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 commutativity of the target Hamiltonian, sparsity of interactions, and prior knowledge of the 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 the 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 (n5/3η2/3+n4/3η2/3)no(1) gates to simulate electronic structure in the plane-wave basis with n spin orbitals and η electrons, improving the best previous result in second quantization up to a negligible factor while outperforming the first-quantized simulation when n=η2−o(1). 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.

Faster Coherent Quantum Algorithms for Phase, Energy, and Amplitude Estimation

“…Which method of Hamiltonian simulation is best depends on the particular physical system involved. Hamiltonian simulation using the Trotter approximation can perform exceedingly well in many situations [SHC20]. However, in our analysis we must be agnostic to the particular Hamiltonian in question, and furthermore need a unified method for comparing the performance.…”

“…However, phase estimation only delivers a quadratic speedup for estimation and the exponential speedup for linear algebra can sidestep the QFT [CKS15,GSLW18]. When applied to energies, phase estimation also requires Hamiltonian simulation, a quantum subroutine that is an entire subject of study in its own right: it requires recent innovations to apply optimally in a black-box setting [BCK15, LC1606, LC1610, LC17, GSLW18] and optimal Hamiltonian simulation for specific systems is still being actively studied [SHC20,Cam20]. Furthermore, phase estimation demands median amplification to guarantee an accurate answer.…”

Exploiting fermion number in factorized decompositions of the electronic structure Hamiltonian

“…Several previous works have estimated the resources required for phase estimation based on product-formula decompositions (also known as Trotterization) [8][9][10][11][12][13][14]. It was recently shown by Su, Huang, and Campbell [15] that knowledge about the number of fermions present in a chemical system can be exploited to improve the asymptotic performance of Trotterization. That work introduced an error metric, termed the fermionic seminorm, to bound the Trotter error.…”

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.…”