Applications of Locality and Asymmetry to Quantum Fault-Tolerance

[摘要] Quantum computing sounds like something out of a science-fiction novel.If we can exert control over unimaginably small systems, then we can harness their quantum mechanical behavior as a computational resource.This resource allows for astounding computational feats, and a new perspective on information-theory as a whole.But there;;s a caveat.The events we have to control are so fast and so small that they can hardly be said to have occurred at all.For a long time after Feynman;;s proposal and even still, there are some who believe that the barriers to controlling such events are fundamental.While we have yet to find anything insurmountable, the road is so pockmarked with challenges both experimental and theoretical that it is often difficult to see the road at all.Only a marriage of both engineering and theory in concert can hope to find the way forward.Quantum error-correction, and more broadly quantum fault-tolerance, is an unfinished answer to this question.It concerns the scaling of these microscopic systems into macroscopic regimes which we can fully control, straddling practical and theoretical considerations in its design.We will explore and prove several results on the theory of quantum fault-tolerance, but which are guided by the ultimate goal of realizing a physical quantum computer.In this thesis, we demonstrate applications of locality and asymmetry to quantum fault-tolerance.We introduce novel code families which we use to probe the behavior of thresholds in quantum subsystem codes.We also demonstrate codes in this family that are well-suited to efficiently correct asymmetric noise models, and determine their parameters.Next we show that quantum error-correcting encodings are incommensurate with transversal implementations of universal classical-reversible computation.Along the way, we resolve an open question concerning almost information-theoretically secure quantum fully homomorphic encryption, showing that it is impossible.Finally, we augment a framework for transversally mapping between stabilizer subspace codes, and discuss prospects for fault-tolerance.

[发布机构] University of Michigan

[效力级别] Quantum Error-Correcting Codes

[关键词] Quantum Computing;Quantum Error-Correcting Codes;Quantum Fault-Tolerance;Mathematics;Physics;Science;Mathematics