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arXiv:2605.00979

Universality of Quantum Gates in Particle and Symmetry Constrained Subspaces

Andreas Stergiou, Nicolas PD Sawaya

quant-phcond-mat.str-elhep-thSubmitted May 1, 2026• Announced May 5, 2026• 0 citations• Revised Oct 2, 2026 (v2)
v1 · May 1, 2026v2 · Oct 2, 2026
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Abstract

Simulating physical systems on near-term quantum computers often requires preparing states within constrained subspaces, like those with fixed particle number or spin. We use Lie algebraic techniques to prove that hardware-efficient gates are universal for state preparation in these subspaces. The key mechanism is Pauli $Z$ dressing: commutators of overlapping gates produce Pauli $Z$ operators on shared qubits, acting as spectator projectors that decompose multi-plane rotations into single-plane generators spanning the full $\mathfrak{so}(w)$ algebra, where $w$ is the dimension of the constrained subspace, thereby guaranteeing universality for real state preparation. Adding independent complex phases extends this to $\mathfrak{su}(w)$, enabling arbitrary complex state preparation. We provide a computationally efficient Jacobian criterion for verifying that a circuit can explore any direction on the target manifold from almost any parameter configuration. Our findings are applicable to many problem areas, including Fermi-Hubbard models, Bose-Hubbard models, and molecular electronic structure. We apply our framework to two physical settings: we prove the completeness of the binary encoded multi-level particles ansatz on the conserved-particle-number subspace, and we construct symmetry-preserving circuits for the fuzzy sphere regularisation of the 3D Ising conformal field theory (CFT). For the latter, we variationally prepare the ground and excited states and extract some CFT data.

Comments
40 pages, 9 figures. v2: new sections on scaling with system size, simulation under a noise model, and extraction of an OPE coefficient; new appendix with explicit commutator identities; references updated. Code used in this work is available at https://github.com/andstergiou/fuzzy-ising-vqe

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Simulating physical systems on near-term quantum computers often requires preparing states within constrained subspaces, like those with fixed particle number or spin. We use Lie algebraic techniques to prove that hardware-efficient gates are universal for state preparation in these subspaces. The key mechanism is Pauli ZZZ dressing: commutators of overlapping gates produce Pauli ZZZ operators on shared qubits, acting as spectator projectors that decompose multi-plane rotations into single-plane generators spanning the full so(w)\mathfrak{so}(w)so(w) algebra, where www is the dimension of the constrained subspace, thereby guaranteeing universality for real state preparation. Adding independent complex phases extends this to su(w)\mathfrak{su}(w)su(w), enabling arbitrary complex state preparation. We provide a computationally efficient Jacobian criterion for verifying that a circuit can explore any direction on the target manifold from almost any parameter configuration. Our findings are applicable to many problem areas, including Fermi-Hubbard models, Bose-Hubbard models, and molecular electronic structure. We apply our framework to two physical settings: we prove the completeness of the binary encoded multi-level particles ansatz on the conserved-particle-number subspace, and we construct symmetry-preserving circuits for the fuzzy sphere regularisation of the 3D Ising conformal field theory (CFT). For the latter, we variationally prepare the ground and excited states and extract some CFT data.
Universal gate sets for quantum computing have been known for decades, yet no universal gate set has been proposed for particle-conserving unitaries, which are the operations of interest in quantum chemistry. In this work, we show that controlled single-excitation gates in the form of Givens rotations are universal for particle-conserving unitaries. Single-excitation gates describe an arbitrary U ( 2 ) rotation on the two-qubit subspace spanned by the states | 01 ⟩ , | 10 ⟩ , while leaving other states unchanged – a transformation that is analogous to a single-qubit rotation on a dual-rail qubit. The proof is constructive, so our result also provides an explicit method for compiling arbitrary particle-conserving unitaries. Additionally, we describe a method for using controlled single-excitation gates to prepare an arbitrary state of a fixed number of particles. We derive analytical gradient formulas for Givens rotations as well as decompositions into single-qubit and CNOT gates. Our results offer a unifying framework for quantum computational chemistry where every algorithm is a unique recipe built from the same universal ingredients: Givens rotations.
In the conventional theory of quantum computation, universality is discussed in terms of properties of unitary gate sets. In many experimental setups, however, we instead have access to a parametrized set of Hamiltonians, which can be exponentiated for any desired time. Accordingly, we formulate a continuous analogue of universality, where adding one expensive Hamiltonian control to a cheap set of local control operators allows for universal computation. We find several concrete sets of 1- or 2-local Pauli generators SSS and nnn-qubit ``magic" Hamiltonians HHH that together generate the full Lie algebra su(2n)\mathfrak{su}(2^n)su(2n), with a special focus on the Ising Hamiltonian JJJ. We propose a method to directly implement any unitary by exponentiating these generator sets, and give bounds on the needed time of application of the magic Hamiltonian HHH. We find that the number of applications of HHH needed to approximate any unitary up to accuracy εεε scales exponentially in the number of qubits as expected from the standard application of the Solovay-Kitaev theorem. We also find phase transitions in the size of the generated Lie algebras as the parameters of the Ising Hamiltonian are varied. We find an algorithmic application of the Chow/Rashevskii technique to implement unitaries corresponding to iterated commutators.
The variational quantum simulation technique on gate-based quantum processors can be made more efficient if symmetries of the Hamiltonian system of interest are factored into the design construction of the underlying parameterized quantum circuits (PQC) needed by the technique. In the first part of this work, we present an easy and practical approach that can be used to construct symmetric quantum circuits. The method we present can directly be adapted for any symmetry, though in the present work we focus explicitly on the examples of Z 2 , Z 3 , and particle number conservation. We show how to construct symmetric gates that respects the aforementioned symmetries. In the second part, we focus on numerical analysis of parameterized particle-conserving gates. In variational quantum simulation, for example, the variational quantum eigensolver (VQE), the symmetric gates of the PQC are parameterized. Since there are often many possible inequivalent parametrizations, it is often not immediately obvious which gate parametrization would be most efficient in general for a wide range of Hamiltonians that respect the symmetry. We streamline our analysis to particle-conserving circuits and Hamiltonians. We numerically analyze the efficiency of different particle-conserving gate parametrizations by studying two examples, namely, a physics problem of learning the ground state of Heisenberg spin chains, and an artificial problem of learning the random arbitary states sampled within the particle number conserved subspace.
Simulating fermionic systems on a quantum computer requires representing fermionic states using qubits. The complexity of many simulation algorithms depends on the complexity of implementing rotations generated by fermionic creation-annihilation operators, and the space depends on the number of qubits used. While standard fermion encodings like Jordan-Wigner are space optimal for arbitrary fermionic systems, physical symmetries like particle conservation can reduce the number of physical configurations, allowing improved space complexity. Such space saving is only feasible if the gate overhead is small, suggesting a (quantum) data structures problem, wherein one would like to minimize space used to represent a fermionic state, while still enabling efficient rotations. We define a structure which naturally captures mappings from fermions to systems of qubits. We then instantiate it in two ways, giving rise to two new second-quantized fermion encodings of F fermions in M modes. An information theoretic minimum of I: = ⌈log₂ binom(M,F)⌉ qubits is required for such systems, a bound we nearly match over the entire parameter regime. 1) Our first construction uses I + o(I) qubits when F = o(M), and allows rotations generated by creation-annihilation operators in O(I) gates and O(log M log log M) depth. 2) Our second construction uses I + O(1) qubits when F = Θ(M), and allows rotations generated by creation-annihilation operators in O(I³) gates. In relation to comparable prior work, the first represents a polynomial improvement in both space and gate complexity (against Kirby et al. 2022), and the second represents an exponential improvement in gate complexity at the cost of only a constant number of additional qubits (against Harrison et al. or Shee et al. 2022), in the described parameter regimes.
Preserving spin symmetry in variational quantum algorithms is essential for producing physically meaningful electronic wavefunctions. Implementing spin-adapted transformations on quantum hardware, however, is challenging because the corresponding fermionic generators translate into noncommuting Pauli operators. In this work, we introduce an exact and computationally efficient factorization of spin-adapted unitaries derived from fermionic double excitation and deexcitation rotations. These unitaries are expressed as ordered products of exponentials of Pauli operators. Our method exploits the fact that the elementary operators in these generators form small Lie algebras. By working in the adjoint representation of these algebras, we reformulate the factorization problem as a low-dimensional nonlinear optimization over matrix exponentials. This approach enables precise numerical reparametrization of the unitaries without relying on symbolic manipulations. The proposed factorization provides a practical strategy for constructing symmetry-conserving quantum circuits within variational algorithms. It preserves spin symmetry by design, reduces implementation cost, and ensures the accurate representation of electronic states in quantum simulations of molecular systems.
Simulating fermionic systems on a quantum computer requires representing fermionic states using qubits. The complexity of many simulation algorithms depends on the complexity of implementing rotations generated by fermionic creation-annihilation operators, and the space depends on the number of qubits used. While standard fermion encodings like Jordan-Wigner are space optimal for arbitrary fermionic systems, physical symmetries like particle conservation can reduce the number of physical configurations, allowing improved space complexity. Such space saving is only feasible if the gate overhead is small, suggesting a (quantum) data structures problem, wherein one would like to minimize space used to represent a fermionic state, while still enabling efficient rotations. We define a structure which naturally captures mappings from fermions to systems of qubits. We then instantiate it in two ways, giving rise to two new second-quantized fermion encodings of FFF fermions in MMM modes. An information theoretic minimum of I:=⌈log⁡(MF)⌉\mathcal{I}:=\lceil\log \binom{M}{F}\rceilI:=⌈log(FM​)⌉ qubits is required for such systems, a bound we nearly match over the entire parameter regime. (1) Our first construction uses I+o(I)\mathcal I+o(\mathcal I)I+o(I) qubits when F=o(M)F=o(M)F=o(M), and allows rotations generated by creation-annihilation operators in O(I)O(\mathcal I)O(I) gates and O(log⁡Mlog⁡log⁡M)O(\log M \log \log M)O(logMloglogM) depth. (2) Our second construction uses I+O(1)\mathcal I+O(1)I+O(1) qubits when F=Θ(M)F=Θ(M)F=Θ(M), and allows rotations generated by creation-annihilation operators in O(I3)O(\mathcal I^3)O(I3) gates. In relation to comparable prior work, the first represents a polynomial improvement in both space and gate complexity (against Kirby et al. 2022), and the second represents an exponential improvement in gate complexity at the cost of only a constant number of additional qubits (against Harrison et al. or Shee et al. 2022), in the described parameter regimes.