Number-Preserving Ansatz in Quantum Simulation

• Number-Preserving Ansatz is a design principle that restricts state preparation to fixed-particle sectors, ensuring exact conservation of the total particle number.
• It is applied in various settings such as HOPS, variational quantum circuits, and nuclear simulations to mitigate leakage into unphysical sectors and maintain symmetry.
• NPA methods outperform penalty-based strategies by directly constraining the accessible state manifold, offering scalable and accurate implementations across fermionic, bosonic, and first-quantized models.

Number-Preserving Ansatz (NPA) denotes a class of symmetry-adapted truncations and variational constructions in which total particle number is conserved exactly, or in which computation is restricted to a fixed-particle-number sector. In the hierarchy of pure states (HOPS), the $n$-particle approximation ($n$PA) is explicitly described as “also known in the literature as the Number-Preserving Ansatz” and truncates the hierarchy by retaining only auxiliary states with nonzero environmental excitations on no more than $n$ molecules (Zhang et al., 2018). In variational quantum simulation, closely related number-preserving circuits preserve total particle number by construction, so that optimization remains in the physical sector throughout (Dev et al., 17 Aug 2025).

1. Terminology and scope

Within the provided literature, “Number-Preserving Ansatz” is not a single formalism but a recurring design principle: enforce exact particle-number conservation at the level of truncation, generators, gates, or state preparation. The shared objective is to avoid variational leakage into unphysical sectors and to make fixed-$N$ observables well defined.

The acronym “NPA” is overloaded. In quantum nonlocality, it usually denotes the Navascués-Pironio-Acín hierarchy for quantum correlations (Ishizaka, 15 Feb 2025). In nuclear shell-model work, it can also denote the Nucleon Pair Approximation (Lei et al., 2022). In the present sense, however, NPA refers to number-preserving constructions in open-system dynamics, variational quantum eigensolvers, bosonic circuit design, and symmetry-preserving operator approximations.

A persistent technical distinction is between exact symmetry preservation and penalty enforcement. Exact symmetry preservation restricts the state manifold itself; penalty methods instead modify the objective while still allowing excursions outside the desired sector. This distinction reappears across fermionic, bosonic, and nuclear applications.

2. The $n$-particle approximation in HOPS

In HOPS, the system-bath dynamics is described by a stochastic hierarchy of auxiliary wavefunctions $\psi_t^{(\vec{k})}(\mathbf{z})$ indexed by a multi-index $\vec{k}$. For molecular aggregates with $N$ molecules and $J$ bath modes per molecule, the hierarchy is organized as

$$\vec{k} = \big\{ \vec{k}_1, \ldots, \vec{k}_N \big\}, \qquad \vec{k}_\ell = (k_{\ell 1}, \ldots, k_{\ell J}).$$

The full hierarchy reported in the source is

$n$0

and the $n$1PA truncates this hierarchy by keeping only those tuples for which $n$2 for no more than $n$3 molecules (Zhang et al., 2018).

For $n$4, the allowed tuples are those where at most one $n$5 is nonzero:

$n$6

The paper also notes a further restriction often used in practice: only those auxiliary states are retained where the molecule carrying the electronic excitation is one of those with nonzero vibrational excitation. This reduces the number of equations by a factor of $n$7 (Zhang et al., 2018).

The source contrasts $n$8PA with triangular truncation (TT) and the $n$9-mode approximation (nMA):

Scheme	Restriction
TT	Keep states with $n$0
$n$1PA	Keep states with no more than $n$2 nonzero $n$3
$n$4MA	Keep states with at most $n$5 nonzero $n$6 in the full tuple

In the absorption example with a chain of $n$7 molecules and $n$8 modes per molecule, converged TT at $n$9 requires $N$0 auxiliary states, whereas the reported counts are 156 for 1MA, 2236 for 1PA, 5500 for 2MA, approximately 70,000 for 3MA, approximately 150,000 for 2PA, and “3PA/4MA” as “Very close to convergence” (Zhang et al., 2018). In the FMO example with $N$1, $N$2, and $N$3, TT uses 82,000 auxiliary states; the paper reports 140 for 1MA, 875 for 1PA, 3,700 for 2MA, 17,000 for 2PA, and “3MA/3PA” as “Practically indistinguishable from TT, limited by stochastic sampling noise” (Zhang et al., 2018). The paper states that nMA generally performs slightly better than nPA, but both provide accurate results for a small number of auxiliary equations.

3. Number-preserving circuit ansätze in variational simulation of fermions

In the Fermi-Hubbard setting, the Number-Preserving Ansatz is implemented as a variational circuit whose elementary two-qubit block preserves excitation number exactly. The fundamental gate is reported as

$N$4

which acts trivially on $N$5, applies a phase on $N$6, and rotates the single-excitation subspace spanned by $N$7 and $N$8 (Dev et al., 17 Aug 2025). The onsite interaction is represented by

$N$9

The circuit structure mirrors the Hamiltonian Variational Ansatz (HVA), but each gate can be assigned its own independent variational parameter. The source expresses the layered variational state as

$n$0

It then distinguishes two constructions. In “pure” NPA, each nontrivial two-qubit gate corresponding to a hopping term is replaced by the generalized number-preserving unitary above. In the hybrid NPA+HVA construction, the ansatz retains the same HVA gates, but, “similar to NPA principles,” each gate is assigned a unique variational parameter (Dev et al., 17 Aug 2025).

This design is used to compute the ground, first, and second excited state energies of $n$1 and $n$2 Hubbard lattices. The workflow reported in the source employs a hybrid optimization strategy in which COBYLA is used for coarse convergence and L-BFGS for fine-tuning. Excited states are obtained by adding a penalty term imposing orthogonality to lower-energy states. The paper states that this strategy “allows the circuit to reach the ground state with a low circuit depth of just two layers” and supports analysis of charge and spin excitation gaps through phase diagrams (Dev et al., 17 Aug 2025).

A central motivation is that particle-number conservation keeps the variational search inside the intended electron-number sector. The source associates this with “symmetry protection” and with reliable extraction of sector-resolved observables such as the charge gap

$n$3

It also notes practical limitations: excited-state errors increase for larger $n$4 and for higher excitations, and circuit size and parameter count scale up quickly with system size and the number of variational layers (Dev et al., 17 Aug 2025).

4. Structured fixed-$n$5 ansätze in nuclear quantum simulation

A structurally different, but conceptually aligned, number-conserving ansatz appears in quantum simulation of a cranked Nilsson $n$6 pairing Hamiltonian on a fixed deformation grid. There, the many-body Routhian is mapped to qubits via the Jordan–Wigner transformation and minimized using VQE in a truncated active space $n$7. The ansatz is a “structured, number-conserving singles-and-doubles ansatz”: double excitations implement pair transfer, while singles are restricted to the nonzero Coriolis-coupling graph of the active Nilsson basis (Abhishek et al., 1 Apr 2026).

The generators are given as

$n$8

for pair-transfer doubles, and

$n$9

for single excitations. Each operator conserves total particle number by construction, and the reference state is already prepared in the desired $\psi_t^{(\vec{k})}(\mathbf{z})$0 sector. For $\psi_t^{(\vec{k})}(\mathbf{z})$1, the ansatz yields 42 parameters, decomposed in the source as 28 doubles and 14 singles (Abhishek et al., 1 Apr 2026).

Exact number conservation has an immediate diagnostic consequence. The conventional pairing gap

$\psi_t^{(\vec{k})}(\mathbf{z})$2

vanishes identically because $\psi_t^{(\vec{k})}(\mathbf{z})$3 in a fixed-$\psi_t^{(\vec{k})}(\mathbf{z})$4 state. The paper therefore introduces

$\psi_t^{(\vec{k})}(\mathbf{z})$5

described as “a scalar measure of off-diagonal pair coherence rather than a BCS gap” (Abhishek et al., 1 Apr 2026).

The reported application concerns even-even $\psi_t^{(\vec{k})}(\mathbf{z})$6Zr. The source states that $\psi_t^{(\vec{k})}(\mathbf{z})$7Zr shows “a stable oblate minimum at $\psi_t^{(\vec{k})}(\mathbf{z})$8,” that $\psi_t^{(\vec{k})}(\mathbf{z})$9Zr “exhibits the strongest rotational evolution,” and that $\vec{k}$0Zr “retains a robust prolate minimum with the largest neutron pairing coherence.” It simultaneously emphasizes that these results “reflect the present truncated model rather than converged spectroscopy,” and that comparisons between $\vec{k}$1 and $\vec{k}$2 show stable trends but visible shifts, so “no active-space convergence is claimed” (Abhishek et al., 1 Apr 2026).

5. Bosonic and first-quantized generalizations

The same design principle extends beyond fermionic VQE. For bosonic systems, the binary encoded multilevel particles circuit ansatz (BEMPA) preserves particle count by construction using carefully positioned symmetry-preserving 2- and 3-qubit gates acting on significant figure blocks in a binary encoding (Bahrami et al., 2024). The relevant two-qubit block is

$\vec{k}$3

with generator

$\vec{k}$4

while the three-qubit block is

$\vec{k}$5

with

$\vec{k}$6

The paper benchmarks BEMPA on the Bose-Hubbard Hamiltonian and reports that it finds ground-state eigenvalues within “drastically shorter runtimes compared to penalty-based strategies methods” over a parameter range spanning Mott-insulator and superfluid regimes (Bahrami et al., 2024).

That comparison makes explicit a recurring methodological divide. The penalty formulation uses

$\vec{k}$7

which, as the source states, “does not prevent the quantum state from exploring outside the correct subspace; it only penalizes such behavior” (Bahrami et al., 2024). By contrast, number-preserving circuit design constrains the accessible state manifold from the outset.

A more algebraic generalization appears in first-quantized state preparation. A 2025 construction uses the Jordan–Schwinger Lie algebra homomorphism together with the Schur–Weyl decomposition to map any polynomial-size superposition of occupation-number configurations to a first-quantized representation while maintaining exact particle number (Baker et al., 8 Oct 2025). The occupation-to-weight correspondence is

$\vec{k}$8

and the first-quantized space decomposes as

$\vec{k}$9

Operationally, the algorithm prepares an encoded superposition of Schur labels and then applies the inverse quantum Schur transform. The stated runtime is $N$0 for $N$1 configurations of $N$2 particles over $N$3 modes to accuracy $N$4, and the construction “applies universally to fermions, bosons, and Green’s paraparticles in arbitrary single-particle bases” (Baker et al., 8 Oct 2025).

A plausible implication is that the number-preserving ansatz idea has two complementary realizations: direct restriction of the variational circuit, and exact embedding of fixed-$N$5 occupation data into a symmetry-adapted representation.

6. Symmetry-preserving operator truncations and recurrent distinctions

The number-preserving principle also appears at the operator level in particle-number-breaking many-body theories. The particle-number-conserving normal-ordered $N$6-body approximation (PNOkB) is designed for calculations that use Bogoliubov reference states but require the effective operator to preserve U(1) symmetry (Ripoche et al., 2019). Its defining construction retains only terms with equal numbers of creation and annihilation operators and recursively corrects lower-body fields:

$N$7

The source states that $N$8 commutes with particle number by construction, in contrast with the naive normal-ordered truncation nNOkB, which retains particle-number-violating anomalous terms (Ripoche et al., 2019).

The numerical test used there compares single and double particle-number projection. For a truly number-conserving operator, the source states that the two projected energies are identical and that

$N$9

should equal 1. The paper reports $J$0 along all isotopic chains for PNOkB, whereas $J$1 for nNOkB (Ripoche et al., 2019).

Several recurrent distinctions follow directly from the cited literature. First, exact number conservation can invalidate conventional broken-symmetry diagnostics: in fixed-$J$2 states, $J$3, so the conventional BCS gap vanishes identically and must be replaced by an off-diagonal coherence measure (Abhishek et al., 1 Apr 2026). Second, penalty enforcement is not equivalent to a number-preserving ansatz because it leaves the optimizer free to explore outside the target sector (Bahrami et al., 2024). Third, the term “NPA” must be interpreted contextually, since the same acronym also denotes the Navascués-Pironio-Acín hierarchy in Bell nonlocality and the Nucleon Pair Approximation in nuclear shell-model truncation (Ishizaka, 15 Feb 2025, Lei et al., 2022).

Taken together, these usages define Number-Preserving Ansatz not as a single algorithm but as a symmetry principle with several concrete realizations: truncating hierarchies by fixed support over excited sites, building excitation-preserving quantum circuits, constructing fixed-$J$4 diagnostics, encoding bosonic particle conservation into compact gate sets, mapping occupation data into symmetry-adapted first-quantized states, and normal-ordering operators so that particle number remains exact despite symmetry-broken references.