We present a symmetry-adapted qubit encoding with complete active space (SAE-CAS) for quantum chemistry on fault-tolerant and near-term quantum processors. Building on exact-symmetry encodings, we extend symmetry-adapted mappings to approximate $Z$-symmetries corresponding to frozen-core and virtual orbitals, thereby reducing qubit requirements without significant loss of accuracy. We derive the mapping from the second-quantised Hamiltonian to active-space qubit Hamiltonians, prove its equivalence to the canonical CAS Hamiltonian with frozen-core and virtual-orbital projection, and integrate it with point-group and spin-parity symmetry encodings via affine Clifford transformations to maximise qubit reduction while preserving the target symmetry sector. The same framework also accommodates the Bravyi--Kitaev mapping, yielding an SAE-CAS-BK variant that is unitarily equivalent to SAE-CAS. Numerical benchmarking on nine small molecules using UCCSD and a hardware-efficient shifted-circular-alternating (HE-SCA) ansatz shows that SAE-CAS reduces qubit counts and Pauli-operator weight, yields shallower circuits with fewer parameters, and often accelerates VQE convergence; with HE-SCA it consistently reaches CAS reference energies in cases where JW-CAS does not converge within the tested budgets. We provide an open-source implementation in the Python package QuantumSymmetry. SAE-CAS offers a route to resource-efficient molecular simulations on fault-tolerant and near-term quantum processors.
We present a symmetry-adapted qubit encoding with complete active space (SAE-CAS) for quantum chemistry on fault-tolerant and near-term quantum processors. Building on exact-symmetry encodings, we extend symmetry-adapted mappings to approximate Z-symmetries corresponding to frozen-core and virtual orbitals, thereby reducing qubit requirements without significant loss of accuracy. We derive the mapping from the second-quantised Hamiltonian to active-space qubit Hamiltonians, prove its equivalence to the canonical CAS Hamiltonian with frozen-core and virtual-orbital projection, and integrate it with point-group and spin-parity symmetry encodings via affine Clifford transformations to maximise qubit reduction while preserving the target symmetry sector. The same framework also accommodates the Bravyi--Kitaev mapping, yielding an SAE-CAS-BK variant that is unitarily equivalent to SAE-CAS. Numerical benchmarking on nine small molecules using UCCSD and a hardware-efficient shifted-circular-alternating (HE-SCA) ansatz shows that SAE-CAS reduces qubit counts and Pauli-operator weight, yields shallower circuits with fewer parameters, and often accelerates VQE convergence; with HE-SCA it consistently reaches CAS reference energies in cases where JW-CAS does not converge within the tested budgets. We provide an open-source implementation in the Python package QuantumSymmetry. SAE-CAS offers a route to resource-efficient molecular simulations on fault-tolerant and near-term quantum processors.
Abstract A symmetry-adapted fermion-to-spin mapping or encoding that is able to store information about the occupancy of the n spin-orbitals of a molecular system into a lower number of n − k qubits in a quantum computer (where the number of reduced qubits k ranges from 2 to 5 depending on the symmetry of the system) is introduced. This mapping reduces the computational cost of a quantum computing simulation and at the same time enforces symmetry constraints. These symmetry-adapted encodings (SAEs) can be explicitly seen as a block-diagonalization of the Jordan–Wigner qubit Hamiltonian, followed by an orthogonal projection. We provide the form of the Clifford tableau for a general class of fermion-to-qubit encodings, and then use it to construct the map that block-diagonalizes the Hamiltonian in the SAEs. The algorithm proposed does not require any further computations to obtain this map, which is derived directly from the character table of the molecular point group. An implementation of the algorithm is presented as an open-source Python package, QuantumSymmetry, a user guide and code examples. QuantumSymmetry uses open-source quantum chemistry software PySCF for Hartree–Fock calculations, and is compatible with quantum computing toolsets OpenFermion and Qiskit. QuantumSymmetry takes arbitrary user input such as the molecular geometry and atomic basis set to construct the qubit operators that correspond in the appropriate SAE to fermionic operators on the molecular system, such as the second-quantized electronic structure Hamiltonian. QuantumSymmetry is used to produce numerical examples of variational quantum algorithm simulations to find the ground state energy for a number of example molecules, for both Unitary Coupled Clusters with Singles and Doubles and Adaptive Derivative Assembled Pseudo-Trotter Variational Quantum Eigensolver ansätze. We show that, beyond the advantage given by the lower qubit count, the proposed encodings consistently result in shallower and less complex circuits with a reduced number of variational parameters that are able to reach convergence faster and without any loss of computed accuracy.
We introduce periodic symmetry-adapted encoding (SAE) for qubit-efficient simulations of crystalline materials, extending the molecular SAE framework developed in earlier work to periodic systems. The method constructs a Gamma-point supercell active-space Hamiltonian from k-point Hartree-Fock data and identifies independent Boolean symmetry generators arising from spin parity, crystal translations, and point-group operations. Each generator removes one qubit while preserving the Hamiltonian spectrum in the target symmetry sector. The inclusion of crystal translations increases the maximum number of independent generators, and hence the number of removable qubits, from five in molecular SAE to eight in periodic SAE. Across ten crystals spanning cubic, hexagonal, trigonal, and tetragonal structures, the method removes four to eight qubits. In CsCl, all eight generators are realised, reducing CAS(6,7) from 14 qubits to 6. Complete-spectrum comparisons verify target-sector isospectrality, while noiseless UCCSD-VQE reaches the fixed-particle FCI energies throughout the suite. The resulting circuits reduce unoptimised logical CNOT counts by up to 99.7% relative to the full Jordan-Wigner encoding. The method is implemented in the open-source QuantumSymmetry package.
This work describes resource‑efficient quantum algorithms for molecular physics that exploit geometric structure and correspondences between physical and qubit symmetries. It begins with a review of classical approaches to the molecular electronic‑structure problem and of the variational quantum eigensolver in quantum computing. We then discuss the symmetry‑adapted encodings we introduced to reduce qubit requirements by eliminating redundant degrees of freedom arising from point‑group and other Boolean symmetries in the quantum simulation of molecular systems. Building on these encodings, we describe a qubit‑based symmetry‑adapted complete active‑space method that permits the simulation of larger molecular systems in a resource‑efficient manner on near‑term quantum devices. We present the QuantumSymmetry software package and illustrate its application to representative molecular systems. Next, we address electron‑molecule scattering by introducing a quantum computing algorithm to solve the inner‑region problem within the R‑matrix framework. In particular, we propose a novel simultaneous diagonalisation algorithm with subspace optimisation that requires only a limited number of Hamiltonian evaluations and could be generalised to a wide range of problems. The final chapter derives the explicit form of qubit operators for particle‑number projection. Altogether, these developments advance practical quantum simulations of complex molecular phenomena by leveraging symmetry and geometry in quantum‑algorithm design.
Accurate determination of ground-state energies for molecules remains a challenge in quantum chemistry and a cornerstone for progress in fields such as drug discovery and materials design. The Variational Quantum Eigensolver (VQE) represents a leading hybrid quantum-classical paradigm for addressing this challenge; however, its widespread realization is limited by noise and the restricted scalability of current quantum hardware. Achieving efficient simulations on Noisy Intermediate-Scale Quantum (NISQ) devices and forthcoming Fault-Tolerant Application-Scalable Quantum (FASQ) systems demands a detailed understanding of how computational resources scale with molecular complexity and fermion-to-qubit encodings. In this study, resource requirements for VQE implementations employing the Unitary Coupled Cluster Singles and Doubles (UCCSD) ansatz are systematically analyzed. The molecular Hamiltonian is formulated in second quantization and mapped to qubit operators through the Jordan-Wigner (JW), Bravyi-Kitaev (BK), and Parity (Pa) transformations. Hamiltonian reduction strategies, including Z2 tapering and frozen-core approximations, are examined to assess their effect on quantum resource scaling. The analysis reveals that appropriate transformations, when combined with symmetry-based reductions, can substantially reduce qubit counts by up to ≈50% and quantum gate counts by up to ≈27.5× and Hamiltonian Pauli string counts by up to ≈2.75×, relative to the corresponding unreduced Hamiltonian representations within the same active-space configuration for the representative set of molecular systems under study. These findings provide practical circuit-level insights for executing chemically relevant simulations on NISQ hardware, while establishing physical-resource baselines that may inform future logical-level analyses targeting FASQ systems.
Quantum computation has the potential to revolutionize quantum chemistry through major speedups in computation times and an exponential reduction in computational resources. Here, we combine the symmetry-adapted Jordan-Wigner encoding based on the full Boolean symmetry group Z2k with our new implementation of the Xia-Bian-Kais (XBK) method for improving the efficiency of electronic structure theory calculations on quantum annealers, particularly by reducing the number of qubits needed to achieve the same accuracy. By providing a more extensive symmetry-adapted encoding (SAE) than previous work, we are able to simulate molecules larger than those previously reported that have been studied using methods developed for quantum annealers and without using an active space. We calculated the potential energy surfaces of H2, LiH, He2, H2O, O2, N2, Li2, F2, CO, BH3, NH3, and CH4, with the largest molecule in the STO-6G basis set requiring 16 qubits with our SAE, and compared them with full configuration interaction results. The application of SAE to the XBK method provides an exponential reduction in the size of the Hilbert space and scales well with the size of the problem. It does not introduce significant additional errors for even or large values of a key variational parameter that determines the number of ancilla qubits used in the XBK method's Hamiltonian embedding, or for certain molecules such as He2 and H2O. We provide an explanation for this behavior and a recommendation on the usage of our method. In addition, we briefly discuss the potential of extracting electronic excited states from our method.
Simulating molecular systems on quantum computers requires efficient mappings from Fermionic operators to qubit operators. Traditional mappings such as Jordan-Wigner or Bravyi-Kitaev often produce high-weight Pauli terms, increasing circuit depth and measurement complexity. Although several local qubit mappings have been proposed to address this challenge, most are specialized for structured models like the Hubbard Hamiltonian and perform poorly for realistic chemical systems with dense two-body interactions. In this work, we utilize a suite of techniques to construct compact and noise-resilient Fermion-to-qubit mappings suitable for general molecular Hamiltonians. Building on the Generalized Superfast Encoding (GSE) and other similar works, we demonstrate that it outperforms prior encodings in both accuracy and hardware efficiency for molecular simulations. Our improvements include path optimization within the Hamiltonian's interaction graph to minimize operator weight, introduction of multi-edge graph structures for enhanced error detection without added circuit depth, and a stabilizer measurement framework that directly maps logical terms and stabilizers to the Z-basis using Clifford simulation. Applying these methods to simulations of (H2)2 and (H2)3 systems yields significantly improved absolute and correlation energy estimates under realistic hardware noise, with further accuracy gains achieved by increasing code distance. We also propose a [[2N, N, 2]] variant of GSE compatible with square-lattice and (quasi-)linear hardware topologies, demonstrating a twofold reduction in RMSE for orbital rotations on IBM Kingston hardware. These results establish GSE as a very attractive mapping for molecular quantum simulations.