Bulletin of the American Physical Society
APS March Meeting 2022
Volume 67, Number 3
Monday–Friday, March 14–18, 2022; Chicago
Session G00: Poster Session I (2pm- 5pm CST)
Tuesday, March 15, 2022
Room: McCormick Place Exhibit Hall F1
Abstract: G00.00242 : Concentration for Trotter error*
This work considers the concentration aspects of Trotter error: we show quantitatively that the Trotter error exhibits 2-norm scaling ``typically'', with the existing estimates in 1-norm being for the ``worst'' cases. For general k-local Hamiltonians, we obtain gate count estimates for input states drawn from a 1-design ensemble (e.g. computational basis states). Our gate count depends on the number of terms in the Hamiltonian but replaces the 1-norm quantity by its analog in 2-norm, giving significant speedup for systems with large connectivity. Our concentration results generalize to Hamiltonians with fermionic terms and when the input state is restricted to a low-particle number subspace. Further, when the Hamiltonian itself has random coefficients, such as the SYK models, we show the stronger result that the 2-norm behavior persists even for the worst input state.
Our main technical tool is a family of simple but versatile inequalities from non-commutative martingales called uniform smoothness. We use them to derive Hypercontractivity, i.e. p-norm estimates for low-degree polynomials, which implies concentration via Markov's inequality. In terms of optimality, we give examples that simultaneously match our p-norm estimates and the spectral norm estimates. This shows our improvement is due to asking a qualitatively different question than the one asked in the spectral norm bounds. Our results give evidence that product formulas in practice may work much better than expected.
*Caltech RA fellowship
The American Physical Society (APS) is a non-profit membership organization working to advance the knowledge of physics.
1 Physics Ellipse, College Park, MD 20740-3844
Editorial Office 1 Research Road, Ridge, NY 11961-2701 (631) 591-4000
Office of Public Affairs 529 14th St NW, Suite 1050, Washington, D.C. 20045-2001 (202) 662-8700