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arXiv:2510.07278

Universal initial state preparation for first quantized quantum simulations

Jack S. Baker, Saxena, Gaurav, Thi Ha Kyaw

Published Oct 8, 2025• 0 citations
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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 deterministically prepare an superposition of target Schur labels and apply the inverse quantum Schur transform. For $L$ configurations of $N$ particles over $d$ modes prepared to accuracy $ε$, the most efficient variant of the algorithm runs with non-Clifford gate complexity $\mathrm{poly}(L, N, \log d, \log ε^{-1})$. The protocol applies universally to fermions, bosons, and Green's paraparticles in arbitrary single-particle bases. Resource estimates establish practicality within leading first-quantized pipelines, with statistics-aware specializations promising further reductions.

DOI
10.48550/arxiv.2510.07278

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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 su(d)\mathfrak{su}(d)su(d) weight states within the Schur--Weyl decomposition. Operationally, we deterministically prepare an superposition of target Schur labels and apply the inverse quantum Schur transform. For LLL configurations of NNN particles over ddd modes prepared to accuracy εεε, the most efficient variant of the algorithm runs with non-Clifford gate complexity poly(L,N,log⁡d,log⁡ε−1)\mathrm{poly}(L, N, \log d, \log ε^{-1})poly(L,N,logd,logε−1). The protocol applies universally to fermions, bosons, and Green's paraparticles in arbitrary single-particle bases. Resource estimates establish practicality within leading first-quantized pipelines, with statistics-aware specializations promising further reductions.
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The quantum simulation of real molecules and materials is one of the most highly anticipated applications of quantum computing. Algorithms for simulating electronic structure using a first-quantized plane-wave representation are especially promising due to their asymptotic efficiency. However, previous proposals for preparing initial states for these simulation algorithms scale poorly with the size of the basis set. We address this shortcoming by showing how to efficiently map states defined in a Gaussian-type orbital basis to a plane-wave basis with a scaling that is logarithmic in the number of plane waves. Our key technical result is a proof that molecular orbitals constructed from Gaussian-type basis functions can be compactly represented in a plane-wave basis using matrix product states. While we expect that other approaches could achieve the same logarithmic scaling with respect to basis-set size, our proposed state-preparation technique is also highly efficient in practice. For example, in a series of numerical experiments on small molecules, we find that our approach allows us to prepare an approximation to the Hartree-Fock state using orders of magnitude fewer non-Clifford gates than a naive approach. By resolving the issue of state preparation, our work allows for the first quantum simulation of molecular systems the end-to-end complexity of which is truly sublinear in the basis-set size.
Simulating physical systems on near-term quantum computers often requires preparing states within constrained subspaces, like those with fixed particle number or spin. We use Lie algebraic techniques to prove that hardware-efficient gates are universal for state preparation in these subspaces. The key mechanism is Pauli ZZZ dressing: commutators of overlapping gates produce Pauli ZZZ operators on shared qubits, acting as spectator projectors that decompose multi-plane rotations into single-plane generators spanning the full so(w)\mathfrak{so}(w)so(w) algebra, where www is the dimension of the constrained subspace, thereby guaranteeing universality for real state preparation. Adding independent complex phases extends this to su(w)\mathfrak{su}(w)su(w), enabling arbitrary complex state preparation. We provide a computationally efficient Jacobian criterion for verifying that a circuit can explore any direction on the target manifold from almost any parameter configuration. Our findings are applicable to many problem areas, including Fermi-Hubbard models, Bose-Hubbard models, and molecular electronic structure. We apply our framework to two physical settings: we prove the completeness of the binary encoded multi-level particles ansatz on the conserved-particle-number subspace, and we construct symmetry-preserving circuits for the fuzzy sphere regularisation of the 3D Ising conformal field theory (CFT). For the latter, we variationally prepare the ground and excited states and extract some CFT data.
While quantum computers are capable of simulating many quantum systems efficiently, the simulation algorithms must begin with the preparation of an appropriate initial state. We present a method for generating physically relevant quantum states on a lattice in real space. In particular, the present algorithm is able to prepare general pure and mixed many-particle states of any number of particles. It relies on a procedure for converting from a second-quantized state to its first-quantized counterpart. The algorithm is efficient in that it operates in time that is polynomial in all the essential descriptors of the system, the number of particles, the resolution of the lattice, and the inverse of the maximum final error. This scaling holds under the assumption that the wave function to be prepared is bounded or its indefinite integral is known and that the Fock operator of the system is efficiently simulatable.
While quantum computers are capable of simulating many quantum systems efficiently, the simulation algorithms must begin with the preparation of an appropriate initial state. We present a method for generating physically relevant quantum states on a lattice in real space. In particular, the present algorithm is able to prepare general pure and mixed many-particle states of any number of particles. It relies on a procedure for converting from a second-quantized state to its first-quantized counterpart. The algorithm is efficient in that it operates in time that is polynomial in all the essential descriptors of the system, the number of particles, the resolution of the lattice, and the inverse of the maximum final error. This scaling holds under the assumption that the wave function to be prepared is bounded or its indefinite integral is known and that the Fock operator of the system is efficiently simulatable.
Symmetry is fundamental in the description and simulation of quantum systems. Leveraging symmetries in classical simulations of many-body quantum systems can result in significant overhead due to the exponentially growing size of some symmetry groups as the number of particles increases. Quantum computers hold the promise of achieving exponential speedup in simulating quantum many-body systems; however, a general method for utilizing symmetries in quantum simulations has not yet been established. In this work, we present a unified framework for incorporating symmetry group transforms on quantum computers to simulate many-body systems. The core of our approach lies in the development of efficient quantum circuits for symmetry-adapted projection onto irreducible representations of a group or pairs of commuting groups. We provide resource estimations for common groups, including the cyclic and permutation groups. Our algorithms demonstrate the capability to prepare coherent superpositions of symmetry-adapted states and to perform quantum evolution across a wide range of models in condensed-matter physics and ab initio electronic structure in quantum chemistry. Specifically, we execute a symmetry-adapted quantum subroutine for small molecules in first-quantization on noisy hardware and demonstrate the emulation of symmetry-adapted quantum phase estimation for preparing coherent superpositions of quantum states in various irreducible representations of a symmetry group. In addition, we present a discussion of open problems regarding treating symmetries in digital quantum simulations of many-body systems, paving the way for future systematic investigations into leveraging symmetries quantumly for practical quantum advantage. The broad applicability and rigorous resource estimation for symmetry transformations make our framework appealing for achieving provable quantum advantage on fault-tolerant quantum computers, especially for symmetry-related properties.