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

Abstract:We construct Clifford+$T$ circuits implementing arbitrary $n$-qubit unitaries with worst-case $T$-count $\widetilde{O}(N^{8/7})$ at constant operator-norm error, where $N=2^n$, improving the previous exponent $5/4$. The construction uses $\widetilde{O}(N^{6/7})$ 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 $\widetilde{O}(N^{8/7}L^{3/7}+NL)$ $T$ gates and $\widetilde{O}(N)$ 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.

Quantum Physics (quant-ph); Computational Complexity (cs.CC)

arXiv:2610.18589 [quant-ph]

(or arXiv:2610.18589v1 [quant-ph] for this version)

https://doi.org/10.48550/arXiv.2610.18589 Focus to learn more arXiv-issued DOI via DataCite (pending registration)

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