# Publications by Alexander Lvovsky

## Reinforcement learning enhanced quantum-inspired algorithm for combinatorial optimization

Machine Learning: Science and Technology IOP Publishing **2** (2020) 025009

Quantum hardware and quantum-inspired algorithms are becoming increasingly popular for combinatorial optimization. However, these algorithms may require careful hyperparameter tuning for each problem instance. We use a reinforcement learning agent in conjunction with a quantum-inspired algorithm to solve the Ising energy minimization problem, which is equivalent to the Maximum Cut problem. The agent controls the algorithm by tuning one of its parameters with the goal of improving recently seen solutions. We propose a new Rescaled Ranked Reward (R3) method that enables a stable single-player version of self-play training and helps the agent escape local optima. The training on any problem instance can be accelerated by applying transfer learning from an agent trained on randomly generated problems. Our approach allows sampling high quality solutions to the Ising problem with high probability and outperforms both baseline heuristics and a black-box hyperparameter optimization approach.

## Comprehensive model and performance optimization of phase-only spatial light modulators

Measurement Science and Technology IOP Publishing **31** (2020) 125202

Several spurious effects are known to degrade the performance of phase-only spatial light modulators. We introduce a comprehensive model that takes into account the major ones: curvature of the back panel, pixel crosstalk and the internal Fabry–Perot cavity. To estimate the model parameters with high accuracy, we generate blazed grating patterns and acquire the intensity response curves of the first and second diffraction orders. The quantitative model is used to generate compensating holograms, which can produce optical modes with high fidelity.

## Quantum-enhanced interferometry with large heralded photon-number states

NPJ QUANTUM INFORMATION **6** (2020) ARTN 89

© 2020, The Author(s). Quantum phenomena such as entanglement can improve fundamental limits on the sensitivity of a measurement probe. In optical interferometry, a probe consisting of N entangled photons provides up to a N enhancement in phase sensitivity compared to a classical probe of the same energy. Here, we employ high-gain parametric down-conversion sources and photon-number-resolving detectors to perform interferometry with heralded quantum probes of sizes up to N = 8 (i.e. measuring up to 16-photon coincidences). Our probes are created by injecting heralded photon-number states into an interferometer, and in principle provide quantum-enhanced phase sensitivity even in the presence of significant optical loss. Our work paves the way toward quantum-enhanced interferometry using large entangled photonic states.

## Fully reconfigurable coherent optical vector–matrix multiplication

Optics Letters Optical Society of America **45** (2020) 5752-5755

Optics is a promising platform in which to help realize the next generation of fast, parallel, and energy-efficient computation. We demonstrate a reconfigurable free-space optical multiplier that is capable of over 3000 computations in parallel, using spatial light modulators with a pixel resolution of only 340×340. This enables vector–matrix multiplication and parallel vector–vector multiplication with vector size of up to 56. Our design is, to the best of our knowledge, the first to simultaneously support optical implementation of reconfigurable, large-sized, and real-valued linear algebraic operations. Such an optical multiplier can serve as a building block of special-purpose optical processors such as optical neural networks and optical Ising machines.

## Exploratory combinatorial optimization with reinforcement learning

Proceedings of the AAAI Conference on Artificial Intelligence Association for the Advancement of Artificial Intelligence **34** (2020)

Many real-world problems can be reduced to combinatorial optimization on a graph, where the subset or ordering of vertices that maximize some objective function must be found. With such tasks often NP-hard and analytically intractable, reinforcement learning (RL) has shown promise as a framework with which efficient heuristic methods to tackle these problems can be learned. Previous works construct the solution subset incrementally, adding one element at a time, however, the irreversible nature of this approach prevents the agent from revising its earlier decisions, which may be necessary given the complexity of the optimization task. We instead propose that the agent should seek to continuously improve the solution by learning to explore at test time. Our approach of exploratory combinatorial optimization (ECO-DQN) is, in principle, applicable to any combinatorial problem that can be defined on a graph. Experimentally, we show our method to produce state-of-the-art RL performance on the Maximum Cut problem. Moreover, because ECO-DQN can start from any arbitrary configuration, it can be combined with other search methods to further improve performance, which we demonstrate using a simple random search.

## Experimental quantum homodyne tomography via machine learning

Optica Optical Society of America **7** (2020) 448-454

Complete characterization of states and processes that occur within quantum devices is crucial for understanding and testing their potential to outperform classical technologies for communications and computing. However, solving this task with current state-of-the-art techniques becomes unwieldy for large and complex quantum systems. Here we realize and experimentally demonstrate a method for complete characterization of a quantum harmonic oscillator based on an artificial neural network known as the restricted Boltzmann machine. We apply the method to optical homodyne tomography and show it to allow full estimation of quantum states based on a smaller amount of experimental data compared to state-of-the-art methods. We link this advantage to reduced overfitting. Although our experiment is in the optical domain, our method provides a way of exploring quantum resources in a broad class of large-scale physical systems, such as superconducting circuits, atomic and molecular ensembles, and optomechanical systems.

## Optical Eratosthenes' sieve for large prime numbers

Optics Express Optical Society of America **28** (2020) 11965-11973

We report the first experimental demonstration of a prime number sieve via linear optics. The prime numbers distribution is encoded in the intensity zeros of the far field produced by a spatial light modulator hologram, which comprises a set of diffraction gratings whose periods correspond to all prime numbers below 149. To overcome the limited far field illumination window and the discretization error introduced by the spatial light modulator finite spatial resolution, we rely on additional diffraction gratings and sequential recordings of the far field. This strategy allows us to optically sieve all prime numbers below 1492 = 22201.

## Engineering Schrodinger cat states with a photonic even parity detector

Quantum Verein zur Förderung des Open Access Publizierens in den Quantenwissenschaften **4** (2020)

When two equal photon-number states are combined on a balanced beam splitter, both output ports of the beam splitter contain only even numbers of photons. Consider the time-reversal of this interference phenomenon: the probability that a pair of photon-number-resolving detectors at the output ports of a beam splitter both detect the same number of photons depends on the overlap between the input state of the beam splitter and a state containing only even photon numbers. Here, we propose using this even-parity detection to engineer quantum states containing only even photon-number terms. As an example, we demonstrate the ability to prepare superpositions of two coherent states with opposite amplitudes, i.e. two-component Schrödinger cat states. Our scheme can prepare cat states of arbitrary size with nearly perfect fidelity. Moreover, we investigate engineering more complex even-parity states such as four-component cat states by iteratively applying our even-parity detector.

## Darkness of two-mode squeezed light in ?-type atomic system

New Journal of Physics IOP Publishing **22** (2020) 13014

We show that, under certain circumstances, an optical field in a two-mode squeezed vacuum (TMSV) state can propagate through a lossy atomic medium without degradation or evolution. Moreover, the losses give rise to that state when a different state is initially injected into the medium. Such a situation emerges in a Λ-type atomic system, in which both optical transitions are driven by strong laser fields that are two-photon resonant with the respective signal modes. Then the interactions of the two signal modes with the ground-state atomic coherence interfere destructively, thereby ensuring the preservation of the TMSV with a particular squeezing parameter. This mechanism permits unified interpretation of recent experimental results and predicts new phenomena.

## Entanglement of macroscopically distinct states of light

Optica Optical Society of America **6** (2019) 1425-1430

Schrödinger’s famous Gedankenexperiment has inspired multiple generations of physicists to think about apparent paradoxes that arise when the logic of quantum physics is applied to macroscopic objects. The development of quantum technologies enabled us to produce physical analogues of Schrödinger’s cats, such as superpositions of macroscopically distinct states as well as entangled states of microscopic and macroscopic entities. Here we take one step further and prepare an optical state which, in Schrödinger’s language, is equivalent to a superposition of two cats, one of which is dead and the other alive, but it is not known in which state each individual cat is. Specifically, the alive and dead states are, respectively, the displaced single photon and displaced vacuum (coherent state), with the magnitude of displacement being on a scale of 10^8 photons. These two states have significantly different photon statistics and are therefore macroscopically distinguishable.

## Quantum technologies in Russia

Quantum Science and Technology IOP Publishing **4** (2019) 40501

Remarkable advancements in the ability to create, manipulate, and measure quantum systems are paving the way to build next generations of devices based on quantum physics. Quantum technologies in Russia are on the list of strategically important cross-cutting directions in the framework of the National Technology Initiative programs and the Digital Economy National Program. The broad focus includes quantum computing and simulation, quantum communications, quantum metrology and sensing. This paper reviews existing research on quantum science and technologies in Russia and summarizes the main goals for the next few years that form the basis of an upcoming major national initiative.

## Measuring fluorescence into a nanofiber by observing field quadrature noise

Optics Letters Optical Society of America **44** (2019) 1678-1681

We perform balanced homodyne detection of the electromagnetic field in a single-mode tapered optical nanofiber surrounded by rubidium atoms in a magneto-optical trap. Resonant fluorescence of atoms into the nanofiber mode manifests itself as increased quantum noise of the field quadratures. The autocorrelation function of the homodyne detector's output photocurrent exhibits exponential fall-off with a decay time constant of 26.3±0.6 ns, which is consistent with the theoretical expectation under our experimental conditions. To the best of our knowledge, this is the first experiment in which fluorescence into a tapered optical nanofiber has been observed and measured by balanced optical homodyne detection.

## Annealing by simulating the coherent Ising machine

Optics Express Optical Society of America **27** (2019) 10288-10295

The coherent Ising machine (CIM) enables efficient sampling of low-lying energy states of the Ising Hamiltonian with all-to-all connectivity by encoding the spins in the amplitudes of pulsed modes in an optical parametric oscillator (OPO). The interaction between the pulses is realized by means of measurement-based optoelectronic feedforward, which enhances the gain for lower-energy spin configurations. We present an efficient method of simulating the CIM on a classical computer that outperforms the CIM itself, as well as the noisy mean-field annealer in terms of both the quality of the samples and the computational speed. It is furthermore advantageous with respect to the CIM in that it can handle Ising Hamiltonians with arbitrary real-valued node coupling strengths. These results illuminate the nature of the faster performance exhibited by the CIM and may give rise to a new class of quantum-inspired algorithms of classical annealing that can successfully compete with existing methods.

## Entanglement and teleportation between polarization and wave-like encodings of an optical qubit

Nature Communications Nature Research **9** (2018) 3672

Light is an irreplaceable means of communication among various quantum information processing and storage devices. Due to their different physical nature, some of these devices couple more strongly to discrete, and some to continuous degrees of freedom of a quantum optical wave. It is therefore desirable to develop a technological capability to interconvert quantum information encoded in these degrees of freedom. Here we generate and characterize an entangled state between a dual-rail (polarization-encoded) single-photon qubit and a qubit encoded as a superposition of opposite-amplitude coherent states. We furthermore demonstrate the application of this state as a resource for the interfacing of quantum information between these encodings. In particular, we show teleportation of a polarization qubit onto a freely propagating continuous-variable qubit.

## Quantum computers put blockchain security at risk

Nature Springer Nature **563** (2018) 465-467

## Two-level masers as heat-to-work converters

Proceedings of the National Academy of Sciences National Academy of Sciences **115** (2018) 9941-9944

Heat engines, which cyclically transform heat into work, are ubiquitous in technology. Lasers and masers may be viewed as heat engines that rely on population inversion or coherence in the active medium. Here we put forward an unconventional paradigm of a remarkably simple and robust electromagnetic heat-powered engine that bears basic differences to any known maser or laser: The proposed device makes use of only one Raman transition and does not rely on population inversion or coherence in its two-level working medium. Nor does it require any coherent driving. The engine can be powered by the ambient temperature difference between the sky and the ground surface. Its autonomous character and “free” power source make this engine conceptually and technologically enticing.

## Quantum-secured blockchain

Quantum Science and Technology IOP Publishing **3** (2018) 035004

Blockchain is a distributed database which is cryptographically protected against malicious modifications. While promising for a wide range of applications, current blockchain platforms rely on digital signatures, which are vulnerable to attacks by means of quantum computers. The same, albeit to a lesser extent, applies to cryptographic hash functions that are used in preparing new blocks, so parties with access to quantum computation would have unfair advantage in procuring mining rewards. Here we propose a possible solution to the quantum era blockchain challenge and report an experimental realization of a quantum-safe blockchain platform that utilizes quantum key distribution across an urban fiber network for information-theoretically secure authentication. These results address important questions about realizability and scalability of quantum-safe blockchains for commercial and governmental applications.

## Optical nanofiber temperature monitoring via double heterodyne detection

AIP ADVANCES **8** (2018) ARTN 055005

## Quantum Physics An Introduction Based on Photons

Springer, 2018

The book uses a mathematically simple physical system – photon polarization – as the visualization tool, permitting the student to see the entangled beauty of the quantum world from the very first pages.

## Generating and breeding optical Schrodinger's cat states

FOURTH INTERNATIONAL CONFERENCE ON QUANTUM TECHNOLOGIES (ICQT-2017) **1936** (2018) ARTN 020018