Optimal ancilla-free Clifford+T approximation of z-rotations

Given a gate set S universal for quantum computing, the problem of decomposing a unitary operator U into a circuit over S is known as the synthesis problem. This problem can be solved exactly, if U belongs to the set of circuits generated by S. Otherwise, it can be solved approximately, by finding a circuit U ′ such that ||U ′ −U || < for some chosen precision > 0. The synthesis problem is important for quantum computing because it significantly impacts the resources required to run a quantum algorithm. Indeed, a logical circuit, to be executed by a quantum computer, must be compiled into some universal gate set and then implemented fault-tolerantly according to an error correcting scheme. The complexity of the final physical circuit therefore crucially depends on the chosen synthesis method. In fact, in view of the considerable resources required for most quantum algorithms on interesting problem sizes, a universal gate set can be realistically considered for practical quantum computing only if, in addition to an efficient fault-tolerant implementation in some error correcting scheme, it comes equipped with a good synthesis algorithm. Here, a good synthesis algorithm is one that is efficient (e.g., runs in polynomial time) and generates a circuit whose gate count is as low as possible. Recall that the Clifford+T gate set consists of all the Clifford operators together with the following T -gate, or π/8 gate: