26th Australian Institute of Physics Congress and the Australian and New Zealand Conference on Optics and Photonics

Processed Product Formulae for More Accurate Quantum Simulation

Not scheduled

20m

Belinda Hutchinson Building (The University of Sydney )

Contributed Oral AIP | Quantum Science and Technology (QST)

Speaker

Abhishek Rawat (Macquarie University)

Description

Simulating how quantum systems evolve in time is one of the most promising applications of quantum computers, spanning chemistry, materials science, and fundamental physics. The original and still one of the most efficient tools for this is the product formula. It approximates the time-evolution operator as a sequence of simpler exponentials that map directly to quantum gates. For structured fermionic and lattice Hamiltonians, its asymptotic scaling rivals and sometimes beats qubitization and quantum signal processing. The open challenge is reaching that promise at the lowest practical cost. Higher-order product formulae reduce the total simulation cost by enabling larger time steps for a given error tolerance, a critical advantage for early fault-tolerant quantum devices.

For decades, the dominant approach constructed higher-order formulae as products of the symmetric Trotter formula, coupling the coefficients of each exponential and restricting the achievable accuracy. We drop this constraint, allowing fully independent coefficients, and combine it with a processing technique that lowers per-step cost for long time simulation: a short, repeated core does most of the work, while a one-time correction at the start and end cancels the leftover error. We search this enlarged space by solving the simultaneous non-linear equations from random initializations, then refine the best candidates with optimization techniques like Nelder-Mead simplex and CMA-ES.

We optimize the eigenvalue-error constant- the error that dominates long simulations and governs energy and phase estimation. Our 4th-order formula achieves nearly 5× lower error constant, cutting total circuit complexity by 43%. At 6th order we obtain an order-of-magnitude error reduction, cutting circuit complexity by 33%. At 8th order, our processed formula surpasses the best previously reported processed result and is, to our knowledge, the first to use independent coefficients at this order. We expect these formulae to provide the fastest product-formula approach in any realistic simulation regime.