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Theory of overparametrization in quantum neural networks

Martin Larocca,Nathan Ju,Diego García-Martín,Patrick J. Coles,M. CerezoSep 27 2021 quant-phcs.LGstat.ML arXiv:2109.11676v1

The prospect of achieving quantum advantage with Quantum Neural Networks (QNNs) is exciting. Understanding how QNN properties (e.g., the number of parameters MM) affect the loss landscape is crucial to the design of scalable QNN architectures. Here, we rigorously analyze the overparametrization phenomenon in QNNs with periodic structure. We define overparametrization as the regime where the QNN has more than a critical number of parameters McMc that allows it to explore all relevant directions in state space. Our main results show that the dimension of the Lie algebra obtained from the generators of the QNN is an upper bound for McMc, and for the maximal rank that the quantum Fisher information and Hessian matrices can reach. Underparametrized QNNs have spurious local minima in the loss landscape that start disappearing when M≥McM≥Mc. Thus, the overparametrization onset corresponds to a computational phase transition where the QNN trainability is greatly improved by a more favorable landscape. We then connect the notion of overparametrization to the QNN capacity, so that when a QNN is overparametrized, its capacity achieves its maximum possible value. We run numerical simulations for eigensolver, compilation, and autoencoding applications to showcase the overparametrization computational phase transition. We note that our results also apply to variational quantum algorithms and quantum optimal control.

Computational phase transition in Quantum Approximate Optimization Algorithm — the difference between hard and easy

Bingzhi Zhang,Quntao ZhuangSep 29 2021 quant-phcond-mat.dis-nncond-mat.stat-mechcs.CC arXiv:2109.13346v1

Quantum Approximate Optimization algorithm (QAOA) is one of the candidates to achieve a near-term quantum advantage. As QAOA seems only capable of solving optimization problems, there is a folklore that QAOA cannot see the difference between easy problems such as 2-SAT and hard problems such as 3-SAT — although 2-SAT is in the polynomial-time (P) class, its optimization version is also nondeterministic polynomial-time (NP)-hard. In this paper, we show that the folklore is not true. We find a computational phase transition in QAOA when solving a variant of 3-SAT — the amplitude of gradient and the success probability achieve their minimum at the well-known SAT-UNSAT phase transition. On the contrary, for 2-SAT, such a phenomena is absent at SAT-UNSAT phase transition and the success probability is unity for a reasonable circuit depth. In solving the NP-hard optimization versions of SAT, we identify quantum advantages over a classical approximate algorithm at quite a shallow depth of p=4 for the problem size of n=10n=10.

Progress towards analytically optimal angles in quantum approximate optimisation

D. Rabinovich,R. Sengupta,E. Campos,V. Akshay,J. BiamonteSep 27 2021 quant-phmath-phmath.MP arXiv:2109.11566v1

The Quantum Approximate Optimisation Algorithm is a pp layer, time-variable split operator method executed on a quantum processor and driven to convergence by classical outer loop optimisation. The classical co-processor varies individual application times of a problem/driver propagator sequence to prepare a state which approximately minimizes the problem’s generator. Analytical solutions to choose optimal application times (called angles) have proven difficult to find, whereas outer loop optimisation is resource intensive. Here we prove that optimal Quantum Approximate Optimisation Algorithm parameters for p=1p=1 layer reduce to one free variable and in the thermodynamic limit, we recover optimal angles. We moreover demonstrate that conditions for vanishing gradients of the overlap function share a similar form which leads to a linear relation between circuit parameters, independent on the number of qubits. Finally, we present a list of numerical effects, observed for particular system size and circuit depth, which are yet to be explained analytically.

Parent Hamiltonian as a benchmark problem for variational quantum eigensolvers

Fumiyoshi Kobayashi,Kosuke Mitarai,Keisuke FujiiSep 27 2021 quant-ph arXiv:2109.11759v1

Variational quantum eigensolver (VQE), which attracts attention as a promising application of noisy intermediate-scale quantum devices, finds a ground state of a given Hamiltonian by variationally optimizing the parameters of quantum circuits called ansatz. Since the difficulty of the optimization depends on the complexity of the problem Hamiltonian and the structure of the ansatz, it has been difficult to analyze the performance of optimizers for the VQE systematically. To resolve this problem, we propose a technique to construct a benchmark problem whose ground state is guaranteed to be achievable with a given ansatz by using the idea of parent Hamiltonian of low-depth parameterized quantum circuits. We compare the convergence of several optimizers by varying the distance of the initial parameters from the solution and find that the converged energies showed a threshold-like behavior depending on the distance. This work provides a systematic way to analyze optimizers for VQE and contribute to the design of ansatz and its initial parameters.

Design of quantum optical experiments with logic artificial intelligence

Alba Cervera-Lierta,Mario Krenn,Alán Aspuru-GuzikSep 29 2021 quant-phcs.AIcs.LO arXiv:2109.13273v1

Logic artificial intelligence (AI) is a subfield of AI where variables can take two defined arguments, True or False, and are arranged in clauses that follow the rules of formal logic. Several problems that span from physical systems to mathematical conjectures can be encoded into these clauses and be solved by checking their satisfiability (SAT). Recently, SAT solvers have become a sophisticated and powerful computational tool capable, among other things, of solving long-standing mathematical conjectures. In this work, we propose the use of logic AI for the design of optical quantum experiments. We show how to map into a SAT problem the experimental preparation of an arbitrary quantum state and propose a logic-based algorithm, called Klaus, to find an interpretable representation of the photonic setup that generates it. We compare the performance of Klaus with the state-of-the-art algorithm for this purpose based on continuous optimization. We also combine both logic and numeric strategies to find that the use of logic AI improves significantly the resolution of this problem, paving the path to develop more formal-based approaches in the context of quantum physics experiments.

Scalable quantum state tomography with artificial neural networks

Tobias Schmale,Moritz Reh,Martin GärttnerSep 29 2021 quant-phcond-mat.quant-gasphysics.comp-ph arXiv:2109.13776v1

Modern day quantum simulators can prepare a wide variety of quantum states but extracting observables from the resulting “quantum data” often poses a challenge. We tackle this problem by developing a quantum state tomography scheme which relies on approximating the probability distribution over the outcomes of an informationally complete measurement in a variational manifold represented by a convolutional neural network. We show an excellent representability of prototypical ground- and steady states with this ansatz using a number of variational parameters that scales polynomially in system size. This compressed representation allows us to reconstruct states with high classical fidelities outperforming standard methods such as maximum likelihood estimation. Furthermore, it achieves a reduction of the root mean square errors of observables by up to an order of magnitude compared to their direct estimation from experimental data.

Variational Quantum-Based Simulation of Waveguide Modes

Wei-Bin Ewe,Dax Enshan Koh,Siong Thye Goh,Hong-Son Chu,Ching Eng PngSep 28 2021 quant-phphysics.comp-ph arXiv:2109.12279v1

Variational quantum algorithms are one of the most promising methods that can be implemented on noisy intermediate-scale quantum (NISQ) machines to achieve a quantum advantage over classical computers. This article describes the use of a variational quantum algorithm in conjunction with the finite difference method for the calculation of propagation modes of an electromagnetic wave in a hollow metallic waveguide. The two-dimensional (2D) waveguide problem, described by the Helmholtz equation, is approximated by a system of linear equations, whose solutions are expressed in terms of simple quantum expectation values that can be evaluated efficiently on quantum hardware. Numerical examples are presented to validate the proposed method for solving 2D waveguide problems.

Variational learning of quantum ground states on spiking neuromorphic hardware

Robert Klassert,Andreas Baumbach,Mihai A. Petrovici,Martin GärtnerOct 01 2021 quant-phcond-mat.dis-nncs.ETcs.NE arXiv:2109.15169v1

We train a neuromorphic hardware chip to approximate the ground states of quantum spin models by variational energy minimization. Compared to variational artificial neural networks using Markov chain Monte Carlo for sample generation, this approach has the advantage that the neuromorphic device generates samples in a fast and inherently parallel fashion. We develop a training algorithm and apply it to the transverse field Ising model, showing good performance at moderate system sizes (N≤10N≤10). A systematic hyperparameter study shows that scalability to larger system sizes mainly depends on sample quality which is limited by parameter drifts on the analog neuromorphic chip. The learning performance shows a threshold behavior as a function of the number of variational parameters of the ansatz, with approximately 5050 hidden neurons being sufficient for representing critical ground states up to N=10N=10. The 6+1-bit resolution of the network parameters does not limit the reachable approximation quality in the current setup. Our work provides an important step towards harnessing the capabilities of neuromorphic hardware for tackling the curse of dimensionality in quantum many-body problems.

Quantum walk-based vehicle routing optimisation

Tavis Bennett,Edric Matwiejew,Sam Marsh,Jingbo B. WangOct 01 2021 quant-phphysics.comp-ph arXiv:2109.14907v1

This paper demonstrates the applicability of the Quantum Walk-based Optimisation Algorithm(QWOA) to the Capacitated Vehicle Routing Problem (CVRP). Efficient algorithms are developedfor the indexing and unindexing of the solution space and for implementing the required alternatingphase-walk unitaries, which are the core components of QWOA. Results of numerical simulationdemonstrate that the QWOA is capable of producing convergence to near-optimal solutions for arandomly generated 8 location CVRP. Preparation of the amplified quantum state in this exampleproblem is demonstrated to produce high-quality solutions, which are more optimal than expectedfrom classical random sampling of equivalent computational effort.

Error-mitigated photonic variational quantum eigensolver using a single-photon ququart

Donghwa Lee,Jinil Lee,Seongjin Hong,Hyang-Tag Lim,Young-Wook Cho,Sang-Wook Han,Hyundong Shin,Junaid ur Rehman,Yong-Su KimSep 30 2021 quant-ph arXiv:2109.14244v1

We report the experimental resource-efficient implementation of the variational quantum eigensolver (VQE) using four-dimensional photonic quantum states of single-photons. The four-dimensional quantum states are implemented by utilizing polarization and path degrees of freedom of a single-photon. Our photonic VQE is equipped with the quantum error mitigation (QEM) protocol that efficiently reduces the effects of Pauli noise in the quantum processing unit. We apply our photonic VQE to estimate the ground state energy of He–H++ cation. The simulation and experimental results demonstrate that our resource-efficient photonic VQE can accurately estimate the bond dissociation curve, even in the presence of large noise in the quantum processing unit.

Hybrid Quantum Classical Graph Neural Networks for Particle Track Reconstruction

Cenk Tüysüz,Carla Rieger,Kristiane Novotny,Bilge Demirköz,Daniel Dobos,Karolos Potamianos,Sofia Vallecorsa,Jean-Roch Vlimant,Richard ForsterSep 28 2021 quant-phcs.LG arXiv:2109.12636v1

The Large Hadron Collider (LHC) at the European Organisation for Nuclear Research (CERN) will be upgraded to further increase the instantaneous rate of particle collisions (luminosity) and become the High Luminosity LHC (HL-LHC). This increase in luminosity will significantly increase the number of particles interacting with the detector. The interaction of particles with a detector is referred to as “hit”. The HL-LHC will yield many more detector hits, which will pose a combinatorial challenge by using reconstruction algorithms to determine particle trajectories from those hits. This work explores the possibility of converting a novel Graph Neural Network model, that can optimally take into account the sparse nature of the tracking detector data and their complex geometry, to a Hybrid Quantum-Classical Graph Neural Network that benefits from using Variational Quantum layers. We show that this hybrid model can perform similar to the classical approach. Also, we explore Parametrized Quantum Circuits (PQC) with different expressibility and entangling capacities, and compare their training performance in order to quantify the expected benefits. These results can be used to build a future road map to further develop circuit based Hybrid Quantum-Classical Graph Neural Networks.

Predicting Dynamics of Transmon Qubit-Cavity Systems with Recurrent Neural Networks

Nima LeclercSep 30 2021 cond-mat.dis-nnquant-ph arXiv:2109.14471v1

Developing accurate and computationally inexpensive models for the dynamics of open-quantum systems is critical in designing new qubit platforms by first understanding their mechanisms of decoherence and dephasing. Current models based on solutions to master equations are not sufficient in capturing the non-Markovian dynamics at play and suffer from large computational costs. Here, we present a method of overcoming this by using a recurrent neural network to obtain effective solutions to the Lindblad master equation for a coupled transmon qubit-cavity system. We present the training and testing performance of the model trained a simulated dataset and demonstrate its ability to map microscopic dissipative mechanisms to quantum observables.

A Quantum-Classical Hybrid Method for Image Classification and Segmentation

Sayantan Pramanik,M Girish Chandra,C V Sridhar,Aniket Kulkarni,Prabin Sahoo,Vishwa Chethan D V,Hrishikesh Sharma,Ashutosh Paliwal,Vidyut Navelkar,Sudhakara Poojary,Pranav Shah,Manoj NambiarSep 30 2021 quant-ph arXiv:2109.14431v1

Enormous activity in the Quantum Computing area has resulted in considering them to solve different difficult problems, including those of applied nature, together with classical computers. An attempt is made in this work to nail down a pipeline consisting of both quantum and classical processing blocks for the task of image classification and segmentation in a systematic fashion. Its efficacy and utility are brought out by applying it to Surface Crack segmentation. Being a sophisticated software engineering task, the functionalities are orchestrated through our in-house Cognitive Model Management framework.

Ionization energies in lithium and boron atoms using the Variational Quantum Eigensolver algorithm

Rene Villela,V. S. Prasannaa,B. P. DasSep 28 2021 physics.atom-phphysics.chem-phquant-ph arXiv:2109.12583v1

The classical-quantum hybrid Variational Quantum Eigensolver algorithm is the most widely used approach in the Noisy Intermediate Scale Quantum era to obtain ground state energies of atomic and molecular systems. In this work, we extend the scope of properties that can be calculated using the algorithm by computing the first ionization energies of Lithium and Boron atoms. We check the precision of our ionization energies and the observed many-body trends and compare them with the results from calculations carried out on traditional

Deep learning of polarization transfer in liquid crystals with application to quantum state preparation

Dominik Vašinka,Martin Bielak,Michal Neset,Miroslav JežekSep 28 2021 quant-phphysics.optics arXiv:2109.12436v1

Accurate control of light polarization represents a core building block in polarization metrology, imaging, and optical and quantum communications. Voltage-controlled liquid crystals offer an efficient way of polarization transformation. However, common twisted nematic liquid crystals are notorious for lacking an accurate theoretical model linking control voltages and output polarization. An inverse model, which would predict control voltages required to prepare a target polarization, is even more challenging. Here we explore a family of models fitted to a finite set of experimentally acquired calibration data. We report the direct and inverse models based on deep neural networks, radial basis functions, and linear interpolation. We demonstrate one order of magnitude improvement in accuracy using deep learning compared to the radial basis function method and two orders of magnitude improvement compared to the linear interpolation. Errors of the deep neural network model also decrease faster than the other methods with an increasing number of training data. The best direct and inverse models reach the average infidelities of 4⋅10−44⋅10−4 and 2⋅10−42⋅10−4, respectively, which is the accuracy level not reported yet. Furthermore, we demonstrate local and remote preparation of an arbitrary single-photon polarization state using the deep learning models. The results will impact the application of twisted-nematic liquid crystals, increasing their control accuracy across the board. The presented method can be used for optimal classical control of complex photonic devices and quantum circuits beyond interpolation.

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