In the first part of this talk, I will introduce a new picture of quantum dynamics—the gauge picture—that could be thought of as resulting from "gauging" the global unitary invariance in Schrodinger's picture. In the gauge picture, the wavefunction is replaced with many "local" wavefunctions (which are analogous to a Higgs field) that are related by unitary gauge connections. A nice feature of the gauge picture is that the local wavefunction evolution only depends on nearby Hamiltonian terms. This results in a kind of "wavefunction growth" analogous to operator growth in the Heisenberg picture. [1]
In the gauge picture, each local wavefunction is associated with its own Hilbert space. As an application of the gauge picture, we consider truncating the dimension of these local Hilbert spaces. This is analogous to e.g. only keeping the lowest-energy atomic orbitals for doing calculations, but with the advantage that we can now keep more orbitals near the local Hilbert space and keep less orbitals far away for better efficiency. The result of this Hilbert space truncation can be viewed as a new way to generalize tensor networks in 1D to higher dimensions. We call the result of this truncation a "quantum gauge network." Via the gauge picture, we obtain a simple algorithm for approximately simulating quantum dynamics in any spatial dimension. Unlike tensor networks such as PEPS, quantum gauge networks boast the advantage that for fixed bond dimension, the computational cost does not increase with the number of spatial dimensions, and encoding fermionic wavefunctions is just as simple as bosonic wavefunctions. [2]
[1] "The Gauge Picture of Quantum Dynamics." arXiv:2210.09314
[2] "Quantum Gauge Networks: A New Kind of Tensor Network." Quantum 7, 1113 (2023).
