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arXiv:2006.13498 · Physical Review A

Quantum system compression: A Hamiltonian guided walk through Hilbert space

Robert L. Kosut, Tak‐San Ho, HERSCHEL A. RABITZ

Published Jan 12, 2021• 5 citations
arXivHTMLPDF

Abstract

We present a systematic study of quantum system compression for the evolution of generic many-body problems. The necessary numerical simulations of such systems are seriously hindered by the exponential growth of the Hilbert-space dimension with the number of particles. For a constant Hamiltonian system of Hilbert-space dimension $n$ with frequencies ranging from ${f}_{min}$ to ${f}_{max}$, we show via a proper orthogonal decomposition that for a run time $T$ the dominant dynamics are compressed in the neighborhood of a subspace whose dimension is the smallest integer larger than the time-bandwidth product $\mathrm{\ensuremath{\Delta}}=({f}_{max}\ensuremath{-}{f}_{min})T$. We also show how the distribution of initial states can further compress the system dimension. Under the stated conditions, the time-bandwidth estimate reveals the existence of an effective compressed model whose dimension is derived solely from system properties and not dependent on the particular implementation of a variational simulator, such as a machine learning system, or quantum device, or possibly even specially adapting traditional methods of solving the time-dependent Schr\"odinger equation. However, finding an efficient solution procedure $\mathit{is}$ dependent on the simulator implementation, which is not discussed in this paper. In addition, we show that the compression rendered by the proper orthogonal decomposition encoding method can be further strengthened via a multilayer autoencoder. Finally, we present numerical illustrations to affirm the compression behavior in time-varying Hamiltonian dynamics in the presence of external fields. The essential time-bandwidth product is also simply estimated for a wide class of physical systems, where typically localized high-frequency motion occurs at or around each of the many particles, and with low-frequency dynamics associated with globally distributed characteristic degrees of freedom. This estimate for the bandwidth has a generic character indicating the wide significance of expected quantum system dynamics compression. We also discuss the potential implications of the findings for machine learning tools to efficiently solve the many-body or other high-dimensional Schr\"odinger equations.

DOI
10.1103/physreva.103.012406

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We present a systematic study of quantum system compression for the evolution of generic many-body problems. The necessary numerical simulations of such systems are seriously hindered by the exponential growth of the Hilbert-space dimension with the number of particles. For a constant Hamiltonian system of Hilbert-space dimension nnn with frequencies ranging from fmin{f}_{min}fmin​ to fmax{f}_{max}fmax​, we show via a proper orthogonal decomposition that for a run time TTT the dominant dynamics are compressed in the neighborhood of a subspace whose dimension is the smallest integer larger than the time-bandwidth product \ensuremathΔ=(fmax\ensuremath−fmin)T\mathrm{\ensuremath{\Delta}}=({f}_{max}\ensuremath{-}{f}_{min})T\ensuremathΔ=(fmax​\ensuremath−fmin​)T. We also show how the distribution of initial states can further compress the system dimension. Under the stated conditions, the time-bandwidth estimate reveals the existence of an effective compressed model whose dimension is derived solely from system properties and not dependent on the particular implementation of a variational simulator, such as a machine learning system, or quantum device, or possibly even specially adapting traditional methods of solving the time-dependent Schr"odinger equation. However, finding an efficient solution procedure is\mathit{is}is dependent on the simulator implementation, which is not discussed in this paper. In addition, we show that the compression rendered by the proper orthogonal decomposition encoding method can be further strengthened via a multilayer autoencoder. Finally, we present numerical illustrations to affirm the compression behavior in time-varying Hamiltonian dynamics in the presence of external fields. The essential time-bandwidth product is also simply estimated for a wide class of physical systems, where typically localized high-frequency motion occurs at or around each of the many particles, and with low-frequency dynamics associated with globally distributed characteristic degrees of freedom. This estimate for the bandwidth has a generic character indicating the wide significance of expected quantum system dynamics compression. We also discuss the potential implications of the findings for machine learning tools to efficiently solve the many-body or other high-dimensional Schr"odinger equations.
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Abstract Simulating quantum dynamics to extract time-evolving observables constitutes a central challenge in quantum computing, with both fundamental significance and broad practical applications. Classical approaches suffer from the exponential scaling of Hilbert space, while existing quantum algorithms face limitations from deep circuits and sequential error accumulation on near-term devices. This work introduces a physics-informed quantum subspace method for the efficient estimation of dynamical properties of quantum systems. The core innovation is a globally physics-informed loss function that incorporates the time-dependent Schrödinger equation as a physics-based penalty. This enforces quantum evolution constraints directly during optimization, thereby circumventing the error accumulation inherent in stepwise simulations. By strategically relaxing the normalization constraint, we obtain convexified loss functions whose optimization reduces to solving a linear system, guaranteeing global convergence and significantly mitigating the convergence issues and barren plateaus common in variational quantum algorithms. Theoretically, we prove that under suitable conditions the true dynamical solution can be approximated with high accuracy within a subspace whose dimension scales only as O ( T log ⁡ ( 1 / ε ) ) , thus breaking the curse of dimensionality in classical simulation. Numerical experiments demonstrate that the proposed method outperforms conventional Trotterization and variational quantum benchmarks in terms of computational cost, convergence speed, and robustness against measurement noise, offering a viable and efficient pathway for practical dynamical simulation on noisy intermediate-scale quantum hardware.
The exact description of many-body quantum systems represents one of the major challenges in modern physics, because it requires an amount of computational resources that scales exponentially with the size of the system. Simulating the evolution of a state, or even storing its description, rapidly becomes intractable for exact classical algorithms. Recently, machine learning techniques, in the form of restricted Boltzmann machines, have been proposed as a way to efficiently represent certain quantum states with applications in state tomography and ground state estimation. Here, we introduce a practically usable deep architecture for representing and sampling from probability distributions of quantum states. Our representation is based on variational auto-encoders, a type of generative model in the form of a neural network. We show that this model is able to learn efficient representations of states that are easy to simulate classically and can compress states that are not classically tractable. Specifically, we consider the learnability of a class of quantum states introduced by Fefferman and Umans. Such states are provably hard to sample for classical computers, but not for quantum ones, under plausible computational complexity assumptions. The good level of compression achieved for hard states suggests these methods can be suitable for characterizing states of the size expected in first generation quantum hardware.
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Solving quantum dynamics is an exponentially difficult problem. Thus, an exact numerical solution is inaccessible for any condensed matter system.\nA promising approach is to divide the system into a quantum subsystem containing degrees of freedom which are of greater interest or those which have more profound quantum character (e.g., have smaller mass) and a classical bath containing the rest of the system.\nImposing such a partition and treating the bath classically results in quantum-classical dynamics. The quantum-classical Liouville equation is a general equation in the Hilbert space of quantum degrees of freedom while it resides in the phase space of the classical degrees of freedom.\nAny numerical solution to this equation requires representation of the quantum subsystem in some basis. Solutions to this equation have been already proposed in the subsystem, adiabatic and force bases, each with its own cons and pros.\n\nIn this work, the quantum-classical equations of motion are cast in the subsystem basis and subsequently mapped to a number of fictitious harmonic oscillators.\nThe result is quantum-classical dynamics in the mapping basis which treats both quantum and classical degrees of freedom on the same footing, i.e., in phase space. Neglecting a portion of the back reaction of the quantum-subsystem to classical bath results in an expression for the time evolution of an operator (density matrix) equal to its Poisson bracket with the Hamiltonian.\nThis equation can be solved in terms of characteristics to provide a computationally tractable method for calculating quantum-classical dynamical properties. The expressions for expectation values and correlation functions in this formalism are derived.\nCalculations on spin-boson system, barrier crossing models---the so called Tully models---and the Fenna-Mathews-Olson pigments show very good agreement between the results of this method and numerical solutions to the Schrödinger equation.