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Fault-tolerant quantum computation (FTQC) is essential to robustly implementquantum algorithms and thus to maximize advantages of quantum computers. Inthis context, a quantum circuit is decomposed into universal gates that can befault-tolerantly implemented, for example, Clifford+$T$ gates. Here, $T$ gateis usually regarded as an essential resource for quantum computation becauseits action cannot be simulated efficiently on classical computers. Practically,it is highly likely that only a limited number of $T$ gates are available inthe near future due to its experimental difficulty of fault-tolerantimplementation. In this paper, considering this Pre-FTQC era, we investigateClifford+$T$ decomposition with a limited budget of $T$ gates and propose a newmodel of the error of such decomposition. More concretely, we propose analgorithm to generate the most accurate Clifford+$T$-gate decomposition of agiven single-qubit rotation gate with a fixed number of $T$ gates. We alsopropose to model the error of Clifford+$T$ decomposition using well-knowndepolarizing noise by considering the average of the effects of the error. Wenumerically verified our model successfully explains the decomposition errorfor a wide variety of molecules using our decomposition algorithm. Thus, weshed light on a first-stage application of quantum computers from a practicalpoint of view and fuel further research towards what quantum computation canachieve in the upcoming future.

Clifford+$T$-gate Decomposition with Limited Number of $T$ gates, its  Error Analysis, and Performance of Unitary Coupled Cluster Ansatz in Pre-FTQC  Era