# Quantum Memristors in Quantum Photonics

## Abstract

We propose a method to build quantum memristors in quantum photonic platforms. We firstly design an effective beam splitter, which is tunable in real-time, by means of a Mach-Zehnder-type array with two equal 50:50 beam splitters and a tunable retarder, which allows us to control its reflectivity. Then, we show that this tunable beam splitter, when equipped with weak measurements and classical feedback, behaves as a quantum memristor. Indeed, in order to prove its quantumness, we show how to codify quantum information in the coherent beams. Moreover, we estimate the memory capability of the quantum memristor. Finally, we show the feasibility of the proposed setup in integrated quantum photonics.

Circuit elements whose dynamics intrinsically depends on their past evolution Ch71 (); pieee97/1717 (); nano24/255201 () promise to induce a novel approach in information processing and neuromorphic computing UMM (); Sc11 (); Le10 (); YSS13 () due to their passive storage capabilities. These history-dependent circuit elements can be roughly classified as purely dissipative, such as memristors, or both dissipative and non-dissipative, such as memcapacitors and meminductors nano24/255201 (); SPN16 (). A classical memristor is a resistor whose resistance depends on the record of electrical charges which crossed through it Ch71 (). The information about the electrical history is contained in the physical configuration of the memristor, summarized in its internal state variable , such that in the (voltage-controlled) memristor --relationship appears as nano24/255201 (),

(1a) | |||||

(1b) |

where the state-variable dynamics, encoded in the real-valued function , and the state-variable-dependent conductance function , lead to a characteristic pinched hysteresis loop of a memristor when a periodic driving is applied CK76 (). Memristor technology is currently a promising paradigm to replace in the medium term computing architectures based on transistors for certain specific tasks, because of the low energy consumption Ch13 (). Indeed, they are energetically more efficient JKKCH16 (), and the presence of memory seems to make them more powerful for key machine learning tasks, such as image recognition Ch13 (); WL95 (); GKLvHW96 (); S-GMPIL-B13 (); BVKISP15 ().

The quantization of these devices with memory, especially the memristor, is a complicated challenge which has only recently been achieved PEdVSS16 (); JDdVSE17 (); SPN16 (). The difficulty lies on the fact that it is necessary to engineer an open quantum system whose classical limit corresponds to the general dynamics given by Eqs. (1). This question was addressed in Ref. PEdVSS16 () by replacing the memristor by a tunable resistor, a weak-measurement protocol and classical feedback acting on the system-resistor coupling. It was proven that this system behaves in the classical limit as the dynamics given by Eqs. (1). Additionally, it was proven that the dynamics of this composed system is genuinely quantum and, therefore, might be used for quantum information tasks. Afterwards, an implementation of quantum memristors in superconducting circuits was proposed JDdVSE17 (), making use of the fact that a memristive behavior emerges naturally in the presence of Josephson junctions Jo74 ().

However, a memristive behavior can be extended to a more general framework beyond charges and fluxes. Indeed, a memristive behavior is characterized by non-Markovian history-dependent dynamics, which produces the characteristic pinched hysteresis loops when observables of the input and output states are depicted PdV11 (); JW09 (). Additionally, we call these devices quantum memristors when the dynamics cannot be described for all states by means of a classical process. They can be used as building blocks for the simulation of complex non-Markovian quantum dynamics or for an efficient (in terms of resources) codification of quantum machine leaning protocols, mimicking the classical case Ch13 (); WL95 (); GKLvHW96 (); S-GMPIL-B13 (); BVKISP15 ().

In this Article, we construct the first quantum memristor in quantum photonics, codifying the quantum information into different quantum states of the photons as depicted in Fig. 1. By showing that the fundamental elements which constitute a quantum memristor, namely a tunable dissipative element, weak measurements and classical feedback PEdVSS16 (), can be straightforwardly constructed in quantum photonics, we study the dynamics of different initial quantum states and demonstrate the presence of prototypical hysteresis loops. We also compute the persistence of the memory of these devices.

In order to build a photonics quantum memristor, we start by constructing a beam splitter with regulable transmittivity (and reflectivity), which will play the role of a tunable coupling to the environment. To construct a beam splitter with a transmittivity and reflectivity dependent on an in-time controllable parameter, we use a Mach-Zehnder array with two beam splitters, a retarder, which introduces a phase between arms, and two final compensating phase-shifters (see Fig. 2). It can be proven that this construction is equivalent to a beam splitter with an arbitrary reflectivity. Indeed,

(2) |

Here, is the effective reflectivity of the beam splitter operator, defined as , and are the annihilation operators in paths and , respectively, and the annihilation operator in path after the first beam splitter. Notice that the phase introduced by the retarder can be straightforwardly controlled in time and, thus, this construction is suitable for our purposes. In order to prove this result, we only need the expression of a 50:50 beam-splitter with transmitted and reflected phases and , respectively, so

where the parameters are , , and , which is an invertible system. The phase difference between the transmitted and reflected phases is , which proves Eq. (2). Now that we can construct an in-time tunable beam splitter, let us study the effect on different initial states, analyzing the hysteretical response and the quantum dynamics of such systems.

Coherent states. — Let us start by studying the output of two coherent states after an arbitrary beam splitter KSBK02 (), which is given by . Now, we must prove that, when equipped with measurements and feedback, this system shows a pinched hysteresis behavior, and thus, that it can be a memristor. Let us consider as input states a coherent state in beam and vacuum in beam , so the outcome states are and . For the sake of simplicity, we have assumed that , something which can be achieved with an additional phase shifter in the outcome of beam . As we are working with coherent states, we will consider as an independent variable of the input beam . Assuming a displacement in the -direction, then . A memristor is a dissipative element, so we will consider the intensity of the light (photon number) in the outcome beam as the dependent variable, while the outcome beam can be understood as losses into a zero-temperature bath. Provided that there is no entanglement between the two paths, we will make use of outcome beam to measure and update the value of reflectivity of the beam splitter by means of the phase . On the other hand, . With all these elements in hand, we can write the following equations for the memristor,

(3a) | |||||

(3b) |

In our case, we pumped the system with a periodic coherent state fulfilling . As , we have that , which may be interpreted as the transmitted intensity per unit of initial displacement. The function , which updates the reflectivity of the beam splitter, can be chosen freely. In our case, for illustrative proposes, we will select a linear behavior of . In Fig. 3, we have depicted the resulting hysteresis loop when plotting vs . As expected, this is a pinched hysteresis loop with an enclosed area decreasing with the frequency of the driving, which shows that this system behaves as a memristor in these variables. As discussed below in detail, this dynamics is deeply related with refractive optical bistability, which makes use of an optical mechanism to change the refractive index inversely to the intensity of the light source Bo08 (), but the classical feedback in our proposal is flexible and we can change the refractive index arbitrarily.

The area of the hysteresis loop has been proposed as a natural measure of the persistence of the memory in the memristor JDdVSE17 (). For the general memristor given by Eqs. (1), this area is given by

(4) | |||||

where the first term in the integrand corresponds to the response related to the selected memory variable and the second term accounts for any remaining explicit time dependence of the conductance. Applying this result to Eqs. (3), we obtain that the area is given by

where is the Bessel function of second order. This formula is valid for , since the curve suffers additional crosses for smaller frequencies and it must be computed more carefully (in fact, if , , each bubble suffers crossings, generating sub-loops). For large frequencies, this area decreases polynomially as , so the area decays slowly and memristive behavior is resilient for a large window of frequencies, as expected.

The problem with coherent states is that a beam splitter cannot change the entanglement degree of any state codified in them KSBK02 (), so the dynamics is essentially classical SPGWC10 (); ICS13 (); MCD13 (); USLS14 (); Fetal15 (); JWdC16 ().

Squeezed states.— Let us consider the situation in which the inputs are a squeezed state with squeezing and a vacuum state BR02 (). We can compute the output modes in the Heisenberg picture, which are given for a beam splitter with transmitted and reflective phases and by

In this case, it is straightforward to compute the number of photons in both output beams, so that and . As independent variable, we choose the variance , which characterizes a squeezed state and is given by , where we have taken for the vacuum. Hence, the function is

(5) |

with , squeezing in the quadrature. We choose , where the sign is chosen depending on the angle, and drive the squeezing below the vacuum variance, so , with . Hence, . The pinched hysteresis loop is depicted in Fig. 4.

The area can be computed again by using Eq. (4), but in this case, there is no analytical expression. However, the asymptotic expression of the area for large frequencies and strong squeezing can be computed as . Therefore, vanishes in the high-frequency regime as , which agrees with Fig. 4 and also shows resilience of te memory with the frequency.

The dynamics of the beam splitter is quantum when working with squeezed states, and it cannot be simulated by an equivalent classical dynamics. The final state without updating the reflectivity of the beam splitter is given by and the quantum information may be codified in continuous variables BvL05 ().

Fock states.— A paradigmatic case of quantum states to codify quantum information corresponds to Fock states. Let us consider a qubit state encoded in a superposition in channel and the vacuum in input channel , as before. Then, the beam splitter yields and KSBK02 (). Hence, by linearity, the quantum superposition yields

(6) | |||||

Let us now prove the memristive behavior of this construction. In order to get it, we have that the mean value of in the initial state is given by , while the intensity of light coming out through channel is given by . Straightforwardly, one obtains that . Let us consider the same dynamical equation for the internal variable as in the previous cases . Then, the hysteresis loops generated, which are valid for frequencies , , are depicted in Fig. 5. For lower frequencies crosses appear, which generates sub-loops. Differently to the previous cases, the hysteresis loops are not pinched, which means that the memristor is not passive and the energy introduced only vanishes when the initial state is . Additionally, this fact also reflects in the fact that the area asymptotically approaches to a constant for high frequencies . This robustness could be a useful resource for quantum information processing.

Let us study the final state in channel when we measure projectively the number of photons in channel given by Eq. (6). If we measure Fock state , then the final state in channel is simply . Otherwise, if we measure Fock state in channel , the final state in channel is , which means that the amplitudes of the initial qubit may be modified by means of the reflectivity of the beam splitter.

Therefore, this is a natural candidate to be considered for (digital qubit-based) quantum information processing, by creating multiple copies of the same initial state. A possible relevant application could be the simulation of non-Markovian quantum dynamics by considering quantum feedback and Fock states in input channel GZ04 (). The reflectivity of the beam splitter should be updated depending on the environment that we want to simulate.

Optical bistability.— Optical bistability is a property of optical devices to show two resonant states, both stable and dependent on the input state. In other words, two different output intensities are possible for a given input intensity, and we need to know the previous states in order to determine which is the right one (non-Markovianity) Bo08 (). This property is characterized by the presence of a hysteresis loop when the output intensity is plotted versus a periodic input intensity. Particularly, refractive bistability makes use of changes in the refractive index of the optical device depending inversely on the intensity of the source light to produce such hysteresis loops with coherent states. This is exactly what we are producing here with different quantum input states, in such a way that the dynamics shows optical bistability / memristive behavior, but must be described quantum mechanically due to the presence of entanglement. Consequently, quantum information and processing may be encoded in this process, combined with the intrinsic memory (non-Markovianity) of the quantum memristor. Indeed, our system shows all the main elements required for the quantum memristor described in Ref. PEdVSS16 ().

Implementation in quantum photonics.— The technology for the implementation of our proposal in photonic quantum technology, as for example integrated quantum photonics, is currently available. Indeed, fully reconfigurable two-qubit gates based on integrated photonic chips have been developed Shadbolt () and their basic principles, as dynamically-tunable directional couplers, can be directly applied to produce our basic unit of quantum memristor. Moreover, the ease of fabrication of a wide variety of chip designs will allow for establishing a network of quantum memristors based on photonic-chip technology, which could give rise to scalable neuromorphic quantum computing.

Conclusions.— By using the fundamental elements for the quantization of a memristor, namely a tunable dissipative environment, weak measurements and classical feedback, we have extended the concept of quantum memristor from superconducting circuits to quantum photonics, showing that all these elements are present in current technology. We have studied the dynamics of this photonics quantum memristor with respect to different paradigmatic initial quantum states, showing the prototypical hysteresis loops, computing the corresponding area, and proving that these dynamics are quantum for squeezed states and Fock states. Finally, we have briefly discussed the implementation in integrated quantum photonics. In a long term vision, we expect that these quantum devices can be use as building blocks for quantum machine learning and neural networks La17 (); BWPRWL17 (); P-OBR-GB17 (); LA-RM-GSS17 () and in the simulation of quantum artificial life A-RSLS14 (); A-RSLS16 ().

We thank G. Gatti, R. Sweke and F. Flamini for fruitful discussions. This work was supported by Spanish MINECO/FEDER FIS2015-69983-P, Ramón y Cajal Grant RYC-2012-11391, and Basque Government IT986-16.

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