Derby, Charles; (2023) Compact fermion to qubit mappings for quantum simulation. Doctoral thesis (Ph.D), UCL (University College London).

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Text Charles Derby Corrected Thesis 17095701 - 15-02-2023.pdf - Accepted Version Download (1MB) | Preview

Abstract

Fermions are one of two types of particles that make up matter in the universe, characterised by many-body wavefunctions that are antisymmetric under particle exchange. Electrons, which underpin many physical systems of interest, are included in this group, so the ability to accurately simulate fermionic physics would be a great asset to research. However, the antisymmetric nature of these particles means that classical simulation of systems of multiple fermions is, in general, infeasible due to sign problems. This infeasibility extends even to simplified systems such as the Fermi-Hubbard model on a 2D grid. Simulation of fermions on a quantum device would avoid this problem entirely. A requisite step in simulating fermions on a quantum computer is mapping a many-body fermionic system onto qubits through a fermionic encoding. Significant properties of fermionic encodings include their qubit to fermionic mode ratio and the weight of their encoded fermionic interaction operators. Both affect the runtime of quantum simulation algorithms so it is ideal to minimise these quantities. This thesis presents the novel ``compact'' encoding which outperforms all previous local encodings in these metrics. The construction of the encoding is shown for a number of interaction graph structures and its general properties are explored. Special attention is given to a remarkable feature where low weight undetectable noise on the encoding corresponds to a natural noise process on fermionic systems, indicating that it may have utility in simulation even on imperfect, noisy quantum devices. An interesting feature of the compact encoding and others is an apparent link to topological error correcting codes like the toric code. Inspection of the compact encoding for a cubic lattice reveals a link to an apparently novel 3D topological code with some unusual properties. This size of its codespace and code distance are calculated and the exact form of its logical operators and syndromes are shown. Excitations with fermionic character exist in this code, consistent with the other codes linked with fermionic encodings, pointing to a possible unifying picture for local fermionic encodings. \end{abstract} \begin{impactstatement} Quantum computers have the potential for a wide reaching impact. The development of a fault tolerant quantum computer would allow hitherto infeasible problems to be tackled computationally. A significant application, which has been a primary motivator for the field since its inception, is the simulation of other quantum systems. Systems containing many fermions are of particular interest. Not only because they are fundamentally difficult to simulate with normal computers but because they include systems of electrons, the particles which underpin almost all of chemistry. Simulating these systems on a quantum computer is not a simple task however, as there must be a procedure to map the physics of many indistinguishable fermions onto the physics of stationary qubits, two fundamentally different systems. This procedure is called a \textit{fermionic encoding} and the main subject of this thesis is an example of this. The content of this thesis could benefit researchers in a number of fields. It adds to the rich zoo of fermionic encodings and may provide inspiration for further results in the field, it also highlights a possible link between the seemingly disparate local fermionic encodings which may pave the way to a more unified general theory of representing fermions on qubits. The encoding presented in this work has favourable properties for simulation on noisy devices so it may benefit research groups working on near term quantum hardware by providing the means to perform interesting fermionic simulation experiments. The content of the last chapter may also be of interest to the error correction community as it provides an example of an apparently unclassified topological code, this may lead to the development of new classes of code. This research may also yield benefits outside of academia. The quantum simulation of electronic systems would lead to greater understanding of chemical reactions such as Nitrogen fixing and materials such as superconductors and batteries. This understanding could lead to improvements in efficiency or the development of new substances which would be invaluable to industries including agriculture, transportation and battery production.

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