Executive Summary : | Fault-tolerant quantum computing is concerned with answering the broad question: How can one design a large-scale quantum computer to operate even in the presence of a small amount of noise from the environment? We propose to work on a promising approach to fault-tolerant quantum computing known as magic state distillation. In fault-tolerant quantum computing, there are two quantities one wishes to optimize the threshold, which roughly states how much noise can be tolerated by the system, and the overhead, which roughly corresponds to how many additional qubits (or qudits) are required to achieve fault tolerance. Small improvements to the threshold or overhead could lead to exponential progress and/or reduction in cost. We therefore believe it is worthwhile for theorists to explore and investigate as many routes to fault-tolerant quantum computation as possible. We specifically wish to explore and investigate magic state distillation routines for qudits of odd-prime dimension. The ideas for this proposal originate from a 2014 paper in Nature by Howard et al., which identified contextuality – an abstract generalization of non-locality – as a necessary and possibly-sufficient condition for quantum computing, via a theoretical study of qudit magic state distillation. In a widely-celebrated series of papers, researchers at the Dayalbagh Educational Institute, Uttar Pradesh (DEI) led by Revered Prof. P.S. Satsangi, presented a powerful framework for modeling quantum systems using graph theory, and applied it to qudits of odd prime dimension, laying emphasis on contextuality. Motivated by these works, and supported by an DST-Early Career Research Award, the PI of the proposed project (S Prakash) discovered that a qutrit error-correcting code based on the ternary Golay code is particularly well-suited for magic state distillation, and possesses a remarkably high threshold to noise. Contextuality sets a theoretical bound on the best threshold achievable by a qudit magic state distillation routine. The ternary Golay code, gives rise to the best known threshold of any qubit or qutrit distillation routine, 0.38. However, the threshold is not near the theoretical bound, 0.75, for qutrits, associated with contextuality. Do codes with better thresholds exist? We would also like to search for codes with low overhead, and apply resource theories of fault tolerant quantum computation, which are closely related to contextuality, to derive and better understand theoretical bounds on the best possible overheads of magic state distillation routines. It is hoped that the graph-theoretic physical systems modeling developed at DEI will play a key role answering these questions. Computational searches will also be invoked. |