Tensor network approaches to open quantum systems, higher dimensional lattices and random quantum circuits

Allen, James

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https://hdl.handle.net/2142/127143 Copy

Description

Title

Tensor network approaches to open quantum systems, higher dimensional lattices and random quantum circuits

Author(s)

Allen, James

Issue Date

2024-08-27

Director of Research (if dissertation) or Advisor (if thesis)

Clark, Bryan K

Doctoral Committee Chair(s)

Vishveshwara, Smitha

Committee Member(s)

• Eckstein, James N
• Stone, Michael

Department of Study

Physics

Discipline

Physics

Degree Granting Institution

University of Illinois at Urbana-Champaign

Degree Name

Ph.D.

Degree Level

Dissertation

Keyword(s)

• Tensor Networks
• Open Quantum Systems
• Neutral Atoms
• DMRG
• Kagome Lattice
• PEPS
• Random Quantum Circuits
• Approximate k-designs
• Approximate t-designs
• Weingarten Calculus

Abstract

In this thesis, we will start by introducing some tensor network methods used to simulate complex physical system. First, we will simulate a neutral atom open quantum system, to develop a relationship between the noise, system size, and performance of a quantum approximate optimization algorithm (QAOA) iteration on this system. In order to help maintain the physicality of the density matrix representing the system, we develop a purity-preserving truncation technique that performs better than a naïve matrix product operator representation while being far more efficient than approaches based on matrix product density operators. We find that for most noise sources, the effect of noise on a QAOA iteration does not scale with system size, although the failure rate of the circuit increases. Next, we will find the ground state of Kagome lattices over different spin values, and identify different properties of the phases found there. In particular, we look at the Jz-J2 phase diagram of the XXZ model on a Kagome lattice for spins 1/2 to 2. For each spin value above 1, we identify a narrow intermediate phase between the q=0 and root(3)-by-root(3) phases that has nematic ordering in their next-nearest neighbour bonds. For spin 1, we also find a wide intermediate phase around the Heisenberg point that has no consistent order parameter. In addition, we will develop a new method of time evolving two-dimensional tensor networks in the form of projected entangled pair states (PEPS), based on converting the PEPS to an isometric tensor network state for each update. This method may be more efficient than a full update based on the boundary MPS method, but the numerical data is inconclusive on whether it has advantages over a simple update. In addition, we will also demonstrate some theoretical and numerical results on random quantum circuits (RQC). We first develop a scheme for numerically computing the anticoncentration of two-dimensional brickwork circuits for a large system size limit and depths up to 15. This approach is based on taking the statistical mechanics model of RQCs and interpreting a depth-wise slice of the system as the matrix product operator in an infinite DMRG algorithm. We find that for these depths, the two-dimensional brickwork maintains the log-depth anticoncentration relation found in one-dimensional brickworks by previous works. We also find that the statistical mechanics model of two-dimensional brickworks naturally hosts a quasiparticle model at very low local Hilbert space dimensions. Next, we show a generalization of results on the epsilon-approximate t-design depths of RQCs to a much wider class of architectures. Previous results were successful at proving that the approximate t-design depth is linear in system size for one-dimensional brickwork architectures, but were unable to generalize these results to arbitrary architectures. We develop an approach based on bounding subleading singular values for each layer in a particular connected block of gates, which naturally translates to a "cluster-merging picture" in which a hierarchy of graphs is connected over time. With this, we were able to translate established one-dimensional brickwork bounds to any possible architecture, preserving system size linearity in the vast majority of them. Finally, we will develop results on bounding the spectral gap of these circuits, which helps lower the t dependence of the approximate t-design depths. In particular, we show that the spectral gap of a one-dimensional brickwork is independent of t for all t <= q, where q is the local Hilbert space dimension of the original circuit. Thus, the epsilon-approximate t-design depth in the small epsilon limit has no t dependence at all. We accomplish this by reducing the spectral gap of a one-dimensional brickwork of any size to a spectral gap of an operator on an effective three-site system. Then, we bound this operator by simplifying its domain according to irreducible representations of the symmetric group, and obtain bounds on the resulting matrix product.

Graduation Semester

2024-12

Type of Resource

Thesis

Handle URL

https://hdl.handle.net/2142/127143

Copyright and License Information

Copyright 2024 James Allen

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