Nearly tight Trotterization of interacting electrons

Abstract

We consider simulating quantum systems on digital quantum computers. We show that the performance of quantum simulation can be improved by simultaneously exploiting the commutativity of Hamiltonian, the sparsity of interactions, and the prior knowledge of initial state. We achieve this using Trotterization for a class of interacting electrons that encompasses various physical systems, including the plane-wave-basis electronic structure and the Fermi-Hubbard model. We estimate the simulation error by taking the transition amplitude of nested commutators of Hamiltonian terms within the η-electron manifold. We develop multiple techniques for bounding the transition amplitude and expectation of general fermionic operators, which may be of independent interest. We show that it suffices to use O n 5/3 η 2/3 + n 4/3 η 2/3 gates to simulate electronic structure in the plane-wave basis with n spin orbitals and η electrons up to a negligible factor, improving the best previous result in second quantization while outperforming the first-quantized simulation when n = O η 2 . We also obtain an improvement for simulating the Fermi-Hubbard model. We construct concrete examples for which our bounds are almost saturated, giving a nearly tight Trotterization of interacting electrons.

Fault-Tolerant Quantum Simulations of Chemistry in First Quantization

“…This argument was first made in the context of second quantized quantum simulations in [30]. Today, the most efficient second quantized algorithm for simulating plane wave electronic structure Hamiltonians with N plane waves and η electrons has Toffoli complexity (N 5/3 /η 2/3 + N 4/3 η 2/3 )N o (1) with space complexity O(N log N ) [29], where this complexity assumes that particle density is held fixed as the system size grows. We compare the scaling of all prior quantum algorithms for plane wave electronic structure in more detail in Table I.…”

“…Su et al [29] Yet tighter Trotter bounds O(N log N ) (η/Ω 1/3 + N 1/3 /Ω 2/3 )N 4/3+o(1) / 1+o (1) TABLE I. Best quantum algorithms for phase estimating chemistry in a plane wave basis.…”

“…Unlike other entries here, the scaling of the Kivlichan et al [25] algorithm is empirically observed for specific systems (with some systems scaling somewhat better and some systems scaling somewhat worse) as opposed to rigorous upper-bounds. The O(N log N ) space complexity of [3,12,29] results from a Fourier transform based method of computing the potential operator discussed in [3]. If those papers were instead to use the Trotter steps introduced in [24] or [25], the algorithms would have N + O(log N ) space complexity but gate complexity that is worse by approximately a factor of N .…”

“…The expectation that quantum simulation might enable an exponential speedup for physical problems stretches back to Feynman's original proposal for quantum computers [4,5] and was first concretely proposed for solving the the central problem of quantum chemistry by Aspuru-Guzik et al [6]. Since then, progress has accelerated on multiple fronts: improvements in general purpose simulation frameworks [2,3,[7][8][9][10][11][12][13][14][15], more efficient algorithms for quantum chemistry [16][17][18][19][20][21][22][23][24][25][26][27][28][29], more advantageous-toquantum-simulate representations of molecular Hamiltonians [30][31][32][33][34][35][36][37][38] and an improved understanding of the problems that genuinely require a quantum computer versus those that can be classically computed [39,40].…”

Fast-forwarding quantum evolution

“…We also consider the problem of fast-forwarding in subspaces, which concerns the case where quantum evolution occurs only in a certain subspace of the full Hilbert space. This notion of subspace fast-forwarding is useful for simulating physical systems because often the evolution occurs in certain subspaces of, for example, low energies or certain preserved symmetries [22,23,24]. Moreover, since certain quantum systems (e.g., bosonic systems) cannot be directly simulated on a digital quantum computer, we provide a definition of fast-forwarding that is relative to a specified set of observables.…”