arXiv subjects

Oleksandr Kyriienko

Publications and source records attributed to Oleksandr Kyriienko.

At least 55 records · Page 3Linked to original sources

arXiv2023

Physics-Informed Quantum Machine Learning: Solving nonlinear differential equations in latent spaces without costly grid evaluations

We propose a physics-informed quantum algorithm to solve nonlinear and multidimensional differential equations (DEs) in a quantum latent space. We suggest a strategy for building quantum models as state overlaps, where exponentially large sets of independent basis functions are used for implicitly representing solutions. By measuring the overlaps between states which are representations of DE terms, we construct a loss that does not require independent sequential function evaluations on grid points. In this sense, the solver evaluates the loss in an intrinsically parallel way, utilizing a global type of the model. When the loss is trained variationally, our approach can be related to the differentiable quantum circuit protocol, which does not scale with the training grid size. Specifically, using the proposed model definition and feature map encoding, we represent function- and derivative-based terms of a differential equation as corresponding quantum states. Importantly, we propose an efficient way for encoding nonlinearity, for some bases requiring only an additive linear increase of the system size $\mathcal{O}(N + p)$ in the degree of nonlinearity $p$. By utilizing basis mapping, we show how the proposed model can be evaluated explicitly. This allows to implement arbitrary functions of independent variables, treat problems with various initial and boundary conditions, and include data and regularization terms in the physics-informed machine learning setting. On the technical side, we present toolboxes for exponential Chebyshev and Fourier basis sets, developing tools for automatic differentiation and multiplication, implementing nonlinearity, and describing multivariate extensions. The approach is compatible with, and tested on, a range of problems including linear, nonlinear and multidimensional differential equations.

quant-ph↗

arXiv2023

Quantum Chebyshev Transform: Mapping, Embedding, Learning and Sampling Distributions

We develop a paradigm for building quantum models in the orthonormal space of Chebyshev polynomials. We show how to encode data into quantum states with amplitudes being Chebyshev polynomials with degree growing exponentially in the system size. Similar to the quantum Fourier transform which maps computational basis space into the phase (Fourier) basis, we describe the quantum circuit for the mapping between computational and Chebyshev spaces. We propose an embedding circuit for generating the orthonormal Chebyshev basis of exponential capacity, represented by a continuously-parameterized shallow isometry. This enables automatic quantum model differentiation, and opens a route to solving stochastic differential equations. We apply the developed paradigm to generative modeling from physically- and financially-motivated distributions, and use the quantum Chebyshev transform for efficient sampling of these distributions in extended computational basis.

quant-ph↗

arXiv2023

Quantum manipulation of a two-level mechanical system

We consider a nonlinearly coupled electromechanical system, and develop a quantitative theory for two-phonon cooling. In the presence of two-phonon cooling, the mechanical Hilbert space is effectively reduced to its ground and first excited states, allowing for quantum operations at the level of individual phonons and preparing nonclassical mechanical states with negative Wigner functions. We propose a scheme for performing arbitrary Bloch sphere rotations, and derive the fidelity in the specific case of a $π$-pulse. We characterise detrimental processes that reduce the coherence in the system, and demonstrate that our scheme can be implemented in state-of-the-art electromechanical devices.

quant-ph↗

arXiv2022

Robust polaritons in magnetic monolayers of CrI3

We show that the regime of strong-light matter coupling with remarkable magnetic properties can be realized in systems based on monolayers of chromium triiodide (CrI3). This two-dimensional material combines the presence of ferromagnetic ordering with the possibility of forming strongly-bound excitonic complexes even at room temperature. Using microscopic first-principle calculations we reveal a rich spectrum of optical transitions, corresponding to both Wannier- and Frenkel-type excitons, including those containing electrons with a negative effective mass. We show that excitons of different polarizations efficiently hybridize with a photonic mode of a planar microcavity, and due to the peculiar selection rules polariton modes become well resolved in circular polarizations. The combination of very strong optical oscillator strength of excitons and cavity confinement leads to large values of the Rabi splitting, reaching 35 meV for a single monolayer, and giant Zeeman splitting between polariton modes of up to 20 meV. This makes CrI3 an excellent platform for magnetopolaritonic applications.

cond-mat.mes-hall↗

arXiv2022

Unsupervised quantum machine learning for fraud detection

We develop quantum protocols for anomaly detection and apply them to the task of credit card fraud detection (FD). First, we establish classical benchmarks based on supervised and unsupervised machine learning methods, where average precision is chosen as a robust metric for detecting anomalous data. We focus on kernel-based approaches for ease of direct comparison, basing our unsupervised modelling on one-class support vector machines (OC-SVM). Next, we employ quantum kernels of different type for performing anomaly detection, and observe that quantum FD can challenge equivalent classical protocols at increasing number of features (equal to the number of qubits for data embedding). Performing simulations with registers up to 20 qubits, we find that quantum kernels with re-uploading demonstrate better average precision, with the advantage increasing with system size. Specifically, at 20 qubits we reach the quantum-classical separation of average precision being equal to 15%. We discuss the prospects of fraud detection with near- and mid-term quantum hardware, and describe possible future improvements.

quant-ph↗

arXiv2022

Attractive trion-polariton nonlinearity due to Coulomb scattering

We theoretically investigate the nonlinearity of trion-polaritons in a two-dimensional material that arises from Coulomb interaction between quasiparticles. To evaluate the interaction constant, we solve a three-body Wannier equation precisely by expanding trion wavefunctions into a Gaussian basis. Using these wavefunctions, we calculate the trion-polariton interaction energies for the exchange processes, resolving the outstanding question of trion-trion scattering. We find that the nonlinearity is the result of the competition between different scattering channels. Such a cancellation effect is sensitive to wavefunction overlaps and depends on material parameters. Most importantly, our result shows that the nonlinear interaction between trion-polaritons is attractive, and is fivefold stronger than exciton-polariton interaction. Our work thus describes the regime where trion-polaritons offer the prospects for attractive fluids of light in monolayers of transition metal dichalcogenides.

cond-mat.mes-hall↗

arXiv2022

Theory for Coulomb scattering of trions in 2D materials

We develop a theoretical description of Coulomb interactions between trions (charged excitons) that define a nonlinear optical response in doped two-dimensional semiconductors. First, we formulate a microscopic theory of trion-trion interactions based on composite nature of these particles, and account for all possible exchange processes. Next, we calculate numerically the trion binding energies and corresponding three-body wavefunctions using a basis set with high expressivity. Then, using the obtained wavefunctions we calculate the matrix elements of two-trion scattering, and compare the contributions coming from direct and exchange terms. Finally, we find that the considered scattering gives significant contribution to the optical nonlinearity in monolayers of transition metal dichalcogenides. In particular, this can lead to an attractive interaction in doped monolayers. Our theory opens a route for studying nonlinear properties of trion-polaritons inaccessible before.

cond-mat.mes-hall↗

arXiv2022

Nonlinear interactions of dipolar excitons and polaritons in MoS2 bilayers

Nonlinear interactions between excitons strongly coupled to light are key for accessing quantum many-body phenomena in polariton systems. Atomically-thin two-dimensional semiconductors provide an attractive platform for strong light-matter coupling owing to many controllable excitonic degrees of freedom. Among these, the recently emerged exciton hybridization opens access to unexplored excitonic species, with a promise of enhanced interactions. Here, we employ hybridized interlayer excitons (hIX) in bilayer MoS2 to achieve highly nonlinear excitonic and polaritonic effects. Such interlayer excitons possess an out-of-plane electric dipole as well as an unusually large oscillator strength allowing observation of dipolar polaritons(dipolaritons) in bilayers in optical microcavities. Compared to excitons and polaritons in MoS2 monolayers, both hIX and dipolaritons exhibit about 8 times higher nonlinearity, which is further strongly enhanced when hIX and intralayer excitons, sharing the same valence band, are excited simultaneously. This gives rise to a highly nonlinear regime which we describe theoretically by introducing a concept of hole crowding. The presented insight into many-body interactions provides new tools for accessing few-polariton quantum correlations.

cond-mat.mes-hall↗

arXiv2022

Quantum Kernel Methods for Solving Differential Equations

We propose several approaches for solving differential equations (DEs) with quantum kernel methods. We compose quantum models as weighted sums of kernel functions, where variables are encoded using feature maps and model derivatives are represented using automatic differentiation of quantum circuits. While previously quantum kernel methods primarily targeted classification tasks, here we consider their applicability to regression tasks, based on available data and differential constraints. We use two strategies to approach these problems. First, we devise a mixed model regression with a trial solution represented by kernel-based functions, which is trained to minimize a loss for specific differential constraints or datasets. Second, we use support vector regression that accounts for the structure of differential equations. The developed methods are capable of solving both linear and nonlinear systems. Contrary to prevailing hybrid variational approaches for parametrized quantum circuits, we perform training of the weights of the model classically. Under certain conditions this corresponds to a convex optimization problem, which can be solved with provable convergence to global optimum of the model. The proposed approaches also favor hardware implementations, as optimization only uses evaluated Gram matrices, but require quadratic number of function evaluations. We highlight trade-offs when comparing our methods to those based on variational quantum circuits such as the recently proposed differentiable quantum circuits (DQC) approach. The proposed methods offer potential quantum enhancement through the rich kernel representations using the power of quantum feature maps, and start the quest towards provably trainable quantum DE solvers.

quant-ph↗

arXiv2022

Protocols for Trainable and Differentiable Quantum Generative Modelling

We propose an approach for learning probability distributions as differentiable quantum circuits (DQC) that enable efficient quantum generative modelling (QGM) and synthetic data generation. Contrary to existing QGM approaches, we perform training of a DQC-based model, where data is encoded in a latent space with a phase feature map, followed by a variational quantum circuit. We then map the trained model to the bit basis using a fixed unitary transformation, coinciding with a quantum Fourier transform circuit in the simplest case. This allows fast sampling from parametrized distributions using a single-shot readout. Importantly, simplified latent space training provides models that are automatically differentiable, and we show how samples from distributions propagated by stochastic differential equations (SDEs) can be accessed by solving stationary and time-dependent Fokker-Planck equations with a quantum protocol. Finally, our approach opens a route to multidimensional generative modelling with qubit registers explicitly correlated via a (fixed) entangling layer. In this case quantum computers can offer advantage as efficient samplers, which perform complex inverse transform sampling enabled by the fundamental laws of quantum mechanics. On a technical side the advances are multiple, as we introduce the phase feature map, analyze its properties, and develop frequency-taming techniques that include qubit-wise training and feature map sparsification.

quant-ph↗

arXiv2021

Quantum Model-Discovery

Quantum computing promises to speed up some of the most challenging problems in science and engineering. Quantum algorithms have been proposed showing theoretical advantages in applications ranging from chemistry to logistics optimization. Many problems appearing in science and engineering can be rewritten as a set of differential equations. Quantum algorithms for solving differential equations have shown a provable advantage in the fault-tolerant quantum computing regime, where deep and wide quantum circuits can be used to solve large linear systems like partial differential equations (PDEs) efficiently. Recently, variational approaches to solving non-linear PDEs also with near-term quantum devices were proposed. One of the most promising general approaches is based on recent developments in the field of scientific machine learning for solving PDEs. We extend the applicability of near-term quantum computers to more general scientific machine learning tasks, including the discovery of differential equations from a dataset of measurements. We use differentiable quantum circuits (DQCs) to solve equations parameterized by a library of operators, and perform regression on a combination of data and equations. Our results show a promising path to Quantum Model Discovery (QMoD), on the interface between classical and quantum machine learning approaches. We demonstrate successful parameter inference and equation discovery using QMoD on different systems including a second-order, ordinary differential equation and a non-linear, partial differential equation.

quant-ph↗

arXiv2021

Generalized quantum circuit differentiation rules

Variational quantum algorithms that are used for quantum machine learning rely on the ability to automatically differentiate parametrized quantum circuits with respect to underlying parameters. Here, we propose the rules for differentiating quantum circuits (unitaries) with arbitrary generators. Unlike the standard parameter shift rule valid for unitaries generated by operators with spectra limited to at most two unique eigenvalues (represented by involutory and idempotent operators), our approach also works for generators with a generic non-degenerate spectrum. Based on a spectral decomposition, we derive a simple recipe that allows explicit derivative evaluation. The derivative corresponds to the weighted sum of measured expectations for circuits with shifted parameters. The number of function evaluations is equal to the number of unique positive non-zero spectral gaps (eigenvalue differences) for the generator. We apply the approach to relevant examples of two-qubit gates, among others showing that the fSim gate can be differentiated using four measurements. Additionally, we present generalized differentiation rules for the case of Pauli string generators, based on distinct shifts (here named as the triangulation approach), and analyse the variance for derivative measurements in different scenarios. Our work offers a toolbox for the efficient hardware-oriented differentiation needed for circuit optimization and operator-based derivative representation.

quant-ph↗

arXiv2021

Quantum Quantile Mechanics: Solving Stochastic Differential Equations for Generating Time-Series

We propose a quantum algorithm for sampling from a solution of stochastic differential equations (SDEs). Using differentiable quantum circuits (DQCs) with a feature map encoding of latent variables, we represent the quantile function for an underlying probability distribution and extract samples as DQC expectation values. Using quantile mechanics we propagate the system in time, thereby allowing for time-series generation. We test the method by simulating the Ornstein-Uhlenbeck process and sampling at times different from the initial point, as required in financial analysis and dataset augmentation. Additionally, we analyse continuous quantum generative adversarial networks (qGANs), and show that they represent quantile functions with a modified (reordered) shape that impedes their efficient time-propagation. Our results shed light on the connection between quantum quantile mechanics (QQM) and qGANs for SDE-based distributions, and point the importance of differential constraints for model training, analogously with the recent success of physics informed neural networks.

quant-ph↗

arXiv2021

Quantum simulation and ground state preparation for the honeycomb Kitaev model

We propose a quantum protocol that allows preparing a ground state (GS) of the honeycomb Kitaev model. Our approach efficiently uses underlying symmetries and techniques from topological error correction. It is based on the stabilization procedure, the developed centralizer ansatz, and utilizes the vortex basis description as the most advantageous for qubit-based simulations. We demonstrate the high fidelity preparation of spin liquid ground states for the original Kitaev model, getting the exact GS for N=24 spins using 230 two-qubit operations. We then extend the variational procedure to non-zero magnetic fields, studying observables and correlations that reveal the phase transition. Finally, we perform dynamical simulation, where the ground state preparation opens a route towards studies of strongly-correlated dynamics, and a potential quantum advantage.

quant-ph↗

arXiv2021

Hamiltonian operator approximation for energy measurement and ground state preparation

The Hamiltonian operator plays a central role in quantum theory being a generator of unitary quantum dynamics. Its expectation value describes the energy of a quantum system. Typically being a non-unitary operator, the action of the Hamiltonian is either encoded using complex ancilla-based circuits, or implemented effectively as a sum of Pauli string terms. Here, we show how to approximate the Hamiltonian operator as a sum of propagators using a differential representation. The proposed approach, named Hamiltonian operator approximation (HOA), is designed to benefit analog quantum simulators, where one has direct access to simulation of quantum dynamics, but measuring separate circuits is not possible. We describe how to use this strategy in the hybrid quantum-classical workflow for performing energy measurements. Benchmarking the measurement scheme, we discuss the relevance of the discretization step size, stencil order, number of shots, and noise. We also use HOA to prepare ground states of complex material science models with direct iteration and quantum filter diagonalization, finding the lowest energy for the 12-qubit Hamiltonian of hydrogen chain H$_6$ with $10^{-5}$ Hartree precision using $11$ time-evolved reference states. The approach is compared to the variational quantum eigensolver, proving HOA beneficial for systems at increasing size corresponding to noisy large scale quantum devices. We find that for Heisenberg model with twelve or more spins our approach may outperform variational methods, both in terms of the gate depth and the total number of measurements.

quant-ph↗

arXiv2021

Few-photon all-optical phase rotation in a quantum-well micropillar cavity

Photonic platforms are an excellent setting for quantum technologies because weak photon-environment coupling ensures long coherence times. The second key ingredient for quantum photonics is interactions between photons, which can be provided by optical nonlinearities in the form of cross-phase-modulation (XPM). This approach underpins many proposed applications in quantum optics and information processing, but achieving its potential requires strong single-photon-level nonlinear phase shifts and also scalable nonlinear elements. In this work we show that the required nonlinearity can be provided by exciton-polaritons in micropillars with embedded quantum wells. These combine the strong interactions of excitons with the scalability of micrometer-sized emitters. We observe XPM up to $3 \pm 1$ mrad per particle using laser beams attenuated to below single photon average intensity. With our work serving as a first stepping stone, we lay down a route for quantum information processing in polaritonic lattices.

physics.optics↗

arXiv2021

Solving nonlinear differential equations with differentiable quantum circuits

We propose a quantum algorithm to solve systems of nonlinear differential equations. Using a quantum feature map encoding, we define functions as expectation values of parametrized quantum circuits. We use automatic differentiation to represent function derivatives in an analytical form as differentiable quantum circuits (DQCs), thus avoiding inaccurate finite difference procedures for calculating gradients. We describe a hybrid quantum-classical workflow where DQCs are trained to satisfy differential equations and specified boundary conditions. As a particular example setting, we show how this approach can implement a spectral method for solving differential equations in a high-dimensional feature space. From a technical perspective, we design a Chebyshev quantum feature map that offers a powerful basis set of fitting polynomials and possesses rich expressivity. We simulate the algorithm to solve an instance of Navier-Stokes equations, and compute density, temperature and velocity profiles for the fluid flow in a convergent-divergent nozzle.

quant-ph↗

arXiv2020

Reply to "Comment on "Quantum Time Crystals from Hamiltonians with Long-Range Interactions""

In the note by Khemani et al. [arXiv:2001.11037] the authors express conceptual disagreement with our recent paper on quantum time crystals [Phys. Rev. Lett. 123, 210602]. They criticise the idealized nature of the considered quantum time crystal, and make several points about properties of Hamiltonians presented in our work. In this reply we answer one-by-one all questions raised in the discussion. As for the ideological dispute, it brightly highlights a bizarre nature of time crystalline order in closed quantum systems, and we offer a different vision for the development of the field.

quant-ph↗