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arXiv:2203.01478 · Physical Review C

Simulating excited states of the Lipkin model on a quantum computer

Manqoba Q. Hlatshwayo, Yinu Zhang, Herlik Wibowo, Ryan LaRose, Denis Lacroix A, Елена Литвинова

Published Aug 18, 2022• 31 citations
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Abstract

We simulate the excited states of the Lipkin model using the recently proposed quantum equation of motion (qEOM) method. The qEOM generalizes the EOM on classical computers and gives access to collective excitations based on quasiboson operators ${\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{O}}_{n}^{\ifmmode\dagger\else\textdagger\fi{}}(\ensuremath{\alpha})$ of increasing configuration complexity $\ensuremath{\alpha}$. We show, in particular, that the accuracy strongly depends on the fermion to qubit encoding. Standard encoding leads to large errors, but the use of symmetries and the Gray code reduces the quantum resources and improves significantly the results on current noisy quantum devices. With this encoding scheme, we use IBM quantum machines to compute the energy spectrum for a system of $N=2$, 3, and 4 particles, and compare the accuracy against the exact solution. We found that the results of the approach with $\ensuremath{\alpha}=2$, an analog of the second random phase approximation (SRPA), are, in principle, more accurate than with $\ensuremath{\alpha}=1$, which corresponds to the random phase approximation (RPA), but the SRPA is more amenable to noise for large coupling strengths. Thus, the proposed scheme shows potential for achieving higher spectroscopic accuracy by implementations with higher configuration complexity, if a proper error mitigation method is applied.

DOI
10.1103/physrevc.106.024319

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We simulate the excited states of the Lipkin model using the recently proposed quantum equation of motion (qEOM) method. The qEOM generalizes the EOM on classical computers and gives access to collective excitations based on quasiboson operators O\ifmmode^\elseˆ\fin\ifmmode†\else\textdagger\fi(\ensuremathα){\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{O}}_{n}^{\ifmmode\dagger\else\textdagger\fi{}}(\ensuremath{\alpha})O\ifmmode^\elseˆ\fi​n\ifmmode†\else\textdagger\fi​(\ensuremathα) of increasing configuration complexity \ensuremathα\ensuremath{\alpha}\ensuremathα. We show, in particular, that the accuracy strongly depends on the fermion to qubit encoding. Standard encoding leads to large errors, but the use of symmetries and the Gray code reduces the quantum resources and improves significantly the results on current noisy quantum devices. With this encoding scheme, we use IBM quantum machines to compute the energy spectrum for a system of N=2N=2N=2, 3, and 4 particles, and compare the accuracy against the exact solution. We found that the results of the approach with \ensuremathα=2\ensuremath{\alpha}=2\ensuremathα=2, an analog of the second random phase approximation (SRPA), are, in principle, more accurate than with \ensuremathα=1\ensuremath{\alpha}=1\ensuremathα=1, which corresponds to the random phase approximation (RPA), but the SRPA is more amenable to noise for large coupling strengths. Thus, the proposed scheme shows potential for achieving higher spectroscopic accuracy by implementations with higher configuration complexity, if a proper error mitigation method is applied.
We investigate the quantum equation of motion (qEOM), a hybrid quantum-classical algorithm for computing excitation properties of a fermionic many-body system, with a particular emphasis on the strong-coupling regime. The method is designed as a stepping stone towards building more accurate solutions for strongly coupled fermionic systems, such as medium-heavy nuclei, using quantum algorithms to surpass the current barrier in classical computation. Approximations of increasing accuracy to the exact solution of the Lipkin-Meshkov-Glick Hamiltonian with N=8N=8N=8 particles are studied on digital simulators and IBM quantum devices. Improved accuracy is achieved by applying operators of growing complexity to generate excitations above the correlated ground state, which is determined by the variational quantum eigensolver. We demonstrate explicitly that the qEOM exhibits a quantum benefit due to the independence of the number of required quantum measurements from the configuration complexity. Postprocessing examination shows that quantum device errors are amplified by increasing configuration complexity and coupling strength. A detailed error analysis is presented, and error mitigation based on zero noise extrapolation is implemented.
Problems in quantum chemical simulations, especially achieving accurate excited-state potential energy surfaces, are among the primary applications to achieve quantum utility. On near-term quantum hardware, variants of the variational quantum eigensolver (VQE) algorithms are the primary choice for chemistry simulation. In this study, a combination of leading ground and excited state quantum algorithms for general excited states, namely, ADAPT-VQE/LUCJ and q-sc-EOM, are utilized to calculate accurate excited state potential energy surfaces in challenging bond-breaking scenarios and compared with the classical scalable EOM-CCSD method. This work investigates avenues toward quantum utility in excited-state quantum chemistry using the q-sc-EOM approach. We assess its accuracy while mitigating major scaling bottlenecks through the Davidson algorithm and basis rotation grouping, reducing the measurement scaling from O(N12^{12}12) to O(N5^{5}5), and implementing the method on quantum hardware with various error mitigation strategies to reduce gate and measurement errors in excited states. The hardware implementation of the q-sc-EOM algorithm, augmented by mitigation of M3 readout error and symmetry projection, produces reasonably accurate excited-state energies with gate noise identified as the predominant source of error. This paves the way for accurate and scalable, generally applicable quantum excited-state methods with potential for quantum utility while identifying critical problems that require advancements.
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.
Simulation of electronic structure is one of the most promising applications on noisy intermediate-scale quantum (NISQ) era devices. However, NISQ devices suffer from a number of challenges like limited qubit connectivity, short coherence times, and sizable gate error rates. Thus, desired quantum algorithms should require shallow circuit depths and low qubit counts to take advantage of these devices. Here, we attempt to reduce quantum resource requirements for molecular simulations on a quantum computer while maintaining the desired accuracy with the help of classical quantum chemical theories of canonical transformation and explicit correlation. In this work, compact ab initio Hamiltonians are generated classically, in the second quantized form, through an approximate similarity transformation of the Hamiltonian with (a) an explicitly correlated two-body unitary operator with generalized pair excitations that remove the Coulombic electron-electron singularities from the Hamiltonian and (b) a unitary one-body operator to efficiently capture the orbital relaxation effects required for accurate description of the excited states. The resulting transcorrelated Hamiltonians are able to describe both the ground and the excited states of molecular systems in a balanced manner. Using the variational quantum eigensolver (VQE) method based on the unitary coupled cluster with singles and doubles (UCCSD) ansatz and only a minimal basis set (ANO-RCC-MB), we demonstrate that the transcorrelated Hamiltonians can produce ground state energies comparable to the reference CCSD energies with the much larger cc-pVTZ basis set. This leads to a reduction in the number of required CNOT gates by more than 3 orders of magnitude for the chemical species studied in this work. Furthermore, using the quantum equation of motion (qEOM) formalism in conjunction with the transcorrelated Hamiltonian, we are able to reduce the deviations in the excitation energies from the reference EOM-CCSD/cc-pVTZ values by an order of magnitude. The transcorrelated Hamiltonians developed here are Hermitian and contain only one- and two-body interaction terms and thus can be easily combined with any quantum algorithm for accurate electronic structure simulations.
Quantum chemistry is a promising application in the era of quantum computing since the unique effects of quantum mechanics that take exponential growing resources to simulate classically are controllable on quantum computers. Fermionic degrees of freedom can be encoded efficiently onto qubits and allow for algorithms such as the quantum equation-of-motion method to find the entire energy spectrum of a quantum system. In this paper, we propose the reduced quantum equation-of-motion method by reducing the dimensionality of its generalized eigenvalue equation, which results in half the measurements required compared to the quantum equation-of-motion method, leading to speed up the algorithm and less noise accumulation on real devices. In particular, we analyze the performance of our method on two noise models and calculate the excitation energies of a bulk silicon and gallium arsenide using our method on an IBM quantum processor. Our method is fully robust to the uniform depolarizing error, and we demonstrate that the selection of suitable atomic orbital complexity could increase the robustness of our algorithm under real noise. We also find that taking the average of multiple experiments tends toward the correct energies due to the fluctuations around the exact values. Such noise resilience of our approach could be used on current quantum devices to solve quantum chemistry problems.
Solving challenging problems in quantum chemistry is one of the most promising applications of quantum computers. Within the quantum algorithms proposed for problems in excited state quantum chemistry, subspace-based quantum algorithms, including quantum subspace expansion (QSE), quantum equation of motion (qEOM) and quantum self-consistent equation-of-motion (q-sc-EOM), are promising for pre-fault-tolerant quantum devices. The working equation of QSE and qEOM requires solving a generalized eigenvalue equation with associated matrix elements measured on a quantum computer. Our careful analytical and numerical analysis of the standard and generalized eigenvalue problems, especially in the context of excited-state methods, shows that the errors in eigenvalues magnify drastically with an increase in the condition number of the overlap matrix when a generalized eigenvalue equation is solved in the presence of statistical sampling errors. This makes methods such as QSE unstable to errors that are unavoidable when using quantum computers. Further, at very high condition numbers of the overlap matrix, the QSE's working equation could not be solved without any additional steps in the presence of sampling errors, as it becomes ill-conditioned. It was possible to use the thresholding technique in this case to solve the equation, but the solutions achieved had missing excited states, which may be a problem for future chemical studies. We also show that excited-state methods that have an eigenvalue equation as the working equation, such as q-sc-EOM, do not have the problems associated with the condition number and could be generally more stable to errors, and therefore, more suitable candidates for excited-state quantum chemistry calculations using quantum computers.