Exact Unitary Synthesis: Simultaneously Optimal in Size and Depth-Ancilla Tradeoff
Abstract
Unitary synthesis underlies universal quantum computation by translating unitary transformations into circuits of elementary gates. We show that for any budget m≥ 0 of ancilla, every n-qubit unitary can be exactly implemented using a quantum circuit consisting of O(4n) CNOT and single-qubit gates in O(n+4n/(n+m)) layers attaining the asymptotically optimal depth--ancilla tradeoff while simultaneously achieving optimal circuit size. The key idea is to encode computational basis states as wavefunctions sharing a Gaussian envelope, converting unitary action on amplitudes into coordinate transformations that admit shallow circuit implementations. The continuous encoding is then discretized, with approximation errors removed by exact residual correction. The depth-ancilla tradeoff follows by recursively applying the quantum Shannon decomposition, synthesizing its smaller unitary factors, and reusing ancillary qubits.
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BibTeX
@misc{yuan2026exact,
title = {{Exact Unitary Synthesis: Simultaneously Optimal in Size and Depth-Ancilla Tradeoff}},
author = {Pei Yuan and Wei Zi and Shengyu Zhang},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.07010},
eprint = {2610.07010},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.07010},
}
RIS
TY - UNPB
TI - Exact Unitary Synthesis: Simultaneously Optimal in Size and Depth-Ancilla Tradeoff
AU - Pei Yuan
AU - Wei Zi
AU - Shengyu Zhang
PY - 2026
DA - 2026/10/04
PB - arXiv
DO - 10.48550/arxiv.2610.07010
UR - https://arxiv.org/abs/2610.07010
AB - Unitary synthesis underlies universal quantum computation by translating unitary transformations into circuits of elementary gates. We show that for any budget m≥ 0 of ancilla, every n-qubit unitary can be exactly implemented using a quantum circuit consisting of O(4^n) CNOT and single-qubit gates in O(n+4^n/(n+m)) layers attaining the asymptotically optimal depth--ancilla tradeoff while simultaneously achieving optimal circuit size. The key idea is to encode computational basis states as wavefunctions sharing a Gaussian envelope, converting unitary action on amplitudes into coordinate transformations that admit shallow circuit implementations. The continuous encoding is then discretized, with approximation errors removed by exact residual correction. The depth-ancilla tradeoff follows by recursively applying the quantum Shannon decomposition, synthesizing its smaller unitary factors, and reusing ancillary qubits.
ER -