Simulating Time-Dependent Hamiltonians

Handling Hamiltonians that change during the evolution, from driven systems to interaction-picture and Dyson-series methods.

When the Hamiltonian changes

Many problems have a Hamiltonian H(t) that varies in time: driven quantum systems, adiabatic sweeps, systems in time-dependent external fields, and interaction-picture reformulations. The simple exponential e^(-iHt) no longer applies, because H at different times need not commute. The correct propagator is the time-ordered exponential U = T exp(-i integral H(s) ds).

Time-slicing

The basic approach divides [0, t] into short intervals over which H(s) is nearly constant, applies a product formula for each slice using the frozen Hamiltonian, and chains them in order. The slice width must be small enough that the variation of H across it is below the error budget, which sets the number of slices by how fast H changes.

The Dyson series

The time-ordered exponential expands as the Dyson series, the time-dependent generalization of the Taylor series, with nested time-ordered integrals. Truncating it and implementing the result as a linear combination of unitaries gives a time-dependent analog of Taylor-series simulation, with logarithmic precision scaling. The integrals are handled by discretizing the time argument on ancilla registers.

• Time-slicing with frozen H per slice: simple, error set by slice width.
• Truncated Dyson series via LCU: better precision scaling, more ancillas.
• Interaction picture: factor out a fast, easily-simulated part.
• Magnus expansion: exponentiate an averaged effective Hamiltonian per slice.

The interaction picture

When H(t) = H_0 + V(t) with H_0 large but easy to simulate, transforming to the interaction picture removes H_0 from the evolution, leaving only the (smaller) driven part V to simulate expensively. This can reduce the effective norm and hence the gate count substantially, a technique used to simulate systems with a large, simple background Hamiltonian plus a weaker interesting perturbation.

Uses

Time-dependent simulation underlies adiabatic state preparation, the modeling of laser-driven chemistry and pump-probe spectroscopy, and Floquet engineering of periodically driven systems. It is a direct extension of the static toolkit, product formulas, LCU, and QSVT all have time-dependent generalizations, and it broadens quantum simulation to the many physical situations where the governing Hamiltonian is not constant.