Application of resource theory based on free Clifford+kT computation to early fault-tolerant quantum computing

Abstract: Recent advances in quantum hardware are bringing fault-tolerant quantum computing (FTQC) closer to reality. In the early stage of FTQC, however, the numbers of available logical qubits and high-fidelity $T$ gates remain limited, making it crucial to optimize the quantum resource usage. In this work, we aim to study the simulation cost of general quantum states under the constraint that only $k$ $T$ gates can be used, alongside an unlimited number of Clifford gates. Inspired by the notion of robustness of magic (RoM) which quantifies the cost of quantum-circuit simulation using stabilizer states ($k = 0$), we introduce its generalization, which we call Clifford+$kT$ robustness, treating Clifford+$kT$ states as free resources. We explore theoretical properties of Clifford$+kT$ robustness and in particular derive a lower bound that reveals the (in)efficiency of quantum-circuit simulation using Clifford$+kT$ states. Through numerical computations, we also evaluate Clifford+$kT$ robustness for key resource states for universal quantum computation, such as tensor products of the magic states. Our results allow to assess the sampling-cost reduction achieved by the use of Clifford+$kT$ states instead of stabilizer states, providing practical guidance for efficient resource usage in the early-FTQC era.