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  • Open Access

Low-Depth Quantum Symmetrization

Zhenning Liu, Andrew M. Childs, and Daniel Gottesman

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Phys. Rev. X 16, 031056 – Published 31 August, 2026

DOI: https://doi.org/10.1103/9qhy-ms2y

Abstract

Quantum symmetrization is the task of transforming a nonstrictly increasing list of n integers into an equal superposition of all permutations of the list (or more generally, performing this operation coherently on a superposition of such lists). This task plays a key role in initial state preparation for first-quantized simulations. Motivated by an application to fermionic systems, various algorithms have been proposed to solve a weaker version of symmetrization in which the input list is strictly increasing, but the general symmetrization problem with repetitions in the input list has not been well studied. We present the first efficient quantum algorithms for the general symmetrization problem. If m is the greatest possible value of the input list, our first algorithm symmetrizes any single classical input list using O˜(logn) depth and O(nlogn+logm) ancilla qubits, and our second algorithm symmetrizes an arbitrary superposition of input lists using O˜(log3n) depth and O(nlogn) ancilla qubits. Our algorithms enable efficient simulation of bosonic quantum systems in first quantization (in particular, we describe a simulation of the Bose-Hubbard model with complexity poly(n,logm)) and can prepare (superpositions of) Dicke states of any Hamming weight in O˜(logn) depth (respectively, O˜(log3n) depth) using O(nlogn) ancilla qubits. We also propose an O˜(log3n)-depth quantum algorithm to transform second-quantized states to first-quantized states. Using this algorithm, QFT-based quantum telescope arrays can image brighter photon sources, extending quantum interferometric imaging systems to a new regime.

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First-quantized simulation of bosons on a quantum computer requires quantum symmetrization: transforming an ordered list of particle locations into an equal superposition of all its rearrangements, producing a state that is unchanged when identical particles are exchanged. Existing efficient symmetrization algorithms work only when every boson occupies a different mode, and therefore exclude the repeated occupations fundamental to bosonic systems. We develop low-depth algorithms that remove this restriction by combining parallel sorting networks with a new way of organizing permutations when entries repeat. We also develop low-depth algorithms for converting between first- and second-quantized descriptions of many-particle states. As applications, our methods provide logarithmic-depth circuits for preparing Dicke states of any weight and their superpositions, and extend quantum interferometric imaging schemes to process multiple photons simultaneously. Together, these results make first-quantized methods applicable to bosonic systems with repeated occupations, a defining feature that previous fast symmetrization methods could not accommodate.

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Phys. Rev. X 16, 031056– Published 31 August, 2026
  • Received 2 May 2025
  • Revised 11 March 2026
  • Accepted 20 July 2026
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