Matrix product state approximations: Bringing theory closer to practice

Maxwell T. West, Azar C. Nakhl, Jamie Heredge +4 more

Quantum machine learning (QML) is emerging as an application of quantum computing with the potential to deliver quantum advantage, but its realization for practical applications remains impeded by challenges. Among these, a key barrier is the computationally expensive task of encoding classical data into a quantum state, which could erase any prospective speedups over classical algorithms. In this study, we implement methods for the efficient preparation of quantum states representing encoded image data using variational, genetic, and matrix product state algorithms. Our results show that these methods can approximately prepare states to a level suitable for QML using circuits 2 orders of magnitude shallower than a standard state preparation implementation, thus drastically reducing circuit depth and gate count without unduly sacrificing classification accuracy. Additionally, the QML models trained and evaluated on approximately encoded data displayed an increased robustness to adversarially generated input data perturbations. This partial alleviation of adversarial vulnerability, made possible by the retention of the meaningful large-scale features of the data despite the “drowning out” of adversarial perturbations, constitutes a considerable benefit for approximate state preparation in addition to lessening the requirements of the quantum hardware. Our results, based on simulations and experiments on IBM quantum devices, highlight a promising pathway for the future implementation of accurate and robust QML models on complex datasets relevant for practical applications, bringing the possibility of NISQ-era QML advantage closer to reality.

Ulrich Schollwöck

Robert König, Volkher B. Scholz

Matrix product states (MPSs) illustrate the suitability of tensor networks for the description of interacting many-body systems: ground states of gapped 1D systems are approximable by MPSs, as shown by Hastings [M. B. Hastings, J. Stat. Mech. (2007) P08024]. By contrast, whether MPSs and more general tensor networks can accurately reproduce correlations in critical quantum systems or quantum field theories has not been established rigorously. Ample evidence exists: entropic considerations provide restrictions on the form of suitable ansatz states, and numerical studies show that certain tensor networks can indeed approximate the associated correlation functions. Here, we provide a complete positive answer to this question in the case of MPSs and 2D conformal field theory: we give quantitative estimates for the approximation error when approximating correlation functions by MPSs. Our work is constructive and yields an explicit MPS, thus providing both suitable initial values and a rigorous justification of variational methods.

Fernando G. S. L. Brandão, Aram W. Harrow

Artemiy Burov, Oliver Nagl, Clément Javerzac‐Galy

While the recent demonstration of accurate computations of classically intractable simulations on noisy quantum processors brings quantum advantage closer, there is still the challenge of demonstrating it for practical problems. Here we investigate the application of noisy intermediate-scale quantum devices for simulating nuclear magnetic resonance (NMR) experiments in the high-field regime. In this work, the NMR interactions are mapped to a quantum device via a product formula with minimal resource overhead, an approach that we discuss in detail. Using this approach, we show the results of simulations of liquid-state proton NMR spectra on relevant molecules with up to 11 spins, and up to a total of 47 atoms, and compare them with real NMR experiments. Despite current limitations, we show that a similar approach will eventually lead to a case of quantum utility, a scenario where a practically relevant computational problem can be solved by a quantum computer but not by conventional means. We provide an experimental estimation of the amount of quantum resources needed for solving larger instances of the problem with the presented approach. The polynomial scaling we demonstrate on real processors is a foundational step in bringing practical quantum computation closer to reality.

Jesse J. Osborne