Quantum field theories are an essential framework in modeling fundamental interactions in nature, yet reliable simulations consume large portions of the world's most powerful supercomputers. In this Dissertation, we explore an alternative simulation paradigm on fault-tolerant quantum hardware, discussing both algorithmic and model advancements. In the former, we construct state-of-the-art algorithmic subroutines that take advantage of useful properties of the Hamiltonian describing the theory to achieve an exponential improvement in resources over prior quantum algorithms. In the latter, we simplify the structure of the Hamiltonian, making it more amenable to quantum simulation. First, we present an improved regularization of the gauge group using braided fusion categories, which evokes a convenient anyon description of the Hamiltonian. Second, we represent the Hamiltonian in Dirac's light-front formalism, which maps simulation to an eigenvalue estimation problem. These improvements motivate the use of quantum computers for simulation with significantly fewer resources than their conventional counterparts.
Quantum field theories are an essential framework in modeling fundamental interactions in nature, yet reliable simulations consume large portions of the world's most powerful supercomputers. In this Dissertation, we explore an alternative simulation paradigm on fault-tolerant quantum hardware, discussing both algorithmic and model advancements. In the former, we construct state-of-the-art algorithmic subroutines that take advantage of useful properties of the Hamiltonian describing the theory to achieve an exponential improvement in resources over prior quantum algorithms. In the latter, we simplify the structure of the Hamiltonian, making it more amenable to quantum simulation. First, we present an improved regularization of the gauge group using braided fusion categories, which evokes a convenient anyon description of the Hamiltonian. Second, we represent the Hamiltonian in Dirac's light-front formalism, which maps simulation to an eigenvalue estimation problem. These improvements motivate the use of quantum computers for simulation with significantly fewer resources than their conventional counterparts.
Many important physical and mathematical problems are difficult to solve on classical computers, resulting in difficulty in finding new advanced materials, simulating quantum field theories and solving large combinatorial optimisation tasks. Quantum computers have shown great promise in being able to solve these (and many other) problems faster than their classical counterparts, and therefore should be thoroughly investigated as a significant tool for enabling the acceleration of scientific progress. This work develops methods to tackle several problems across the spectrum of quantum algorithms, providing methods suitable for the near-term hardware, to the future where large fault-tolerant quantum computers exist. Therefore, the works in this thesis range from efficiently making use of limited quantum hardware to taking advantage of hardware benefits (e.g. excess qubits) as quantum computation advances. Covered topics include: an alternative paradigm for the optimisation of variational circuits and quantum neural networks; techniques for improving energy estimation of quantum states through the collection and processing of large amounts of randomised measurement data; the use of entanglement in guiding the automatic discovery of quantum circuits; investigating the role of symmetries in adiabatic state preparation of molecules and the large scale parallelisation of a common fault-tolerant subroutine, potentially producing large runtime speed-ups for the simulation of chemistry on quantum computers. Although there are many challenges remaining from both hardware and algorithms perspectives before the large-scale utility of quantum computers can be demonstrated, the contributions of this work make steps towards the efficient utilisation of quantum hardware that will be available in the coming years.
The framework of this thesis is fault-tolerant quantum algorithms. Grover's algorithm and quantum walks are described in Chapter 2. We start by highlighting the central role that rotations play in quantum algorithms, explaining Grover's, why it is optimal, and how it may be extended. Key subroutines explained in this area are amplitude amplification and quantum walks, which will constitute useful parts of other algorithms. In the third chapter, in contrast, we turn to the exponential advantages promised by the Fourier transform in the context of the hidden subgroup problem. However, since this application is restricted to cryptography, we later explore its use in quantum linear algebra problems. Here we explain the development of the original quantum linear solver algorithm, its improvements, and finally the dequantization techniques that would often restrict the quantum advantage to polynomial. Chapter 4 is concerned with quantum simulation. We will review classical quantum chemistry techniques, and then focus on Hamiltonian simulation and ground state preparation as the key problems to be solved. Hamiltonian simulation, in particular, will enable the use of quantum phase estimation which computes the eigenvalues or energies of a given quantum state. Given the tradition of our group with error correction, we could not end this thesis without dedicating a final chapter to this topic. Here we explain the most important quantum error correction codes, the surface and color codes, and one extension of the latter, gauge color codes. They will show the complexity of implementing non-Clifford quantum gates, therefore validating their consideration as the bottleneck metric.
To maximize the value of fault-tolerant quantum computers, it is essential to develop concrete applications beyond well-established domains such as chemistry and condensed-matter physics. Here we construct and compile quantum algorithms to simulate the structure of atomic nuclei -- a topic that has received relatively little attention from the quantum computing community despite its similarities to the electronic structure problem in chemistry -- via effective shell-model Hamiltonians and no-core-shell-model Hamiltonians with three-body interactions derived from chiral effective field theory. Furthermore, we provide quantum resource estimates, in terms of Toffoli gate and qubit counts, for these algorithms, which, to our knowledge, are the first such estimates for fault-tolerant quantum simulation of atomic nuclei. Notably, the estimates for 32Mg and 219At shell-model Hamiltonians are comparable to recent estimates of Femoco simulations, a standard benchmark in chemistry. For no-core-shell-model Hamiltonians suitable for light nuclei (up to 40Ca or so), we find that resource requirements are significantly higher, suggesting that more bespoke strategies are required to make such simulations practicable. Throughout this work, we draw upon the similarities between nuclear and electronic structure problems, while also highlighting challenges that are specific to the former. We hope this work will spur long-term collaborations between the nuclear and quantum computing community with the ultimate goal of realizing useful nuclear simulations on quantum computers.
Quantum computers have the potential to push the limits of computation in areas such as quantum chemistry, cryptography, optimization, and machine learning. Even though many quantum algorithms show asymptotic improvement compared to classical ones, the overhead of running quantum computers limits when quantum computing becomes useful. Thus, by optimizing components of quantum algorithms, we can bring the regime of quantum advantage closer. My work focuses on developing efficient subroutines for quantum computation. I focus specifically on algorithms for scalable, fault-tolerant quantum computers. While it is possible that even noisy quantum computers can outperform classical ones for specific tasks, high-depth and therefore fault-tolerance is likely required for most applications. In this thesis, I introduce three sets of techniques that can be used by themselves or as subroutines in other algorithms. The first components are coherent versions of classical sort and shuffle. We require that a quantum shuffle prepares a uniform superposition over all permutations of a sequence. The quantum sort is used within the shuffle and as well as in the next algorithm in this thesis. The quantum shuffle is an essential part of state preparation for quantum chemistry computation in first quantization. Second, I review the progress of Hamiltonian simulations and give a new algorithm for simulating time-dependent Hamiltonians. This algorithm scales polylogarithmic in the inverse error, and the query complexity does not depend on the derivatives of the Hamiltonian. A time-dependent Hamiltonian simulation was recently used for interaction picture simulation with applications to quantum chemistry. Next, I present a fully quantum Boltzmann machine. I show that our algorithm can train on quantum data and learn a classical description of quantum states. This type of machine learning can be used for tomography, Hamiltonian learning, and approximate quantum cloning.
Quantum computers promise to simulate complex quantum field theories that are intractable for classical computers, potentially revealing new physics in strongly interacting systems. Current approaches for simulating Yang-Mills gauge theories face significant technical barriers due to complex group structures and complicated truncation schemes that vary drastically between different theories. Here we show that the orbifold lattice formulation provides a universal framework for quantum simulation of Yang-Mills theories with arbitrary gauge groups and dimensions. Our approach reduces all theories to the same simple Hamiltonian form, enabling implementation with standard quantum gates regardless of system complexity. We demonstrate explicit quantum circuits using only controlled-NOT and single-qubit operations, with concrete resource estimates for time evolution algorithms. This universal framework simplifies quantum simulation of gauge theories and enables systematic scaling to larger systems on fault-tolerant quantum computers.
Quantum computing offers a new, orthogonal direction for investigating fundamental physics, extending beyond classical numerical methods and conventional observables. Realizing this potential requires directly confronting the noise limiting currently available quantum computers. Progress rests on advancing algorithms, interpreting their results, and managing their errors together. This thesis presents several advancements in the use of quantum simulation and quantum information to probe fundamental physics.
The first is in the use of quantum computers to simulate collisions in quantum field theories. Central to these simulations are new wavepacket preparation, time evolution, and error mitigation techniques, which allow for simulations with some of the largest effective circuit volumes to date. These methods enable the first quantum simulation providing numerical evidence for inelastic particle production, a key process in fundamental physics.
The second advancement centers on the role quantum-information-theoretic quantities play in physical processes. Beyond mere correlations with the physics of the process, entanglement and magic are shown to probe the interactions present in scattering and hadronization dynamics.
A precision study requires a complete quantification of algorithmic and hardware uncertainties, an outstanding goal as quantum simulations mature. The third advancement in this thesis addresses error management. A framework minimizing the effect of algorithmic errors in analog quantum simulations is presented. In a step toward fault tolerance, error detection in encoded quantum simulations is shown to improve estimation of local observables relative to unencoded runs.
Together, the developments in this thesis mark practical progress toward fault-tolerant quantum simulations of fundamental physics capable of scientific discovery.