(PHD, 2022)

Abstract:

Random circuit simulation, the task of replicating the output of a randomly chosen noiseless quantum computation, has been proposed as a path toward achieving quantum advantage: it is believed to be easy for quantum devices, but hard for classical ones. This thesis scrutinizes both sides of this belief. On the one hand, we investigate whether the task is classically hard—we find that, in certain non-trivial cases, it can actually be easy, complicating a potential general proof of hardness. On the other hand, we investigate whether the task can be easily accomplished on realistic quantum devices, which are subject to substantial noise rates—we find that, indeed, a version of the circuit simulation task can be salvaged even on a noisy quantum device performing the computation with low fidelity, as long as the noise meets certain conditions. Thus, this thesis emphasizes that, to construct a strong argument of quantum advantage via random circuit simulation on noisy quantum hardware, the core theoretical challenge remains proving lower bounds on the classical complexity of the task; doing so will require new ideas to circumvent the barriers presented by our work.

A key analytical technique we utilize for each of our results is the statistical mechanics method for random quantum circuits, which maps random quantum circuits made from local Haar-random gates to partition functions of classical statistical mechanical systems. This thesis demonstrates the utility of this method by applying it in several new ways. In some cases, we use it for heuristic reasoning about the behavior of random quantum circuits. In others, we go further and perform rigorous calculations of the resulting partition function, leading to precise technical conclusions about random quantum circuits, such as sharp bounds on the number of random gates needed to achieve the anti-concentration property.

]]>

(BS, 2017)

Abstract: It has been recently noted in a paper by Brandao et al. that the structure of a linear program in a classical semidefinite programming algorithm lends itself to quantization, such that the classical algorithm may experience a quantum speedup if the step of solving a linear program is replaced with the preparation of a Gibbs state of classical Hamiltonian on a quantum computer, where the Hamiltonian is given by a linear combination of the semidefinite program’s constraint matrices. The quantum speedup would be exponential if the complexity of the Gibbs sampler used to execute the update step is polynomial in system size. The Gibbs samplers with explicitly defined runtimes are exponential in system size; however, while the quantum Metropolis sampling algorithm by Temme et al. does not have a runtime bounded explicitly in system size, the algorithm heuristically runs in polylogarithmic time. Since the inverse spectral gap of the quantum Metropolis map varies inversely with the running time of the algorithm, we simulate the behavior of the quantum Metropolis map’s spectral gap as a function of system size and row sparsity. We also examine how different definitions of fixed row sparsity affect the spectral gap’s behavior when the system size is increased linearly. While more numerical evidence is needed to draw a definitive conclusion, the current results appear to indicate that for system sizes ranging from three to ten qubits, if fixed row sparsity is defined as a fixed polynomial function of the system size, then the quantum Metropolis spectral gap behaves as a polynomial function of system size.

]]>