Abstract
Memristive devices are a class of physical systems with historydependent dynamics characterized by signature hysteresis loops in their input–output relations. In the past few decades, memristive devices have attracted enormous interest in electronics. This is because memristive dynamics is very pervasive in nanoscale devices, and has potentially groundbreaking applications ranging from energyefficient memories to physical neural networks and neuromorphic computing platforms. Recently, the concept of a quantum memristor was introduced by a few proposals, all of which face limited technological practicality. Here we propose and experimentally demonstrate a novel quantumoptical memristor (based on integrated photonics) that acts on singlephoton states. We fully characterize the memristive dynamics of our device and tomographically reconstruct its quantum output state. Finally, we propose a possible application of our device in the framework of quantum machine learning through a scheme of quantum reservoir computing, which we apply to classical and quantum learning tasks. Our simulations show promising results, and may break new ground towards the use of quantum memristors in quantum neuromorphic architectures.
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Main
The term memristor (that is, a memory resistor) was introduced in the early 1970s in a seminal paper by L. Chua^{1}. He hypothesized the existence of the memristor as the fourth fundamental passive circuit element—the other three being resistor, capacitor and inductor. The name ‘memory resistor’ reflects the main property of such a device, namely, the fact that its resistance depends on the history of its input; hence, this device retains a memory of the past states. Chua’s work remained mostly unnoticed until 2008, when Struckov^{2} famously reported finding the ‘missing memristor’. Later works have disputed this conceptual line by raising doubts on whether Chua’s memristor is even physical^{3} and questioning its status as a ‘fundamental’ element^{4}. A wider framework is now commonly adopted, which also includes memcapacitors and meminductors, as it has been shown that all the memory elements can be actually derived from Kubo’s response theory^{5,6}.
Notwithstanding the conceptual intricacies, Struckov’s study^{2} undeniably sparked enormous interest among experimentalists. In fact, it was quickly realized that memristors could potentially revolutionize electronics by allowing the storage of information without a power source and by enabling logic operations^{7}, as well as being able to mimic the behaviour of neural synapses^{8,9}. These observations opened up a whole field of applications in nontraditional computing, namely, physical neural networks and neuromorphic architectures^{10,11,12,13,14,15,16,17}. Moreover, the formalism of memristive devices^{18} applies to a wide variety of physical systems and allows to extend these concepts well beyond the electronic domain.
In this work, we provide the first experimental demonstration of a quantum memristive device, which we simply refer to as a ‘quantum memristor’ for consistency with the previous literature^{19,20,21}, even though it is clear that these are not ideal memristors. Our device is based on a laserwritten integrated photonic circuit that is fully reconfigurable by means of integrated phase shifters, and is able to produce memristive dynamics on singlephoton states through a scheme of measurement and classical feedback. Additionally, through numerical simulations, we show a possible application of our quantum memristor in the framework of quantum reservoir computing.
From classical to quantum memristors
We start by discussing the recent quantum memristor proposals^{19,20,21} and by analysing the issues in their definition and the challenges in their implementation.
In its most general formulation, a classical memristive device is defined by the following coupled equations:
where u and y denote the input and output variables, respectively; and s denotes a state variable; all of them are implicitly assumed to depend on time t. What distinguishes this from a generic dynamical system is equation (1), where u multiplies the function f(). This implies that when the input u is zero, the output y is zero, too—except for particular cases where f() may go to infinity^{6}; therefore, the input–output characteristic is typically a figureofeight hysteresis curve that always crosses the origin (Fig. 1e). In the case of the electronic memristor, u and y correspond to the current and voltage, and the zerocrossing property simply reflects the passive nature of a resistor (that is, zero current implies zero voltage).
At an intuitive level, we want a quantum memristor to show similar features, in addition to providing a genuinely quantum behaviour that may be employed for manipulating quantum information. Here we propose to define the quantum memristor as a device that provides the following features:

1.
Memristive behaviour in the classical limit, that is, showing the dynamics of equations (1) and (2) when the expectation values of the quantum observables are considered.

2.
Quantum coherent processing, that is, the ability to coherently map a quantum input state onto an output state.
This is an appropriate definition because it captures the inherent conceptual and technological challenge of the device. In fact, point (1) requires nonMarkovian behaviour, which cannot be achieved by simply performing unitary operations on closed quantum systems, which, in turn, is precisely the type of processing that ensures quantum coherence, as required in point (2). In other words, achieving memristive behaviour requires the design of an open quantum system where the quantum device interacts with an environment (for example, through a measurement process). This, in practice, is always associated with some level of decoherence, but a device with no quantum coherence (for example, one that collapses all quantum superpositions) would be no different than a classical memristor, an issue that previous proposals touched upon but did not fully discuss^{19,20}. The apparent contradiction can be overcome by engineering the interaction with the environment to be strong enough to provide memristivity but weak enough to sufficiently preserve quantum coherence.
In Fig. 1, we summarize and compare the main properties of classical and quantum memristors.
A photonic quantum memristor
The possibility of realizing a quantum memristor in the photonic domain was first pointed out in another study^{21}, but the scheme suffered from conceptual and technical drawbacks that severely hindered practical implementations. Here we go beyond the original proposal by introducing a substantially improved scheme suitable for realization in integrated optics. A detailed comparison of our scheme with respect to the original one is provided in Supplementary Section I.
To illustrate the basic principle, let us consider the beamsplitter represented in Fig. 2a, whose reflectivity R(t) is tunable and dynamically controlled by an active feedback based on singlephoton detection at the output mode D. When a quantum state with photonnumber expectation value 〈n_{in}(t)〉 is sent to input mode A at time t, the expectation value 〈n_{out}(t)〉 at mode C is
The temporal dynamics of the device is determined by the choice of feedback, that is, the update rule for R(t). Assuming that 〈n_{in}(t)〉 takes values between zero and 〈n〉_{max}, we choose the following relation:
Evidently, equations (3) and (4) satisfy the form required by equations (1) and (2) and therefore define a memristive device with R(t) as the state variable. In fact, Supplementary Section J shows that equation (3) has a close formal analogy with Struckov’s memristor^{2}, which inspired the choice of equation (4). Note that these two equations apply to any input state, which may also be classical light, and thus, they do not define a quantum memristor per se.
However, consider now an input state ψ_{in}(t)〉 in the quantum superposition
where ∣α(t)∣^{2} + ∣β(t)∣^{2} = 1, and 0〉_{A} and 1〉_{A} represent the vacuum and singlephoton state in mode A, respectively. In the singlephoton case, 〈n_{in}(t)〉 = ∣β(t)∣^{2} and 〈n〉_{max} = 1. If the photon is detected in D, the output at mode C is just the vacuum state 0〉_{C}. However, when the photon is not detected in D, then the output state ψ_{out,C}(t)〉 at mode C is projected onto
(N is the normalization factor) which is still a quantum superposition, thus proving that this device provides a genuine quantum behaviour. Intuitively, assuming the user only has access to output C, the overall output state is given by the statistical mixture of both cases, weighted by their respective probability:
(a formal derivation is provided in Supplementary Section A). The purity of this state can be calculated as
and is shown as a function of reflectivity R and input variable ∣β∣^{2} (Fig. 2b). The fact that the state is not fully mixed (except for the case of ∣β∣^{2} = 1, R = 0.5) shows that the device is capable of preserving some measure of quantum coherence, thus satisfying the requirements of a quantum memristor.
Implementation and results
The input state proposed in equation (5) encodes a qubit as a superposition of two energy levels. Despite offering an intuitive picture and a ready comparison across different quantum platforms, this type of encoding (also known as singlerail encoding) is highly impractical in linear optics^{22,23}. A more natural approach in quantum photonics is path encoding (also known as a dual rail), where the qubit is represented by a single photon being present in either of the two spatial modes. In Supplementary Section B, we show how the singlerail protocol presented earlier has a straightforward dualrail equivalent. Practically, one just needs to introduce an additional spatial mode that does not go through the beamsplitter. Figure 2c–e summarizes the steps from the basic concept to the final integrated photonic processor.
The photonic quantum memristor processor is realized by femtosecondlaser micromachining^{24,25}. All the sections are fully configurable by means of thermal phase shifters^{26,27} featuring novel thermal isolation structures that strongly reduce the power consumption and thermal crosstalk^{28}. Fabrication details are reported in Supplementary Section C.
The reflectivity of the onchip quantum memristor stage is externally set by a microcontroller, which approximates the solution of equation (4) by performing a timewindow integration of the form
where T is the width of the integration window (Supplementary Section D provides the derivation). A challenge in implementing this operation is that the measurement of the expectation value 〈n_{in}〉 itself requires some form of windowed integration of the input signal. Such a window needs to be large enough to collect meaningful photon statistics; since it is much smaller than T, on the time scale of the memristor, 〈n_{in}〉 can be considered to be an instantaneous quantity. Our solution, along with a full description of the experimental setup, is detailed in Supplementary Section E, where we show that 〈n_{in}〉 is estimated within a time window of approximately 100 ms, corresponding to a few hundred photon counts on average.
A stream of single photons is coupled via a singlemode fibre to the upper mode of the chip (Fig. 2e) and using the integrated statepreparation stage, the input number of photons to the memristor is varied in time as
where T_{osc} is the oscillation period. The dynamics of the device is determined by the ratio T/T_{osc}. We refer to the highfrequency regime when the input oscillates many times within an integration window, that is, T ≫ T_{osc}, and conversely to the lowfrequency regime when T ≪ T_{osc}.
An upper bound to f_{osc} = 1/T_{osc} is given by the response of the thermal phase shifters of the chip, which can be modelled as lowpass filters with a cutoff frequency f_{cut} ≃ 5 Hz. Notably, we observed that when f_{osc} approaches this frequency range, it causes an additional memristive behaviour (Supplementary Section F). In Fig. 3, we instead report our results when keeping a constant f_{osc} = 0.1 Hz (well below f_{cut}) and varying the integration time T. The device shows a hysteresis figure pinched at the origin, which reduces to a linear relation at higher frequencies and to a nonlinear one at lower frequencies. This is precisely Chua’s definition of a memristive device^{18}.
For further demonstrating the functionality of the quantum memristor, we have characterized the output state with respect to the input state and reflectivity R. As an example, for ∣β∣^{2} = 0.3 and R = 0.7, we experimentally reconstruct the density matrix with a fidelity of F = 99.7% to the theoretical one. The purity of the state is measured to be Tr(ρ_{out,EXP}^{2}) = 0.66, which matches the theoretical value of Tr(ρ_{out,THEO}^{2}) = 0.67, showing that our quantum memristor does not substantially introduce additional decoherence. Supplementary Section G provides details of the reconstruction together with 16 output states with an average fidelity of F = 98.8%.
A memristorbased quantum reservoir computer
Neural networks are known to be very effective in computational tasks where a small amount of information (for example, whether an image represents a cat or a dog) needs to be extracted from highdimensional data (for example, an image matrix of thousands of pixels). Typical neural approaches to these problems involve densely connected, multilayer structures like the schematic shown in Fig. 4a. Although proven to be extremely effective, training these networks requires an iterative optimization of thousands—sometimes, millions—of parameters, which, in turn, requires very large amounts of highquality training data and computational time. This issue represents the main limiting factor for the scalability of these architectures.
Reservoir computing^{29,30} addresses this challenge by having the input data processed through a fixed nonlinear highdimensional system (a reservoir). This reservoir maps the data such that the output only requires an elementary readout network for being interpreted, for example, a linear classifier (Fig. 4b). One key advantage of this approach is that only the readout network needs to be trained, which requires minimal resources in both time and data. Secondly, reservoirs can be implemented on physical systems rather than computer models, which promises even further speedups^{31}. Classical physical reservoirs have been demonstrated on a variety of platforms, including classical memristors^{16,32} and classical optics^{33,34,35}. Considerable interest has been recently devoted to quantum reservoirs^{36,37,38,39,40}. Here we propose and numerically evaluate a quantum photonic reservoir based on quantum memristors.
Figure 4c shows a schematic of the working principle of a quantum reservoir computer. In this simulated example, the input information is encoded as the quantum states represented by three photons that can occupy nine different optical modes. A fixed matrix of beamsplitters with randomly assigned reflectivity scrambles the information across all the optical modes, which is then fed into the input ports of three quantum memristors. The outputs of the quantum memristors are scrambled again before reaching an array of photon counters. Note that the system is inherently resilient to photon losses, as the detectors always herald threefold events. In the end, this detected output signal is fed into the readout network. It has been shown^{40} that reservoir computing provides excellent performances when having access to (1) high dimensionality, (2) nonlinearity resources and (3) shortterm memory. Here we propose a quantum reservoir that combines passive optical networks with our demonstrated quantum memristors. The photonic network gives access to a large Hilbert space that grows exponentially with the size of the quantum system. In contrast, the nonlinearity and shortterm memory are provided by quantum memristors. This is a key difference with respect to the scheme shown elsewhere^{36}, where the nonlinearity and memory arise from the dynamics of the ensemble of solidstate qubits.
Image classification by sequential data analysis
Reservoir computing is naturally suited for interpreting timedependent data. Image classification, although usually regarded as a static task, can be reframed as a timedependent task when considering images as pixels whose arrangement is defined by an ordered sequence of columns. Such an approach provides the advantage that the instantaneous input dimension is greatly reduced, as it only needs to encode one column at a time rather than the whole pixel matrix. A second, more practical advantage is that very high quality image databases are available. We consider here a subset of the MNIST handwritten digit database^{41} representing digits ‘0’, ‘3’ and ‘8’ (chosen for their columnwise similarity). Each image is cropped to 18 pixels × 12 pixels, and the columns are encoded one at a time into the quantum reservoir via a simple amplitudeencoding scheme (Supplementary Section H). At each step, the state of the quantum memristors is updated via a discretetime equivalent of equation (9). The output corresponding to the last column is finally interpreted by the linear readout network, which is composed of approximately 1,600 tunable parameters. After training on 1,000 different images over 15 epochs, we achieve a classification accuracy of 95% on a never before seen test set of 1,000 images evenly split across the chosen digits.
Remarkably, our analysis shows that high accuracy was achieved on this threedigit classification task by only using an extremely small training set of just 1,000 images, using a very small physical reservoir containing only three quantum memristors and a very small readout network. Although comparing the performances of neural networks is challenging as they tend to be very case specific, reported classical schemes require more resources for similar tasks. The authors of another study^{32}, who implemented a similar scheme, reported a simulated 91% accuracy on the tendigit classification using 14,000 training images and a reservoir containing 88 classical memristors. In another study^{42}, an accuracy of 92% was reached with threelayered reservoirs, 60,000 training images and approximately 500,000 tunable parameters. It, thus, seems plausible to conclude that our scheme is more resource efficient than these reported classical ones. Whether such efficiency reflects a genuine quantum advantage associated to the quantum reservoir remains to be discussed. Although numerical evidence has often been reported, a full proof of quantum advantage is still an active field of research.
Therefore, for obtaining insights into the quantum advantage of quantum memristors, we have compared the performance of our quantum reservoir computation with the one obtained when using only classical information as the input (Fig. 5a). This was achieved by encoding the input information with coherent classical light, rather than single photons, and by keeping all other conditions the same. The resulting accuracy for distinguishing the three digits dropped to approximately 71%, which indicates a superior performance of the quantum case. Also, when switching off the feedback loop of the quantum memristor (which eliminates both nonlinearity and memory from the reservoir), the performance drops to 34%, which is essentially random guessing for a threelabel classification task.
Entanglement detection
Naturally, a quantum reservoir is also suited for quantum tasks that are inaccessible with classical resources. For demonstrating this potential, we took the same quantum reservoir computer as the classical image classification. As an exemplary task for quantum applications, we analysed the capability of detecting quantum entanglement as a twoway discrimination problem between separable and maximally entangled quantum states. Here 100 copies of each state are sequentially fed to the quantum reservoir, and the state of the quantum memristors is updated based on the measurement statistics collected from this sequence. For this specific task, the nonlinearity rather than the memory of quantum memristors is exploited for increasing the complexity of the map performed by the quantum reservoir. By training on a set of just 1,000 randomly generated pure states, we obtain a discrimination accuracy of 98% (Fig. 5b), which indicates that the network has effectively learned to generate a relatively highperforming entanglement detection protocol with no user input.
Conclusions
We have designed an optical memristive element that allows the transmission of coherent quantum information as a superposition of single photons on spatial modes. We have realized the prototype of such a device on a glassbased, laserwritten photonic processor and thereby provided what is, to the best of our knowledge, the first experimental demonstration of a quantum memristor. We have then designed a memristorbased quantum reservoir computer and tested it numerically on both classical and quantum tasks, achieving strong performance with very limited physical and computational resources and, most importantly, no architectural change from one to the other.
Our demonstrated quantum memristor is feasible in practice and readily scalable to larger architectures using integrated quantum photonics, with immediate feasibility in the noisy intermediatescale quantum regime. The only hard limit for larger scalability—as with most quantum photonic applications—is the achievable singlephoton rate. A foreseeable advancement would be the integration of optical and electronic components within the same chip (rather than using external electronics), which is conceivable using current semiconductor technology. Additionally, the frequency at which our quantum memristor operates can be easily improved. For laserwritten circuits, highfrequency operations are readily available at the expense of higherpower consumption^{28}, whereas other photonic platforms routinely enable frequencies even in the gigahertz regime^{43}. For exploiting these frequencies, however, the photon detection rate must be improved as well. The vast development of quantum photonics technology shows that such performances are in reach by using customized fast detectors and bright singlephoton sources using quantum dots^{44}.
We emphasize that our results are not restricted to photonic quantum systems, and would be equally applicable to other platforms such as superconducting qubits^{19,20}. On the other hand, our photonic implementation offers a particularly simple and robust approach that relies on a mature technological platform, and may even provide the missing nonlinear element for recently proposed quantumoptical neural networks^{45}. Given the recent progress in photonic circuits for neuromorphic applications^{46}, we envisage our device to play a key role in future photonic quantum neural networks.
Data availability
All the relevant data to replicate the experiment are included in the main text and Supplementary Information. Raw measurement data are available at https://doi.org/10.5281/zenodo.5833624.
Code availability
The code for the the quantum reservoir simulation is available at https://github.com/QCmonk/Qmemristor.
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Acknowledgements
We are thankful to V. Saggio, T. Strömberg, P. Schiansky, B. Dakić, B. Peterson, L. Rozema, G. Zanin, I. A. Calafell, G. Bellomia and the PoliFAB staff (https://www.polifab.polimi.it/) for advice and support. P.W. acknowledges support from the research platform TURIS; the European Commission through UNIQORN (no. 820474); EPIQUS (no. 899368); the Austrian Science Fund (FWF) through CoQuS (W12104); BeyondC (F7113); Research Group 5 (FG5); the AFOSR via PhoQuGraph (FA86552017030); and the Austrian Federal Ministry for Digital and Economic Affairs, the National Foundation for Research, Technology and Development, and the Christian Doppler Research Association. R.O. acknowledges financial support from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (project CAPABLE, no. 742745). P.W. and R.O. acknowledge financial support from the European Commission through HiPhoP (no. 731473).
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M.S. and F.M. designed the quantum memristor and the experimental setup. M.S. and M.A. implemented the experiment and performed the data analysis. S.P., F.C., A.C. and R.O. designed and realized the integrated photonic processor. J.M. designed and simulated the memristorbased quantum reservoir computer. R.O. and P.W. supervised the project. All the authors contributed to writing the paper.
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M.S., F.M. and P.W. are named as inventors on a patent application for a quantumoptical memristor by the University of Vienna (application no. EP21172766.4; status, pending). The other authors declare no competing interests.
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Spagnolo, M., Morris, J., Piacentini, S. et al. Experimental photonic quantum memristor. Nat. Photon. 16, 318–323 (2022). https://doi.org/10.1038/s41566022009735
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DOI: https://doi.org/10.1038/s41566022009735
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