Universality of quantum gates in particle and symmetry constrained subspaces

1 Introduction

Quantum computers promise to provide a pathway to explore previously inaccessible regimes in physics, including strongly-coupled systems and complex many-body dynamics. Due to the limited coherence times and gate fidelities of near-term noisy devices, algorithms typically rely on shallow, parametrised quantum circuits. A central challenge in these variational quantum algorithms (VQAs), such as the variational quantum eigensolver (VQE) [1, 2], is the design of the state preparation ansatz. The ansatz must be expressive enough to reach the desired physical states while remaining hardware-efficient.

In many physical contexts, the relevant Hilbert space is restricted by conservation laws, such as particle number conservation or total spin. Preparing an arbitrary state within such a constrained subspace usually involves complicated unitaries. For example, controlled gates or operations involving long non-local Pauli mappings, such as Jordan-Wigner strings, can prepare exact superpositions. However, their physical implementation demands a prohibitive number of native device gates, drastically increasing circuit depth. Instead, simpler hardware-efficient gates that omit these non-local strings are often employed, relying entirely on adjacent or nearby interactions.

In this work, we use Lie algebraic techniques to establish a mathematical foundation for universal state preparation with hardware-efficient gates within constrained subspaces. A constrained subspace of dimension $w$, spanned by $w$ computational basis states regarded as coordinate axes of ${ℝ}^w$, carries a natural geometric structure: unit-normalised real superpositions live on the sphere $S^{w-1}$, acted on by the rotation group $\mathit{𝑆𝑂}(w)$ with Lie algebra $\mathit{𝔰𝔬}(w)$. We show in Proposition 1 that hardware-efficient 2-qubit “multi-plane” rotation gates, which simultaneously rotate in every plane connecting basis states differing only on the two active qubits, generate the full $\mathit{𝔰𝔬}(w)$ algebra. The key mechanism, which we term Pauli $Z$ dressing, is that commutators of overlapping gates produce Pauli $Z$ operators on shared qubits, which act as spectator projectors and allow the extraction of “single-plane” generators rotating between individual pairs of basis states. Connectivity of the graph of constant-particle-number basis states under pairwise swaps then ensures that these generators span the complete algebra, so circuits composed of such gates can universally prepare any real state within the constrained subspace. When each gate is equipped with an independent complex phase, we show in section 5 that the same spectator mechanism extends to the imaginary hopping generators, promoting the algebra from $\mathit{𝔰𝔬}(w)$ to the full $\mathit{𝔰𝔲}(w)$ and enabling universal complex state preparation.

Unlike standard universality proofs for discrete gate sets, which establish dense subgroups via the Solovay-Kitaev theorem [3, 4], our approach exploits the continuous parametrisation of variational gates. Drawing on the Lie algebra rank condition from quantum control theory [5], we show that spanning the Lie algebra guarantees that finite-depth circuits can reach general elements of the relevant connected Lie group exactly. Theorem 1 below provides the essential mathematical foundation for this. Although this continuous approach has been employed previously [6–9], our focus here is on spanning specific constrained subspaces of a Hilbert space.

While Theorem 1 establishes universal reachability in principle, verifying that a specific, finite-depth ansatz spans the target space in practice is a separate challenge. To partly address this, we provide a computationally efficient local criterion, Lemma 1, showing that a minimal circuit of $w-1$ generators has a Jacobian of rank $w-1$ at almost every parameter value — meaning the circuit can explore any direction on $S^{w-1}$ from almost any parameter configuration — provided the Jacobian achieves this rank at a single reference point. Although surjective parametrisations of the target sphere exist, critical points of any circuit map are topologically unavoidable, and for hardware-efficient circuits these can partition the reachable region into disconnected patches. Crossing such boundaries generally requires over-parametrising the circuit.

Our framework applies immediately to any fermionic system with conserved particle number, such as the Fermi-Hubbard model, one of the most widely studied targets for variational quantum simulation [10–12]. We present detailed applications to two settings that involve additional algebraic structure beyond the basic constant-Hamming-weight case. In section 3, we treat the binary encoded multi-level particles ansatz (BEMPA) [13] for bosonic simulation. The BEMPA gate set involves two families of gates ($Â$ and $\widehat{B}$) acting at different binary significance levels; we prove that their commutators activate the same projection mechanism, establishing completeness on the conserved-particle-number subspace (Proposition 2). In section 4, we study the fuzzy sphere regularisation of conformal field theories (CFTs) [14], specifically the 3D Ising model at criticality. Here the physically relevant subspace carries an additional $S_z=0$ constraint on the total azimuthal angular momentum. Two-qubit gates alone cannot change orbital occupancies while preserving this symmetry, so we consider 4-qubit gates that simultaneously transfer two excitations between orbitals. We then prove (Proposition 3) that the combined 2- and 4-qubit gate set generates the full orthogonal group on the symmetry-constrained subspace, enabling universal state preparation for this model.

The fuzzy sphere Hamiltonian we examine has a finite-dimensional spectrum whose low-lying eigenvalues map, via the state-operator correspondence, to the scaling dimensions of the 3d Ising CFT. Finding the ground and excited states of this Hamiltonian is closely analogous to quantum chemistry problems, and already developed algorithms like VQE and the variational quantum deflation (VQD) algorithm for excited states [15] can be readily applied, as can quantum subspace methods [16]. Because exactly diagonalising this Hamiltonian classically becomes unmanageable for large numbers of modes, quantum simulation holds immense value. We demonstrate the practical construction of minimal state-preparation circuits tailored to the model’s constraints and extract the ground state and two low-lying excited states.

The 3D Ising model at criticality is a particularly attractive target because high-precision data exist through the numerical conformal bootstrap [17]. This provides a rare setting where both strong coupling and reliable independent results coexist, making it an ideal benchmark for quantum simulation: agreement with bootstrap data validates the quantum algorithms, while discrepancies can reveal shortcomings of current devices and guide future improvements.

The remainder of this paper is organised as follows. In section 2 we develop the mathematical framework: we prove the completeness of hardware-efficient gates on constant-Hamming-weight subspaces (Proposition 1) and provide a computationally efficient Jacobian criterion for verifying that a circuit can explore any direction in the target manifold from almost any point (Lemma 1). Section 3 applies this framework to bosonic simulation with binary encoding, establishing the completeness of the BEMPA ansatz (Proposition 2). Section 4 treats the fuzzy sphere regularisation of the 3D Ising CFT: we review the physical setup, construct symmetry-preserving circuits that generate the full orthogonal group on the $S_z=0$ subspace (Proposition 3), and present variational quantum simulation results for the ground and excited states. We then extract an OPE coefficient from the variationally prepared states, examine how the required circuit size grows with the number of electrons, and determine the gate error rate at which noise begins to dominate the finite-size error. In section 5 we show how independent complex phases extend all preceding results from $\mathit{𝔰𝔬}(w)$ to $\mathit{𝔰𝔲}(w)$. We conclude in section 6. Appendix A verifies explicitly the commutator identities on which the three propositions rest, and appendix B collects the elementary gate decompositions used in this work.

2 Lie algebraic framework for constrained state preparation

Efficient state preparation is a critical subroutine in variational quantum algorithms. In many physical systems of interest, such as electronic structure and the fuzzy sphere regularisation discussed later, the target states are constrained by symmetries like particle number conservation or total spin. While controlled operations could be used to prepare generic superpositions within these constrained subspaces [18], they often lead to prohibitively deep circuits on near-term hardware. Instead, hardware-efficient ansatze employing nearest-neighbour or sparsely connected 2-, 3- or 4-qubit gates are preferred.

Here, we provide mathematical results demonstrating how these simpler hardware-efficient gates can achieve universal state preparation within constrained subspaces, drawing on the Lie algebraic properties of the gates. We focus on the subspace of states of constant Hamming weight in this section, and we provide extensions to further cases in the application sections below. We formulate our essential theoretical results as propositions to guide the construction of suitable quantum circuits.

2.1 Completeness of hardware-efficient ansatze

Suppose we want to generate an arbitrary superposition in a subspace of constant Hamming weight $n_{\text{q}}$ using $N_{\text{q}}$ qubits. The dimension of this subspace is

$$w=\left(\genfrac{}{}{0.0pt}{}{N_{\text{q}}}{n_{\text{q}}}\right).$$

(2.1)

Let us think of these $w$ basis states as the axes ${𝐞}_0,{𝐞}_1,\dots ,{𝐞}_{w-1}$ of ${ℝ}^w$. A unit-normalised, time-reversal symmetric state can be represented by a vector $𝐯$ in ${ℝ}^w$ that can be written as a linear combination of these $w$ states with real coefficients whose squares add up to unity:

$$𝐯=v_0{𝐞}_0+v_1{𝐞}_1+\cdots +v_{w-1}{𝐞}_{w-1},v_0^2+v_1^2+\cdots +v_{w-1}^2=1.$$

(2.2)

In other words, the states we are interested in live on the unit sphere $S^{w-1}$ in ${ℝ}^w$. To navigate this subspace, we must apply operations from the special orthogonal group $\mathit{𝑆𝑂}(w)$, which has dimension $w(w-1)∕2$.

In this geometric language, the question we’re interested in is how to get to a generic location on $S^{w-1}$ starting from a specified location, e.g. ${𝐞}_0$. Of course this is possible with the use of $w\times w$ orthogonal matrices with unit determinant (elements of $\mathit{𝑆𝑂}(w)$). However, we would like to achieve this without using gates that grow with $w$.

A common approach is the use of Givens rotations,

$$G(𝜃)=\left(\begin{array}{cccc}\hfill 1\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill \cos <!--\; FUNCTION\; APPLICATION\; -->𝜃\hfill & \hfill -\sin <!--\; FUNCTION\; APPLICATION\; -->𝜃\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill \sin <!--\; FUNCTION\; APPLICATION\; -->𝜃\hfill & \hfill \cos <!--\; FUNCTION\; APPLICATION\; -->𝜃\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 1\hfill \end{array}\right),$$

(2.3)

which are understood to be acting on two qubits only, and we use the ordering $|00⟩,|01⟩,|10⟩,|11⟩$ for the corresponding column of basis states on which the matrix (2.3) acts. The two qubits on which the Givens rotation is acting need to be specified, for example,

where the first qubit is indexed by 0, the second by 1, and so on.

From a geometric perspective, a Givens rotation can be thought of as a rotation in the plane defined by two axes. For example, if ${𝐞}_0$ is the axis representing $|010101⟩$ and ${𝐞}_1$ the axis representing $|001101⟩$, then (2.4c) represents a rotation by angle $𝜃$ in the plane defined by ${𝐞}_0,{𝐞}_1$:

The essential complication is that a Givens rotation acts simultaneously across all possible configurations of the spectator qubits. At some point, this will result in a rotation in more than one plane simultaneously, for example if $G_{45}$ is applied on the last of (2.4). One can avoid this with the use of controlled gates, but the increased circuit depth renders them undesirable. As we will see, efficient quantum circuits can be constructed leveraging the algebraic structure of the generators of multi-plane rotations: their commutators produce intermediate $Z$ operators that act as projectors on the spectator qubits, allowing one to isolate single-plane rotations.

Let us start with a fundamental theorem.¹

This theorem reduces our problem to showing that a given set of generators is sufficient to generate the desired Lie algebra. For those less familiar with formal group theory, we note that Lie algebra is analogous to quantum operator algebra, and elements of the corresponding Lie group are analogous to exponentials and products of exponentials of the quantum operators.

The omission of $Z$-strings means the interacting $G$-gate applies identical pairwise-swap transitions across all possible configurations of spectator qubits. We call $L_{\mathit{𝑖𝑗}}$ a multi-plane generator because it simultaneously rotates in every 2-dimensional plane connecting pairs of basis states that differ only in the occupations of qubits $i$ and $j$.

The $A$-gate [20, 21],

$$A(𝜃)=\left(\begin{array}{cccc}\hfill 1\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill \cos <!--\; FUNCTION\; APPLICATION\; -->𝜃\hfill & \hfill \sin <!--\; FUNCTION\; APPLICATION\; -->𝜃\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill \sin <!--\; FUNCTION\; APPLICATION\; -->𝜃\hfill & \hfill -\cos <!--\; FUNCTION\; APPLICATION\; -->𝜃\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 1\hfill \end{array}\right),$$

(2.8)

applies the $G$-gate proper rotation combined with a local reflection matrix that selectively applies a $-1$ phase to the $|10⟩$ state,

$$A_{\mathit{𝑖𝑗}}(𝜃)=A_{\mathit{𝑖𝑗}}(0)G_{\mathit{𝑖𝑗}}(-𝜃),$$

(2.9)

where

$$A_{\mathit{𝑖𝑗}}(0)=I-2\left(\frac{I-Z_i}{2}\right)\left(\frac{I+Z_j}{2}\right).$$

(2.10)

Unlike $L_{\mathit{𝑖𝑗}}$, the generator $E_{\mathit{𝑥𝑦}}$ rotates in exactly one plane: the single plane spanned by $|x⟩$ and $|y⟩$. When $|x⟩$ and $|y⟩$ differ on exactly two qubits, $i$ and $j$, the elementary generator can be expressed explicitly as

$$E_{\mathit{𝑥𝑦}}=\frac{i}{2}{L}_{\mathit{𝑖𝑗}}\prod _{k\ne i,j}\frac{I+{(-1)}^{x_k}{Z}_k}{2},$$

(2.12)

where $x_i=1,x_j=0$, and $x_k$ is the value of qubit $k$ in $|x⟩$. The product of projectors pins every spectator qubit to its value in $|x⟩$ (and $|y⟩$, since they agree on spectators).

A related theorem regarding the universality of Hamming-weight preserving gates has recently appeared in [22]. There, the authors analytically evaluate the Lie algebra generated by continuous one-parameter families of quantum gates and prove that for purely continuous hardware-efficient operators to span the full special unitary Lie algebra $\mathit{𝔰𝔲}(w)$, their generator must inherently include two-body interaction diagonal terms that break free-fermion integrability. Our formalisation in Proposition 1 is fully complementary to their result: while pure $G$-gates (which represent pure single-particle hopping) are mathematically insufficient to approximate arbitrary complex unitary matrices, as accurately dictated by the conditions established in [22], we establish that they precisely span the special orthogonal algebra $\mathit{𝔰𝔬}(w)$. This orthogonal universality guarantees that such minimal hardware-efficient circuits can access any arbitrary real quantum state within the symmetry-constrained subspace, which constitutes the essential requirement for the variational preparation of ground states of real Hamiltonians.

2.2 Conditions for spanning circuits

For practical scaling on near-term devices, it is desirable to have a robust algebraic technique for confirming if a specialised ansatz spans the required manifold with the minimal number of parameters. The most efficient preparation circuit mandates exactly $w-1$ independent parameters to traverse the manifold. An ansatz that fails to span the target manifold may be unable to reach the true ground state even in principle, so circuit completeness is not merely a convenience but a fundamental computational necessity. Approximating the ground state energy of a generic local Hamiltonian is QMA-complete [23], a complexity class believed to be strictly harder than BQP.

The criterion of Lemma 1 provides a powerful, computationally inexpensive classical method to test gate configurations. In the minimal case $p=w-1$, a full-rank Jacobian at a reference point certifies that the image of the circuit map

$$\begin{array}{ccc}\hfill U:T^{w-1}& \to S^{w-1}\hfill & \hfill \\ \hfill 𝜃& \mapsto U(𝜃)|{\psi }_0⟩\hfill \end{array}$$

(2.16)

explores a full $(w-1)$-dimensional patch of the target manifold at generic parameter values. This local condition can be used to show, for example, that a circuit composed exclusively of continuous proper $G$-gates restricted only to adjacent qubits cannot generate the intermediate $Z$ operators necessary to span the subspace of constant Hamming weight. Thus, to achieve universality across the full subspace simply with proper parametrised $G$-gates, the circuit must include gates acting selectively on non-adjacent qubits.

The dimensional gap arises because generic states in $S^{w-1}$ are not Slater determinants (e.g., the equal superposition $\frac{1}{\sqrt{2}}(|1100⟩+|0011⟩)$ lies strictly outside ${\mathrm{Gr}}_{ℝ}(2,4)$). A direct Pauli computation provides a complementary structural perspective on the restriction of Corollary 1. The commutator of two adjacent generators produces a non-adjacent generator dressed by a $Z$-string on the shared qubit. Every iterated commutator of adjacent generators has the form $L_{\mathit{𝑖𝑗}}{Z}_{i+1}\cdots Z_{j-1}$, and since commutators of two such operators again have this form, they close into a Lie subalgebra; this is precisely the free-fermion algebra of Corollary 1. This subalgebra cannot produce the non-contiguous dressed operators $L_{\mathit{𝑖𝑗}}{Z}_k$ (with $k\notin \{i+1,\dots ,j-1\}$) required by Proposition 1 to span the full $\mathit{𝔰𝔬}(w)$.

It is worth noting that the specific $A$-gate circuits proposed by Gard et al. [21], which use repeating blocks of strictly adjacent $A$-gates, appear to bypass this limitation but only if the input state is of the Néel type (i.e. $|0101\dots ⟩$). Discrete reflections $A_{\mathit{𝑖𝑗}}(0)$ in adjacent $A$-gates, when applied to the alternating occupation pattern of the Néel state, appear to break the free-fermion integrability that would otherwise constrain adjacent-only continuous rotations. While adjacent $A$-gates are not generically universal for arbitrary state-to-state mappings across the full algebra, they might be sufficient to construct complete state-preparation ansatze when anchored to the correct starting point, as suggested in [21]. A rigorous proof of this statement, however, remains elusive.

While Lemma 1 ensures that a minimal circuit can explore a local neighbourhood of states, it does not guarantee that every possible state in the subspace can be reached. Any smooth mapping from the circuit’s periodic parameters to the target sphere must encounter “critical points” where the circuit loses its ability to move in certain directions. This is due to a fundamental topological difference: the parameter space (a torus) has “holes” or cycles that the sphere lacks, making it impossible for any smooth map between them to have full-rank Jacobian everywhere. In practice, hardware-efficient circuits are even more restricted because each gate acts on several pairs of states simultaneously. This structure often partitions the reachable states into separate “islands” or patches, meaning that a single minimal circuit may be “trapped” within one region and unable to reach states beyond certain topological boundaries.

For variational algorithms such as VQE, which target low-lying excitations near a known reference state, local coverage of a single patch is typically sufficient. Achieving strict global reachability of arbitrary superpositions, however, generally requires over-parametrising the circuit, for instance by repeating the minimal gate sequence, to supply the extra degrees of freedom needed to cross these boundaries. The construction of [21] appears to achieve a surjective map from $T^w$ to $S^{w-1}$ precisely by introducing one additional parameter beyond the minimal count.

3 Application to bosonic simulation with binary encoding

The mathematical framework developed in section 2 for dynamically generating the full orthogonal group on a constrained subspace has a variety of extensions. One of them is the simulation of bosonic systems with a conserved particle number. Several considerations related to the quantum simulation of bosonic and phononic problems have been previously studied [24–30].

Recently, the binary encoded multi-level particles ansatz (BEMPA) [13] was proposed to simulate such systems using standard binary encoding for the bosonic modes. In this encoding, the available states of a bosonic mode are represented by a qubit register where the $k$-th qubit corresponds to a weight of $2^{k-1}$. To enforce global particle conservation, the total sum of these weights across all modes must be constant.

The BEMPA circuit relies on two fundamental building blocks, whose generators we denote ${Ĝ}_A$ and ${Ĝ}_B$ (using hats to distinguish them from the $G$-gates of section 2). The $Â$ gate acts on two qubits of the same significance $2^{k-1}$ (belonging to different bosonic modes) and performs an exact rotation in the subspace spanned by $|01⟩$ and $|10⟩$. Its generator is ${Ĝ}_A=i(|01⟩⟨10|-|10⟩⟨01|)$, which in Pauli form reads

$${Ĝ}_A=\frac{1}{2}(X\otimes Y-Y\otimes X).$$

(3.1)

The $\widehat{B}$ gate acts on three qubits, transferring amplitude between two qubits of significance $2^{k-1}$ and one qubit of significance $2^k$; its generator is ${Ĝ}_B=i(|001⟩⟨110|-|110⟩⟨001|)$. The corresponding Pauli string representation is given by

$${Ĝ}_B=\frac{1}{4}(X\otimes X\otimes Y-Y\otimes Y\otimes Y-X\otimes Y\otimes X-Y\otimes X\otimes X).$$

(3.2)

Crucially, just as the $G$ gate is applied directly to the qubit registers without intervening non-local Jordan-Wigner $Z$-strings, so too are the BEMPA $Â$ and $\widehat{B}$ gates.

4 Application to the fuzzy sphere regularisation of the 3D Ising model

We now turn our attention to the physical context that strongly motivated the preceding algebraic developments: the non-perturbative simulation of conformal field theories on quantum computers. We consider the non-commutative fuzzy sphere geometry as a regularisation of the 3D Ising critical point [14]. This approach is particularly advantageous because, unlike lattice regularisations, it preserves the rotation symmetry that is an essential ingredient of any CFT in three dimensions.

4.1 Lowest Landau level physics and fuzzy sphere

Consider electrons (charged particles) on a sphere with a magnetic monopole of charge $4\mathit{𝜋𝑠}$, $2s\in \mathbb{Z}$, at its centre. The purpose of the monopole is to create a radial magnetic field. If we neglect electron-electron interactions, then due to their interaction with the magnetic field the electrons will occupy Landau levels [31]. As it turns out, the interaction Hamiltonian is given in terms of the $\mathit{𝑆𝑈}(2)$ Casimir operator and, consequently, its eigenvalue problem is reduced to obtaining the irreducible representations of $\mathit{𝑆𝑈}(2)$. For electrons in the lowest Landau level (LLL) in the presence of the monopole, such a representation is furnished by the functions²

$$\begin{array}{cc}\begin{array}{rl}{\mathrm{\Phi }}_m(𝜃,\varphi )=& {(-1)}^m\sqrt{\frac{(2s+1)!}{4\pi (s-m)!(s+m)!}}{e}^{\mathit{𝑖𝑚𝜙}}{\cos <!--\; FUNCTION\; APPLICATION\; -->}^{s-m}\left(\frac{𝜃}{2}\right){\sin <!--\; FUNCTION\; APPLICATION\; -->}^{s+m}\left(\frac{𝜃}{2}\right),\\ & m=-s,-s+1,\dots ,s,\end{array}& \end{array}$$

(4.1)

where $𝜃,\varphi$ are coordinates on the sphere. These are obtained as the $l=s$ case of ${}_s{Y}_{\mathit{𝑙𝑚}}(𝜃,\varphi )$ spin-weighted or monopole spherical harmonics [32–34]. There are obviously $2s+1$ orbitals in the LLL, so $N=2s+1$ electrons will fill the LLL.

The Landau orbitals (4.1) are in one-to-one correspondence with states on the fuzzy sphere non-commutative geometry [35, 36]. Here the fuzzy sphere of radius $R$ is a fuzzy manifold whose coordinates ${\widehat{x}}_i,i=1,2,3$, satisfy $\mathit{𝑆𝑈}(2)$ commutation relations,

$$[{\widehat{x}}_i,{\widehat{x}}_j]=i\frac{R}{s}{𝜀}_{\mathit{𝑖𝑗𝑘}}{\widehat{x}}_k,$$

(4.2)

where ${𝜀}_{\mathit{𝑖𝑗𝑘}}$ is the fully antisymmetric Levi-Civita symbol with ${𝜀}_{123}=1$. Thus, fuzzy sphere geometry and LLL physics are equivalent.

To realise a phase transition in the same universality class as the 3D Ising model, which must involve the breaking of a ${\mathbb{Z}}_2$ global symmetry, let us assign a ${\mathbb{Z}}_2$ isospin quantum number to each electronic mode, which we will denote by an up or down arrow. This effectively doubles the number of electrons that can fit in the LLL, since now there are two distinct electron states that can occupy each Landau orbital. With the additional isospin quantum number, $N=2s+1$ electrons provide a half-filling of the LLL.

So far the electrons we have considered have been interacting only with the background magnetic field of the monopole. Let us now add electron-electron interactions and assume that all electrons remain in the LLL. To represent the interactions, let us consider second-quantisation creation and annihilation operators $c_{\mathit{𝑚\uparrow }}^{†},c_{\mathit{𝑚\downarrow }}^{†}$ and $c_{\mathit{𝑚\uparrow }},c_{\mathit{𝑚\downarrow }}$, respectively, for $m=-s,-s+1,\dots ,s$.³ For brevity of notation we may define the 2-vector ${𝐜}_m={(c_{\mathit{𝑚\uparrow }}{c}_{\mathit{𝑚\downarrow }})}^T$, in terms of which we construct the Hamiltonian

$$H=\sum _{m_1,\dots ,m_4=-s}^s{V}_{m_1,m_2,m_3,m_4}[({𝐜}_{m_1}^{†}{𝐜}_{m_4}^{})({𝐜}_{m_2}^{†}{𝐜}_{m_3}^{})-({𝐜}_{m_1}^{†}{\sigma }^z{𝐜}_{m_4}^{})({𝐜}_{m_2}^{†}{\sigma }^z{𝐜}_{m_3}^{})]-h\sum _{m=-s}^s{𝐜}_m^{†}{\sigma }^x{𝐜}_m^{},$$

(4.3)

where ${\sigma }^x$ and ${\sigma }^z$ are the first and third Pauli matrices, respectively, $h$ is a transverse magnetic field and the interaction potential is given by

$$V_{m_1,m_2,m_3,m_4}={\delta }_{m_1+m_2,m_3+m_4}\sum _{l=0}^{2s}{V}_l(4s-2l+1)\left(\begin{array}{ccc}\hfill s\hfill & \hfill s\hfill & \hfill 2s-l\hfill \\ \hfill m_1\hfill & \hfill m_2\hfill & \hfill -m_1-m_2\hfill \end{array}\right)\times \left(\begin{array}{ccc}\hfill s\hfill & \hfill s\hfill & \hfill 2s-l\hfill \\ \hfill m_3\hfill & \hfill m_4\hfill & \hfill -m_3-m_4\hfill \end{array}\right),$$

(4.4)

where $V_l$ are coefficients that depend on the specific type of electron-electron interactions one uses and $\left(\begin{array}{ccc}{j}_1& j_2& j_3\\ m_1& m_2& m_3\end{array}\right)$is the Wigner $3j$ symbol. One can easily check that $H$ preserves particle number and is invariant under the ${\mathbb{Z}}_2$ transformation

$$c_{\mathit{𝑚\uparrow }}\leftrightarrow c_{\mathit{𝑚\downarrow }}.$$

(4.5)

By virtue of being placed on a sphere, it also enjoys an $\mathit{𝑆𝑈}(2)$ rotation symmetry, with ${𝐜}_m,m=-s,-s+1,\dots ,s$, forming a spin-$s$ representation of $\mathit{𝑆𝑈}(2)$.

Following [14], we choose

$$V_0\ne 0,V_1=1,V_l=0\text{ for }l>1,$$

(4.6)

which describe ultra-local density-density interactions on the physical sphere. One can then see that by tuning the couplings $V_0$ and $h$ the system will move between different phases. For $h=0$ there will be two degenerate ground states, namely

$$|{\mathrm{\Psi }}_{\uparrow }⟩=\prod _{m=-s}^s{c}_{\mathit{𝑚\uparrow }}^{†}|0⟩,|{\mathrm{\Psi }}_{\downarrow }⟩=\prod _{m=-s}^s{c}_{\mathit{𝑚\downarrow }}^{†}|0⟩,$$

(4.7)

corresponding to a phase in which the ${\mathbb{Z}}_2$ symmetry is spontaneously broken. For $h\gg V_0$ the transverse magnetic field term in (4.3) will dominate, so that there will be a single ground state,

$$|{\mathrm{\Psi }}_x⟩=\prod _{m=-s}^s(c_{\mathit{𝑚\uparrow }}^{†}+c_{\mathit{𝑚\downarrow }}^{†})|0⟩,$$

(4.8)

corresponding to a phase with unbroken ${\mathbb{Z}}_2$ symmetry. One then expects that there will be a phase transition between these regimes lying in the Ising universality class. This is attained on a line in the $V_0$-$h$ plane. The point

$$V_0=4.75,h=6.32$$

(4.9)

lies on that critical line and yields surprisingly small finite-size effects [14].

4.2 State-operator correspondence

Let us briefly describe how conformal field theories allow the extraction of physical observables from the fuzzy sphere construction. The interested reader can find more details in standard references such as [37].

It is widely believed that the 3D Ising model at criticality is a CFT. CFTs are quantum field theories that are invariant under the conformal group, whose generators include translations $P_{\mu }$, rotations $M_{\mathit{𝜇𝜈}}$, dilatations $D$, and special conformal transformations $K_{\mu }$. The fundamental degrees of freedom of a CFT are captured by operators appearing in correlation functions. Any operator $𝒪$ has a well-defined scaling dimension ${\mathrm{\Delta }}_{𝒪}$, which is revealed by the action of $D$ on it and which governs its 2-point correlation function:

$$⟨𝒪(x)𝒪(0)⟩=\frac{1}{|x{|}^{2{\mathrm{\Delta }}_{𝒪}}}.$$

(4.10)

Computing ${\mathrm{\Delta }}_{𝒪}$ for the operators of a given CFT is a highly non-trivial problem, especially in strongly-coupled cases like the 3D Ising model.

One crucial feature of conformal transformations is that they can map flat space ${ℝ}^3$ to the “cylinder” $S^2\times ℝ$. Under this mapping, the generator of dilatations $D$ on ${ℝ}^3$ becomes the generator of time translations on $S^2\times ℝ$, i.e. the Hamiltonian on $S^2$. Operators on ${ℝ}^3$ give rise to states on $S^2$ and vice versa; this is the state-operator correspondence. The scaling dimensions of the operators are therefore identical to the eigenvalues of the $S^2$ Hamiltonian.

The fuzzy sphere construction is based on the idea that $H$ of (4.3) with the parameter choices (4.6) and (4.9) is precisely such a Hamiltonian. Extracting its energy spectrum therefore gives direct access to the low-lying scaling dimensions of the 3D Ising CFT. Moreover, the identification of eigenstates enables the determination of 3-point function coefficients, which constitute additional CFT observables. As explained in [38, 39], this is achieved by computing ratios of appropriate amplitudes and becomes precise in the limit of infinite system size. We carry this out explicitly for $f_{\mathit{𝜎𝜎𝜖}}$ in section 4.6, using the states prepared variationally in section 4.5.

4.3 Classical extraction via exact diagonalisation

The second-quantisation Hamiltonian (4.3) can be conveniently constructed using OpenFermion [40], which provides methods for performing computations with fermionic creation and annihilation operators directly. Since $H$ preserves particle number and total azimuthal angular momentum $S_z$, it is block-diagonal in these quantum numbers. Working at half-filling and restricting to the $S_z=0$ sector reduces the Hilbert space dimension considerably.⁴ This sector is sufficient because every spin-$\ell$ multiplet in the spectrum contains an ${\ell }_z=0$ component that appears in this sector.

To find the spectrum we can proceed with exact diagonalisation (ED) for this small system size. To extract operator dimensions, the ground-state energy is subtracted from all energies, so that the ground state corresponds to the identity operator with scaling dimension zero. The shifted spectrum is then rescaled by an overall factor, fixed by demanding that the leading ${\mathbb{Z}}_2$-even state with spin two has energy exactly $3$, as required for the energy-momentum tensor, $T_{\mathit{𝜇𝜈}}$, of the 3D Ising CFT. Table 1 shows the resulting low-lying spectrum for $N=4$ electrons, compared with conformal bootstrap results [42].

The primary limitation of this classical approach is the rapid growth of the Hilbert space dimension with system size. While the $N=4$ case is easily tractable, the dimension grows quickly: for example, the $N=40$ case requires a Hilbert space of dimension 2,944,055,592 (for the $S_z=0$ sector). On a quantum computer, by contrast, we would need only 80 qubits (two per fermionic mode). This provides strong motivation for exploring quantum simulation of this problem, as it holds the promise of reaching system sizes far beyond the reach of any classical method. Cases requiring even more fermionic modes, such as the Heisenberg model recently studied in [43–45] and others [46–58], would also benefit greatly from a realisation on a quantum computer.

4.4 Gates for preserving spin symmetries

For the most efficient implementation of the fuzzy sphere model, states must not only be half-filled but also maintain a total azimuthal angular momentum $S_z=0$. This requires unitary operations to commute with the $S_z$ operator. Symmetry constraints such as these are commonly encountered in a variety of problems.

The total azimuthal angular momentum on the fuzzy sphere is the sum of the magnetic quantum numbers $m$ of all occupied orbitals:

$$S_z=\sum _k{m}_k\frac{I-Z_k}{2}.$$

(4.11)

For any parametrised continuous rotation to unconditionally preserve the $S_z=0$ constraint, its Lie algebra generator must commute with the $S_z$ operator.

If we consider the 2-qubit $G$-gate, we have

$$[L_{\mathit{𝑖𝑗}},S_z]=i(m_i-m_j)(X_i{X}_j+Y_i{Y}_j).$$

(4.12)

This commutator vanishes if and only if $m_i=m_j$. For our encoding, this restricts $G^{(2)}$ gates to swapping the two isospins ($\uparrow$ and $\downarrow$) of the same $m$ orbital. Such operations cannot change overall orbital occupancies and are thus individually insufficient to explore the full $S_z=0$ subspace.

To achieve transitions that change orbital occupancies while simultaneously preserving $S_z$, we must introduce 4-qubit interaction gates, denoted $G^{(4)}$, which act on an ordered quartet of qubits $(i,j,k,l)$ and generate the transition $|1100⟩\leftrightarrow |0011⟩$. This operation preserves $S_z$ provided $m_i+m_j=m_k+m_l$. The continuous parametrised gate is given by

$$G_{\mathit{𝑖𝑗𝑘𝑙}}^{(4)}(𝜃)=\exp <!--\; FUNCTION\; APPLICATION\; -->\left[i\frac{𝜃}{8}{L}_{\mathit{𝑖𝑗𝑘𝑙}}\right],$$

(4.13)

with the explicit generator

$$\begin{array}{cc}\begin{array}{rl}{L}_{\mathit{𝑖𝑗𝑘𝑙}}& =X_i{X}_j{X}_k{Y}_l+X_i{X}_j{Y}_k{X}_l-X_i{Y}_j{X}_k{X}_l-Y_i{X}_j{X}_k{X}_l\\ & +X_i{Y}_j{Y}_k{Y}_l+Y_i{X}_j{Y}_k{Y}_l-Y_i{Y}_j{X}_k{Y}_l-Y_i{Y}_j{Y}_k{X}_l.\end{array}& \end{array}$$

(4.14)

Analogous to the 2-qubit gate, the improper reflection form of the rotation is

$$A_{\mathit{𝑖𝑗𝑘𝑙}}^{(4)}(𝜃)=A_{\mathit{𝑖𝑗𝑘𝑙}}^{(4)}(0)G_{\mathit{𝑖𝑗𝑘𝑙}}^{(4)}(-𝜃),$$

(4.15)

where the fixed reflection component

$$A_{\mathit{𝑖𝑗𝑘𝑙}}^{(4)}(0)=I-2\left(\frac{I-Z_i}{2}\right)\left(\frac{I-Z_j}{2}\right)\left(\frac{I+Z_k}{2}\right)\left(\frac{I+Z_l}{2}\right)$$

(4.16)

operates as the identity on all states except $|1100⟩$, to which it applies a $-1$ phase. For clarity of presentation, we will use $G^{(2)}$ for (2.3) and $A^{(2)}$ for (2.8) from now on.

4.5 Variational quantum simulation

The Python code used in this subsection and below can be found in a dedicated GitHub repository, andstergiou/fuzzy-ising-vqe, which is archived on Zenodo [59].

For the $N=4$ case we have $s=3∕2$, with the magnetic quantum number taking the values $m=-\frac{3}{2},-\frac{1}{2},\frac{1}{2},\frac{3}{2}$. Each value of $m$ accommodates two isospin states, giving $N_{\text{q}}=8$ qubits in total. Restricting to the half-filled $S_z=0$ subspace leaves $w_0=18$ allowed basis states.

Using Proposition 3 and Lemma 1, we find that a circuit with exactly $w_0-1=17$ independently parametrised gates has a full-rank Jacobian at a reference point, establishing that the Jacobian has rank $w_0-1$ at almost every parameter value. This local criterion does not, however, certify that the circuit map $f:T^{w_0-1}\to S^{w_0-1}$ is globally surjective. Numerical optimisation over the circuit parameters (minimising $\parallel f(𝜃)-𝐲{\parallel }^2$ for random targets $𝐲\in S^{w_0-1}$) reveals that minimal 17-parameter circuits generically miss a small fraction of the sphere: the best such circuit reached 47 out of 50 random targets, with unreachable targets plateauing at residual costs $\sim 10^{-4}$, clearly separated from the $\sim 10^{-15}$ costs of reachable targets. Adding two extra parameters closes this gap: the 19-parameter $G$-gate circuit shown in figure 1 achieves $100∕100$ random-target reachability across multiple independent seeds, with maximum cost $\sim 10^{-14}$, providing strong numerical evidence for global surjectivity. Even though it is not necessarily minimal or unique, we will use the circuit of figure 1 for our VQE tests. Its resource cost is given in appendix B.

Key: spanning 2-wire boxes $=G^{(2)}$; single-wire boxes joined by vertical links $=G^{(4)}$

Since our system involves fermionic creation and annihilation operators, we use OpenFermion [40] to simulate it. After a Jordan-Wigner transformation, the resulting Hamiltonian is imported into PennyLane [60], and the circuit in figure 1 is constructed using PennyLane’s SingleExcitation and DoubleExcitation rotations. Our setup exploits all symmetries of the problem. Besides the optimised circuit depth obtained, the benefits of using symmetries for convergence of VQE can also be considerable [61].

For our variational quantum simulation of the fuzzy sphere Hamiltonian for $N=4$ electrons we use a classical state vector simulator (lightning.qubit backend of PennyLane). For the ground state, the cost function is the energy expectation value,

$$C_0(𝜃)=⟨\psi (𝜃)|H|\psi (𝜃)⟩.$$

(4.19)

Optimisation proceeds in two phases: a coarse phase using the Adam adaptive gradient method [62] (step size $0.1$, up to $500$ iterations) to rapidly navigate the energy landscape and escape local minima, followed by a fine phase using gradient descent (step size $0.02$) for precise convergence. Starting from a random initialisation, the VQE converges to the ground-state energy to within $1.2\times 10^{-7}$ in absolute value.

For the excited states we use the variational quantum deflation (VQD) method [15], augmenting the cost function with penalty terms that penalise overlaps with the previously found eigenstates:

$$C_k(𝜃)=⟨\psi (𝜃)|H|\psi (𝜃)⟩+\sum _{j<k}{\beta }_j|⟨\psi (𝜃)|{\psi }_j⟩{|}^2,$$

(4.20)

where $|{\psi }_j⟩$ is the $j$-th eigenstate already prepared and ${\beta }_j>0$ is a penalty strength chosen to lift the corresponding eigenvalue above the target. We use ${\beta }_0=10$ for the first excited state and ${\beta }_0=30$, ${\beta }_1=20$ for the second excited state; each VQD step employs the same two-phase Adam ($1000$ iterations, step size $0.1$) then gradient-descent (step size $0.01$) strategy.

The convergence curves are displayed in figure 2, and the final energies are compared with ED in table 2. All three eigenvalues are recovered to within $4\times 10^{-7}$ in absolute energy. Moreover, the overlap matrix $|⟨{\psi }_k({𝜃}^{\ast })|{\varphi }_j⟩{|}^2$, where $|{\varphi }_j⟩$ are the exact eigenvectors, equals the identity matrix to ten significant figures, confirming that the circuit faithfully prepares each target eigenstate.

4.6 Extracting an OPE coefficient

The energy spectrum is not the only CFT observable accessible from the fuzzy sphere. As anticipated in section 4.2, 3-point function coefficients follow from ratios of matrix elements of a local operator [38, 39]. It is worth verifying that the variational pipeline reaches these too, since they probe the prepared eigenvectors rather than merely the eigenvalues, and are therefore a considerably more stringent test of the ansatz.

The construction is particularly simple at the north pole of the sphere. From (4.1) the Landau orbital ${\mathrm{\Phi }}_m(𝜃,\varphi )$ carries a factor ${\sin <!--\; FUNCTION\; APPLICATION\; -->}^{s+m}(𝜃∕2)$, which vanishes at $𝜃=0$ unless $m=-s$. Any local bilinear evaluated at the north pole therefore collapses onto the single orbital $m=-s$. Choosing the ${\mathbb{Z}}_2$-odd channel gives

$${\mathrm{\Lambda }}_{\sigma }\propto {𝐜}_{-s}^{†}{\sigma }^z{𝐜}_{-s}^{}=n_{-\mathit{𝑠\uparrow }}-n_{-\mathit{𝑠\downarrow }},$$

(4.21)

which commutes with both the particle number and $S_z$, and which is diagonal in the computational basis. Thus, it can be read off from a single $Z$-basis measurement of the two qubits encoding the $m=-s$ orbital.

Expanding the lattice operator in CFT operators, ${\mathrm{\Lambda }}_{\sigma }=c_{\sigma }\sigma +\dots$, the state-operator correspondence gives $⟨𝟙|{\mathrm{\Lambda }}_{\sigma }|\sigma ⟩=c_{\sigma }$ from the 2-point normalisation and $⟨𝜖|{\mathrm{\Lambda }}_{\sigma }|\sigma ⟩=c_{\sigma }{f}_{\mathit{𝜎𝜎𝜖}}$ from the 3-point function. The unknown overlap $c_{\sigma }$ cancels in the ratio,

$$f_{\mathit{𝜎𝜎𝜖}}=\frac{⟨𝜖|{\mathrm{\Lambda }}_{\sigma }|\sigma ⟩}{⟨𝟙|{\mathrm{\Lambda }}_{\sigma }|\sigma ⟩},$$

(4.22)

and, conveniently, the three states appearing here are exactly the ground state and the two excited states already prepared in table 2: $|𝟙⟩$ is the ground state, $|\sigma ⟩$ the first excited state and $|𝜖⟩$ the second, as identified by their spin and ${\mathbb{Z}}_2$ quantum numbers in table 1. No further variational optimisation is needed.

Evaluating (4.22) on the VQE/VQD states of table 2 gives

$$f_{\mathit{𝜎𝜎𝜖}}{|}_{\text{VQE/VQD}}=0.96141,f_{\mathit{𝜎𝜎𝜖}}{|}_{\text{ED}}=0.96144,$$

(4.23)

the two agreeing to four significant figures. The most accurate determination of this quantity is given by the conformal bootstrap value $f_{\mathit{𝜎𝜎𝜖}}=1.0518537(41)$ [63]. The residual $8.6\%$ discrepancy is a finite-size effect of the $N=4$ truncation, not a failure of the variational preparation: the ED value computed in the same $18$-dimensional subspace carries the identical deviation. It is larger than the corresponding errors on the low-lying scaling dimensions in table 1, which is expected, since OPE coefficients are known to converge more slowly with system size than the spectrum [38]. The point of the exercise is that our symmetry-constrained variational method reproduces the exact subspace answer for an observable built from the eigenvectors, and does so using only measurements the circuit already supports.

4.7 Scaling with system size

The results above are for $N=4$ alone, and a single system size cannot distinguish polynomial from exponential growth of the gate count. Since the near-term relevance of the construction depends on how that count grows, we now examine $N=2,4,6$, for which the $S_z=0$ half-filled subspace has dimensions $w_0=4,18,120$ on $4,8,12$ qubits respectively.

We first verify that the gate set of section 4 remains adequate as $N$ grows. The configuration graph on the $S_z=0$ subspace, with edges given by the allowed $G^{(2)}$ and $G^{(4)}$ transitions, is connected for every case we examined, including $N=8$ (where $w_0=1064$), confirming the connectivity step in the proof of Proposition 3. We note in passing that connectivity of this graph is necessary but not sufficient: for $N=2$ the two $G^{(2)}$ gates alone already connect all four basis states, yet they generate only states that factorise across the two orbitals and so cannot reach the entangled ground state. This is an instance of the same dimensional gap identified in Corollary 1, here between the $2$-dimensional set of reachable factorised states and the $3$-sphere on which the sector lives.

To measure the required circuit size we use a layered ansatz of fixed structure. One layer consists of all $N$ $G^{(2)}$ gates together with a set of $G^{(4)}$ gates added until the configuration graph is connected; the layer is then repeated, and truncated at the required length, to give a circuit with $p$ parameters. For each $p$ we minimise the infidelity $1-|⟨{\psi }_{\text{gs}}|\psi (𝜃)⟩{|}^2$ with the exact ground state from many random starts, and record the smallest $p$ at which a chosen infidelity is reached. Optimising the infidelity directly, rather than the energy, isolates the expressivity of the circuit from the trainability of the VQE landscape; we found that greedy operator-selection schemes stall in local minima well before exhausting the expressivity of the ansatz, so the fixed-structure multi-start protocol gives a more faithful measure. Table 3 collects the results.

As we see from these results, for essentially exact preparation the required parameter count tracks the dimension of the subspace: $p(10^{-8})=3,19,141$ against $w_0-1=3,17,119$. Since $w_0$ grows exponentially with $N$, so does the cost of exact ground state preparation with this gate set. Our completeness theorems guarantee that the target is reachable at finite depth, but they carry no promise that the depth is small, and the data show that it is not. The picture is more favourable if only moderate accuracy is demanded: $p(10^{-2})$ grows from $12$ to $42$ between $N=4$ and $N=6$, a factor of $3.5$, while $w_0$ grows by a factor of $6.7$. Approximate preparation thus scales appreciably better than the subspace dimension, though still steeply, and it is in this regime that variational preparation on near-term devices is plausible.

The over-parametrisation noted in section 4.5 is a separate question, since it concerns reaching arbitrary states in the sector rather than one particular ground state, and it is the one on which the topological obstruction of section 2.2 bears. Repeating the random-target test used there, the smallest layered circuit that reaches every member of a set of random targets to infidelity below $10^{-8}$ has

$$p=4,19,124\text{for }N=2,4,6,$$

(4.24)

against the minimal counts $w_0-1=3,17,119$. At $N=4$ this independently reproduces the two extra parameters found in section 4.5. The over-parametrisation therefore persists and grows slowly in absolute terms, by $1,2,5$ parameters, but shrinks steadily relative to the minimal count, from $33\%$ to $12\%$ to $4\%$. The topological obstruction does not disappear as $N$ increases, but neither does it become a leading contribution to the circuit size; the dominant cost is the minimal count $w_0-1$ itself.

It is worth recording that the physics improves with $N$ even as the circuits become more expensive. Identifying the energy-momentum tensor and normalising on it as in section 4.3, the $N=6$ system gives ${\mathrm{\Delta }}_{\sigma }=0.52180$ and ${\mathrm{\Delta }}_{𝜖}=1.38471$, against $0.53014$ and $1.38210$ at $N=4$ and the bootstrap values $0.518$ and $1.413$. The error on ${\mathrm{\Delta }}_{\sigma }$ falls from $2.3\%$ to $0.73\%$.

4.8 Simulation under a noise model

All results so far are for noiseless state vector simulation, which leaves open the question of what gate fidelity would actually be required to extract the spectrum on hardware. To answer this question, we repeat the $N=4$ calculation with the state represented as a density matrix rather than a state vector, as the noise channels map the pure ansatz state to a mixed one. The circuit of figure 1 is decomposed into its $182$ CNOT gates and $196$ single-qubit rotations, and we apply a two-qubit depolarising channel of rate $\gamma$ after every CNOT together with a single-qubit depolarising channel of rate $\gamma ∕10$ after every single-qubit gate. The VQE and VQD optimisations are re-run in full at each value of $\gamma$, rather than reusing the noiseless parameters. Scaling dimensions are obtained with the $T_{\mathit{𝜇𝜈}}$ normalisation held fixed at its exact value, which isolates the effect of the noise on the measured gaps.

Because the ansatz preserves both particle number and $S_z$ exactly, any population that the noise moves out of the half-filled $S_z=0$ sector is detectable in a computational-basis measurement and can be discarded. We therefore also report the symmetry-post-selected estimator, in which the density matrix is projected back onto the physical sector and renormalised before the energies are read off. This costs nothing beyond the measurements already performed, and is available precisely because the circuit is symmetry-preserving by construction.

The results are shown in figure 3. The noise-induced error grows essentially linearly in $\gamma$ throughout the regime in which the calculation remains meaningful, as expected when the dominant effect is the accumulation of independent errors over a fixed number of gates. Reading off the crossing with the finite-size error gives

$${\gamma }^{\ast }\approx 2.0\times 10^{-4}\text{(raw)},{\gamma }^{\ast }\approx 6.5\times 10^{-4}\text{(symmetry post-selected)},$$

(4.25)

so that below these rates the $N=4$ truncation, rather than the hardware, limits the accuracy of the extracted spectrum. We should be clear that this is a demanding target: it is roughly an order of magnitude beyond typical current two-qubit gate error rates, and it is a direct consequence of the $182$ CNOT gates in the circuit. Symmetry post-selection improves things by a factor of about three in the tolerable error rate, and by a factor of between three and four in the error at fixed $\gamma$ throughout the regime where it is useful, which is a concrete practical benefit of insisting that the ansatz preserve the symmetry exactly rather than approximately.

The method degrades rapidly above $\gamma =10^{-3}$, where the weight remaining in the physical sector has fallen to $0.85$ and the raw ${\mathrm{\Delta }}_{\sigma }$ is underestimated by $12\%$, although post-selection still recovers it to within $3.7\%$. Beyond the range plotted the breakdown is complete: by $\gamma =3\times 10^{-3}$ the weight is $0.63$, the extracted spectrum has lost all resemblance to the CFT, and post-selection no longer helps at all, since the surviving population in the physical sector is itself close to uniform. We note that re-optimising the parameters at each $\gamma$, rather than reusing the noiseless ones, does not improve the extracted dimensions: to leading order the noise acts as a global depolarising channel, which rescales all energies by a common factor and adds a $𝜃$-independent constant, leaving the optimum where it was and contracting all gaps uniformly.

5 Extension to $\mathit{𝑆𝑈}(w)$ with independent complex phases

The results of section 2 generate the real orthogonal group $\mathit{𝑆𝑂}(w)$, the natural setting for states with purely real amplitudes. Many physical applications, e.g. systems lacking time-reversal symmetry and complex ground-state preparation, require the full special unitary group $\mathit{𝑆𝑈}(w)$. We now show that the Pauli $Z$ dressing mechanism extends directly to $\mathit{𝑆𝑈}(w)$ once the gates carry independent complex phases.

Replacing the real Givens rotation $G(𝜃)$ with a parametrised $\mathit{𝑆𝑈}(2)$ rotation on the active $\{|01⟩,|10⟩\}$subspace introduces an additional phase $\varphi$:

$$G(𝜃,\varphi )=\left(\begin{array}{cccc}\hfill 1\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill \cos <!--\; FUNCTION\; APPLICATION\; -->𝜃\hfill & \hfill -e^{\mathit{𝑖𝜙}}\sin <!--\; FUNCTION\; APPLICATION\; -->𝜃\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill e^{-\mathit{𝑖𝜙}}\sin <!--\; FUNCTION\; APPLICATION\; -->𝜃\hfill & \hfill \cos <!--\; FUNCTION\; APPLICATION\; -->𝜃\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 1\hfill \end{array}\right).$$

(5.1)

In exponential form,

$$G_{\mathit{𝑖𝑗}}(𝜃,\varphi )=\exp <!--\; FUNCTION\; APPLICATION\; -->\left[i\frac{𝜃}{2}(\cos <!--\; FUNCTION\; APPLICATION\; -->\varphi L_{\mathit{𝑖𝑗}}^a-\sin <!--\; FUNCTION\; APPLICATION\; -->\varphi L_{\mathit{𝑖𝑗}}^s)\right],$$

(5.2)

where $L_{\mathit{𝑖𝑗}}^a\equiv L_{\mathit{𝑖𝑗}}=X_i{Y}_j-Y_i{X}_j$ is the real (antisymmetric) hopping generator defined earlier, and $L_{\mathit{𝑖𝑗}}^s=X_i{X}_j+Y_i{Y}_j$ is the corresponding imaginary (symmetric) hopping generator. When $\varphi$ varies independently across different gates in the circuit, the continuous generators span both hopping directions.

The proof of Proposition 1 rests on the overlapping commutator (2.13), whose $Z$ spectator on the shared qubit enables the isolation of single-plane generators via dressing. We now show that the same spectator structure emerges for every combination of $L^s$ and $L^a$. Using $[X,Y]=2\mathit{𝑖𝑍}$ and the tensor product identity $[A_i{B}_j,C_j{D}_k]=A_i[B_j,C_j]D_k$, we compute

The first line is the commutator used in the proof of Proposition 1; the remaining two extend it. Every commutator produces a Pauli $Z$ on the shared qubit, regardless of whether the constituent generators are real or imaginary. Commuting two generators of the same type ($\mathit{𝑎𝑎}$ or $\mathit{𝑠𝑠}$) yields a spectator-dressed real generator $L^{a}Z$, while commuting generators of different types ($\mathit{𝑠𝑎}$ or $\mathit{𝑎𝑠}$) yields a spectator-dressed imaginary generator $L^{s}Z$.

Because the spectator $Z$ emerges uniformly in all cases, the Lie algebraic generation carries through unchanged with $L^s$ in place of $L^a$. Concretely, isolating single-plane rotations as in the $\mathit{𝔰𝔬}(w)$ case applied to $L^a$ yields the single-plane antisymmetric generators $E_{\mathit{𝑥𝑦}}^a=|x⟩⟨y|-|y⟩⟨x|$; the same operations applied to $L^s$ isolate single-plane symmetric generators $E_{\mathit{𝑥𝑦}}^s=i(|x⟩⟨y|+|y⟩⟨x|)$. The algebra $\mathit{𝔰𝔲}(w)$ requires three types of generators for each pair $(x,y)$: the antisymmetric $E_{\mathit{𝑥𝑦}}^a$, the symmetric $E_{\mathit{𝑥𝑦}}^s$, and a diagonal generator $E_{\mathit{𝑥𝑦}}^d=i(|x⟩⟨x|-|y⟩⟨y|)$. The first two are thus provided structurally by $L^a$ and $L^s$ respectively, and the third follows from their commutator:

$$[E_{\mathit{𝑥𝑦}}^a,E_{\mathit{𝑥𝑦}}^s]=2E_{\mathit{𝑥𝑦}}^d.$$

(5.4)

Therefore, the diagonal generators require no additional circuit elements. The set $\{E_{\mathit{𝑥𝑦}}^a,E_{\mathit{𝑥𝑦}}^s,E_{\mathit{𝑥𝑦}}^d\}$ over all pairs $(x,y)$ therefore spans the full $\mathit{𝔰𝔲}(w)$ Lie algebra on the target subspace.

This result connects to the criteria of Yan et al. [22], whose Theorem 1 proves that hardware-efficient generators (without non-local Jordan-Wigner $Z$-strings) can span $\mathit{𝔰𝔲}(w)$ on the constant Hamming-weight subspace only if they break free-fermion integrability via a two-body diagonal $Z_i{Z}_j$ component. At the level of a single link, the same-site commutator

$$[L_{\mathit{𝑖𝑗}}^a,L_{\mathit{𝑖𝑗}}^s]=4i(Z_i-Z_j)$$

(5.5)

is single-body and remains within the free-fermion algebra. This means that the complex phase alone does not break integrability. In our framework, it is the Pauli $Z$ dressing that does so: dressing a hopping term with $Z$-string projectors produces effective generators containing multi-body diagonal components $Z_i{Z}_j$ and higher, exactly the terms demanded by the Yan et al. condition. The independent phase $\varphi$ then supplies the second hopping direction within this already non-integrable structure.

The same logic applies to further symmetry-constrained sectors. For the $S_z=0$ subspace of section 4, recall that the 4-qubit gates $G_{\mathit{𝑖𝑗𝑘𝑙}}^{(4)}(𝜃)$ have a real generator $L_{\mathit{𝑖𝑗𝑘𝑙}}^a$ driving $|1100⟩⟨0011|-|0011⟩⟨1100|$. An independent phase $\varphi$ now adds the imaginary generator $L_{\mathit{𝑖𝑗𝑘𝑙}}^s$. The overlapping commutators of these 4-qubit generators produce spectator $Z$-strings by exactly the same mechanism as the 2-qubit case, so the dressing procedure of Proposition 3 isolates both real and imaginary single-plane generators across the connected graph of the $S_z=0$ subspace, completing the $\mathit{𝔰𝔲}(w_0)$ algebra. As a consistency check with the Yan et al. criterion, the same-site commutator $[L_{\mathit{𝑖𝑗𝑘𝑙}}^s,L_{\mathit{𝑖𝑗𝑘𝑙}}^a]$ is proportional to $|1100⟩⟨1100|-|0011⟩⟨0011|$, which expands into single-body ($Z_i$) and three-body ($Z_i{Z}_j{Z}_k$) Pauli terms, the latter manifestly breaking free-fermion integrability.

6 Conclusion

We have established a Lie algebraic framework for proving that hardware-efficient quantum gates, applied without non-local Jordan-Wigner strings, generate the full orthogonal group on constrained subspaces of fixed particle number and symmetry. Central to this is the mechanism of Pauli $Z$ dressing: the emergence of Pauli $Z$ operators through commutators of overlapping generators, which act as spectator projectors and allow multi-plane rotations to be decomposed into elementary single-plane generators $E_{\mathit{𝑥𝑦}}$. We formalised this in three propositions: Proposition 1 for the constant Hamming weight subspace, Proposition 2 for the BEMPA ansatz in bosonic systems with conserved particle number, and Proposition 3 for further symmetry-constrained sectors such as $S_z=0$. Because these propositions rely on continuously parametrised generators spanning the relevant Lie algebras, they guarantee exact reachability across the target connected Lie group. This distinguishes our completeness results from traditional quantum gate universality proofs, which rely on discrete gate sets and only guarantee asymptotic density. The local Lemma 1 provides a computationally efficient sufficient criterion for establishing that a given minimal circuit can explore any direction on the target manifold from a generic parameter configuration, though it does not settle global reachability of every point in the manifold.

We applied this framework to two physical settings. For the BEMPA ansatz, Proposition 2 establishes that the $Â$ and $\widehat{B}$ gates generate the full algebra for bosonic systems with conserved particle number. For the fuzzy sphere regularisation of the 3D Ising CFT, we constructed explicit circuits with 19 parameters that span the full relevant subspace for $N=4$ electrons, with numerical evidence for global surjectivity, and demonstrated agreement of the extracted spectrum between ED and a VQE simulation. Beyond the spectrum, the same variationally prepared states yield the OPE coefficient $f_{\mathit{𝜎𝜎𝜖}}$ from a ratio of amplitudes requiring no additional circuitry. Extending the study to $N=2$ and $N=6$ shows that the parameter count needed for exact ground state preparation tracks the dimension $w_0$ of the constrained subspace, and therefore grows exponentially with $N$, so that our completeness results should not be read as a promise of shallow circuits; approximate preparation is appreciably cheaper. Under a depolarising noise model the extracted scaling dimensions are limited by the hardware rather than by finite size once the two-qubit error rate exceeds ${\gamma }^{\ast }\approx 2\times 10^{-4}$, which symmetry post-selection relaxes by roughly a factor of three.

We further showed in section 5 that the framework extends from $\mathit{𝑆𝑂}(w)$ to the full special unitary group $\mathit{𝑆𝑈}(w)$ once each gate carries an independent complex phase. The key observation is that the overlapping commutators producing spectator $Z$ operators, the mechanism underlying all three propositions, emerge identically for the imaginary hopping generators $L_{\mathit{𝑖𝑗}}^s=X_i{X}_j+Y_i{Y}_j$ as for the real generators $L_{\mathit{𝑖𝑗}}^a=X_i{Y}_j-Y_i{X}_j$. The Pauli $Z$ dressing therefore isolates both real and imaginary single-plane generators for every pair of basis states, spanning the full $\mathit{𝔰𝔲}(w)$. This is consistent with the criteria of Yan et al. [22]: the integrability-breaking diagonal components their theorem demands arise from the Pauli $Z$ dressing itself, while the independent phases supply the additional hopping directions.

We suspect that Pauli $Z$ dressing is, in fact, a universal structural feature of hardware-efficient ansatze rather than a property specific to the systems studied here. The mechanism relies on a single algebraic ingredient: commutators of overlapping multi-qubit generators produce $Z$ operators on shared qubits, which then serve as spectator projectors. This ingredient is available whenever the gate set contains generators with overlapping support on at least one qubit, a condition satisfied by essentially any non-trivial hardware-efficient circuit topology. It would be interesting to determine whether there exist natural constrained subspaces for which the $Z$ dressing mechanism fails to generate the full algebra, or whether completeness is instead guaranteed for any connected gate graph acting on a connected configuration space. Such a classification could provide a unifying criterion for the expressibility of hardware-efficient ansatze across many fields of study. For example, establishing exactly what set of independent complex phases enables $\mathit{𝔰𝔲}(w)$ completeness across different symmetry-constrained physical models, and classifying their minimal gate sets and circuit topologies, remains a valuable target for future work.

More broadly, the systematic construction of minimal circuits, i.e. those with the minimum number of independent parameters, that are locally surjective onto a given target manifold remains an open problem. While the Jacobian criterion of Lemma 1 can verify specific ansatze locally, it offers no guidance on how to find one. A constructive procedure that, given a target subspace and a set of available gates, produces a minimal-depth spanning circuit, or establishes that no such circuit of a given topology exists, would significantly accelerate the deployment of variational algorithms on near-term hardware.

On the applications side, the fuzzy sphere construction is particularly promising for larger system sizes, where classical methods become impractical but the qubit count remains modest. At these sizes, noise-aware circuit optimisation and error mitigation strategies become essential. The noise analysis of section 4.8 suggests that the symmetry structure is itself a resource in this respect, since exact conservation makes leakage out of the physical sector detectable at no additional measurement cost; the algebraic framework developed here could inform the design of circuits that are simultaneously minimal in depth and robust to hardware errors.

Acknowledgments

AS is grateful to Jake Belton and Nadav Drukker for useful discussions and collaboration in the initial stages of this project, and to Petr Kravchuk for insightful conversations. Claude Code and Gemini were both used in this work for coding tests and claim verification. They also aided in the writing of this paper. AS is supported by the Royal Society under grant URF\R1\211417 and by STFC under grant ST/X000753/1. NS acknowledges use of resources of the National Energy Research Scientific Computing Center, a DOE Office of Science User Facility supported by the Office of Science of the U.S. Department of Energy under Contract No. DE-AC02-05CH11231 using NERSC award NERSC DDR-ERCAP0038154.

Appendix

A Commutator identities

The proofs of Propositions 1, 2 and 3 all rest on commutators of overlapping generators producing Pauli $Z$ operators on the shared qubits. In this appendix we verify these identities explicitly. They are all instances of a single elementary statement about Pauli strings, which also explains why a $Z$ is produced in every case.

The $Z$ operators are therefore not an accident of any particular gate set: two hardware-efficient generators built from $X$ and $Y$ only, overlapping on an odd number of qubits with mismatched Paulis there, necessarily produce a $Z$ on precisely those qubits. Equivalently, writing ${\sigma }^{\pm }=\frac{1}{2}(X\pm \mathit{𝑖𝑌})$, so that $L_{\mathit{𝑖𝑗}}=2i{(|01⟩⟨10|-|10⟩⟨01|)}_{\mathit{𝑖𝑗}}$, the shared qubit is acted on by ${\sigma }^{+}{\sigma }^{-}=\frac{1}{2}(I+Z)$ or ${\sigma }^{-}{\sigma }^{+}=\frac{1}{2}(I-Z)$ depending on the order of the product, and the commutator retains exactly the $Z$ part.

The 2-qubit identity (2.13). With $L_{\mathit{𝑖𝑘}}=X_i{Y}_k-Y_i{X}_k$ and $L_{\mathit{𝑘𝑗}}=X_k{Y}_j-Y_k{X}_j$, the overlap is $S=\{k\}$. Of the four pairs of terms, the two with matching Pauli on $k$ drop out by Lemma 2. The surviving pairs are $(X_i{Y}_k,X_k{Y}_j)$, with $(p_k,q_k)=(Y,X)$ and hence ${\eta }_k=-1$, and $(-Y_i{X}_k,-Y_k{X}_j)$, with $(p_k,q_k)=(X,Y)$ and hence ${\eta }_k=+1$. They contribute $-2i{X}_i{Y}_j{Z}_k$ and $(-1)(-1)(+1)2i{Y}_i{X}_j{Z}_k$ respectively, so that

$$[L_{\mathit{𝑖𝑘}},L_{\mathit{𝑘𝑗}}]=-2i{X}_i{Y}_j{Z}_k+2i{Y}_i{X}_j{Z}_k=-2i(X_i{Y}_j-Y_i{X}_j)Z_k=-2i{L}_{\mathit{𝑖𝑗}}{Z}_k.$$

(A.3)

The BEMPA identity (3.3). Here

$${Ĝ}_{A;\mathit{𝑖𝑘}}=\frac{1}{2}(X_i{Y}_k-Y_i{X}_k),{Ĝ}_{B;\mathit{𝑘𝑗𝑙}}=\frac{1}{4}(X_k{X}_j{Y}_l-Y_k{Y}_j{Y}_l-X_k{Y}_j{X}_l-Y_k{X}_j{X}_l),$$

(A.4)

where in ${Ĝ}_B$ the shared qubit $k$ carries the first Pauli of each string. The two operators overlap only on $k$, so Lemma 2 applies to each of the $2\times 4=8$ pairs of terms. Four pairs have matching Pauli on $k$ ($d=0$) and vanish; the surviving four are

Summing the four contributions,

$$[{Ĝ}_{A;\mathit{𝑖𝑘}},{Ĝ}_{B;\mathit{𝑘𝑗𝑙}}]=-\frac{i}{4}(X_i{X}_j{Y}_l-Y_i{Y}_j{Y}_l-X_i{Y}_j{X}_l-Y_i{X}_j{X}_l)Z_k=-i{Ĝ}_{B;\mathit{𝑖𝑗𝑙}}{Z}_k,$$

(A.5)

which is (3.3). Note that the four surviving terms reassemble precisely into ${Ĝ}_B$ with $k$ replaced by $i$: the sign ${\eta }_k$ supplied by Lemma 2 is exactly compensated by the relative sign carried by the corresponding term of ${Ĝ}_A$, so the internal sign pattern of ${Ĝ}_B$ is transported unchanged.

Two overlapping $\widehat{B}$ gates generate the 4-body generators of section 4 by the same mechanism,

$$[{Ĝ}_{B;\mathit{𝑖𝑗𝑘}},{Ĝ}_{B;\mathit{𝑙𝑚𝑘}}]=\frac{i}{8}{L}_{\mathit{𝑖𝑗𝑙𝑚}}{Z}_k,$$

(A.6)

where the two strings share only the highest-significance qubit $k$.

The identities of Proposition 3. For the mixed 2-body/4-body commutator the overlap is again a single qubit, $S=\{j\}$, carrying the first Pauli of $L_{\mathit{𝑗𝑘𝑙𝑚}}$. Splitting $L_{\mathit{𝑗𝑘𝑙𝑚}}$ according to that Pauli,

$$\begin{array}{cc}\begin{array}{rl}{L}_{\mathit{𝑗𝑘𝑙𝑚}}& =X_j(X_k{X}_l{Y}_m+X_k{Y}_l{X}_m-Y_k{X}_l{X}_m+Y_k{Y}_l{Y}_m)\\ & +Y_j(-X_k{X}_l{X}_m+X_k{Y}_l{Y}_m-Y_k{X}_l{Y}_m-Y_k{Y}_l{X}_m),\end{array}& \end{array}$$

(A.7)

the term $X_i{Y}_j$ of $L_{\mathit{𝑖𝑗}}$ pairs only with the $X_j$ block ($\eta =-1$) and $-Y_i{X}_j$ only with the $Y_j$ block ($\eta =+1$). The eight surviving contributions are

$$\begin{array}{cc}\begin{array}{rl}[L_{\mathit{𝑖𝑗}},L_{\mathit{𝑗𝑘𝑙𝑚}}]=-2i{Z}_j[& X_i{X}_k{X}_l{Y}_m+X_i{X}_k{Y}_l{X}_m-X_i{Y}_k{X}_l{X}_m-Y_i{X}_k{X}_l{X}_m\\ & +X_i{Y}_k{Y}_l{Y}_m+Y_i{X}_k{Y}_l{Y}_m-Y_i{Y}_k{X}_l{Y}_m-Y_i{Y}_k{Y}_l{X}_m],\end{array}& \end{array}$$

(A.8)

and the bracket is exactly $L_{\mathit{𝑖𝑘𝑙𝑚}}$, so $[L_{\mathit{𝑖𝑗}},L_{\mathit{𝑗𝑘𝑙𝑚}}]=-2i{L}_{\mathit{𝑖𝑘𝑙𝑚}}{Z}_j$.

For two 4-body generators overlapping on two qubits, $S=\{k,l\}$, the even-$d$ part of Lemma 2 does the essential work: every pair of terms that disagrees on both shared qubits has $d=2$ and therefore commutes, and pairs agreeing on both have $d=0$ and also commute. Only pairs disagreeing on exactly one shared qubit survive, producing a single $Z$ on that qubit. Collecting the terms that disagree on $k$ alone and on $l$ alone gives the two symmetric families

$$[L_{\mathit{𝑖𝑗𝑘𝑙}},L_{\mathit{𝑘𝑙𝑚𝑛}}]=-4i{L}_{\mathit{𝑖𝑗𝑚𝑛}}{Z}_k-4i{L}_{\mathit{𝑖𝑗𝑚𝑛}}{Z}_l=-4i{L}_{\mathit{𝑖𝑗𝑚𝑛}}(Z_k+Z_l),$$

(A.9)

as quoted in the proof of Proposition 3.

B Elementary gate decompositions

In this appendix, we provide decompositions of the gates used in this work into elementary single-qubit rotations and CNOT gates. Throughout, $R_k(\alpha )=\exp <!--\; FUNCTION\; APPLICATION\; -->(-\mathit{𝑖𝛼}{\sigma }_k∕2)$ for $k=x,y,z$, and $H$ denotes the Hadamard gate. Note the half-angle convention: $R_y(\alpha )|0⟩=\cos <!--\; FUNCTION\; APPLICATION\; -->(\alpha ∕2)|0⟩+\sin <!--\; FUNCTION\; APPLICATION\; -->(\alpha ∕2)|1⟩$, so the argument of $R_y$ must be $2𝜃$ to achieve a net rotation by $𝜃$ in a 2-dimensional subspace.

2-qubit Givens rotation $G_{\mathit{𝑖𝑗}}^{(2)}(𝜃)$. The Givens rotation acts as a rotation by $𝜃$ in the $\{|01⟩,|10⟩\}$ subspace while leaving $|00⟩$ and $|11⟩$ invariant. This is essentially the SingleExcitation rotation of PennyLane. The Hadamard on $q_i$ followed by a CNOT computes the parity of the two qubits into $q_j$; the two $R_y(-𝜃)$ rotations then act with opposite effective signs in the even- and odd-parity sectors, producing a net rotation only in the active subspace; see figure 4.

2-qubit $A$-gate $A_{\mathit{𝑖𝑗}}^{(2)}(𝜃)$. Since $A(𝜃)=A(0)G(-𝜃)$ and the discrete reflection matrix $A(0)=\mathrm{diag}<!--\; FUNCTION\; APPLICATION\; -->(1,1,-1,1)$ flips the sign of the determinant in the active subspace, it is impossible to decompose $A_{\mathit{𝑖𝑗}}^{(2)}(𝜃)$ using only two CNOTs and single-qubit $\mathit{𝑆𝑈}(2)$ rotations. The reflection $A(0)$ admits a single-CNOT implementation as $(I\otimes R_y(\pi ∕2))\text{CNOT}(I\otimes R_y(-\pi ∕2))$, which combined with $G^{(2)}(-𝜃)$ yields a 3-CNOT decomposition; see figure 5.

BEMPA $\widehat{B}$ gate. The 3-qubit $\widehat{B}$ gate rotates in the $\{|001⟩,|110⟩\}$ subspace. Its decomposition follows the same compute-rotate-uncompute pattern as $G^{(4)}$, adapted to three qubits; see figure 6.

4-qubit Givens rotation $G_{\mathit{𝑖𝑗𝑘𝑙}}^{(4)}(𝜃)$. The 4-qubit gate rotates in the $\{|1100⟩,|0011⟩\}$ subspace. It is essentially the DoubleExcitation rotation of PennyLane. The strategy is to use a cascade of CNOT gates to map the Hamming distance of the active basis states from four down to one, apply a multi-controlled rotation, and uncompute; see figure 7.

4-qubit $A$-gate $A_{\mathit{𝑖𝑗𝑘𝑙}}^{(4)}(𝜃)$. Since $A^{(4)}(𝜃)=A^{(4)}(0)G^{(4)}(-𝜃)$ and $A^{(4)}(0)$ applies a $-1$ phase solely to $|1100⟩$, the circuit reuses the compute-rotate-uncompute structure of $G^{(4)}$ but replaces the multi-controlled rotation. In the mapped basis, the two active states $|1110⟩$ (from $|1100⟩$) and $|1111⟩$ (from $|0011⟩$) are distinguished by $q_l$. The combined rotation and reflection require the single-qubit gate $R_y(2𝜃)Z$ on $q_l$ (which acts as $\left[\begin{array}{cc}\cos <!--\; FUNCTION\; APPLICATION\; -->𝜃& \sin <!--\; FUNCTION\; APPLICATION\; -->𝜃\\ \sin <!--\; FUNCTION\; APPLICATION\; -->𝜃& -\cos <!--\; FUNCTION\; APPLICATION\; -->𝜃\end{array}\right]$), with an additional $C^3(-I)$ phase to account for the $A^{(4)}(0)$ reflection; see figure 8.

Decomposition of multi-controlled rotations. The multi-controlled $C^m{R}_y(\alpha )$ gates appearing in the circuits above can be decomposed into $2^m$ CNOT gates and $2^m$ single-qubit $R_y$ rotations using the uniformly controlled rotation technique [64]. The construction interleaves CNOT gates — whose controls cycle through the $m$ control qubits in Gray code order — with $R_y(\pm \alpha ∕2^m)$ rotations of alternating sign on the target qubit; see figure 9. More generally, the same technique applies to any multi-controlled single-qubit gate $C^m(U)$ with arbitrary $U\in U(2)$, using $2^m$ CNOTs with general $e^{\mathit{𝑖𝛼}}{R}_z\cdot R_y\cdot R_z$ blocks between them instead of bare $R_y$ rotations [64]. For $A^{(4)}$, the combined gate $C^3(-R_y(2𝜃)Z)$, which absorbs the $C^3(-I)$ phase correction of figure 8, has $U=-R_y(2𝜃)Z\notin \mathit{𝑆𝑈}(2)$ (since $\det <!--\; FUNCTION\; APPLICATION\; -->U=-1$), but is decomposed with the same $2^3=8$ CNOT count; the non-unit determinant is accommodated by a global phase factor $e^{\mathit{𝑖𝛼}}$ in the single-qubit blocks between the CNOTs.

Summary of circuit resources. The circuit depth is the number of sequential time steps when gates acting on disjoint qubits are executed in parallel. Table 4 summarises the total resource counts after decomposing all multi-controlled gates as in figure 9.

Cost of the circuit of figure 1. Assembling the $7$ $G^{(2)}$ and $12$ $G^{(4)}$ gates of figure 1 from the decompositions above gives $182$ CNOT gates in total, with a CNOT-depth of $165$ when gates on disjoint qubits are executed in parallel. The count is dominated by the $G^{(4)}$ gates, at $14$ CNOTs each; these are unavoidable here, since by the argument of section 4 no $2$-qubit gate can change orbital occupancies while preserving $S_z$. We note that the $14$-CNOT decomposition of $G^{(4)}$ used here follows the transparent uniformly-controlled-rotation route of figure 9 and is not optimised. Throughout this work “hardware-efficient” should be read as parameter-economical while remaining exactly symmetry-preserving, rather than as depth-minimal.

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Notes

^1. We thank Petr Kravchuk for suggesting the connection to Yamabe’s theorem presented in the proof.

^2. ${\mathrm{\Phi }}_m(𝜃,\varphi ),m=-s,-s+1,\dots ,s$, furnish a spin-$s$ representation of $\mathit{𝑆𝑈}(2)$.

^3. As an example, the operator $c_{\mathit{𝑠\uparrow }}^{†}$ creates an electron in a state with $S_z$ eigenvalue $s$ and ${\mathbb{Z}}_2$ isospin $\uparrow$ in the LLL.

^4. The dimension of the Hilbert space sufficient for our considerations is given by sequence A125809 of the Online Encyclopedia of Integer Sequences [41].