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For universal quantum computation, a major challenge to overcome forpractical implementation is the large amount of resources required forfault-tolerant quantum information processing. An important aspect isimplementing arbitrary unitary operators built from logical gates within thequantum error correction code. A synthesis algorithm can be used to approximateany unitary gate up to arbitrary precision by assembling sequences of logicalgates chosen from a small set of universal gates that are fault-tolerantlyperformable while encoded in a quantum error-correction code. However, currentprocedures do not yet support individual assignment of base gate costs and manydo not support extended sets of universal base gates. We analysed cost-optimalsequences using an exhaustive search based on Dijkstra's pathfinding algorithmfor the canonical Clifford+$T$ set of base gates and compared them to whenadditionally including $Z$-rotations from higher orders of the Cliffordhierarchy. Two approaches of assigning base gate costs were used. First, costswere reduced to $T$-counts by recursively applying a $Z$-rotation catalystcircuit. Second, costs were assigned as the average numbers of raw (i.e.physical level) magic states required to directly distil and implement thegates fault-tolerantly. We found that the average sequence cost decreases by upto $54\pm 3\%$ when using the $Z$-rotation catalyst circuit approach and by upto $33\pm 2 \%$ when using the magic state distillation approach. In addition,we investigated observed limitations of certain assignments of base gate costsby developing an analytic model to estimate the proportion of sets of$Z$-rotation gates from higher orders of the Clifford hierarchy that are foundwithin sequences approximating random target gates.

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