Optimal Approximation of Single Qubit Rotations within a Quantum Circuit

Fault-tolerant quantum computing typically requires the transpilation of arbitrary quantum circuits into a finite, universal gate set, such as Clifford+T. As a baseline, Diagonal approximation can be used for synthesizing single-qubit Pauli rotations, yielding an approximating sequence with $T$-count that equals $3 \log_2(1/\epsilon)$ for a target precision $\epsilon$. Magnitude Approximation can reduce the $T$-count to only $1 \log_2(1/\epsilon)$ by allowing large residual errors, which are rotations about orthogonal axes. Within a complete quantum circuit, these residual errors can then be absorbed into neighboring gates before they are approximated themselves. Determining the optimal allocation of approximation strategies within a large, multi-qubit circuit presents a significant combinatorial challenge. In this work, we present a linear-time algorithm that guarantees an optimal solution to this problem. We demonstrate that the issue of delegating Magnitude versus Diagonal approximation across a circuit maps formally to a classical 1D Ising model with a spatially varying field. By minimizing the energy of this Hamiltonian, we identify the optimal approximation configuration for each rotation without exponential overhead. Benchmarking our method against standard diagonal approximation on random quantum circuits, we observe an average reduction of 26\% in the total approximating circuit gate count, offering a significant efficiency gain for the implementation of quantum algorithms on near-term and fault-tolerant architectures.