Efficient Quantum Circuits for Coherent Conversion Between General First- and Second-Quantized Many-Body Representations

• URL: http://arxiv.org/abs/2606.25029v1
• Date: Tue, 23 Jun 2026 18:00:04 GMT
• Title: Efficient Quantum Circuits for Coherent Conversion Between General First- and Second-Quantized Many-Body Representations
• Abstract summary: We construct an explicit $Q$, with inverse $Qdagger$, that maps a first-quantized state to its fixed-$N$ occupation-number form.<n>The conversion is symmetry-agnostic at the input yet fully resolved at the output, and it applies uniformly to bosonic, fermionic, and parastatistical sectors.
• Score: 1.6089851562703383
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: Quantum simulation at fixed particle number admits two equivalent descriptions, a first-quantized (particle) representation and a second-quantized (occupation-number) representation. Their quantum resource costs differ sharply across computational tasks, so the ability to convert coherently between them is valuable. We construct an explicit unitary $Q$, with inverse $Q^\dagger$, that maps a first-quantized state to its fixed-$N$ occupation-number form while diagnosing the input's particle-exchange symmetry. The conversion is therefore symmetry-agnostic at the input yet fully resolved at the output, and it applies uniformly to bosonic, fermionic, and parastatistical sectors. At its foundation lies a structural identification that we place at the center of this work: the quantum Schur transform supplied by Schur-Weyl duality is the non-abelian Fourier transform of the commuting pair $(S_N,U(d))$, and the occupation-number representation is its weight basis, retaining only the labels shared by both factors, the irrep $λ$ and the $\mathfrak{u}(d)$ weight. This reduction is lossless for bosons and fermions, while a canonical Gelfand-Tsetlin promise renders it one-to-one for the remaining sectors. Algorithmically, $Q$ composes the strong Schur transform with reversible arithmetic that computes occupations as successive row-sum differences of the Gelfand-Tsetlin pattern, yielding gate complexity $\mathrm{poly}(N,d,\log(1/ε))$. The converted state is prepared efficiently in quantum memory. Any classical algorithm that outputs it explicitly, however, pays a cost set by the sector dimension, which is polynomial of degree $N$ in $d$ at fixed $N$ and exponential in $N$ when $d=Θ(N)$. Finally, an efficient classical sampler for the induced occupation-number distribution would yield one for arbitrary quantum circuits, contrary to standard complexity assumptions.

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