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arXiv:2312.08310

Quantum simulation of highly-oscillatory many-body Hamiltonians for near-term devices

Guannan Chen, Mohammadali Foroozandeh, Chris J. Budd, Pranav Singh

Published Dec 13, 2023• 1 citation
arXivHTMLPDF

Abstract

We develop a fourth-order Magnus expansion based quantum algorithm for the simulation of many-body problems involving two-level quantum systems with time-dependent Hamiltonians, $\mathcal{H}(t)$. A major hurdle in the utilization of the Magnus expansion is the appearance of a commutator term which leads to prohibitively long circuits. We present a technique for eliminating this commutator and find that a single time-step of the resulting algorithm is only marginally costlier than that required for time-stepping with a time-independent Hamiltonian, requiring only three additional single-qubit layers. For a large class of Hamiltonians appearing in liquid-state nuclear magnetic resonance (NMR) applications, we further exploit symmetries of the Hamiltonian and achieve a surprising reduction in the expansion, whereby a single time-step of our fourth-order method has a circuit structure and cost that is identical to that required for a fourth-order Trotterized time-stepping procedure for time-independent Hamiltonians. Moreover, our algorithms are able to take time-steps that are larger than the wavelength of oscillation of the time-dependent Hamiltonian, making them particularly suited for highly-oscillatory controls. The resulting quantum circuits have shorter depths for all levels of accuracy when compared to first and second-order Trotterized methods, as well as other fourth-order Trotterized methods, making the proposed algorithm a suitable candidate for simulation of time-dependent Hamiltonians on near-term quantum devices.

DOI
10.48550/arxiv.2312.08310

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We develop a fourth-order Magnus expansion based quantum algorithm for the simulation of many-body problems involving two-level quantum systems with time-dependent Hamiltonians, H(t)\mathcal{H}(t)H(t). A major hurdle in the utilization of the Magnus expansion is the appearance of a commutator term which leads to prohibitively long circuits. We present a technique for eliminating this commutator and find that a single time-step of the resulting algorithm is only marginally costlier than that required for time-stepping with a time-independent Hamiltonian, requiring only three additional single-qubit layers. For a large class of Hamiltonians appearing in liquid-state nuclear magnetic resonance (NMR) applications, we further exploit symmetries of the Hamiltonian and achieve a surprising reduction in the expansion, whereby a single time-step of our fourth-order method has a circuit structure and cost that is identical to that required for a fourth-order Trotterized time-stepping procedure for time-independent Hamiltonians. Moreover, our algorithms are able to take time-steps that are larger than the wavelength of oscillation of the time-dependent Hamiltonian, making them particularly suited for highly-oscillatory controls. The resulting quantum circuits have shorter depths for all levels of accuracy when compared to first and second-order Trotterized methods, as well as other fourth-order Trotterized methods, making the proposed algorithm a suitable candidate for simulation of time-dependent Hamiltonians on near-term quantum devices.
Efficient simulation of quantum dynamics with time-dependent Hamiltonians is important not only for time-varying systems but also for time-independent Hamiltonians in the interaction picture. Such simulations are more challenging than their time-independent counterparts due to the complexity introduced by time ordering. Existing algorithms that aim to capture commutator-based scaling either exhibit polynomial cost dependence on the Hamiltonian’s time derivatives or are limited to low-order accuracy. In this work, we establish the general commutator-scaling error bounds for the truncated Magnus expansion at arbitrary order, where only Hamiltonian terms appear in the nested commutators, with no time derivatives involved. Building on this analysis, we design a high-order quantum algorithm with explicit circuit constructions. The algorithm achieves cost scaling with the commutator structure in the high-precision regime and depends only logarithmically on the Hamiltonian’s time variation, making it efficient under the standard input model for general time-dependent settings, including Hamiltonians arising in the interaction picture.
Hamiltonian simulation is arguably the most fundamental application of quantum computers. The Magnus operator is a popular method for time-dependent Hamiltonian simulation in computational mathematics, yet its usage requires the implementation of exponentials of commutators, which has previously made it unappealing for quantum computing. The development of commutator-free quasi-Magnus operators (CFQMs) circumvents this obstacle, at the expense of a lack of provable global numeric error bounds. In this work, we establish one such error bound for CFQM-based time-dependent quantum Hamiltonian simulation by carefully estimating the error of each step involved in their definition. This allows us to compare its cost with the alternatives, and show that CFQMs are often the most efficient product-formula technique available by more than an order of magnitude. As a result, we find that CFQMs may be particularly useful to simulate time-dependent Hamiltonians on early fault-tolerant quantum computers.
The evolution of a quantum system under time-dependent driving exhibits phenomena that are absent in its stationary counterpart. However, the high dimensionality and non-commutative nature of quantum dynamics make this a challenging problem. The Magnus expansion provides an analytic framework to approximate the effective dynamics on short time-scales, but computing high-order terms with existing methods is computationally expensive. We introduce a scalable approach that reduces the computational effort to depend only on the degrees of freedom defining the time-dependent control function. We focus specifically on Hamiltonians consisting of a constant drift term and a controllable term. Our method provides a polynomial expression for the Magnus expansion which can be evaluated several orders of magnitude faster than previous techniques, enabling broad applications in the realms of quantum simulation and quantum optimal control. We showcase an application of the method by designing control pulses for the 5-qubit phase gate on a neutral-atom platform utilizing Rydberg atoms.
We consider the numerical propagation of models that combine both quantum and classical degrees of freedom, usually, electrons and nuclei, respectively. We focus, in our computational examples, on the case in which the quantum electrons are modeled with time-dependent density-functional theory, although the methods discussed below can be used with any other level of theory. Often, for these so-called quantum-classical molecular dynamics models, one uses some propagation technique to deal with the quantum part and a different one for the classical equations. While the resulting procedure may, in principle, be consistent, it can however spoil some of the properties of the methods, such as the accuracy order with respect to the time step or the preservation of the geometrical structure of the equations. Few methods have been developed specifically for hybrid quantum-classical models. We propose using the same method for both the quantum and classical particles, in particular, one family of techniques that proves to be very efficient for the propagation of Schrödinger-like equations: the (quasi)-commutator free Magnus expansions. These have been developed, however, for linear systems, yet our problem is nonlinear: formally, the full quantum-classical system can be rewritten as a nonlinear Schrödinger equation, i.e., one in which the Hamiltonian depends on the system itself. The Magnus expansion algorithms for linear systems require the application of the Hamiltonian at intermediate points in a given propagating interval. For nonlinear systems, this poses a problem as this Hamiltonian is unknown due to its dependence on the state. We approximate it by employing a higher order extrapolation using previous steps as input. The resulting technique can then be regarded as a multistep technique or, alternatively, as a predictor corrector formula.
We propose a simple quantum algorithm for simulating highly oscillatory quantum dynamics, which does not require complicated quantum control logic for handling time-ordering operators. To our knowledge, this is the first quantum algorithm that is both insensitive to the rapid changes of the time-dependent Hamiltonian and exhibits commutator scaling. Our method can be used for efficient Hamiltonian simulation in the interaction picture. In particular, we demonstrate that for the simulation of the Schr\odinger equation, our method exhibits superconvergence and achieves a surprising second order convergence rate, of which the proof rests on a careful application of pseudo-differential calculus. Numerical results verify the effectiveness and the superconvergence property of our method.
Quantum dynamics can be simulated on a quantum computer by exponentiating elementary terms from the Hamiltonian in a sequential manner.However, such an implementation of Trotter steps has gate complexity depending on the total Hamiltonian term number, comparing unfavorably to algorithms using more advanced techniques.We develop methods to perform faster Trotter steps with complexity sublinear in the number of terms.We achieve this for a class of Hamiltonians whose interaction strength decays with distance according to power law.Our methods include one based on a recursive block encoding and one based on an average-cost simulation, overcoming the normalization-factor barrier of these advanced quantum simulation techniques.We also realize faster Trotter steps when certain blocks of Hamiltonian coefficients have low rank.Combining with a tighter error analysis, we show that it suffices to use η 1/3 n 1/3 + n 2/3 /η 2/3 n 1+o(1) gates to simulate uniform electron gas with n spin orbitals and η electrons in second quantization in real space, asymptotically improving over the best previous work.We obtain an analogous result when the external potential of nuclei is introduced under the Born-Oppenheimer approximation.We prove a circuit lower bound when the Hamiltonian coefficients take a continuum range of values, showing that generic n-qubit two-local Hamiltonians with commuting terms require at least (n 2 ) gates to evolve with accuracy = (1/poly(n)) for time t = ( ).Our proof is based on a gateefficient reduction from the approximate synthesis of diagonal unitaries within the Hamming weight-2 subspace, which may be of independent interest.Our result thus suggests the use of Hamiltonian structural properties as both necessary and sufficient to implement Trotter steps with lower gate complexity.