Apel, Harriet; (2024) Theory of quantum Hamiltonian simulation and duality. Doctoral thesis (Ph.D), UCL (University College London).

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Abstract

Analogue Hamiltonian simulation is a promising near-term application of quantum computing and has recently been put on a theoretical footing alongside experiencing wide-ranging experimental success. This idea is closely related to the notion of duality, whereby two superficially different theories are somewhat mathematically equivalent. We strengthen the mathematical foundation of these concepts by characterising duality maps on operators describing the observables of finite-dimensional systems. We also characterise the map on states and prove the equivalence of duality formulated in terms of observables, partition functions and entropies. Alongside this abstract investigation, new perturbative tools are established that reduce the resources required for a geometrically local Hamiltonian to simulate a sparse system with long-ranging interactions. As encodings and preserving mappings are prevalent throughout quantum information theory, the tools and concept of Hamiltonian simulation have applications beyond practical analogue simulation. Here we leverage these techniques to construct a 2D simulation of good error correcting codes, at the cost of a polynomial energy penalty. This simulation approximates the energy spectrum and states of the code Hamiltonian, effectively approximating a [[N,\Omega(\sqrt{N}),\Omega(\sqrt{N})]] code in 2D and breaches the barrier set for commuting exact 2D codes. The AdS/CFT holographic correspondence is a key physical duality serving as a tool to provide insight into quantum gravity and strongly interacting field theories. Viewing this relationship as a simulation leads us further towards a complete construction of holographic duality. Our random tensor network based model describes a duality between theories encompassing local Hamiltonians whilst exactly obeying the Ryu-Takayanagi entropy formula for all boundary regions —making it the first model to capture these AdS/CFT features simultaneously. Furthermore, incorporating the “temporal” scaling of the AdS metric by hand restores consistent causality, revealing an implication between security of a quantum cryptographic protocol and Hamiltonian simulation limits.

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