UC Berkeley

Quantum computers exist today that are capable of performing calculations that challenge the largest classical supercomputers. Now the great challenge is to make use of these devices, or their successors, to perform calculations of independent interest. This thesis focuses on a family of algorithms that aim to accomplish this goal, known as variational quantum algorithms, and the challenges that they face. We frame these challenges in terms of two kinds of resources, the number of operations that we can afford to perform in a single quantum circuit, and the overall number of circuit repetitions required. Before turning to our specific contributions, we provide a review of the electronic structure problem, which serves as a central example for our work. We also provide a self-contained introduction to the field of quantum computing and the notion of a variational quantum algorithm.

We begin the main body of the thesis by studying a generalization of the standard unitary coupled cluster ansatz. We present a sparse version of unitary coupled cluster and show that it can accurately represent the ground and excited states of small molecular systems using fewer quantum gates than the full unitary coupled cluster. We then introduce a strategy for representing molecular ground states as linear combinations of parameterized wavefunctions, allowing for a tradeoff between the number of operations required for each circuit and the number of circuit repetitions. We provide circuit primitives that allow for the efficient measurement of the required matrix elements between these wavefunctions. Subsequently, we show how the cost of estimating the energy of a quantum chemical wavefunction on a near-term quantum computer can be dramatically reduced by using a factorization of the two-electron integral

tensor. Furthermore, we explain how this measurement strategy helps mitigate against errors during state preparation and measurement. We then present a Monte Carlo version of a classical algorithm for calculating the partition function of two-dimensional lattice models that shows a similar kind of tradeoff between two resources, albeit in a different context. Finally, we show how ideas based on tensor networks can inform the design of quantum circuits for machine learning tasks. We argue that this application is especially tolerant to noise and present numerical data substantiating this argument. We conclude by explaining explicitly how our work addresses the problems of the limited resources available to near-term quantum computers and offering some optimism about the future of the field.