Representation, Characterization, and Mitigation of Noise in Quantum Processors

Abstract

Quantum computers have the potential to outperform classical computers on several families of important problems, and have a great potential to revolutionize our understanding of computational models. However, the presence of noise deteriorates the output quality from near-term quantum computers and may even offset their advantage over classical computers. Studies on noise in these near-term quantum devices has thus become an important field of research during the past years. This thesis addresses several topics related to this subject including representing, quantifying, and mitigating noise in quantum processors.

To study noise in quantum processors, it is first necessary to ask how noise can be accurately represented. This is the subject of Chapter 2. The conventional way is to use a gate-set, which include mathematical objects assigned to each component of a quantum processor, and compare individual gate-set elements to their ideal images. Here, we present some clarifications on this approach, pointing out that a gauge freedom exists in this representation. We demonstrate with experimentally relevant examples that there exists equally valid descriptions of the same experiment which distribute errors differently among objects in a gate-set, leading to different error rates. This leads us to rethink about the operational meaning to figures of merit for individual gate-set elements. We propose an alternative operational figure of merit for a gate-set, the mean variation error, and develop a protocol for measuring this figure. We performed numerical simulations for the mean variation error, illustrating how it suggests a potential issue with conventional randomized benchmarking approaches.

Next, we study the problem of whether there exist sufficient assumptions under which the gauge ambiguity can be removed, allowing one to obtain error rates of individual gate-set elements in a more conventional manner. We focus on the subset of errors including state preparation and measurement (SPAM) errors, both subject to a gauge ambiguity issue. In Chapter 3, we provide a sufficient assumption that allows a separate SPAM error characterization, and propose a protocol that achieves this in the case of ideal quantum gates. In reality where quantum gates are imperfect, we derived bounds on the estimated SPAM error rates, based on gate error measures which can be estimated independently of SPAM processes. We tested the protocol on a publicly available quantum processor and demonstrated its validity by comparing our results with simulations.

In Chapter 4, we present another protocol capable of separately characterizing SPAM errors, based on a different principle of algorithmic cooling (AC). We propose an alternative AC method called measurement-based algorithmic cooling (MBAC), which assumes the ability to perform (potentially imperfect) projective measurements on individual qubits and is available on various modern quantum computing platforms. Cooling reduces the error on initial states while keeping the measurement operations untouched, thereby breaking the gauge symmetry between the two. We demonstrate that MBAC can significantly reduce state preparation error under realistic assumptions, with a small overhead that can be upper bounded by measurable quantities. Thus, our results can be a valuable tool not only for benchmarking near-term quantum processors, but also for improving the quality of state preparation processes in an algorithmic manner.

The capability of AC for improving initial state quality has inspired us to perform a parallel study on the thermodynamic cost of AC protocols. The motivation is that since cooling a subset of qubits may result in finite energy increase in its environment, applying them in certain platforms that are temperature-sensitive could induce a negative impact on the overall stability. Meanwhile, previous studies on AC have largely focused on subjects like cooling limits, without paying attention to their thermodynamics. Understanding the thermodynamic cost of AC is of both theoretical and practical interest. These results are presented in Chapter 5. After reviewing their procedure, cooling limits, and target state evolution of various AC protocols, we propose two efficiency measures based on the amount of work required, or the amount of heat released. We show how these measures are related to each other and how they can be computed for a given protocol. We then compare the previously studied protocols using both measures, providing suggestions on which ones to use when these protocols are to be carried out experimentally. We also propose improved protocols that are energetically more favorable over the original proposals.

Finally, in Chapter 6, we present a study on a different family of methods aiming at reducing effective noise level in near-term hardware called quantum error mitigation (QEM). The principle behind various QEM approaches is to mimic outputs from the ideal circuit one wants to implement using noisy hardware. These methods recently became popular because many near-term hybrid quantum-classical algorithms only involve relatively shallow depth circuits and limited types of local measurements, implying a manageable cost of performing data processing to alleviate the effect of noise. Using some intuitions built upon classical and quantum communication scenarios, we clarify some fundamental distinctions between quantum error correction (QEC) and QEM. We then discuss the implications of noise invertibility for QEM, and give an explicit construction called Drazin-inverse for non-invertible noise, which is trace preserving while the commonly-used Moore-Penrose pseudoinverse may not be. Finally, we study the consequences of having an imperfect knowledge about the noise, and derive conditions when noise can be reduced using QEM.