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Showing posts from January, 2022

Hamiltonian simulation via the Trotter-Suzuki decomposition

This academic term, some colleagues at Stanford and I are running a journal club on Hamiltonian simulation — the problem of how to use a quantum computer to simulate the time evolution of a physical system. Hamiltonian simulation is a hot topic in research, in part because it's believed that simulating certain systems on quantum computers will allow us to probe aspects of those systems that we don't know how to access with traditional laboratory experiments. The earliest approach to this problem, and one that is still practically useful for certain applications, makes use of the Trotter decomposition and its generalization the Trotter-Suzuki decomposition . These are algorithms for decomposing a time evolution operator that acts simultaneously on the entire quantum system, into a sequence of time evolution operators that act locally on only a few physical sites at a time. Specifically, given a time-independent Hamiltonian $H = \sum_j h_j,$ we would like to find a way to approx

Ward Identities

Ward identities are one of the most fundamental tools for studying quantum field theory, and they're encountered in almost any quantum field theory course. You've almost certainly encountered them before, so why should I bother writing about them? Simply put: despite learning how to derive Ward identities for the first time more than 5 years ago (in my first quantum field theory class, as an undergraduate at UChicago), I didn't really understand why they were important until quite recently. This is a product of my own unique research path — I haven't ever done any research in pure QFT, working instead mostly in quantum information and classical geometry, which means I haven't ever had to really understand what's going on under the hood in field theory. I don't think this oversight is so uncommon, so I'm putting together some basic thoughts on Ward identities in this post. So, what is a Ward identity? On its face, it's an equation that tells you how