Cost-optimal single-qubit gate synthesis in the Clifford hierarchy

Gary J. Mooney1, Charles D. Hill1,2, and Lloyd C. L. Hollenberg1

1School of Physics, University of Melbourne, VIC, Parkville, 3010, Australia
2School of Mathematics and Statistics, University of Melbourne, VIC, Parkville, 3010, Australia

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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+$T$ set of base gates and compared them to when additionally including $Z$-rotations from higher orders of the Clifford hierarchy. Two approaches of assigning base gate costs were used. First, costs were reduced to $T$-counts by recursively applying a $Z$-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 to $54\pm 3\%$ when using the $Z$-rotation catalyst circuit approach and by up to $33\pm 2 \%$ 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 of $Z$-rotation gates from higher orders of the Clifford hierarchy that are found within sequences approximating random target gates.

Performing large scale complex quantum algorithms will require the use of robust error-correction protocols over encoded qubits. However, these protocols present unique challenges that need to be overcome in order to achieve universal quantum computation. In general, many logic gate operations used in quantum algorithms cannot be directly applied to encoded quantum states. Logic gate operations that can be directly applied transversally over the encoded qubits are considered to be easy to implement because they are inherently fault tolerant and hence require relatively small numbers of physical gates. In addition, there are gates that can be fault-tolerantly implemented, but require resource intensive magic state distillation procedures. These gates, together with the transversal gates, form universal sets of base gates that can be used to approximate any unitary gate up to arbitrary precision by using gate synthesis algorithms to generate corresponding sequences of base gates. Within the quantum compilation literature, gate synthesis primarily focuses on the Clifford+T set of universal base gates where the T gate is implemented through magic state distillation. However, other universal sets of base gates exist that can be implemented fault-tolerantly in error correction codes. For instance, by including gates from higher orders of the Clifford hierarchy, where the resource cost of gate implementation increases for increasing orders of the hierarchy.

Although single-qubit synthesis algorithms have been well studied within the quantum compilation literature, they typically do not yet support the variety of available base gates and do not optimise with respect to individual gate costs of implementation. In this work we use an exhaustive search algorithm, based on Dijkstra’s pathfinding algorithm, to find cost-optimal sequences for random target single-qubit gates. It is compatible with arbitrary sets of universal base gates with individually assigned costs, enabling us to explore potential cost benefits from including higher order Clifford hierarchy base gates. We calculate the synthesis cost against error scaling relations and compare between the canonical Clifford+T set of base gates and the sets resulting from accumulatively including Z-rotation gates from increasing orders of the Clifford hierarchy. Two gate implementation approaches were used to assign resource costs to gates from each order of the hierarchy. The first approach reduces the gate costs to T-counts by using a Z-rotation catalyst circuit to implement the gates. This resulted in the scaling factor decreasing by up to 54±3%. The second approach assigns the costs as the average number of raw magic states required to directly distil and implement the gates fault-tolerantly. This approach decreased the scaling factor by up to 33±2%.

The results of our paper demonstrate that the resource requirements of synthesis algorithms could be considerably improved by including higher orders of the Clifford hierarchy as base gates and optimising with respect to individually assigned gate costs.

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[3] Maxwell T. West, Shu-Lok Tsang, Jia S. Low, Charles D. Hill, Christopher Leckie, Lloyd C. L. Hollenberg, Sarah M. Erfani, and Muhammad Usman, "Towards quantum enhanced adversarial robustness in machine learning", Nature Machine Intelligence 5 6, 581 (2023).

[4] G. A. L. White, C. D. Hill, and L. C. L. Hollenberg, "Truncated phase-based quantum arithmetic: Error propagation and resource reduction", Physical Review A 108 5, 052608 (2023).

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