#39 September 11th – 17th

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Critical Points in Hamiltonian Agnostic Variational Quantum Algorithms

Eric R. AnschuetzSep 16 2021 quant-ph arXiv:2109.06957v1

One of the most important properties of classical neural networks is the clustering of local minima of the network near the global minimum, enabling efficient training. This has been observed not only numerically, but also has begun to be analytically understood through the lens of random matrix theory. Inspired by these results in classical machine learning, we show that a certain randomized class of variational quantum algorithms can be mapped to Wishart random fields on the hypertorus. Then, using the statistical properties of such random processes, we analytically find the expected distribution of critical points. Unlike the case for deep neural networks, we show the existence of a transition in the quality of local minima at a number of parameters exponentially large in the problem size. Below this transition, all local minima are concentrated far from the global minimum; above, all local minima are concentrated near the global minimum. This is consistent with previously observed numerical results on the landscape behavior of Hamiltonian agnostic variational quantum algorithms. We give a heuristic explanation as to why ansatzes that depend on the problem Hamiltonian might not suffer from these scaling issues. We also verify that our analytic results hold experimentally even at modest system sizes.

Towards a variational Jordan-Lee-Preskill quantum algorithm

Junyu Liu,Jinzhao Sun,Xiao YuanSep 14 2021 quant-phcs.ITcs.LGhep-thmath.IT arXiv:2109.05547v1

Rapid developments of quantum information technology show promising opportunities for simulating quantum field theory in near-term quantum devices. In this work, we formulate the theory of (time-dependent) variational quantum simulation, explicitly designed for quantum simulation of quantum field theory. We develop hybrid quantum-classical algorithms for crucial ingredients in particle scattering experiments, including encoding, state preparation, and time evolution, with several numerical simulations to demonstrate our algorithms in the 1+1 dimensional λϕ4λϕ4 quantum field theory. These algorithms could be understood as near-term analogs of the Jordan-Lee-Preskill algorithm, the basic algorithm for simulating quantum field theory using universal quantum devices. Our contribution also includes a bosonic version of the Unitary Coupled Cluster ansatz with physical interpretation in quantum field theory, a discussion about the subspace fidelity, a comparison among different bases in the 1+1 dimensional λϕ4λϕ4 theory, and the “spectral crowding” in the quantum field theory simulation.

Realizing Quantum Convolutional Neural Networks on a Superconducting Quantum Processor to Recognize Quantum Phases

Johannes Herrmann,Sergi Masot Llima,Ants Remm,Petr Zapletal,Nathan A. McMahon,Colin Scarato,Francois Swiadek,Christian Kraglund Andersen,Christoph Hellings,Sebastian Krinner,Nathan Lacroix,Stefania Lazar,Michael Kerschbaum,Dante Colao Zanuz,Graham J. Norris,Michael J. Hartmann,Andreas Wallraff,Christopher EichlerSep 14 2021 quant-ph arXiv:2109.05909v1

Quantum computing crucially relies on the ability to efficiently characterize the quantum states output by quantum hardware. Conventional methods which probe these states through direct measurements and classically computed correlations become computationally expensive when increasing the system size. Quantum neural networks tailored to recognize specific features of quantum states by combining unitary operations, measurements and feedforward promise to require fewer measurements and to tolerate errors. Here, we realize a quantum convolutional neural network (QCNN) on a 7-qubit superconducting quantum processor to identify symmetry-protected topological (SPT) phases of a spin model characterized by a non-zero string order parameter. We benchmark the performance of the QCNN based on approximate ground states of a family of cluster-Ising Hamiltonians which we prepare using a hardware-efficient, low-depth state preparation circuit. We find that, despite being composed of finite-fidelity gates itself, the QCNN recognizes the topological phase with higher fidelity than direct measurements of the string order parameter for the prepared states.

Generation of photonic tensor network states with Circuit QED

Zhi-Yuan Wei,J. Ignacio Cirac,Daniel MalzSep 15 2021 quant-ph arXiv:2109.06781v1

We propose a circuit QED platform and protocol to deterministically generate microwave photonic tensor network states. We first show that using a microwave cavity as ancilla and a transmon qubit as emitter is a favorable platform to produce photonic matrix-product states. The ancilla cavity combines a large controllable Hilbert space with a long coherence time, which we predict translates into a high number of entangled photons and states with a high bond dimension. Going beyond this paradigm, we then consider a natural generalization of this platform, in which several cavity–qubit pairs are coupled to form a chain. The photonic states thus produced feature a two-dimensional entanglement structure and are readily interpreted as radial plaquetteradial plaquette projected entangled pair states, which include many paradigmatic states, such as the broad class of isometric tensor network states, graph states, string-net states.

Generative Quantum Learning of Joint Probability Distribution Functions

Elton Yechao Zhu,Sonika Johri,Dave Bacon,Mert Esencan,Jungsang Kim,Mark Muir,Nikhil Murgai,Jason Nguyen,Neal Pisenti,Adam Schouela,Ksenia Sosnova,Ken WrightSep 15 2021 quant-ph arXiv:2109.06315v1

Modeling joint probability distributions is an important task in a wide variety of fields. One popular technique for this employs a family of multivariate distributions with uniform marginals called copulas. While the theory of modeling joint distributions via copulas is well understood, it gets practically challenging to accurately model real data with many variables. In this work, we design quantum machine learning algorithms to model copulas. We show that any copula can be naturally mapped to a multipartite maximally entangled state. A variational ansatz we christen as a `qopula’ creates arbitrary correlations between variables while maintaining the copula structure starting from a set of Bell pairs for two variables, or GHZ states for multiple variables. As an application, we train a Quantum Generative Adversarial Network (QGAN) and a Quantum Circuit Born Machine (QCBM) using this variational ansatz to generate samples from joint distributions of two variables for historical data from the stock market. We demonstrate our generative learning algorithms on trapped ion quantum computers from IonQ for up to 8 qubits and show that our results outperform those obtained through equivalent classical generative learning. Further, we present theoretical arguments for exponential advantage in our model’s expressivity over classical models based on communication and computational complexity arguments.

Quadratic Quantum Speedup for Perceptron Training

Pengcheng Liao,Barry C. Sanders,Tim ByrnesSep 13 2021 quant-ph arXiv:2109.04695v1

Perceptrons, which perform binary classification, are the fundamental building blocks of neural networks. Given a data set of size~NN and margin~γγ (how well the given data are separated), the query complexity of the best-known quantum training algorithm scales as either (\nicefracN−−√γ2)log(\nicefrac1γ2)(\nicefracNγ2)log⁡(\nicefrac1γ2) or \nicefracNγ−−√\nicefracNγ, which is achieved by a hybrid of classical and quantum search. In this paper, we improve the version space quantum training method for perceptrons such that the query complexity of our algorithm scales as \nicefracNγ−−−−−−−−−−√\nicefracNγ. This is achieved by constructing an oracle for the perceptrons using quantum counting of the number of data elements that are correctly classified. We show that query complexity to construct such an oracle has a quadratic improvement over classical methods. Once such an oracle is constructed, bounded-error quantum search can be used to search over the hyperplane instances. The optimality of our algorithm is proven by reducing the evaluation of a two-level AND-OR tree (for which the query complexity lower bound is known) to a multi-criterion search. Our quantum training algorithm can be generalized to train more complex machine learning models such as neural networks, which are built on a large number of perceptrons.

Quantifying the Impact of Precision Errors on Quantum Approximate Optimization Algorithms

Gregory Quiroz,Paraj Titum,Phillip Lotshaw,Pavel Lougovski,Kevin Schultz,Eugene Dumitrescu,Itay HenSep 13 2021 quant-ph arXiv:2109.04482v1

The quantum approximate optimization algorithm (QAOA) is a hybrid quantum-classical algorithm that seeks to achieve approximate solutions to optimization problems by iteratively alternating between intervals of controlled quantum evolution. Here, we examine the effect of analog precision errors on QAOA performance both from the perspective of algorithmic training and canonical state- and observable-dependent QAOA-relevant metrics. Leveraging cumulant expansions, we recast the faulty QAOA as a control problem in which precision errors are expressed as multiplicative control noise and derive bounds on the performance of QAOA. We show using both analytical techniques and numerical simulations that errors in the analog implementation of QAOA circuits hinder its performance as an optimization algorithm. In particular, we find that any fixed precision implementation of QAOA will be subject to an exponential degradation in performance dependent upon the number of optimal QAOA layers and magnitude of the precision error. Despite this significant reduction, we show that it is possible to mitigate precision errors in QAOA via digitization of the variational parameters, therefore at the cost of increasing circuit depth. We illustrate our results via numerical simulations and analytic and empirical error bounds as a comparison. While focused on precision errors, our approach naturally lends itself to more general noise scenarios and the calculation of error bounds on QAOA performance and broader classes of variational quantum algorithms.

Solving Rubik’s Cube via Quantum Mechanics and Deep Reinforcement Learning

Sebastiano Corli,Lorenzo Moro,Davide E. Galli,Enrico PratiSep 16 2021 quant-ph arXiv:2109.07199v1

Rubik’s Cube is one of the most famous combinatorial puzzles involving nearly 4.3×10194.3×1019 possible configurations. Its mathematical description is expressed by the Rubik’s group, whose elements define how its layers rotate. We develop a unitary representation of such group and a quantum formalism to describe the Cube from its geometrical constraints. Cubies are describedby single particle states which turn out to behave like bosons for corners and fermions for edges, respectively. When in its solved configuration, the Cube, as a geometrical object, shows symmetrieswhich are broken when driven away from this configuration. For each of such symmetries, we build a Hamiltonian operator. When a Hamiltonian lies in its ground state, the respective symmetry of the Cube is preserved. When all such symmetries are preserved, the configuration of the Cube matches the solution of the game. To reach the ground state of all the Hamiltonian operators, we make use of a Deep Reinforcement Learning algorithm based on a Hamiltonian reward. The Cube is solved in four phases, all based on a respective Hamiltonian reward based on its spectrum, inspired by the Ising model. Embedding combinatorial problems into the quantum mechanics formalism suggests new possible algorithms and future implementations on quantum hardware.

Avoiding symmetry roadblocks and minimizing the measurement overhead of adaptive variational quantum eigensolvers

V. O. Shkolnikov,Nicholas J. Mayhall,Sophia E. Economou,Edwin BarnesSep 14 2021 quant-ph arXiv:2109.05340v1

Quantum simulation of strongly correlated systems is potentially the most feasible useful application of near-term quantum computers. Minimizing quantum computational resources is crucial to achieving this goal. A promising class of algorithms for this purpose consists of variational quantum eigensolvers (VQEs). Among these, problem-tailored versions such as ADAPT-VQE that build variational ansätze step by step from a predefined operator pool perform particularly well in terms of circuit depths and variational parameter counts. However, this improved performance comes at the expense of an additional measurement overhead compared to standard VQEs. Here, we show that this overhead can be reduced to an amount that grows only linearly with the number nn of qubits, instead of quartically as in the original ADAPT-VQE. We do this by proving that operator pools of size 2n−22n−2 can represent any state in Hilbert space if chosen appropriately. We prove that this is the minimal size of such “complete” pools, discuss their algebraic properties, and present necessary and sufficient conditions for their completeness that allow us to find such pools efficiently. We further show that, if the simulated problem possesses symmetries, then complete pools can fail to yield convergent results, unless the pool is chosen to obey certain symmetry rules. We demonstrate the performance of such symmetry-adapted complete pools by using them in classical simulations of ADAPT-VQE for several strongly correlated molecules. Our findings are relevant for any VQE that uses an ansatz based on Pauli strings.

Energy Extrapolation in Quantum Optimization Algorithms

Chenfeng Cao,Yunlong Yu,Zipeng Wu,Nic Shannon,Bei Zeng,Robert JoyntSep 17 2021 quant-ph arXiv:2109.08132v1

Quantum annealing and the variational quantum eigensolver are two promising quantum algorithms to find the ground state of complicated Hamiltonians on near-term quantum devices. However, it is necessary to limit the evolution time or the circuit depth as much as possible since otherwise decoherence will degrade the computation. Even when this is done, there always exists a non-negligible estimation error in the ground state energy. Here we propose a scalable extrapolation approach to mitigate this error. With an appropriate regression, we can significantly improve the estimation accuracy for quantum annealing and variational quantum eigensolver for fixed quantum resources. The inference is achieved by extrapolating the annealing time to infinity or extrapolating the variance to zero. The only additional overhead is an increase in the number of measurements by a constant factor. We verified the validity of our method with the transverse-field Ising model. The method is robust to noise, and the techniques are applicable to other physics problems. Analytic derivations for the quadratic convergence feature of the residual energy in quantum annealing and the linear convergence feature of energy variance are given.

Reinforcement Learning vs. Gradient-Based Optimisation for Robust Energy Landscape Control of Spin-1/2 Quantum Networks

I. Khalid,C. A. Weidner,E. A. Jonckheere,S. G. Schirmer,F. C. LangbeinSep 16 2021 quant-phmath.OC arXiv:2109.07226v1

We explore the use of policy gradient methods in reinforcement learning for quantum control via energy landscape shaping of XX-Heisenberg spin chains in a model agnostic fashion. Their performance is compared to finding controllers using gradient-based L-BFGS optimisation with restarts, with full access to an analytical model. Hamiltonian noise and coarse-graining of fidelity measurements are considered. Reinforcement learning is able to tackle challenging, noisy quantum control problems where L-BFGS optimization algorithms struggle to perform well. Robustness analysis under different levels of Hamiltonian noise indicates that controllers found by reinforcement learning appear to be less affected by noise than those found with L-BFGS.

Short Quantum Circuits in Reinforcement Learning Policies for the Vehicle Routing Problem

Fabio Sanches,Sean Weinberg,Takanori Ide,Kazumitsu KamiyaSep 17 2021 quant-phcs.LG arXiv:2109.07498v1

Quantum computing and machine learning have potential for symbiosis. However, in addition to the hardware limitations from current devices, there are still basic issues that must be addressed before quantum circuits can usefully incorporate with current machine learning tasks. We report a new strategy for such an integration in the context of attention models used for reinforcement learning. Agents that implement attention mechanisms have successfully been applied to certain cases of combinatorial routing problems by first encoding nodes on a graph and then sequentially decoding nodes until a route is selected. We demonstrate that simple quantum circuits can used in place of classical attention head layers while maintaining performance. Our method modifies the networks used in [1] by replacing key and query vectors for every node with quantum states that are entangled before being measured. The resulting hybrid classical-quantum agent is tested in the context of vehicle routing problems where its performance is competitive with the original classical approach. We regard our model as a prototype that can be scaled up and as an avenue for further study on the role of quantum computing in reinforcement learning.

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