$T$-efficient unitary synthesis via block-sparse Hamiltonian simulation

Abstract

We construct Clifford+$T$ circuits implementing arbitrary $n$-qubit unitaries with worst-case $T$-count $\tilde{O}(N^{8/7})$ at constant operator-norm error, where $N=2^n$, improving the previous exponent $5/4$. The construction uses $\tilde{O}(N^{6/7})$ clean ancillas. The main idea is to reduce the data addressed by each query to the block-encoding circuit for a sparse piece while controlling the additional simulation work needed to combine the pieces. We simulate a Hermitian logarithm of the target by flattening its traceless part with a two-layer random similarity transformation, randomly partitioning the transformed generator into block-sparse Hermitian pieces, and combining their evolutions using qDRIFT. For a target operator-norm error $\epsilon \in (0,1/2)$, let $L={\log }_2(N/\epsilon )$, the qubit count plus the requested bits of accuracy. By successively correcting the approximation error, we obtain a circuit using $\tilde{O}(N^{8/7}{L}^{3/7}+NL)$ $T$ gates and $\tilde{O}(N)$ clean ancillas whenever $L\le N$. We also prove that, at constant precision, the exponent $8/7$ is optimal within a sequential block-SELECT model.