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arXiv:1907.01493

Symmetry Configuration Mapping for Representing Quantum Systems on Quantum Computers

Sean A. Fischer, Daniel Gunlycke

Published Jul 2, 2019• 4 citations
arXivHTMLPDF

Abstract

Quantum computing has the potential to significantly speed up complex computational tasks, and arguably the most promising application area for near-term quantum computers is the simulation of quantum mechanics. To make the most of our limited quantum computing resources, we need new and more compact algorithms and mappings. Whereas previous work, including recent demonstrations, has focused primarily on general mappings for fermionic systems, we propose instead to construct customized mappings tailored to the considered quantum mechanical systems. Specifically, we take advantage of existing symmetry, which we build into the mappings a priori to obtain optimal compactness. To demonstrate this approach, we have performed quantum computing calculations of the fluorine molecule, in which we have mapped 16 active spin-orbitals to 4 qubits. This is a four-fold reduction in the qubit requirement, as compared to the standard general mappings. Moreover, our compact system-to-qubits mappings are robust against noise that breaks symmetry, thereby reducing non-statistical errors in the computations. Furthermore, many systems, including F2, are described by real Hamiltonians, allowing us to also reduce the number of single-qubit operations in the hardware-efficient ansatz for the quantum variational eigensolver by roughly a factor of three.

DOI
10.48550/arxiv.1907.01493

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Quantum computing has the potential to significantly speed up complex computational tasks, and arguably the most promising application area for near-term quantum computers is the simulation of quantum mechanics. To make the most of our limited quantum computing resources, we need new and more compact algorithms and mappings. Whereas previous work, including recent demonstrations, has focused primarily on general mappings for fermionic systems, we propose instead to construct customized mappings tailored to the considered quantum mechanical systems. Specifically, we take advantage of existing symmetry, which we build into the mappings a priori to obtain optimal compactness. To demonstrate this approach, we have performed quantum computing calculations of the fluorine molecule, in which we have mapped 16 active spin-orbitals to 4 qubits. This is a four-fold reduction in the qubit requirement, as compared to the standard general mappings. Moreover, our compact system-to-qubits mappings are robust against noise that breaks symmetry, thereby reducing non-statistical errors in the computations. Furthermore, many systems, including F2, are described by real Hamiltonians, allowing us to also reduce the number of single-qubit operations in the hardware-efficient ansatz for the quantum variational eigensolver by roughly a factor of three.
Near-term quantum computers may be able to significantly speed up complex computational tasks, but algorithms that make efficient use of quantum resources are needed. Quantum chemistry is widely regarded as a candidate for the first demonstration of quantum advantage with near-term quantum computers. In the present work, we demonstrate how taking advantage of the symmetries of a molecule leads to a reduction in the number of qubits required. This reduction in qubits also leads to a reduction in the number of variational parameters needed to reach chemical accuracy. Furthermore, we show how a simple modification of the hardware-efficient ansatz for the variational quantum eigensolver yields a factor of 3 reduction in the number of parameters with no loss in accuracy for most problems in quantum chemistry.
Compact representations of fermionic Hamiltonians are necessary to perform calculations on quantum computers that lack error-correction. A fermionic system is typically defined within a subspace of fixed particle number and spin while unnecessary states are projected out of the Hilbert space. We provide a bijective mapping using combinatoric ranking to bijectively map fermion basis states to qubit basis states and express operators in the standard spin representation. We then evaluate compact mapping using the Variational Quantum Eigensolver (VQE) with the unitary coupled cluster singles and doubles excitations (UCCSD) ansatz in the compact representation. Compactness is beneficial when the orbital filling is well away from half, and we show at 30 spin orbital H2H_{2}H2​ calculation with only 8 qubits. We find that the gate depth needed to prepare the compact wavefunction is not much greater than the full configuration space in practice. A notable observation regards the number of calls to the optimizer needed for the compact simulation compared to the full simulation. We find that the compact representation converges faster than the full representation using the ADAM optimizer in all cases. Our analysis demonstrates the effect of compact mapping in practice.
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.
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\mathbb{Z}_2Z2​ 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%\approx 50\%≈50% and quantum gate counts by up to ≈27.5×\approx 27.5\times≈27.5× and Hamiltonian Pauli string counts by up to ≈2.75×\approx 2.75\times≈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.
Simulating electronic structure on a quantum computer requires encoding of fermionic systems onto qubits. Common encoding methods transform a fermionic system of NNN spin-orbitals into an NNN-qubit system, but many of the fermionic configurations do not respect the required conditions and symmetries of the system so the qubit Hilbert space in this case may have unphysical states and thus cannot be fully utilized. We propose a generalized qubit-efficient encoding (QEE) scheme that requires the qubit number to be only logarithmic in the number of configurations that satisfy the required conditions and symmetries. For the case of considering only the particle-conserving and singlet configurations, we reduce the qubit count to an upper bound of O(mlog2N)\mathcal{O}(m{log}_{2}N)O(mlog2​N), where mmm is the number of particles. This QEE scheme is demonstrated on an H2{\mathrm{H}}_{2}H2​ molecule in the 6-31G basis set and a LiH\mathrm{LiH}LiH molecule in the STO-3G basis set using fewer qubits than the common encoding methods. We calculate the ground-state energy surfaces using a variational quantum eigensolver algorithm with a hardware-efficient ansatz circuit. We choose to use a hardware-efficient ansatz since most of the Hilbert space in our scheme is spanned by desired configurations so a heuristic search for an eigenstate is sensible. The simulations are performed on IBM Quantum machines and the Qiskit simulator with a noise model implemented from a IBM Quantum machine. Using the methods of measurement error mitigation and error-free linear extrapolation, we demonstrate that most of the distributions of the extrapolated energies using our QEE scheme agree with the exact results obtained by Hamiltonian diagonalization in the given basis sets within chemical accuracy. Our proposed scheme and results show the feasibility of quantum simulations for larger molecular systems in the noisy intermediate-scale quantum (NISQ) era. The number of terms in the Hamiltonian has an upper bound of $\mathcal{O}(\frac{{N}^{2m+1}}{(m\ensuremath{-}1)!\phantom{\rule{0.16em}{0ex}}m!})$ for the QEE scheme while it scales as O(N4)\mathcal{O}({N}^{4})O(N4) for the Jordan-Wigner encoding scheme. Nevertheless, we present several cases where QEE is useful.