Quantum Memristors

P. Pfeiffer,1 I. L. Egusquiza,2 M. Di Ventra,3 M. Sanz,1, ∗ and E. Solano1, 4 1Department of Physical Chemistry, University of the Basque Country UPV/EHU, Apartado 644, E-48080 Bilbao, Spain 2Department of Theoretical Physics and History of Science, University of the Basque Country UPV/EHU, Apartado 644, E-48080 Bilbao, Spain 3Department of Physics, University of California, San Diego, La Jolla, CA 92093, USA 4IKERBASQUE, Basque Foundation for Science, Maria Diaz de Haro 3, 48013 Bilbao, Spain (Dated: January 12, 2017) Technology based on memristors, resistors with memory whose resistance depends on the history of the crossing charges, has lately enhanced the classical paradigm of computation with neuromorphic architectures. However, in contrast to the known quantized models of passive circuit elements, such as inductors, capacitors or resistors, the design and realization of a quantum memristor is still missing. Here, we introduce the concept of a quantum memristor as a quantum dissipative device, whose decoherence mechanism is controlled by a continuous-measurement feedback scheme, which accounts for the memory. Indeed, we provide numerical simulations showing that memory effects actually persist in the quantum regime. Our quantization method, speciﬁcally designed for superconducting circuits, may be extended to other quantum platforms, allowing for memristor-type constructions in different quantum technologies. The proposed quantum memristor is then a building block for neuromorphic quantum computation and quantum simulations of non-Markovian systems.

Although they are often misused terms, the difference between information storage and memory is relevant. While the former refers to save information in a physical device for a future use without changes, a physical system shows memory when its dynamics, usually named non-Markovian [1,2], depend on the past states of the system. Recently, there is a growing interest in memristors, resistors with history- dependent resistance which provide memory effects in form of a resistive hysteresis [3,4]. In this sense, memristors, due to their memory capabilities [5–7], offer novel applications in information processing architectures. A classical memristor is a resistor whose resistance depends on the record of the electrical signals, namely voltage FIG. 1. Artist view of a quantum memristor coupled to a supercon- or charges, applied to it. The information about the electrical ducting circuit in which there is an information ﬂow between the history is contained in the physical conﬁguration of the mem- circuit and the memristive environment. ristor, summarized in its internal state variable µ, which enters the (voltage-controlled) memristor I-V -relationship via [8], represented by a transmission line [15] or a bath of harmonic I(t) = G(µ(t))V (t), (1) oscillators [16]. Yet, previous studies of memory elements µ˙ (t) = f(µ(t),V (t)). (2) in quantized circuits focussed only on non-dissipative com- ponents like memcapacitors and meminductors [17–19]. This The state variable dynamics, encoded in the real-valued func- is due to the fact that the history-dependent damping requires tion f(µ(t),V (t)) and the state variable-dependent conduc- dissipative potentials, which cannot be cast in simple system- tance function G(µ(t)) > 0, leads to a characteristic pinched environment frames [20]. arXiv:1511.02192v2 [quant-ph] 11 Jan 2017 hysteresis loop of a memristor under a periodic driving. In this Article, we propose a design of a quantum memris- In superconducting circuits, electrical signals are quantized tor (Fig.1), closing the gap left by the classical memristor in and can be used to implement quantum simulations [9, 10], the quantization toolbox for circuits [21]. Equation (2) can be perform quantum information tasks [11], or quantum comput- understood as an information extraction process and, hence, ing [12, 13]. However, despite its prospects in classical infor- its effect may be modeled on a circuit by continuous measure- mation processing, memristors have not been considered so ments. Furthermore, the state-dependent resistance in Eq. (1) far in quantized circuits. The quantum regime of electrical sig- is mimicked by a measurement-controlled coupling between nals is accessed by operating superconducting electric circuits the circuit and a bath of harmonic oscillators. Hence, our at cryogenic temperatures. Their behavior is well-described model for a quantum memristor constitutes a special case of by canonical quantization of Lagrange models that reproduce quantum feedback control [22], which is not restricted to su- the classical circuit dynamics [14]. Furthermore, the effects perconducting circuits. As a paradigmatic example of a quan- of dissipative elements like conventional resistors are studied tum memristor, we study a quantum LC circuit shunted by a by coupling the circuit to environmental degrees of freedom memristor (see Fig.2(a)), and address the compatibility be- 2

output have respectively the form τ dρmeas = 2 [q, [q, ρ(t)]] dt −q0 s 2τ + 2 ( q, ρ(t) 2 q ρ(t)) dW, (6) q0 { } − h i r ! 1 q2 M (t) = q(t) + 0 ζ(t) , (7) V C h i 8τ FIG. 2. (a) Scheme of an LC circuit shunted by a memristor, which following our scheme, it is replaced by a tunable dissipative ohmic where A, B = AB + BA is the anticommutator and the environment (depicted by the resistor with a knob to choose a re- mean value{ of} an observable reads A = Tr (ρA). The sistance value between R1 and R2), a weak-measurement protocol h i (depicted by the voltmeter on the right), and a feedback tuning the projection frequency τ is defined as the inverse of the mea- coupling of the system to the dissipative environment depending on surement time needed to determine the mean charge up to the measurement outcome (represented by the grey arrow). (b) Feed- an uncertainty q0, and depends on the measurement strength τ back model of a memristor and implementation in quantum dynamics k = 2 . In the limit τ , we recover the usual projec- q0 → ∞ via a feedback-controlled open quantum system. tive measurement. On the other hand, in the limit τ 0, the measurement apparatus is decoupled from the system,→ obtain- ing no information about it. Finally, the stochasticity of the tween memory effects and quantum properties like coherent measurement output enters via the white noise ζ(t), and the superpositions. Wiener increment dW induces the corresponding probabilis- Quantum memristor dynamics We decompose the influ- − tic update of the quantum state. ence of a quantum memristor on the system into a Markovian A constant resistor with resistance R can be simulated by a tunable dissipative environment, a weak-measurement proto- bath of LC circuits with an ohmic spectral density [14], col, and a classical feedback controlling the coupling of the system to the dissipative environment. Then, the evolution of 2Cγ Ω2 J (ω) = ω . (8) the circuit quantum state ρ includes a back action term due to ohm π Ω2 + ω2 the presence of the weak measurements. In addition, the dy- Here, the relaxation rate of the circuit is 1 and de- namics contains a dissipative contribution, which depends on γ = 2RC Ω the state variable µ, and a Hamiltonian part, so that notes a cut-off frequency. In the high temperature and high kB T cut-off frequency limit, λ := , Ω ω0, with ω0 the typi- ~ cal circuit frequency, the dissipative contribution to the circuit dρ = dρ + dρ + dρ(µ) , (3) H meas damp dynamics can be cast in the Caldeira-Leggett (C-L) master equation [1, 16]. In a sufficiently small time slice dt, in which which essentially plays the role of Eq. (1). Correspondingly, µ remains approximately constant, a quantum memristor acts if the interaction between the state variable and the circuit is like a resistor with resistance G(µ)−1. Therefore, we adapt cast in a measurement, the voltage records M (t) govern the V the C-L form by replacing the constant relaxation rate γ by a state variable dynamics, G(µ) function γ(µ) = 2C , µ˙ (t) = f(µ(t),MV (t)). (4) iγ(µ) dργ(µ) = [ϕ, q, ρ(t) ] dt damp − ~ { } 2Cλγ(µ) Finally, the Hamiltonian part is determined by the circuit [ϕ, [ϕ, ρ(t)]] dt. (9) − ~ structure. In the following, we illustrate the method by shunt- This phenomenological form could in principle be verified in ing an LC circuit with a quantum memristor, as depicted in a two-step procedure. Firstly, by capturing the full joint dy- Fig.2. Its description requires only one degree of freedom, the namics of the circuit and the bath when subjected to a feed- top node flux ϕ, and its conjugate momentum q [14], which back control of their interaction Hamiltonian. Secondly, by corresponds to the charge on the capacitor connected to the extracting the circuit dynamics after tracing out the bath de- top node. Therefore, we have grees of freedom, as depicted in Fig.2(b). However, the re- i quired feedback is non-Markovian, i.e. comprises voltage val- dρH = [H(ϕ, q), ρ(t)] dt . (5) ues from the past, and these systems are generally not analyt- −~ ically tractable [22]. Still, the C-L form represents a plausible The measurement of the voltage applied to the quantum mem- choice, if the ordering trelax tcontrol texchange dt ristor implies a monitoring of the node charge q, since the volt- holds for a time-coarse graining (See Supplementary material age determines the charge on the capacitor with capacitance for a detailed argumentation). C, q = CV . Therefore, according to the theory of continuous Consider now an observer that has no access to the mea- measurements [22, 32], the state update and the measurement surement output MV (t). From the observer’s point of view, 3

0.15

0.1 Damping Rate

0.05 tf =6⇡ 0 5 10 15 1 t t

0.9

0.8

0.7

0.6

0.5

0.4

0.3

0.2

0.1

0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 t0 =0

(a) τ = 0.005 (b) τ = 0.2 (c) τ = 4

FIG. 3. Hysteresis plots of the memristor for the unconditioned evolution with increasing projection frequencies τ. The comparison with the classical hysteresis curve (black, dashed) shows the collapse in the case of a very strong or very weak measurement. The inset shows the evolution of the damping rate and depict several of the underlying stochastic trajectories corresponding to one realisation of the conditioned dynamics. The parameters are γ0 = 0.1, = 0.5, λ = 10, ν = 0.1, the initial conditions are hϕi (0) = 20, hqi (0) = 0,Vϕ = 0.5,Vq = 0.5,Cϕ,q = 0, µ(0) = 0 and the average was obtained by generating 3000 trajectories of the stochastic dynamics via the Euler algorithm with dt = 10−3 the system evolves with density matrix ρ¯ = ρ , where the measurement contribution to the dynamics, preserve the denotes the average over all realizations ofhh theii Wiener Gaussianity of an initial state. Therefore, it suffices to fol- noise.hh·ii We shall use the term unconditioned state to refer to low the evolution of the first and second moments of flux ρ¯. The evolution of the unconditioned state is determined by and charge (the charge and flux variances are defined as 2 2 2 2 the ensemble average of Eq. (3). Clearly, we do not generally Vq = q q , Vϕ = ϕ ϕ and the covariance − h1 i − h i obtain a closed system for ρ¯ since γ(µ)ρ , which appears reads Cϕ,q = 2 ϕ, q ϕ q ). This, together with the in the ensemble average of the dissipativehh ii term, does not in damping functionh{γ(µ)}iand − theh i state h i variable dynamics, fully principle factorize. Now, when it does factorize and γ(µ) determine the dynamics of the conditioned state ρ (see [32]), is constant, the evolution equation of ρ¯ is of Lindbladhh form,ii d ϕ = q dt + √8τC dW, (10) describing a memoryless quantum Markov process. Thus, it h i h i ϕ,q provides us with a witness for non-Markovianity. d q = ϕ dt 2γ(µ) q dt + √8τVqdW, (11) To sum up, the quantum-memristor dynamics given by in- h i − h i − h i 2 dVϕ = 2Cϕ,qdt + 2τ(1 4C )dt, (12) crements of the quantum state in Eqs. (5), (17a) and (9), − ϕ,q dV = 2C dt 4γ(µ)(V λ) dt 8τV 2dt,(13) together with the state variable update in Eq. (4), evolves q − ϕ,q − q − − q via two coupled, non-linear stochastic differential equations. dC = (V V )dt C (2γ(µ) + 8τV )dt, (14) ϕ,q q − ϕ − ϕ,q q Their form is designed to mimic the memory effect due to dW a memristor, by means of a damping rate which depends on dµ = ν q dt + , (15) h i √8τ partial information of the history of the quantum state. Un- γ(µ) = γ (1 + cos(µ)). (16) fortunately, this complex dynamics prevents an analytical ap- 0 proach, so we treat it numerically. The frequency ν determines the rate of change of the state Hysteresis in a quantum memristor We present a numer- − variable per unit of charge q0. If the unitless feedback param- ical study of the dynamics of Gaussian states in an LC circuit eter [0, 1] vanishes, the damping rate equals the constant shunted by a quantum memristor with linear state variable dy- ∈ γ0, and the system reduces to an LC circuit coupled to a con- namics and the memristance of a Josephson junction [24, 25]. stant resistor and a voltmeter. The specific form chosen for For convenience, charge and flux are expressed in units of q γ(µ) is inspired in Josephson junction physics (see [25]), but their vacuum fluctuations, ϕ = ~ and q = √ ω C, is only determined here for definiteness. 0 ω0C 0 ~ 0 with C the capacitance. Furthermore, frequency is expressed The set of Eqs. (10)-(16) depends on the projection fre- in units of the circuit frequency ω = √1 , where L is the quency τ, which is a free parameter, since it is not deter- 0 LC inductance of the coil. Hence, the Hamiltonian reads mined by Eqs. (1)-(2). Indeed, if we consider the classical limit corresponding to charging the capacitor for q , ~ h i → ∞ H = (q2 + ϕ2). we recover Eqs. (1)-(2) for every positive τ. Therefore, there 2 is an infinite family of quantum memristors producing the This quadratic Hamiltonian, as well as the damping and same classical memristive dynamics. However, in the low 4 charging regime, the area of the hysteresis loop in the I-V - supercurrents are cancelled in a Josephson junction, and the characteristic of the quantum memristor in the unconditioned parameters explored there are achievable with current technol- evolution changes with τ. As the voltage is proportional to ogy. This shows, at the very least, that a quantum memristor the charge on the capacitor and the conductance is propor- will soon be experimentally feasible. tional to the damping, to obtain the hysteresis loop means to Conclusion We have introduced quantum memristors and − plot V q vs. IM = γ(µ)q . From the set presented a protocol to construct properly the evolution equa- of Eqs.hh (ii10)-( ∝16 hh),ii two sourceshh ii of diffusionhh ii of the state vari- tion for superconducting circuits coupled to quantum memris- able µ can be identified, namely, the noisy measurement out- tors. Our model is not restricted to electric circuits and could put and the stochastic back-action on the first moments (terms also be investigated in other quantum platforms like trapped in Eqs. (10), (11), and (15) proportional to the Wiener incre- ions or quantum photonics. Besides, we have constructively ment dW ). These diffusive terms reduce the hysteresis area, demonstrated the non-Markovian character of the quantum since their physical origins, statistical averaging over multi- memristor dynamics, which allowed us to conjecture that the ple voltage histories and respectively, insufficient information memory effects measured as the area of the hysteresis loop extraction, counteract memory effects. Indeed, once the state are maximized in the classical limit. Due to the impressive variable is spread over a range 2π, the periodicity of the features shown by novel memristor-based computer architec- damping rate function leads to a stationary≥ value of its ensem- tures [5,6, 34], the quantum memristors proposed here may ble average, γ = γ , and the hysteresis loop collapses. be considered as a building block for neuromorphic quantum hh ii 0 In Fig.3, the successive collapse of the hysteresis for a computation and quantum simulation of non-Markovian sys- strong and a weak measurement case is contrasted with the tems. classical hysteresis, which almost coincides with the hysteresis for the optimal choose of τ, balancing information gain and measurement back-action (cf. Supplementary Material). AUTHOR CONTRIBUTIONS According to our model, the dynamics of the quantum LC circuit has acquired a non-Markovian character by coupling P.P., as the first author, has been responsible for the devel- it to a quantum memristor. The question remains, whether opment of this work. M.S., with the support of I.L.E., have characteristics of a genuine quantum system like coherent su- contributed to the mathematical demonstrations, carried out perpositions are affected by the memristive environment. The calculations, and examples. M.S. and E. S. suggested the sem- existence of non-linear terms in Eqs. (10), which in princi- inal ideas. M.V. has helped to improve the ideas and results ple destroy the coherence of the superpositions, makes non- shown in the paper. All authors have carefully proofread the trivial the answer to this question. In any case, one must un- manuscript. E.S. supervised the project throughout all stages. derstand this non-linear behavior as the effective action of the environment-measurement-feedback protocol, i. e. the quantum memristor, onto the system. This paves the way for em- ACKNOWLEDGMENTS ploying these quantum memristors as a natural building block for simulating non-linear dynamics or designing non-linear We would like to acknowledge support from Spanish dissipative gates. In particular, and in view of the success MINECO grant FIS2012-36673-C03-02, Basque Government of the classical memristor proposal [5,6], this opens the door grants IT-472-10 and IT-559-10, UPV/EHU grant UFI 11/55, to a possible development of neuromorphic architectures for PROMISCE and SCALEQIT EU projects. E. S. also acknowl- quantum computing. edges support from a TUM August-Wilhelm Scheer Visiting Similarly, one may wonder about the quantumness of the Professorship and hospitality of Walther-Meiner-Institut and quantum memristor dynamics. Numerically, one can observe TUM Institute for Advanced Study. M. D. acknowledges par- oscillations in the squeezing of the quantum state, so that an tial support from DOE grant DE-FG02-05ER46204. initial Gaussian state, whose variance is squeezed in momentum Vq, periodically changes its squeezing to space, Vϕ, during the evolution. In other words, the system density matrix ADDITIONAL INFORMATION does not commute with itself for different times, which is an evidence of the quantumness of the dynamics [26–29]. The authors declare no competing financial interests. Even though the idea of engineering memristors in the quantum realm seems cumbersome, there are already propos- als for employing the memristive component of the Joseph- APPENDIX son junctions in an asymmetric SQUID in superconducting qubits [24]. Unfortunately, the quantization of that proposed Time scales of the quantum memristor dynamics design is not complete, since it is described by a semiclassical model. More recently, a fully quantum realisation of a su- The Caldeira-Leggett (C-L) model describes the dynamics perconducting quantum memristor has been put forward [25]. of open quantum systems weakly coupled to a bath of har- This proposal exploits quasi-particle induced tunneling when monic oscillators at high temperature. Hence, it assumes, first, 5 that the collection of harmonic oscillators is in a thermal state, IC (capacitor) and IM (quantum memristor) to sum to zero, and, second, that the interaction with the open system is gov- IL + IC + IM = 0, or equivalently, erned by a fixed, weakly interacting Hamiltonian HI [1]. Φ Q In the model of the quantum memristor, the interaction + Q˙ + G(µ) = 0. (18) L C Hamiltonian is controlled by measurement results. The output of measurements is included in Eq. (5) of the manuscript: Here, the inductor flux is Φ, the charge on the capacitor is denoted by Q, and the memconductance G is a function of the state variable µ. Furthermore, the voltages across the inductor τ and the capacitor are the same, dρmeas = 2 [q, [q, ρ(t)]] dt −q0 Q s Φ˙ = . (19) 2τ C + 2 ( q, ρ(t) 2 q ρ(t)) dW, (17a) q0 { } − h i Finally, the second quantum memristor equation (see Eq. (1b) r 2 ! of the manuscript) determines the evolution of the state vari- 1 q0 MV (t) = q(t) + ζ(t) . (17b) able , which in our case is a linear function of the applied C h i 8τ µ voltage There is a measurement result for every interval dt. We as- V Q sume systematically that the control time that is re- µ˙ = ν = ν . (20) tcontrol V0 Q0 quired at each step to fully achieve the necessary tuning of the Here, the role of the memory frequency becomes clear. It interaction Hamiltonian is small when compared to the inter- ν determines the rate of change of the quantum memristor state val dt. Schematically, the interaction Hamiltonian does not variable per unit charge Q on the capacitor. Clearly, it de- change during intervals of length dt and does change during 0 pends on the choice of this unit charge and we chose the intervals of width t , as illustrated in Fig.4. control charge fluctuation ( √2) q = √ ω C of the LC-circuit in We require that during most of the time interval of length 0 ~ 0 the ground state. × dt the bath, and the system interaction with it, be properly de- In the dimensionless notation used in the main body of the scribed by the C-L model. As a first consequence, we need the paper, equations (18),(19) and (20) read thermal structure of the harmonic oscillator bath to be guaran- teed. Therefore, the bath relaxation time scale must be shorter d q(t) = ϕ(t) 2γ(µ(t))q(t), than the control time, dt − − d t t . ϕ(t) = q(t), relax control dt d Secondly, the interaction between the system and the bath has µ(t) = νq(t). d to be determined by the values of the interaction Hamiltonian t in the plateaus, and should not depend on the form of the con- On solving this set of equations, the resulting V -IM curve trol step. Specifically, the time during which there is exchange shows hysteresis in the voltage-current relation of the memris- of excitations between open system and bath needs to be large tor for a classical circuit. The black dotted hysteresis curves compared to the control time, H (t) t t . I control exchange In this case, no excitation is exchanged during the tuning of HI (µ(t +dt)) the interaction Hamiltonian. trelax In summary, our working assumption is the time scale ordering shown in Fig.4, tcontrol

t t t dt. relax control exchange texchange HI (µ(t)) Classical hysteresis and optimal projection frequency dt

t Classical hysteresis The classical hysteresis curve is obtained by treating the FIG. 4. Schematic tuning of the interaction Hamiltonian between LC circuit coupled to the quantum memristor (see Fig. 2 bath and system. The time ordering of bath relaxation, control time of the manuscript) classically. The second Kirchhoff’s law and excitation exchange duration ensures the validity of the Caldeira- Leggett model in every interval dt of the coarse graining of time. requires the currents from the three branches, IL (inductor), 6

6 5

4 8 t Vq

3 8 t Cjq

Out[50]= Coefficients 2 1 8t

Noise 1 D t 0 0 1 2 3 4 t H L

FIG. 5. Dependence of all diffusive terms in ﬁrst moments and state variable on the projection frequency τ. The low noise regime suggested by this image is determined by τ ≈ 0.2, for the values of damping rate γ = 0.1 and thermal frequency λ = 10.

in the q-γ(µ)q plot in Fig. 3 of the manuscript correspond to such solutions, with initial conditions given by the initial expectation values of the corresponding quantum operators q(0) = q (0) and ϕ(0) = ϕ (0), and the same initial state FIG. 6. Blow-up of the region around the origin of Fig. 3a of the h i h i article revealing non pinching at the origin.This is due to the devi- variable value µ(0) = µ0. ation δq(t) from factorization of hhγ(µ)qii. Only one and a half oscillations are shown, for clarity. The black line corresponds to the classical I-V curve. The red bars denote the ensemble estimation error, not including numerical errors due to the Euler Algorithm. Optimal projection frequency

In our model τ, which is the projection frequency of the the stationary values of Eq. (8d) and (8e) are measurement, is a free parameter. It measures the rate in which information is extracted by measurements in the cir- q st 1 2 cuit. In a physical implementation of the feedback scheme in Cϕ,q = 1 + (4τ) 1 , (22) the quantum memristor model, it would be tuned by control- −8τ − p ling the coupling between the open system and the measure- γ2 + 4τ(2γ λ C ) γ V st(τ) = 0 0 − ϕ,q − . (23) ment device. In the classical limit, which corresponds to a q 4τ large charge in the capacitor, the effective classical memristor equations are independent of τ. Inserting these values in Eqs. (8) for the parameters used As discussed in the main text, the projection frequency de- in our simulations, the minimisation of the noise sum yields termines the information gain about the circuit variables as τopt 0.2 (see Fig.5). ≈ well as their perturbation due to the measurement. Both uncertainty in the measurement, on the one hand, and backaction, on the other, lead to diffusion of the state variable. Since the projection frequency impacts on both, an optimal τ would cor- Quantum hysteresis is not-pinched at 0 respond to a measuremen, which extracts sufficient information, while inducing only small fluctuations. In other words, In the unconditioned evolution of the quantum memristor, the optimal τ minimises the state variable variance growth. the q vs. γ(µ)q curve is not necessarily pinched at To determine an approximation of the optimal τ we analyse the origin.hh ii In fact,hh numericallyii evaluating the deviation from the sum of the noise terms in Eqs. (8) in the article, factorisation δq = γ(µ)q γ(µ) q , provided small ( 0.1), but non-zerohh valuesii − for hh the zeroiihh crossingsii of q . 1 The≤ resulting hysteresis observed in the unconditioned evolu-hh ii D(τ) = √8τVq + √8τ Cϕ,q + . (21) | | √8τ tion is depicted in Fig.6. Due to the correlations between damping rate and charge on the capacitor, this corresponds to The charge variance Vq and the covariance Cϕ,q are a function non-zero (quantum average) current for vanishing (quantum of time, but they quickly reach quasi stationary values. There average) voltage. The passivity of the device is still guar- are small variations due to the changes in the damping rate γ. anteed, as the expectation value of the emitted power fulfils If we fix the damping rate at its mean value γ (see Eq. (8g)), I V γ(µ)q2 0. 0 hh M ii ∝ hh ii ≥ 7

Noisy signals and hysteresis tuations of first moments. The noise in the first moments induced by the measurement becomes important for large mea- Both noisy voltage signals and fluctuations of the state vari- surement strengths, say τ > 1/ω0. In this regime, and using able have been studied for classical memristors [8, 30]. How- the adimensional setting of the main text, the noise is domi- nated by the term √8τC with C 1 and hence the first ever, for a classical voltage signal, these two noise sources are ϕ,q ϕ,q ≈ 2 independent and can be simultaneously suppressed, whereas two inequalities in Eq. (24) require in the quantum case of the measurement back-action ties them 8τtC together. The parameter that controls the noise amplitude is q2. 2 t the measurement strength k, or more precisely, the projec- 2 This corresponds to the upper bound for the projection fre- tion frequency τ = q0k , as defined in the main text. For a strong measurement, (large τ 1/ω0), noise in the state quency in Eq. (25) growing linearly with the typical action s hEi q2 variable is suppressed, but the back-action increases the fluc- in units of the planck constant, because s = = 0 q2 = ~ω ~ωC tuations in the first moments of flux and charge, thus leading q2. Correspondingly, the last inequality in Eq. (24) to diffusion of the state variable. On the contrary, for a weak tC 2 tC measurement, τ 1/ω0, first moments evolve in an almost q τ (26) t 2 unperturbed manner, but the loss of information in the voltage 8τ ⇒ 8qt measurement produces a highly fluctuating state variable up- provides the inverse scaling with the typical action s of the date. Finally, the accumulated noise has to be compared to the lower bound on τ in Eq. (25). typical charge on the capacitor, qt, which represents the signal to be recorded, and the threshold 2π, from which on hysteresis collapses. As a Wiener noise contribution of the form DdW Numeric implementation induces a linear growth in time of the variance proportional to Dt2, the condition for the existence of hysteresis up to a time We study Gaussian state dynamics in an LC circuit coupled tC reads (in adimensionalized terms) to a quantum memristor with numerical simulation of the set of Eqs. (8) in the manuscript. These are stochastic differential tC 2 2 8τVqtC , 8τCϕ,qtC , min(qt , 4π ). (24) equations with Gaussian noise and are cast in Itoˆ form. We 8τ use the explicit Euler algorithm [2]. It is based on the discreti- Like all open quantum systems, a quantum memristor is sation of the equations according to subject to decoherence, a process which has been exten- dt ∆t, sively studied in the context of the quantum-to-classical tran- → dW G√∆t, sition [31]. However, a memory-specific feature of the mem- → ristive environment consists in the continuous monitoring of where ∆t has to be chosen sufficiently small and the G are the system with a fine-tuned measurement strength. It is independently, identically, distributed values drawn from a known that continuous measurements are able to provide well- Gaussian distribution with mean 0 and variance 1. localised trajectories in phase space, a characteristic of classical systems [32]. In a linear system, such as the LC circuit, 0.15 the measurement strength has to fulfil two conditions in order Matlab 0.001 Matlab 0.00025 for localisation of trajectories to take place, namely, sufficient Mathematica 0.01 suppression of the variances and a negligible measurement Mathematica 0.001 back action in comparison to the system dynamics. The amplitude of the system dynamics is given by the typical action 0.1 of the system s (in units of ~) and constraints the projection

frequency by [33, Eq. (6)], Damping Rate

2 τ 4s. (25) s 0.05 hEi 2 0 5 10 15 The typical action can be estimated by s = q (with t ~ω0 ≈ t charge unit q0) and therefore the conditions in Eq. (24) for a memory effect coincide with the requirement for a well- FIG. 7. Test of the Matlab code via a comparison of the evolution localised trajectory. Loosely speaking, a localised phase space of the mean value damping rate with a solution obtained in Mathe- trajectory keeps the state variable localised as well, and thus matica. The different lines depict different time increments dt. The allows for memory effects. ensemble sizes are: 3000 (Matlab), 300 (Mathematica). In order to connect the requirement for a memory effect up to time tC in Eq. (24) and the condition for a well-localised Concretely, the simulation is implemented in Matlab (code phase space trajectory in Eq. (25), we first concentrate on fluc- available upon request). All the above results are obtained 8 with a time increment ∆t = 10−3, which in the chosen units 1995 Les Houches Summer School (eds Reynaud, S., Gia- is much smaller than the period of one oscillation in the LC cobino, E. & Zinn-Justin, J.), 351 (Elsevier, 1997). circuit, T = 2π. In order to obtain the hysteresis curves of the [15] Yurke, B. & Denker, J. S. Quantum network theory. Phys. Rev. ensemble we average over 3000 trajectories. A 29, 1419 (1984). [16] Caldeira, A. O. & Leggett, A. J. Quantum tunneling in a dissipative system. Annals of Physics 149, 374-456 (1983). The stability of our implementation with respect to the [17] Di Ventra, M., Pershin, Y. V., & Chua, L. O. Circuit elements choice of the time increment and the sufficient ensemble size with memory: Memristors, memcapacitors, and meminductors. is demonstrated by a comparison of the evolution of the damp- Proceedings of the IEEE 97, 1717 (2009). ing rate γ(µ(t)) for different time increments in Fig.7. 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