Abstract
For universal quantum computation, a major challenge to overcome for practical implementation is the large amount of resources required for fault-tolerant quantum information processing. An important aspect is implementing arbitrary unitary operators built from logical gates within the quantum error correction code. A synthesis algorithm can be used to approximate any unitary gate up to arbitrary precision by assembling sequences of logical gates chosen from a small set of universal gates that are fault-tolerantly performable while encoded in a quantum error-correction code. However, current procedures do not yet support individual assignment of base gate costs and many do not support extended sets of universal base gates. We analysed cost-optimal sequences using an exhaustive search based on Dijkstra’s pathfinding algorithm for the canonical Clifford+Tset of base gates and compared them to when additionally includingZ-rotations from higher orders of the Clifford hierarchy. Two approaches of assigning base gate costs were used. First, costs were reduced toT-counts by recursively applying aZ-rotation catalyst circuit. Second, costs were assigned as the average numbers of raw (i.e. physical level) magic states required to directly distil and implement the gates fault-tolerantly. We found that the average sequence cost decreases by up to54±3%when using theZ-rotation catalyst circuit approach and by up to33±2%when using the magic state distillation approach. In addition, we investigated observed limitations of certain assignments of base gate costs by developing an analytic model to estimate the proportion of sets ofZ-rotation gates from higher orders of the Clifford hierarchy that are found within sequences approximating random target gates.
Highlights
Quantum computing has the potential to solve many real-world problems by using significantly fewer physical resources and computation time than the best known classical algorithms
When including Z-rotation base gates from higher orders of the Clifford hierarchy with T -counts assigned using the Z-rotation catalyst approach, we find that the average cost-optimal sequence T -counts can potentially be reduced by over 50% when output gates are directly applied to target qubits and by over 30% when intermediate magic states are used
The pattern of the data about the lines of best fit for each logical base gate set are similar between plots because for each of the logical base gate errors, the ratios of the base gate cost values between orders of the Clifford hierarchy are similar, the cost optimal sequences will be comparable. (a) Synthesis using logical base gate costs associated with μ = 10−5 logical gate error. (b) Synthesis using logical base gate costs associated with μ = 10−10 logical gate error. (c) Synthesis using logical base gate costs associated with μ = 10−15 logical gate error. (d) Synthesis using logical base gate costs associated with μ = 10−20 logical gate error
Summary
Quantum computing has the potential to solve many real-world problems by using significantly fewer physical resources and computation time than the best known classical algorithms. When including Z-rotation base gates from higher orders of the Clifford hierarchy with T -counts assigned using the Z-rotation catalyst approach, we find that the average cost-optimal sequence T -counts can potentially be reduced by over 50% when output gates are directly applied to target qubits and by over 30% when intermediate magic states are used. When using the alternative approach of assigning costs from direct magic state distillation, we find that by including Z-rotation logical base gates from the fourth order of the Clifford hierarchy, the average cost-optimal sequence costs can be reduced by 30% These cost reductions indicate that a significant amount of resources could be saved by adapting current synthesis algorithms to include higher orders of the Clifford hierarchy and to optimise sequences with respect to individual gate costs. The parameters of the calculation include the maximum sequence cost and separate logical base gate costs for each order of the Clifford hierarchy, which can be readily be extended to specify costs for individual logical base gates
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