---
title: Number-Preserving Ansatz in Quantum Simulation
url: https://www.emergentmind.com/topics/number-preserving-ansatz-npa
type: topic
---

# Number-Preserving Ansatz in Quantum Simulation

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 [1802.04530]. In variational quantum simulation, closely related number-preserving circuits preserve total particle number by construction, so that optimization remains in the physical sector throughout [2508.12307].

## 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 [2502.10746]. In nuclear shell-model work, it can also denote the Nucleon Pair Approximation [2201.10721]. 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
$$
\partial_t \psi_t^{(\vec{k})}(\mathbf{z}) =
\Big[ -i H - \vec{k} \cdot \mathbf{w} + \sum_{\ell j} z^*_{\ell j}(t) L_\ell \Big] \psi_t^{(\vec{k})}(\mathbf{z})
+ \sum_{\ell j} k_{\ell j} p_{\ell j} L_\ell \psi_t^{(\vec{k} - \mathbf{e}_{\ell j})}(\mathbf{z})
- \sum_{\ell j} L_\ell \psi_t^{(\vec{k} + \mathbf{e}_{\ell j})}(\mathbf{z}),
$$
and the $n$PA truncates this hierarchy by keeping only those tuples for which $k_{\ell j} \neq 0$ for no more than $n$ molecules [1802.04530].

For $n=1$, the allowed tuples are those where at most one $\vec{k}_\ell$ is nonzero:
$$
\vec{k} = (\vec{0},\ldots,\vec{0}, \vec{k}_\ell, \vec{0}, \ldots, \vec{0}).
$$
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 $N$ [1802.04530].

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

| Scheme | Restriction |
|---|---|
| TT | Keep states with $\sum_{\ell, j} k_{\ell j} \le D$ |
| $n$PA | Keep states with no more than $n$ nonzero $\vec{k}_\ell$ |
| $n$MA | Keep states with at most $n$ nonzero $k_{\ell j}$ in the full tuple |

In the absorption example with a chain of $N=4$ molecules and $J=3$ modes per molecule, converged TT at $D=13$ requires $5 \times 10^6$ 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” [1802.04530]. In the FMO example with $N=7$, $J=5$, and $D=4$, 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” [1802.04530]. 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
$$
U_{\text{NP}}(\theta, \phi) =
\begin{pmatrix}
1 & 0 & 0 & 0 \\
0 & \cos \theta & i\sin \theta & 0 \\
0 & i\sin \theta & \cos \theta & 0 \\
0 & 0 & 0 & e^{i\phi}
\end{pmatrix},
$$
which acts trivially on $\ket{00}$, applies a phase on $\ket{11}$, and rotates the single-excitation subspace spanned by $\ket{01}$ and $\ket{10}$ [2508.12307]. The onsite interaction is represented by
$$
O(\theta) =
\begin{bmatrix}
1 & 0 & 0 & 0 \\
0 & 1 & 0 & 0 \\
0 & 0 & 1 & 0 \\
0 & 0 & 0 & e^{i\theta}
\end{bmatrix}.
$$

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
$$
|\psi_p\rangle = \prod_{\ell=1}^p \left( \prod_{s} \exp(-i \theta_{s,\ell} H_s) \right) |\psi_0\rangle.
$$
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 [2508.12307].

This design is used to compute the ground, first, and second excited state energies of $4\times 1$ and $2\times 2$ 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 [2508.12307].

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
$$
\Delta_c = E_0(N+1) + E_0(N-1) - 2 E_0(N).
$$
It also notes practical limitations: excited-state errors increase for larger $U/t$ and for higher excitations, and circuit size and parameter count scale up quickly with system size and the number of variational layers [2508.12307].

## 4. Structured fixed-$N$ ansätze in nuclear quantum simulation

A structurally different, but conceptually aligned, number-conserving ansatz appears in quantum simulation of a cranked Nilsson $+$ 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 $M$. 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 [2604.00844].

The generators are given as
$$
U_D(\theta) = \exp\left[\frac{\theta}{2}(a_p^\dagger a_q^\dagger a_s a_r - a_r^\dagger a_s^\dagger a_q a_p) \right],
$$
for pair-transfer doubles, and
$$
U_S(\phi) = \exp\left[\frac{\phi}{2}(a_p^\dagger a_q - a_q^\dagger a_p) \right],
$$
for single excitations. Each operator conserves total particle number by construction, and the reference state is already prepared in the desired $N$ sector. For $M=8$, the ansatz yields 42 parameters, decomposed in the source as 28 doubles and 14 singles [2604.00844].

Exact number conservation has an immediate diagnostic consequence. The conventional pairing gap
$$
\Delta_\kappa = G \left| \sum_k \langle P_k \rangle \right|
$$
vanishes identically because $\langle P_k \rangle = 0$ in a fixed-$N$ state. The paper therefore introduces
$$
\Delta_{\mathrm{coh}} = G \sqrt{\sum_{k \neq l} \left| \langle P_k^\dagger P_l \rangle \right|},
$$
described as “a scalar measure of off-diagonal pair coherence rather than a BCS gap” [2604.00844].

The reported application concerns even-even $^{80,82,84}$Zr. The source states that $^{80}$Zr shows “a stable oblate minimum at $δ^\ast \approx -0.25$,” that $^{82}$Zr “exhibits the strongest rotational evolution,” and that $^{84}$Zr “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 $M=6$ and $M=8$ show stable trends but visible shifts, so “no active-space convergence is claimed” [2604.00844].

## 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 [2402.18768]. The relevant two-qubit block is
$$
\hat{A}(\theta) = \exp(-i \theta \hat{G}_A),
$$
with generator
$$
\hat{G}_A = i |01\rangle \langle 10| - i |10\rangle \langle 01|
\longrightarrow \hat{X}_0 \hat{Y}_1 - \hat{Y}_0 \hat{X}_1,
$$
while the three-qubit block is
$$
\hat{B}(\alpha) = \exp(-i \alpha \hat{G}_B),
$$
with
$$
\hat{G}_B = i|001\rangle \langle 110| - i |110\rangle \langle 001|.
$$
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 [2402.18768].

That comparison makes explicit a recurring methodological divide. The penalty formulation uses
$$
\hat{H}_{\rm eff} = \hat{H} + \eta (\hat{N}_{\rm tot} - N)^2,
$$
which, as the source states, “does not prevent the quantum state from exploring outside the correct subspace; it only penalizes such behavior” [2402.18768]. 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 [2510.07278]. The occupation-to-weight correspondence is
$$
z_i = n_i - n_{i+1}, \qquad \sum_{i=1}^d n_i = N,
$$
and the first-quantized space decomposes as
$$
(\mathbb{C}^d)^{\otimes N}
\cong
\bigoplus_{\lambda \vdash N,\; \ell(\lambda)\leq d} V^{U(d)}_{\lambda} \otimes V^{S_N}_{\lambda}.
$$
Operationally, the algorithm prepares an encoded superposition of Schur labels and then applies the inverse quantum Schur transform. The stated runtime is $\mathrm{poly}(L, N, d, \log \epsilon^{-1})$ for $L$ configurations of $N$ particles over $d$ modes to accuracy $\epsilon$, and the construction “applies universally to fermions, bosons, and Green’s paraparticles in arbitrary single-particle bases” [2510.07278].

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-$N$ 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 $k$-body approximation (PNOkB) is designed for calculations that use Bogoliubov reference states but require the effective operator to preserve U(1) symmetry [1908.00765]. Its defining construction retains only terms with equal numbers of creation and annihilation operators and recursively corrects lower-body fields:
$$
O^{\text{PNOkB}} = \sum_{n=0}^k \tilde{o}^{nn},
\qquad
\tilde{o}^{nn} = \Lambda^{nn} - \sum_{m=n+1}^k \tilde{\Lambda}^{nn(mm)}.
$$
The source states that $O^{\text{PNOkB}}$ commutes with particle number by construction, in contrast with the naive normal-ordered truncation nNOkB, which retains particle-number-violating anomalous terms [1908.00765].

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
$$
Q^{A}_F = \frac{\langle\Phi|P^A F |\Phi\rangle}{\langle\Phi|P^A F P^A |\Phi\rangle}
$$
should equal 1. The paper reports $Q^{A}_F = 1$ along all isotopic chains for PNOkB, whereas $Q^A_F \neq 1$ for nNOkB [1908.00765].

Several recurrent distinctions follow directly from the cited literature. First, exact number conservation can invalidate conventional broken-symmetry diagnostics: in fixed-$N$ states, $\langle P_k\rangle = 0$, so the conventional BCS gap vanishes identically and must be replaced by an off-diagonal coherence measure [2604.00844]. Second, penalty enforcement is not equivalent to a number-preserving ansatz because it leaves the optimizer free to explore outside the target sector [2402.18768]. 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 [2502.10746; 2201.10721].

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-$N$ 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.

Source: https://www.emergentmind.com/topics/number-preserving-ansatz-npa