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Variational Quantum Eigensolver in Compressed Space for Nearest-Neighbour Quadratic Fermionic Hamiltonians.

Guillermo Blázquez-Cruz, Pierre-Luc Dallaire-Demers

Published Oct 18, 2021• 0 citations
OpenAlex

Abstract

Variational quantum eigensolvers (VQE) are one of the possible tasks that can be performed in noisy intermediate-scale quantum (NISQ) computers. While one of the limitations of NISQ platforms is their restricted number of available qubits, the theory of compressable matchgate circuits can be used to circumvent this limitation for certain types of Hamiltonians. We show how VQE algorithms can be used to find the ground state of quadratic fermionic Hamiltonians, providing an expressible ansatz in a logarithmic number of qubits. In particular, for systems of $n$ orbitals encoded to 2-local qubit models with nearest neighbour interactions, the ground state energy can be evaluated with $O\left(\log n\right)$ sets of measurements. This result is invariant of the dimensions in which the $n$ sites are arranged.

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Variational quantum eigensolvers (VQE) are one of the possible tasks that can be performed in noisy intermediate-scale quantum (NISQ) computers. While one of the limitations of NISQ platforms is their restricted number of available qubits, the theory of compressable matchgate circuits can be used to circumvent this limitation for certain types of Hamiltonians. We show how VQE algorithms can be used to find the ground state of quadratic fermionic Hamiltonians, providing an expressible ansatz in a logarithmic number of qubits. In particular, for systems of nnn orbitals encoded to 2-local qubit models with nearest neighbour interactions, the ground state energy can be evaluated with O(log⁡n)O\left(\log n\right)O(logn) sets of measurements. This result is invariant of the dimensions in which the nnn sites are arranged.
Abstract Variational quantum algorithms aim at harnessing the power of noisy intermediate-scale quantum (NISQ) computers, by using a classical optimizer to train a parameterized quantum circuit to solve tractable quantum problems. The variational quantum eigensolver (VQE) is one of the aforementioned algorithms designed to determine the ground-state of many-body Hamiltonians. Here, we apply the VQE to study the ground-state properties of N -component fermions. With such knowledge, we study the persistent current of interacting SU( N ) fermions, which is employed to reliably map out the different quantum phases of the system. Our approach lays out the basis for a current-based quantum simulator of many-body systems that can be implemented on NISQ computers.
Quantum computers have the potential to deliver speed-ups for solving certain important problems that are intractable for classical counterparts, making them a promising avenue for advancing modern computation. However, many quantum algorithms require deep quantum circuits, which are challenging to implement on current noisy devices. To address this limitation, variational quantum algorithms (VQAs) have been actively developed, enabling practical quantum computing in the noisy intermediate-scale quantum (NISQ) era. Among them, the variational quantum eigensolver (VQE) stands out as a leading approach for solving problems in quantum chemistry, many-body physics, and even integer factorization. The VQE algorithm can be implemented on various quantum hardware platforms, including photonic systems, quantum dots, trapped ions, neutral atoms, and superconducting circuits. In particular, photonic platforms offer several advantages: they operate at room temperature, exhibit low decoherence, and support multiple degrees of freedom, making them suitable for scalable, high-dimensional quantum computation. Here we present methodologies for realizing VQE on photonic systems, highlighting their potential for practical quantum computing. We first provide a theoretical overview of the VQE framework, focusing on the procedure for variationally estimating ground state energies. We then explore how photonic systems can implement these processes, showing that a wide variety of problems can be addressed using either multiple qubit states or a single qudit state.
We perform a systematic investigation of variational forms (wave-function Ans\"atze), to determine the ground-state energies and properties of two-dimensional model fermionic systems on triangular lattices (with and without periodic boundary conditions), using the variational quantum eigensolver (VQE) algorithm. In particular, we focus on the nature of the entangler blocks which provide the most efficient convergence to the system ground state inasmuch as they use the minimal number of gate operations, which is key for the implementation of this algorithm in noisy intermediate-scale quantum computers. Using the concurrence measure, the amount of entanglement of the register qubits is monitored during the entire optimization process, illuminating its role in determining the efficiency of the convergence. Finally, we investigate the scaling of the VQE circuit depth as a function of the desired energy accuracy. We show that the number of gates required to reach a solution within an error $\ensuremath{\varepsilon}$ follows the Solovay-Kitaev scaling, $O[{log}_{10}^{c}(1/\ensuremath{\varepsilon})]$, with an exponent $c=1.31\ifmmode\pm\else\textpm\fi{}0.13$.
The variational quantum eigensolver (VQE) is one of the most representative quantum algorithms in the noisy intermediate-scale quantum (NISQ) era, and is generally speculated to deliver one of the first quantum advantages for the ground-state simulations of some nontrivial Hamiltonians. However, short quantum coherence time and limited availability of quantum hardware resources in the NISQ hardware strongly restrain the capacity and expressiveness of VQEs. In this Letter, we introduce the variational quantum-neural hybrid eigensolver (VQNHE) in which the shallow-circuit quantum Ansatz can be further enhanced by classical post-processing with neural networks. We show that the VQNHE consistently and significantly outperforms the VQE in simulating ground-state energies of quantum spins and molecules given the same amount of quantum resources. More importantly, we demonstrate that, for arbitrary postprocessing neural functions, the VQNHE only incurs a polynomial overhead of processing time and represents the first scalable method to exponentially accelerate the VQE with nonunitary postprocessing that can be efficiently implemented in the NISQ era.
Classical simulation of molecular electronic structure is often limited by the rapid increase in computational cost associated with describing electron correlation. The Variational Quantum Eigensolver (VQE) is one hybrid quantum-classical approach ideally adapted to the Noisy Intermediate-Scale Quantum (NISQ) regime. This work applies the VQE algorithm to compute ground state energies of a set of molecules:H2,H3+,LiH,HF,OH−,H2O,BH3,CH4\mathrm{H}_{2}, \mathrm{H}_{3}^{+}, \text{LiH}, \text{HF}, \text{OH}^{-}, \mathrm{H}_{2} \mathrm{O}, \text{BH}_{3}, \text{CH}_{4}H2​,H3+​,LiH,HF,OH−,H2​O,BH3​,CH4​, andNH3\text{NH}_{3}NH3​. The approach utilizes the Unitary Coupled Cluster for Single and Double excitations (UCCSD) ansatz along with the Bravyi-Kitaev transformation for fermionic-to-qubit encoding. Quantum simulation outputs are systematically benchmarked against Full Configuration Interaction (FCI), a numerically exact traditional approach, and Unrestricted Hartree-Fock (UHF). The results establish a robust consistency of the VQE-obtained molecular ground state energies against standard FCI and UHF benchmarks. This research provides a critical benchmark, confirming that VQE has the capability for high-precision molecular simulations and playing the role of a pivotal stepping stone toward revolutionizing applications in drug discovery and materials science.
The Variational Quantum Eigensolver (VQE), as a hybrid quantum-classical algorithm, is an important tool for effective quantum computing in the current noisy intermediate-scale quantum (NISQ) era. However, the traditional hardware-efficient ansatz without taking into account symmetries requires more computational resources to explore the unnecessary regions in the Hilbert space. The conventional Subspace-Search VQE (SSVQE) algorithm, which can calculate excited states, is also unable to effectively handle degenerate states since the loss function only contains the expectation value of the Hamiltonian. In this study, the energy eigenstates of the one-dimensional Fermi-Hubbard model with two lattice sites and the two-dimensional Hubbard model with four lattice sites are calculated. By incorporating symmetries into the quantum circuits and loss function, we find that both the ground state and excited state calculations are improved greatly compared to the case without symmetries. The enhancement in excited state calculations is particularly significant. This is because quantum circuits that conserve the particle number are used, and appropriate penalty terms are added to the loss function, enabling the optimization process to correctly identify degenerate states. The results are verified through repeated simulations.