Improved Diagnostics & Performance for Quantum Error Correction

Abstract

Building large scale quantum computers is one of the most exciting ventures being pursued by researchers in the 21st century. However, the presence of noise in quantum systems poses a major hindrance towards this ambitious goal. Unlike the developmental history of classical computers where noise levels were brought under reasonable threshold levels early on, the field of quantum computing is struggling to do the same. Nonetheless, there have been many significant theoretical and experimental advancements in the past decade. Quantum error correction and fault tolerance in general is believed to be a reliable long term strategy to mitigate noise and perform arbitrarily long quantum computations. Optimizing and assessing the quality of components in fault-tolerance scheme is a crucial task. We address these tasks in this thesis.

In the first part of the thesis, we provide a method to efficiently estimate the performance of a large class of codes called concatenated stabilizer codes. We show how to employ noise tailoring techniques developed for computations at the physical level to circuits protected by quantum error correction to enable this estimation. We also develop a metric called the logical estimator, which is an approximation of the logical infidelity of the code. We show that this metric can be used to guide the selection of the optimal (concatenated stabilizer) code and the optimal (lookup style) decoder for a given device. Moreover, the metric also aids in estimating the resource requirements for a target logical error rate efficiently and reliably.

In the second part, we show how a combination of noise tailoring tools with quantum error correction can improve the performance of concatenated stabilizer codes by several orders of magnitude. These gains in turn bring down the resource overheads for quantum error correction. We explore the gains using concatenated Steane code under a wide variety of physically motivated error models including arbitrary rotations and combinations of coherent and stochastic noise. We also study the variation of gains with the number of levels of concatenation. For the simple case of rotations about a Pauli axis, we show that the gain scales doubly exponentially with the number of levels in the code. We analyze and show the presence of threshold rotation angles below which the gains can be arbitrarily magnified by increasing the number of levels in the code.

The last part of the thesis explores the testing of an important property of error correcting codes - the minimum distance, often referred to as the distance. We operate in the regime of large classical binary linear codes described in terms of their parity check matrices. We are given access to these codes in terms of an oracle which when supplied an index, returns a single column of the parity check matrix corresponding to that index. We derive lower and upper bounds on the query complexity of finding the minimum distance of a given code. We also ask and (partially) answer the same question in the property testing framework. In particular, we provide a tester which queries a sub-linear number of columns of the parity check matrix and certifies whether a code has high distance or is far away from all codes which have high distance. We also provide non-trivial lower bounds for this task. Although this study is done for classical linear codes, it has implications for designing quantum codes which are built using classical codes. This part of the thesis defines the beginning of a significant area of interest encompassing efficiently testing important properties of classical and quantum codes.