Algorithmic Quantum-State Generation for Simulating Quantum Field Theories on a Quantum Computer
Date
2022-01
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Abstract
Simulating a quantum field theory (QFT) on a quantum computer comprises three steps: generating an initial state, simulating time evolution and measuring observables, with the initial-state generation being the most expensive step for the entire simulation. In this thesis, we introduce a general framework for simulating a QFT on a quantum computer, build a foundation for developing high-level quantum algorithms, and employ wavelet representations as a tool for constructing the first two quantum algorithms for initial-state generation in simulating a QFT. We show that our two quantum algorithms are nearly optimal and compare them for two cases: simulating theories with a broken translational invariance and preparing particle states above the ground state at variable length scales. Moreover, we construct two quantum algorithms for preparing one-dimensional Gaussian states, which have applications beyond QFT simulation. Our first algorithm uses a standard state-preparation method, which requires costly arithmetic. We employ novel techniques in our second algorithm to significantly reduce arithmetic operations. In addition to employing wavelets for state generation, we formulate subsystem entanglement entropy for free bosonic and fermionic QFTs in a wavelet basis. We verify the consistency of the wavelet-based formulation for analyzing ground state entanglement in these theories with the conventional lattice-based formulation developed by Calabrese and Cardy. By showing that lattice-based results can hold true in wavelet-based representations of QFTs, we bolster the case for wavelet-based representations as a key tool for analyzing the physics of quantum fields. The last step of a full quantum simulation is to extract simulation outputs by measuring observables on a quantum computer. We build on a reformulation of the standard amplitude estimation and quantum walks for unitary implementation of observables to develop a new approach for estimating expectation values of an observable. Furthermore, we establish a tight lower bound, with respect to a given accuracy, on the query complexity for computing expectation values. Our approach for expectation-value estimation results in an optimal quantum algorithm for measuring observables and is applicable to the last part of a full QFT simulation.
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Keywords
Quantum computing, Quantum algorithms, Quantum entanglement, Quantum field theory, Quantum complexity, Quantum simulation, Quantum measurement, Wavelet representations
Citation
Bagherimehrab, M. (2022). Algorithmic quantum-state generation for simulating quantum field theories on a quantum computer (Doctoral thesis, University of Calgary, Calgary, Canada). Retrieved from https://prism.ucalgary.ca.