Quantum Gauge Networks: A New Kind of Tensor Network

Kevin Slagle

Department of Electrical and Computer Engineering, Rice University, Houston, Texas 77005 USA
Department of Physics, California Institute of Technology, Pasadena, California 91125, USA
Institute for Quantum Information and Matter and Walter Burke Institute for Theoretical Physics, California Institute of Technology, Pasadena, California 91125, USA

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Abstract

Although tensor networks are powerful tools for simulating low-dimensional quantum physics, tensor network algorithms are very computationally costly in higher spatial dimensions. We introduce $\textit{quantum gauge networks}$: a different kind of tensor network ansatz for which the computation cost of simulations does not explicitly increase for larger spatial dimensions. We take inspiration from the gauge picture of quantum dynamics, which consists of a local wavefunction for each patch of space, with neighboring patches related by unitary connections. A quantum gauge network (QGN) has a similar structure, except the Hilbert space dimensions of the local wavefunctions and connections are truncated. We describe how a QGN can be obtained from a generic wavefunction or matrix product state (MPS). All $2k$-point correlation functions of any wavefunction for $M$ many operators can be encoded exactly by a QGN with bond dimension $O(M^k)$. In comparison, for just $k=1$, an exponentially larger bond dimension of $2^{M/6}$ is generically required for an MPS of qubits. We provide a simple QGN algorithm for approximate simulations of quantum dynamics in any spatial dimension. The approximate dynamics can achieve exact energy conservation for time-independent Hamiltonians, and spatial symmetries can also be maintained exactly. We benchmark the algorithm by simulating the quantum quench of fermionic Hamiltonians in up to three spatial dimensions.

Simulating many-particle or many-qubit quantum systems is computationally demanding due to the exponential growth of the Hilbert space dimension with the number of particles or qubits. A class of wavefunction ansatz known as "tensor networks" can efficiently parameterize these enormous Hilbert spaces using a contraction of a grid of tensors. While they have demonstrated notable success in one spatial dimension (via e.g. the "DMRG" algorithm), tensor network algorithms are less efficient and more complicated in two or more spatial dimensions.

Our work initiates the study of a novel wavefunction ansatz termed "quantum gauge network." We show that quantum gauge networks are related to tensor networks in one spatial dimension, but are algorithmically simpler and potentially more efficient in two or more spatial dimensions. Quantum gauge networks make use of a new picture of quantum mechanics, called the "gauge picture," which is briefly described in the featured image. We provide a simple algorithm to approximately simulate the time-evolution of a wavefunction using a quantum gauge network. We benchmark the algorithm on a system of fermions in up to three spatial dimensions. Simulating the three-dimensional system using tensor networks would be extremely challenging. However, further research is needed to better understand quantum gauge network theory and to develop more algorithms, such as a ground state optimization algorithm.

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Cited by

[1] Sayak Guha Roy and Kevin Slagle, "Interpolating between the gauge and Schrödinger pictures of quantum dynamics", SciPost Physics Core 6 4, 081 (2023).

[2] Kevin Slagle, "The Gauge Picture of Quantum Dynamics", arXiv:2210.09314, (2022).

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