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arXiv:2410.15256

Improved Time-independent Hamiltonian Simulation

Nhat A. Nghiem

Published Oct 20, 2024• 0 citations
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Abstract

We describe a simple method for simulating time-independent Hamiltonian $H$ that could be decomposed as $H = \sum_{i=1}^m H_i$ where each $H_i$ can be efficiently simulated. Approaches relying on product formula generally work by splitting the evolution time into segments, and approximate the evolution in each segment by the evolution of composing Hamiltonian $H_i$. This key step incur a constraint, that prohibits a (poly)logarithmic scaling on approximation error. We employ the recently introduced quantum singular value transformation framework to utilize the ability to simulate $H_i$ in an alternative way, which then allows us to construct and simulate the main Hamiltonian $H$ with polylogarithmical scaling on the inverse of desired error, which is a major improvement with respect to product formula approaches.

DOI
10.48550/arxiv.2410.15256

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We describe a simple method for simulating time-independent Hamiltonian HHH that could be decomposed as H=∑i=1mHiH = \sum_{i=1}^m H_iH=∑i=1m​Hi​ where each HiH_iHi​ can be efficiently simulated. Approaches relying on product formula generally work by splitting the evolution time into segments, and approximate the evolution in each segment by the evolution of composing Hamiltonian HiH_iHi​. This key step incur a constraint, that prohibits a (poly)logarithmic scaling on approximation error. We employ the recently introduced quantum singular value transformation framework to utilize the ability to simulate HiH_iHi​ in an alternative way, which then allows us to construct and simulate the main Hamiltonian HHH with polylogarithmical scaling on the inverse of desired error, which is a major improvement with respect to product formula approaches.
Product formula approximations of the time-evolution operator on quantum computers are of great interest due to their simplicity, and good scaling with system size by exploiting commutativity between Hamiltonian terms. However, product formulas exhibit poor scaling with the time ttt and error εεε of simulation as the gate cost of a single step scales exponentially with the order mmm of accuracy. We introduce well-conditioned multiproduct formulas, which are a linear combination of product formulas, where a single step has polynomial cost O(m2log⁡(m))\mathcal{O}(m^2\log{(m)})O(m2log(m)) and succeeds with probability Ω(1/log⁡2(m))Ω(1/\operatorname{log}^2{(m)})Ω(1/log2(m)). Our multiproduct formulas imply a simple and generic simulation algorithm that simultaneously exploits commutativity in arbitrary systems and has a worst-case cost O(tlog⁡2(t/ε))\mathcal{O}(t\log^{2}{(t/ε)})O(tlog2(t/ε)) which is optimal up to poly-logarithmic factors. In contrast, prior Trotter and post-Trotter Hamiltonian simulation algorithms realize only one of these two desirable features. A key technical result of independent interest is our solution to a conditioning problem in previous multiproduct formulas that amplified numerical errors by eΩ(m)e^{Ω(m)}eΩ(m) in the classical setting, and led to a vanishing success probability e−Ω(m)e^{-Ω(m)}e−Ω(m) in the quantum setting.
We propose a new method for simulating certain type of time-dependent Hamiltonian H(t)=∑i=1mγi(t)HiH(t) = \sum_{i=1}^m γ_i(t) H_iH(t)=∑i=1m​γi​(t)Hi​ where γi(t)γ_i(t)γi​(t) (and its higher order derivatives) is bounded, computable function of time ttt, and each HiH_iHi​ is time-independent, and could be efficiently simulated. Our quantum algorithms are based on high-order Runge-Kutta method and forward Euler method, where the time interval is divided into subintervals. Then in an iterative manner, the evolution operator at given time step is built upon the evolution operator at previous time step, utilizing algorithmic operations from the recently introduced quantum singular value transformation framework.
We provide a quantum method for simulating Hamiltonian evolution with complexity polynomial in the logarithm of the inverse error. This is an exponential improvement over existing methods for Hamiltonian simulation. In addition, its scaling with respect to time is close to linear, and its scaling with respect to the time derivative of the Hamiltonian is logarithmic. These scalings improve upon most existing methods. Our method is to use a compressed Lie-Trotter formula, based on recent ideas for efficient discrete-time simulations of continuous-time quantum query algorithms.
In this work we propose an approach for implementing time-evolution of a quantum system using product formulas. The quantum algorithms we develop have provably better scaling (in terms of gate complexity and circuit depth) than a naive application of well-known Trotter formulas, for systems where the evolution is determined by a Hamiltonian with different energy scales (i.e., one part is "large" and another part is "small"). Our algorithms generate a decomposition of the evolution operator into a product of simple unitaries that are directly implementable on a quantum computer. Although the theoretical scaling is suboptimal compared with state-of-the-art algorithms (e.g., quantum signal processing), the performance of the algorithms we propose is highly competitive in practice. We illustrate this via extensive numerical simulations for several models. For instance, in the strong-field regime of the 1D transverse-field Ising model, our algorithms achieve an improvement of one order of magnitude in both the system size and evolution time that can be simulated with a fixed budget of 1000 arbitrary 2-qubit gates, compared with standard Trotter formulas.
Given a quantum Hamiltonian and its evolution time, the corresponding unitary evolution operator can be constructed in many different ways, corresponding to different trajectories between the desired end-points and different series expansions. A choice among these possibilities can then be made to obtain the best computational complexity and control over errors. It is shown how a construction based on Grover's algorithm scales linearly in time and logarithmically in the error bound, and is exponentially superior in error complexity to the scheme based on the straightforward application of the Lie–Trotter formula. The strategy is then extended first to simulation of any Hamiltonian that is a linear combination of two projection operators, and then to any local efficiently computable Hamiltonian. The key feature is to construct an evolution in terms of the largest possible steps instead of taking small time steps. Reflection operations and Chebyshev expansions are used to efficiently control the total error on the overall evolution, without worrying about discretization errors for individual steps. We also use a digital implementation of quantum states that makes linear algebra operations rather simple to perform.
Abstract We study the problem of simulating the dynamics of spin systems when the initial state is supported on a subspace of low energy of a HamiltonianH. This is a central problem in physics with vast applications in many-body systems and beyond, where the interesting physics takes place in the low-energy sector. We analyze error bounds induced by product formulas that approximate the evolution operator and show that these bounds depend on an effective low-energy norm ofH. We find improvements over the best previous complexities of product formulas that apply to the general case, and these improvements are more significant for long evolution times that scale with the system size and/or small approximation errors. To obtain these improvements, we prove exponentially decaying upper bounds on the leakage to high-energy subspaces due to the product formula. Our results provide a path to a systematic study of Hamiltonian simulation at low energies, which will be required to push quantum simulation closer to reality.