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Physical Review C

Quantum simulation of nuclear Hamiltonian with a generalized transformation for Gray code encoding

Pooja Siwach, P. Arumugam

Published Sep 3, 2021• 27 citations
Publisher

Abstract

We present the quantum simulation of the deuteron to calculate its binding energy using the variational quantum eigensolver, which is based on a hybrid quantum-classical approach. Apart from a commonly used Hamiltonian derived from pionless effective field theory, we consider the interaction which can be easily studied with conventional classical methods but the corresponding operator leading to nonzero off-tridiagonal matrix elements. To map the many-body basis states on the qubit states, three encodings are explored, namely, one-hot, Bravyi-Kitaev, and the Gray code. We perform a generalized transformation for many-body operators in Gray code encoding, and simulate the Hamiltonians with nonzero off-tridiagonal matrix elements. Furthermore, the analyses of the relative efficiency of all encodings and corresponding transformations are carried out using the noise model of a real IBM quantum device. We demonstrate that the Gray code is more efficient for a large basis, irrespective of the form of potential and the presence of hardware noise.

DOI
10.1103/physrevc.104.034301

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We present the quantum simulation of the deuteron to calculate its binding energy using the variational quantum eigensolver, which is based on a hybrid quantum-classical approach. Apart from a commonly used Hamiltonian derived from pionless effective field theory, we consider the interaction which can be easily studied with conventional classical methods but the corresponding operator leading to nonzero off-tridiagonal matrix elements. To map the many-body basis states on the qubit states, three encodings are explored, namely, one-hot, Bravyi-Kitaev, and the Gray code. We perform a generalized transformation for many-body operators in Gray code encoding, and simulate the Hamiltonians with nonzero off-tridiagonal matrix elements. Furthermore, the analyses of the relative efficiency of all encodings and corresponding transformations are carried out using the noise model of a real IBM quantum device. We demonstrate that the Gray code is more efficient for a large basis, irrespective of the form of potential and the presence of hardware noise.
Due to the limitations of present-day quantum hardware, it is especially critical to design algorithms that make the best possible use of available resources. When simulating quantum many-body systems on a quantum computer, straightforward encodings that transform many-body Hamiltonians into qubit Hamiltonians use NNN of the available basis states of an NNN-qubit system, whereas 2N{2}^{N}2N are in theory available. We explore an efficient encoding that uses the entire set of basis states, where terms in the Hamiltonian are mapped to qubit operators with a Hamiltonian that acts on the basis states in Gray code order. This encoding is applied to the commonly studied problem of finding the ground-state energy of a deuteron with a simulated variational quantum eigensolver (VQE). It is compared to a standard ``one-hot'' encoding, and various trade-offs that arise are analyzed. The energy distribution of VQE solutions has smaller variance than the one obtained by the one-hot encoding even in the presence of simulated hardware noise, despite an increase in the number of measurements. The reduced number of qubits and a shorter-depth variational Ansatz enables the encoding of larger problems on current-generation machines. This encoding also demonstrates improvements for simulating time evolution of the same system, producing circuits for the evolution operators with reduced depth and roughly half the number of gates compared to a one-hot encoding.
Nuclear lattice effective field theory has become an important framework for quantum many-body calculations in nuclear physics, yet its classical implementation remains increasingly challenging for more general interactions and larger systems. In this work, we develop a quantum-computing framework for a three-dimensional nuclear lattice model. We construct a variational quantum eigensolver framework and systematically compare the Jordan-Wigner and Gray code encodings. Our analysis shows that for the few-body systems considered here, Gray code combined with symmetry reduction yields a substantially more compact qubit representation. Based on this framework, we perform numerical studies for 2H^{2}\mathrm{H}2H, 3H^{3}\mathrm{H}3H, and 4He^{4}\mathrm{He}4He on finite lattices. The calculated ground-state energies exhibit a clear approach toward the corresponding experimental binding energies as the lattice size increases. These results provide a proof-of-principle foundation for future quantum simulations of nuclear many-body problems.
Some of the computational limitations in solving the nuclear many-body problem could be overcome by utilizing quantum computers. The nuclear shell-model calculations providing deeper insights into the properties of atomic nuclei, is one such case with high demand for resources as the size of the Hilbert space grows exponentially with the number of particles involved. Quantum algorithms are being developed to overcome these challenges and advance such calculations. To develop quantum circuits for the nuclear shell-model, leveraging the capabilities of noisy intermediate-scale quantum (NISQ) devices. We aim to minimize resource requirements (specifically in terms of qubits and gates) and strive to reduce the impact of noise by employing relevant mitigation techniques. We achieve noise resilience by designing an optimized ansatz for the variational quantum eigensolver (VQE) based on Givens rotations and incorporating qubit-ADAPT-VQE in combination with variational quantum deflation (VQD) to compute ground and excited states incorporating the zero-noise extrapolation mitigation technique. Furthermore, the qubit requirements are significantly reduced by mapping the basis states to qubits using Gray code encoding and generalizing transformations of fermionic operators to efficiently represent manybody states. By employing the noise-resilient protocols, we achieve the ground and excited state energy levels of 38Ar and 6Li with better accuracy. These energy levels are presented for noiseless simulations, noisy conditions, and after applying noise mitigation techniques. Results are compared for Jordan Wigner and Gray code encoding using VQE, qubit-ADAPT-VQE, and VQD. Our work highlights the potential of noise-resilient protocols to leverage the full potential of NISQ devices in scaling the nuclear shell model calculations.
Based on the Generator Coordinate Method (GCM), we use a Quantum GCM (QuGCM) within a hybrid quantum-classical framework to simulate low-lying eigenstates of nuclear systems on quantum devices. The generator basis states are constructed from Hartree-Fock (HF) reference states, excited via symmetry-adapted unitary coupled-cluster (UCC) operators. These states are prepared as non-orthogonal quantum circuits and measured pairwise to compute the required overlap and Hamiltonian kernels. The resulting data is processed using a classical generalized-eigenvalue solver, following the GCM formalism, to extract the system's energy spectrum. To enhance efficiency and reduce circuit depth, we apply the Adaptive Generator Coordinate Inspired method (ADAPT-GCIM), which iteratively selects generator excitations based on energy gradients, thereby avoiding the need to explore the full Hilbert space. Our implementation is applied to nuclear systems, specifically the deuteron with the Reid68 potential and shell-model Hamiltonians of 6^66Li and 38^{38}38Ar. For each system, both the QuGCM and ADAPT-GCIM methods produce energy spectra in agreement with classical diagonalization results, demonstrating robustness even under noise and limited-depth constraints. Additionally, we compare fermionic encoding strategies, specifically Jordan-Wigner (JW) transformations of one-hot (OH) encoding and Gray code (GC) mappings, and show that GC encoding reduces circuit complexity and improves fidelity during multi-reference state preparation. Our findings indicate that QuGCM and ADAPT-GCIM provide a practical and scalable path toward simulating correlated quantum systems, with lesser vulnerability to noise and better compatibility with the limitations of current quantum hardware.
Abstract We have obtained the energy of the deuteron on a quantum simulator using the variational quantum eigensolver. We have employed realistic two-body interactions, namely, chiral N4LO and AV 18 , thus incorporating the role of tensor forces. These interactions are subsequently evolved to low resolution scales using the similarity renormalization group approach with parameter λ . The deuteron ground state energy has been calculated in the truncated harmonic oscillator basis, using the Qiskit-Aer simulator in both noise-free and noisy cases. The noise models have been taken from the actual IBM quantum hardware, and the results obtained have been extrapolated to the zero noise limit. The number of harmonic oscillator basis states (hence qubits) needed for computing the energy to within 1 percent of the experimental value in the quantum simulator, decreases with decreasing λ . We have analysed the extent of entanglement between oscillator modes using concurrence as the entanglement quantifier. It is seen that the entanglement decreases as λ is lowered from the bare value to ∼1.0 fm −1 independent of the form of the bare interaction and the number of harmonic oscillator basis states.
Calculations of nuclear properties and nuclear models are performed using quantum computing algorithms on simulated and real quantum computers. The models are a realistic calculation of deuteron binding based on effective field theory, and a simplified two-level version of the nuclear shell model. A method of reducing the number of qubits needed for practical calculation is presented, the reduction being with respect to that used when the standard Jordan-Wigner encoding is used. Its efficacy is shown in the case of the deuteron binding and shell model. A version of the variational quantum eigensolver in which all eigenstates in a spectrum are targeted on an equal basis is shown. The method involves finding the minima of the variance of the Hamiltonian, and its ability to find the full spectrum of small version of the simplified shell model is presented.