, which changes the direction
of the hysteresis curve and the relative phase is inverted in the 29th
oscillation (c). We have used the same initial state and system parameters as
in the Fig. 3.

##  Appendix D Quantum memristors coupled via an inductor and a capacitor
simultaneously

Apart from using either an inductor or a capacitor, the coupling
characteristics can be investigated when both the circuit elements are
simultaneously used to couple the QMs. Thus, the system is fully described by
the simplified and the quantized hamiltonian given in the Eqn. (24) and Eqn.
(27) with both the inductive and capacitive coupling contributions. For this
system we study the performance and the entanglement/memristive dynamics in
order to compare it with the other coupling schemes.

The performance, quantified by the form factor
($\mathcal{F}=4\pi\frac{A}{P^{2}}$) is shown in the Fig. 7, for identical and
nonidentical cases. As we can see, the coupling leads to correlated inputs
inducing rise and decay of the form factor. In comparison to the
inductive/capacitive coupling, the oscillation frequency is low. Here the
coupling doesn’t lead to an enhanced performance of the QMs. Moreover, the
nonidentical ones do not show a significant change in their behavior as
compared to the identical case. This observation can be understood by
investigating the interplay of the entanglement and memristive dynamics as
shown in the Fig. 7. Similar to the other cases, the periodic shrink/expansion
of the hysteresis curves and the rise/decay of the concurrence is observed. We
see that there’s no observation of the ESD/ESB phenomena. The entanglement in
this case is very low due to which the maximal/minimal of concurrence/form
factor never coincide and their oscillation frequencies are not comparable.
Therefore, here we observe that the coupling leads to correlated inputs but
there’s no observation of ordered characteristics as in the coupling via a
single inductor/capacitor. Nonetheless, the direction of the hysteresis is
reversed during the time evolution of the system. As can be seen in the Fig. 9
(a), the two QMs initialized out of phase have their relative phases inverted
during the time evolution of the system Fig. 9 (c). This inversion occurs when
the entanglement tends towards a maximum value after undergoing net decay
(compare with Fig. 8).

## References

  * (1) L. Chua, Memristor-The missing circuit element, IEEE Trans. Circuit Theory 18, 507 (1971).
  * (2) L. O. Chua and S. M. Kang, Memristive devices and systems, Proc. IEEE 64, 209 (1976).
  * (3) D. B. Strukov, G. S. Snider, D. R. Stewart, and R. S. Williams, The Missing Memristor Found, Nature 453, 80 (2008).
  * (4) C. D. Schuman, T. E. Potok, R. M. Patton, J. D. Birdwell, M. E. Dean, G. S. Rose, and J. S. Plank, A Survey of Neuromorphic Computing and Neural Networks in Hardware, arXiv:1705.06963 [cs.NE] (2017).
  * (5) Z. Wang, et al., Reinforcement learning with analogue memristor arrays, Nat. Electron. 2, 115 (2019).
  * (6) Z. Liu, et al., Neural signal analysis with memristor arrays towards high-efficiency brain-machine interfaces,  Nat. Commun. 11, 4234 (2020).
  * (7) Z. Wang, et al., In situ training of feed-forward and recurrent convolutional memristor networks, Nat. Mach. Intell. 1, 434 (2019).
  * (8) P. Lin, et al., Three-dimensional memristor circuits as complex neural networks, Nat. Electron. 3, 225 (2020).
  * (9) S. B. Laughlin and T. J. Sejnowski, Communication in Neuronal Networks, Science 301, 1870 (2003).
  * (10) Y. Wang, Y. Liutao, W. Si, R. Huang, and Y. Yang, Memristor-Based Biologically Plausible Memory Based on Discrete and Continuous Attractor Networks for Neuromorphic Systems, Adv. Intell. Syst. 2, 2000001 (2020).
  * (11) P. Yao, H. Wu, B. Gao, J. Tang, Q. Zhang, W. Zhang, J. J. Yang, and H. Qian, Fully hardware-implemented memristor convolutional neural network, Nature 577, 641 (2020).
  * (12) J. Borghetti, G. S. Snider, P. J. Kuekes, J. J. Yang, D. R. Stewart, and R. S. William, “Memristive” switches enable “stateful” logic operations via material implication, Nature 464, 873 (2010).
  * (13) M. Prezioso, F. Merrikh-Bayat, B. D. Hoskins, G. C. Adam, K. K. Likharev, and D. B. Strukov, Training and operation of an integrated neuromorphic network based on metal-oxide memristors, Nature 521, 61 (2015).
  * (14) F. M. Bayat, F. M. Bayat, M. Prezioso, B. Chakrabarti, H. Nili, I. Kataeva, and D. Strukov, Implementation of multilayer perceptron network with highly uniform passive memristive crossbar circuits, Nat. Commun. 9, 2331 (2018).
  * (15) M. Davies, Benchmarks for progress in neuromorphic computing, Nat. Mach. Intell. 1, 386 (2019).
  * (16) R. K. Budhathoki, M. Pd. Sah, S. P. Adhikari, H. Kim, and L. Chua, Composite Behavior of Multiple Memristor Circuits, IEEE Trans. Circuits Syst. I, Reg. Papers. 60, 2688 (2013).
  * (17) L. Luo, X. Hu, S. Duan, Z. Dong, and L. Wang, Multiple memristor series-parallel connections with use in synaptic circuit design, IET Circuits, Devices Syst. 11, 123 (2017).
  * (18) D. Marković and J. Grollier, Quantum neuromorphic computing, Appl. Phys. Lett. 117, 150501 (2020).
  * (19) Y. Cao, G. G. Guerreschi, and A. Aspuru-Guzik, Quantum Neuron: an elementary building block for machine learning on quantum computers, arXiv:1711.11240 [quant-ph] (2017).
  * (20) F. Tacchino, C. Macchiavello, D. Gerace, and D. Bajoni, An artificial neuron implemented on an actual quantum processor, npj Quantum Inf. 5, 26 (2019).
  * (21) Y. Li, Simulation of memristive synapses and neuromorphic computing on a quantum computer, Phys. Rev. Res. 3, 023146 (2021).
  * (22) L. B. Kristensen, M. Degroote, P. Wittek, A. Aspuru-Guzik, and N. T. Zinner, An artificial spiking quantum neuron, npj Quantum Inf. 7, 59 (2021).
  * (23) M. Sanz, L. Lamata, and E. Solano, Quantum Memristors in Quantum Photonics, APL Photonics 3, 080801 (2018).
  * (24) T. Gonzalez-Raya, J. M. Lukens, L. C. Céleri, and M. Sanz, Quantum Memristors in Frequency-Entangled Optical Fields, Materials 13, 864 (2020).
  * (25) M. Spagnolo, J. Morris, S. Piacentini, M. Antesberger, F. Massa, F. Ceccarelli, A. Crespi, R. Osellame, and P. Walther, Experimental quantum memristor, arXiv:2105.04867 [quant-ph] (2021).
  * (26) Y. Li, G. W. Holloway, S. C. Benjamin, G. A. D. Briggs, J. Baugh, and J. A. Mol, Double quantum dot memristor, Phys. Rev. B 96 075446 (2017).
  * (27) P. Pfeiffer, I. L. Egusquiza, M. Di Ventra, M. Sanz and E. Solano, Quantum memristors, Sci. Rep. 6, 29507 (2016).
  * (28) S. N. Shevchenko, Y. V. Pershin, and F. Nori, Qubit-Based Memcapacitors and Meminductors, Phys. Rev. Appl. 6, 014006 (2016).
  * (29) S. N. Shevchenko and D. S. Karpov, Thermometry and Memcapacitance with a Qubit-Resonator System, Phys. Rev. Appl. 10, 014013 (2018).
  * (30) S. Peotta and M. Di Ventra, Superconducting Memristors, Phys. Rev. Appl. 2, 034011 (2014).
  * (31) J. Salmilehto, F. Deppe, M. Di Ventra, M. Sanz, and E. Solano, Quantum Memristors with Superconducting Circuits, Sci. Rep. 7, 42044 (2017).
  * (32) W.-H. Zhou, J. Gao, Z.-Q. Jiao, X.-W. Wang, R.-J. Ren, X.-L. Pang, L.-F. Qiao, C.-N. Zhang, T.-H. Yang, and X.-M. Jin, Timestamp Boson Sampling, arXiv:2009.03327 [quant-ph] (2020).
  * (33) B. D. Josephson, Possible new effects in superconductive tunnelling, Phys. Lett. 1, 251 (1962).
  * (34) A. A. Golubov, M. Yu. Kupriyanov, and E. Il’ichev, The current-phase relation in Josephson junctions, Rev. Mod. Phys. 76, 411 (2004).
  * (35) G. Catelani, J. Koch, L. Frunzio, R. J. Schoelkopf, M. H. Devoret, and L. I. Glazman, Quasiparticle Relaxation of Superconducting Qubits in the Presence of Flux, Phys. Rev. Lett. 106, 077002 (2011).
  * (36) G. Catelani, R. J. Schoelkopf, M. H. Devoret, and L. I. Glazman, Relaxation and frequency shifts induced by quasiparticles in superconducting qubits, Phys. Rev. B. 84, 064517 (2011).
  * (37) V. Ambegaokar and A. Baratoff, Tunneling between superconductors, Phys. Rev. Lett. 10, 486 (1963).
  * (38) Juha Leppäkangas, Michael Marthaler, and Gerd Schön, Phase-dependent quasiparticle tunneling in Josephson junctions: Measuring the cos$\varphi$ term with a superconducting charge qubit, Phys. Rev. B 84, 060505(R) (2011).
  * (39) John M. Martinis, M. Ansmann, and J. Aumentado, Energy Decay in Superconducting Josephson-Junction Qubits from Nonequilibrium Quasiparticle Excitations, Phys. Rev. Lett. 103, 097002 (2009).
  * (40) U. Vool, I. M. Pop, K. Silwa, B. Abdo, C.Wang, T. Brecht, Y. Y. Gao, S. Shankar, M. Hatridge, G. Catelani, M. Mirrahimi, L. Frunzio, R. J. Schoelkopf, L. I. Glazman and M. H. Devoret, Non-Poissonian Quantum Jumps of a Fluxonium Qubit due to Quasiparticle Excitations, Phys. Rev. Lett. 113, 247001 (2014).
  * (41) John M. Martinis, Saving superconducting quantum processors from decay and correlated errors generated by gamma and cosmic rays, npj Quantum Information, 7, 90 (2021) .
  * (42) M. S. Sarandy and D. A. Lidar, Adiabatic approximation in open quantum systems, Phys. Rev. A. 71, 012331 (2005)
  * (43) A. G. Radwan, M. A. Zidan, and K. N. Salama, On the mathematical modeling of memristors, 2010 International Conference on Microelectronics (2010).
  * (44) Z. Biolek, D. Biolek, and V. Biolkova, Computation of the Area of Memristor Pinched Hysteresis Loop, IEEE Trans. Circuits Syst., II, Exp. Briefs 59, 9 (2012).
  * (45) Z. Biolek, D. Biolek, and V. Biolková, Interpreting area of pinched memristor hysteresis loop, Electron. Lett. 50, 74 (2014).
  * (46) W. K. Wootters, Entanglement of Formation of an Arbitrary State of Two Qubits, Phys. Rev. Lett. 80, 2245 (1998).
  * (47) J. R. Johansson, P.D. Nation, and F. Nori, QuTiP 2: A Python framework for the dynamics of open quantum systems, Comp. Phys. Comm. 184, 1234 (2013).
  * (48) X X Yi, DM Tong, L C Kwek and CH Oh, Adiabatic approximation in open systems: an alternative approach,  J. Phys. B: At. Mol. Opt. Phys. 40, 281 (2007).
  * (49) C. E. López, G. Romero, F. Lastra, E. Solano, and J. C. Retamal, Sudden Birth versus Sudden Death of Entanglement in Multipartite Systems, Phys. Rev. Lett. 101, 080503 (2008).
  * (50) Ana Laura Gramajo, Daniel Domínguez, and María José Sánchez, Efficient steady-state-entanglement generation in strongly driven coupled qubits, Phys. Rev. A. 104, 032410 (2021).
  * (51) Mohammed A. Zidan, YeonJoo Jeong, Jihang Lee, Bing Chen, Shuo Huang, Mark J. Kushner, and Wei D. Lu, A general memristor-based partial differential equation solver, Nature Electronics. 1, 411–420 (2008).
  * (52) Miguel Angel Lastras-Montaño and Kwang-Ting Cheng, Resistive random-access memory based on ratioed memristors, Nature Electronics. 1, 466–472 (2018).