Time-dependent Hamiltonian Simulation Using Discrete Clock Constructions

Abstract

In this work we provide a new approach for approximating an ordered operator exponential using an ordinary operator exponential that acts on the Hilbert space of the simulation as well as a finitedimensional clock register. This approach allows us to translate results for simulating time-independent systems to the time-dependent case. Our result solves two open problems in simulation. It first provides a rigorous way of using discrete time-displacement operators to generate time-dependent product and multiproduct formulas. Second, it provides a way to allow qubitization to be employed for certain time-dependent formulas. We show that as the number of qubits used for the clock grows, the error in the ordered operator exponential vanishes, as well as the entanglement between the clock register and the register of the state being acted upon. As an application, we provide a new family of multiproduct formulas (MPFs) for time-dependent Hamiltonians that yield both commutator scaling and poly-logarithmic error scaling. This in turn, means that this method outperforms existing methods for simulating physically-local, time-dependent Hamiltonians. Finally, we apply the formalism to show how qubitization can be generalized to the time-dependent case and show that such a translation can be practical if the second derivative of the Hamiltonian is sufficiently small.

Reducing molecular electronic Hamiltonian simulation cost for linear combination of unitaries approaches

“…All methods that work with the residual ĤR of the interaction picture greatly improved the performance over decomposing Ĥ, decreasing the 1-norm by an order of magnitude. However, the interaction picture methodology requires a time-dependent simulation method, rendering it incompatible with some simulation techniques such as qubitization (although recent work has shown that qubitization can be made to work for restricted families of time-dependent Hamiltonians, existing results preclude the interaction picture [34]). For the time-independent case, the symmetry shift technique here introduced can be employed, which yields a significant decrease of the 1-norm when compared with commonly used techniques [10] and can be included in a tensor hypercontraction framework [11].…”

Calculating Potential Energy Surfaces with Quantum Computers by Measuring Only the Density Along Adiabatic Transitions

“…The discrete-clock construction 27 is a method that approximates the time-evolution of a system as a linear combination of M different k-step second-order Trotter evolution unitaries. The evolution of the Hamiltonian at time t for time step dt is approximated as…”

“…Measured from a Quantum Algorithm. In order to examine its applicability to quantum computing, we implement the discrete clock time evolution algorithm 27 in Tangelo. 28 In the Appendix (Section 6.2), we show that the discrete-clock algorithm works very well for the Fourier grid Hamiltonian basis set 2 used here.…”

Fock-Space Schrieffer–Wolff Transformation: Classically-Assisted Rank-Reduced Quantum Phase Estimation Algorithm

“…A typical illustration of the above techniques is the interaction and Heisenberg pictures widely used in quantum mechanics, which have recently been explored in the context of quantum computing simulations. The interaction-picture-based approach has recently been studied in the context of quantum computing [94][95][96].…”