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Universal initial state preparation for first quantized quantum simulations

Jack S. Baker, Gaurav Saxena, Thi Ha Kyaw

TL;DR

The paper tackles the bottleneck of preparing symmetry-adapted initial states for first-quantized quantum simulations. It introduces a universal protocol that leverages the Jordan--Schwinger map and Schur--Weyl duality to translate occupation-number superpositions into Schur-basis states, which are then prepared via a block-encoded LCU and transformed back with the inverse quantum Schur transform. The approach yields a time complexity of $poly(L, N, d, \log \epsilon^{-1})$ and applies to fermions, bosons, and parabosons in arbitrary bases, with resource estimates showing feasibility alongside leading first-quantized pipelines; the authors also outline routes to further speed up via faster Schur transforms and Schur-basis techniques. This framework unifies first- and second-quantized state preparation and broadens quantum-simulation capabilities to exotic statistics and flexible bases, potentially accelerating practical quantum advantage in chemistry, materials, and fundamental physics.

Abstract

Preparing symmetry-adapted initial states is a principal bottleneck in first-quantized quantum simulation. We present a universal approach that efficiently maps any polynomial-size superposition of occupation-number configurations to the first-quantized representation on a digital quantum computer. The method exploits the Jordan--Schwinger Lie algebra homomorphism, which identifies number-conserving second-quantized operators with their first-quantized action and induces an equivariant bijection between Fock occupations and $\mathfrak{su}(d)$ weight states within the Schur--Weyl decomposition. Operationally, we prepare an encoded superposition of Schur labels via a block-encoded linear combination of unitaries and then apply the inverse quantum Schur transform. The algorithm runs in time $\text{poly}(L, N, d, \log ε^{-1})$ for $L$ configurations of $N$ particles over $d$ modes to accuracy $ε$, and applies universally to fermions, bosons, and Green's paraparticles in arbitrary single-particle bases. Resource estimates indicate practicality within leading first-quantized pipelines; statistics-aware or faster quantum Schur transforms promise further reductions.

Universal initial state preparation for first quantized quantum simulations

TL;DR

The paper tackles the bottleneck of preparing symmetry-adapted initial states for first-quantized quantum simulations. It introduces a universal protocol that leverages the Jordan--Schwinger map and Schur--Weyl duality to translate occupation-number superpositions into Schur-basis states, which are then prepared via a block-encoded LCU and transformed back with the inverse quantum Schur transform. The approach yields a time complexity of and applies to fermions, bosons, and parabosons in arbitrary bases, with resource estimates showing feasibility alongside leading first-quantized pipelines; the authors also outline routes to further speed up via faster Schur transforms and Schur-basis techniques. This framework unifies first- and second-quantized state preparation and broadens quantum-simulation capabilities to exotic statistics and flexible bases, potentially accelerating practical quantum advantage in chemistry, materials, and fundamental physics.

Abstract

Preparing symmetry-adapted initial states is a principal bottleneck in first-quantized quantum simulation. We present a universal approach that efficiently maps any polynomial-size superposition of occupation-number configurations to the first-quantized representation on a digital quantum computer. The method exploits the Jordan--Schwinger Lie algebra homomorphism, which identifies number-conserving second-quantized operators with their first-quantized action and induces an equivariant bijection between Fock occupations and weight states within the Schur--Weyl decomposition. Operationally, we prepare an encoded superposition of Schur labels via a block-encoded linear combination of unitaries and then apply the inverse quantum Schur transform. The algorithm runs in time for configurations of particles over modes to accuracy , and applies universally to fermions, bosons, and Green's paraparticles in arbitrary single-particle bases. Resource estimates indicate practicality within leading first-quantized pipelines; statistics-aware or faster quantum Schur transforms promise further reductions.

Paper Structure

This paper contains 22 sections, 102 equations, 3 figures, 1 table, 3 algorithms.

Table of Contents

  1. Overview
  2. Mathematical Preliminaries
  3. Concrete example in $U(3)$
  4. From Dynkin Weights to Gelfand--Tsetlin Patterns
  5. Non-Clifford resource estimation for the BCH Schur transform
  6. Register encodings
  7. Step 0: top level
  8. Step 1: $U_d$ CG as a recursion in the rank
  9. Step 2: two-level compilation, global error budget, and synthesis tolerances
  10. Step 3: online evaluation of reduced-Wigner coefficients and angle generation
  11. Per-rank and overall toffoli-equivalent cost
  12. Ancilla accounting for arithmetic
  13. Total qubit accounting
  14. Non-Clifford resource estimation for block encoding
  15. LCU with fixed $\lambda$ and $\sigma$
  16. ...and 7 more sections

Figures (3)

  • Figure 1: (a) Schur--Weyl decomposition of $(\mathbb{C}^{4})^{\otimes 3}$ into $U(4)$ irreps $V^{U(4)}_{\lambda}$ with $S_{3}$ multiplicities (in parentheses). Colors encode statistics: fermionic, bosonic, and parastatistics. (b) Fermionic Fock space $\mathcal{F}_{-}(\mathbb{C}^{4})=\bigoplus_{N=0}^{4}\wedge^{N}\mathbb{C}^{4}$. The shaded region highlights the $U(4)$-equivariant bijection induced by the Jordan--Schwinger (JS) realization between the fermionic Schur--Weyl sector in (a) and the corresponding Fock subspaces. (c) Permitted semistandard Young tableaux for $\lambda=(1,1,1)$ with $N=3$ and $d=4$. One tableau is identified with its Fock occupation $\ket{1,1,1,0}$ (2nd-Q) and with the first-quantized Slater determinant $\ket{1\wedge 2\wedge 3}=\tfrac{1}{\sqrt{3!}}\det\!(\ket{1}_r,\ket{2}_r,\ket{3}_r)_{r=1}^{3}$. (d) Universal first-quantized state-preparation pipeline: a classical high performance computer supplies $\sum_i c_i\ket{n}_i$ in 2nd-Q; coefficients are mapped to Schur labels $\sum_{i} c_{i}\ket{\lambda,\mu_i,\sigma}$; a block-encoded LCU implemented via $\text{PREP}\!-\!\text{SEL}\!-\!\text{PREP}^{\dagger}$ with successful post selection (the repeat-until-success variant) prepares the encoded Schur superposition; an inverse quantum Schur transform $U_{\mathrm{Schur}}^{-1}$ produces the target first-quantized state on the system register.
  • Figure 2: Resource estimates for universal first-quantized state preparation. Panels (a), (b) and (d) assume an equal superposition of $L=50$ configurations. All panels use the Cauchy--Schwarz worst-case bound for the LCU $\ell_1$-norm, yielding $\ell_1=\sqrt{L}$. (a) Toffoli-equivalent gate count, $\mathrm{TE}=\#\mathrm{Toffoli}+(\mathrm{T\!-\!count})/7$, versus single-particle dimension $d$ at fixed precision $\epsilon=10^{-4}$, using oblivious amplitude amplification (OAA) for the block encoding. (b) $\mathrm{TE}$ versus target precision $\epsilon$ (legend lists $(N,d)$ tuples); OAA is used for block encoding. (c) Crossover in $\mathrm{TE}$ as a function of the number of configurations $L$: block encoding with OAA and with repeat-until-success (RUS) compared against the inverse Schur-transform cost $\mathrm{TE}(U_{\mathrm{Schur}}^{-1})$. We set $N=10$, $d=50$, and $\epsilon=10^{-4}$. (d) Peak qubit count versus $d$ at $\epsilon=10^{-4}$.
  • Figure S1: Abstract quantum circuit diagram implementing the BCH quantum Schur transform. A cascade of $N - 1$$U_{\text{CG}}$ circuits is applied, resulting in a history (corresponding to $|\sigma \rangle$), young diagram $|\lambda \rangle$ and GT pattern register $| \mu \rangle$.

Theorems & Definitions (3)

  • Definition 1
  • Definition 2
  • Definition 3