Spectral Analysis of Product Formulas for Quantum Simulation

Abstract

We consider Hamiltonian simulation using the first order Lie-Trotter product formula under the assumption that the initial state has a high overlap with an energy eigenstate, or a collection of eigenstates in a narrow energy band. This assumption is motivated by quantum phase estimation (QPE) and digital adiabatic simulation (DAS). Treating the effective Hamiltonian that generates the Trotterized time evolution using rigorous perturbative methods, we show that the Trotter step size needed to estimate an energy eigenvalue within precision using QPE can be improved in scaling from to 1/2 for a large class of systems (including any Hamiltonian which can be decomposed as a sum of local terms or commuting layers that each have real-valued matrix elements). For DAS we improve the asymptotic scaling of the Trotter error with the total number of gates M from O(M −1 ) to O(M −2 ), and for any fixed circuit depth we calculate an approximately optimal step size that balances the error contributions from Trotterization and the adiabatic approximation. These results partially generalize to diabatic processes, which remain in a narrow energy band separated from the rest of the spectrum by a gap, thereby contributing to the explanation of the observed similarities between the quantum approximate optimization algorithm and diabatic quantum annealing at small system sizes. Our analysis depends on the perturbation of eigenvectors as well as eigenvalues, and on quantifying the error using state fidelity (instead of the matrix norm of the difference of unitaries which is sensitive to an overall global phase).

Geometric quantum adiabatic methods for quantum chemistry

“…Therefore, to achieve the same fixed accuracy at larger atomic distances, we may need more discretization steps M and a larger evolution time T . But a recent study shows that the accumulation error in using Trotterization for the adiabatic evolution is not as severe as one would expect if the initial state is an eigenstate [22], as we have also observed in the bottom panel of Fig. 2.…”

Behavior of Analog Quantum Algorithms

“…Our second approach is restricted to an adiabatic anneal where the goal is to maintain the populations of eigenstates, specifically the ground state in our setting. The overall Trotter error bound in this setting was recently tightened by Yi and Crosson [40]. The same oscillatory enhancement found in the case of operator errors can be shown to occur in this setting as well, but the method requires a perturbative limit which does not hold for the QAOA angles.…”

“…The methods in this section closely follow the results of Yi and Crosson [40] who themselves draw inspiration from [6] and [41]. Specifically, this result can be thought of as a modification of their Proposition 1 (proven in their Appendix F) to the setting where the underlying annealing curve has an oscillatory structure.…”

“…The key result of the method of Ref. [40] is a reduction in the scaling of the Trotter error from O(∆t 2 p) down to O( 1 p ) + O( ∆t p ) when trying to simulate an adiabatic evolution. We show that this error scaling does not vanish when an oscillating schedule is considered (for small enough oscillations that the adiabatic theorem still holds) and show that there is an enhancement to the error scaling when the product formula step size matches the oscillation period.…”

“…It is not possible to fully apply this second approach to our setting because of the perturbative ∆t issues. Our product formula enhancement works partially in this setting and inherits the improved p scaling that the adiabatic Trotter method [40] naturally has over the operator error scaling, Eq. ( 33).…”