Clifford+T-gate decomposition with limited number of T gates and performance of Unitary Coupled Cluster ansatz in early-FTQC era

In this work, we propose a method to provide a Clifford+T-gate decomposition of a given quantum gate when the maximum number of T gates available is restricted especially in an early stage of fault-tolerant quantum computation (FTQC). Fault-tolerant realization of universal gates, e.g., Clifford+T gates, is essential for FTQC. Solovay-Kitaev (SK) theorem evaluated the number of T gates needed to obtain a target quantum gate within desired accuracy. On the other hand, in an early FTQC era, we likely need to implement a given quantum circuit with a limited number of T gates. Now, the question is, "what is the best accuracy we can achieve only using the available number of T gates?" We propose a gate-decomposition method based on the algorithm by Ross and Selinger [Quantum Inf. Comput. 16(11-12), 2016], which yields a decomposition of a single qubit z-rotation with the smallest error under a limited budget of T gates. In addition, we propose to average the effects of SK error, the error caused by the SK decomposition, when the given circuit is sufficiently large. With our averaging method, we may model the effects of SK error by the single-qubit depolarizing channel. We also numerically examine these proposals using the unitary coupled cluster (UCC) ansatz, and estimate the number of T gates needed to simulate an approximate ground state of molecules by the UCC ansatz within a given accuracy.