Preparation of Hamming-Weight-Preserving Quantum States with Log-Depth Quantum Circuits

Abstract:Hamming-Weight-Preserving (HWP) states, $\vert\psi_{\mathrm{H}}\rangle=\sum_{\mathrm{HW}(x)=k}\alpha_x\vert x\rangle$, represent a fundamental class of quantum states on $n$ qubits with fixed excitation number $k$, playing a pivotal role in quantum simulation, machine learning, and optimization, with prominent examples including the Bethe eigenstates of integrable models and Configuration Interaction states. However, existing preparation methodologies typically rely on sequential architectures whose depth scales linearly with the subspace dimension, whereas practical deployment on realistic quantum hardware demands shallow circuits to mitigate decoherence. In this work, we present quantum circuits that achieve the information theoretic limit of depth logarithmic in the number of populated computational basis states while maintaining asymptotically optimal size. Specifically, for general HWP states, we provide a construction that matches the lower bounds of $\Theta(\log \binom{n}{k})$ depth and $\Theta(\binom{n}{k})$ size. Furthermore, we propose a resource efficient scheme utilizing only $O(\log k)$ ancillary qubits. For the $k=2$ case, where HWP states naturally admit a graph representation, we show that states whose support graphs are trees or grids admit ancilla free preparation with $\Theta(\log n)$ depth and $\Theta(m)$ size. Overall, our work provides logarithmic depth state preparation algorithms across various regimes while establishing a connection between quantum state synthesis and Krawtchouk polynomials.

arXiv:2508.14470 [quant-ph]

(or arXiv:2508.14470v2 [quant-ph] for this version)

https://doi.org/10.48550/arXiv.2508.14470 Focus to learn more arXiv-issued DOI via DataCite

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