Symmetry conservation with Trotterization and Quantum Phase Estimation

Abstract: Quantum algorithms for quantum chemistry and other many-body fermionic systems work by expressing the Hamiltonian in a basis of qubits and fragmenting the Hamiltonian into a sum of products of Pauli operators whose exponentials are easily encoded on a quantum device. Applying the product of exponentials, known as Trotterization, leads to an error associated with the non-commutativity of operators. This error can lead to breaking the symmetries of the Hamiltonian because the fragments are not symmetry conserving in general. Nonetheless, many algorithms for time evolution rely on Trotterization, including time evolution and quantum phase estimation. We show that we can express the Hamiltonian in terms of Hermitian excitation operators which map to sums of commuting Pauli strings for any encoding and conserve symmetries corresponding to Abelian groups of symmetry operators. Symmetries corresponding to non-Abelian groups, on the other hand, are not fully conserved by Trotterized Hermitian excitation operators, so we developed ``operator kirigami'' to cut the sum of non-commuting operators by orthogonal projection and to fold terms together using unitary rotations. We tested pools of operators for small molecules and basis sets, and found that electron number and spin symmetry conserving pools led to greater errors that decreased for larger molecules and were negated with second-order Trotterization. Our work shows the potential for testing quantum computing algorithms on classical computers by adapting tools used in electronic structure theory with conserved symmetries.

• The paper introduces operator kirigami, a novel method to enforce both Abelian and non-Abelian symmetry in Trotterized quantum simulations.
• It demonstrates that employing Hermitian excitation operator bases preserves electron number and spin symmetry under Trotterization while quantifying energy error trade-offs.
• Numerical tests on molecules like H₂, LiH, and H₂O validate the approach, offering practical insights for scalable quantum chemistry simulation.

Symmetry Conservation in Trotterization and Quantum Phase Estimation

Introduction

The simulation of quantum many-body fermionic systems, crucial for quantum chemistry and materials science, faces exponential Hilbert space scaling with system size. Quantum algorithms, particularly those utilizing Trotterization and Quantum Phase Estimation (QPE), address this challenge by mapping the fermionic Hamiltonian to qubit operator sums, e.g., products of Pauli strings, whose exponentials are efficiently encodable on quantum hardware. However, the non-commutativity intrinsic to most Hamiltonian fragmentations induces Trotter errors, which can result in symmetry violations (e.g., electron number, spin). The presented work rigorously analyzes symmetry conservation within Trotterized quantum algorithms and introduces a novel operator manipulation technique—termed “operator kirigami”—to systematically enforce symmetry, with explicit analysis across Abelian and non-Abelian cases.

Hamiltonian Fragmentation and Symmetry Challenges

Hamiltonian fragmentation schemes serve as the foundation for quantum simulation approaches. Commonly, the electronic structure Hamiltonian is mapped (e.g., via Jordan-Wigner, Bravyi-Kitaev) into sums of Pauli strings. While this provides a tractable route to gate decompositions, a generic random ordering of these non-commuting operators breaks important physical symmetries such as particle number and spin, since Trotterization only approximates time evolution by consecutive applications of non-symmetry-conserving fragments.

Figure 1: Schematic of Hamiltonian decompositions and their symmetry conservation properties with Trotterization. a: Pauli string sum; b: Hermitian excitation operator grouping; c-d: Operator kirigami for non-Abelian symmetry.

To address this, the authors show that by expressing the Hamiltonian in Hermitian excitation operator bases (which correspond to real-valued, commuting Pauli string sums across all common encodings), symmetries corresponding to Abelian groups (e.g., electron number, $S_z$) can be strictly preserved under Trotterization. For non-Abelian cases (notably, total spin $S^2$), simple fragmentation does not suffice; specialized techniques are required.

Operator Kirigami: Systematic Non-Abelian Symmetry Conservation

The core contribution is the introduction of “operator kirigami”—a two-step process:

1. Cutting: Projections isolate commuting and non-commuting operator components.
2. Folding: Non-commuting fragments are consolidated via unitary rotations into a form permitting exact exponential implementation (i.e., no Trotter error, symmetry preserved).

Mathematically, this leverages the tripotency of Hermitian excitation operators and manipulates their BCH commutator structures. In the framework, all terms are recursively expressed in combinations whose exponentiation, along with unitary rotations defined by commutators, precisely preserves the target symmetry sector, as formalized for composite fragments like $\hat{C}_q^{pr}$ and $\hat{B}_q^p$.

This procedure is applicable to all fermion-to-qubit encodings supporting one Pauli string/Majorana operator correspondence (includes all standard encodings in quantum chemistry).

Numerical Analysis: Symmetry Error and Energy Behavior

To validate the theoretical claims, the performance of various fragmentation and ordering schemes was benchmarked on quantum simulations (classically emulated) for small molecules (H$_2$, LiH, H$_2$O) across several basis sets. Operator pools compared:

• Qubit: All unique qubit Pauli strings;
• Slater: Hermitian excitation operators (Abelian symmetry conserving);
• CSF: Pools enforcing configuration state function (non-Abelian) symmetries.

  Figure 2: Time evolution of infidelities and $\braket{\hat{S}^2}$ under different operator pools and random orderings.

Empirical results revealed that:

• Qubit pools (no symmetry enforcement) demonstrated largest symmetry violations, rapid drift in $\langle\hat{S}^2\rangle$ and electron number, and faster accumulation of non-physical population;
• Slater/CSF pools strictly conserved symmetries, but with increased infidelity to exact evolution due to larger commutator sums and the reduced possibility of error cancellation in randomly ordered Pauli fragments.

  Figure 3: Ground state energy differences ($\Delta E$) between full CI solution and Trotterized energy across operator pools for various molecules and Trotter orders.

Notably, with first-order Trotterization, symmetry-conserving pools exhibit energy errors up to an order of magnitude larger than the unstructured Qubit pool for intermediate system sizes, attributed to increased fragment commutator complexity. However, second-order Trotterization (SOT) substantially suppresses these errors, rendering the energy discrepancies comparable across all pools, while retaining enforced symmetry.

Algorithmic and Practical Implications

Theoretically, this work establishes that non-Abelian symmetry conservation in Trotterized quantum algorithms is feasible and systematic for arbitrary fermion-to-qubit encodings using operator kirigami. Practically:

• Quantum resource requirements: Symmetry-conserving pools drastically reduce Hilbert space dimension and operator count for classical simulation, permitting larger-scale pre-quantum validation.
• Quantum hardware translation: On quantum devices, operator kirigami prescribes gate compilation steps to exactly preserve physically-relevant symmetries, eliminating symmetry-induced errors and symmetry-breaking leakage.
• Error analysis: The formalism clarifies that Trotter error suppression is nontrivial for symmetry-adapted bases, guiding operator ordering, Trotter step sizing, and symmetry sector targeting in real devices.

These findings are directly useful for quantum simulation algorithm developers seeking scalable and symmetry-respecting implementations for quantum chemistry and materials science.

Perspectives and Future Developments

While the kirigami method is algorithmically complete for Abelian and spin symmetry in electronic structure Hamiltonians, the approach is extensible to broader symmetry groups (e.g., spatial symmetry, combined electronic and nuclear spins) and is compatible with advanced Hamiltonian simulation or error mitigation techniques. Its explicit mapping between classical electronic structure methods and quantum simulation enables near-term benchmarking and algorithmic innovation. Future work could target

• Automated detection and enforcement of higher symmetries using kirigami-style projections and folds,
• Integration into hybrid quantum-classical and variational simulation frameworks,
• Application on fault-tolerant devices for large-scale chemical simulation with provable error bounds.

Conclusion

This work rigorously addresses symmetry conservation in Trotterized quantum simulation of fermionic systems, introducing operator kirigami as a systematic technique for enforcing Abelian and non-Abelian symmetries. The method achieves exact symmetry conservation or controlled Trotter error, is demonstrated numerically for challenging cases, and provides a foundation for symmetry-preserving, scalable quantum simulation algorithms. The results clarify the trade-offs between symmetry conservation and Trotter error scaling and empower practical quantum algorithm design for chemical applications.