[Paper Review] Quantum Assisted Eigensolver

Kishor Bharti|arXiv (Cornell University)|Sep 23, 2020

Advanced Fiber Laser Technologies1 references19 citations

TL;DR

This paper proposes a hybrid quantum-classical algorithm called the Quantum Assisted Eigensolver (QAE) for approximating the ground state and ground state energy of a Hamiltonian. Unlike variational quantum eigensolvers, QAE eliminates the classical-quantum feedback loop by using quantum hardware solely to compute overlap matrices D and E, which are then used in a classical quadratically constrained quadratic program (QCQP) to find the optimal ansatz coefficients. The method provides efficiently computable lower bounds via convex relaxation and a sufficient condition for global optimality, enabling reliable convergence.

ABSTRACT

We propose a hybrid quantum-classical algorithm for approximating the ground state and ground state energy of a Hamiltonian. Once the Ansatz has been decided, the quantum part of the algorithm involves the calculation of two overlap matrices. The output from the quantum part of the algorithm is utilized as input for the classical computer. The classical part of the algorithm is a quadratically constrained quadratic program with a single quadratic equality constraint. Unlike the variational quantum eigensolver algorithm, our algorithm does not have any classical-quantum feedback loop. Using convex relaxation techniques, we provide an efficiently computable lower bound to the classical optimization program. Furthermore, using results from Bar-On et al. (Journal of Optimization Theory and Applications, 82(2):379--386, 1994), we provide a sufficient condition for a local minimum to be a global minimum. A solver can use such a condition as a stopping criterion.

Motivation & Objective

• To develop a hybrid quantum-classical algorithm for approximating the ground state and energy of a Hamiltonian in the NISQ era.
• To eliminate the classical-quantum feedback loop present in variational quantum algorithms like VQE, which suffer from barren platet avoiding training issues.
• To provide a classical optimization framework based on QCQP with a single quadratic equality constraint, enabling theoretical guarantees and efficient lower bounds.
• To use convex relaxation techniques (Lagrangian and semidefinite relaxation) to compute efficiently computable lower bounds on the ground state energy for a given ansatz.
• To establish a sufficient condition for a local minimum to be globally optimal, enabling solvers to use this as a stopping criterion.

Proposed method

• The algorithm begins with selecting an ansatz as a linear combination of m efficiently preparable quantum states |φⱼ⟩, parameterized by complex coefficients α.
• Quantum hardware computes matrix elements of the form ⟨φⱼ|Uᵢ|φₖ⟩ for each unitary Uᵢ in the Hamiltonian H = Σᵢ βᵢUᵢ, forming the matrix D with entries Dⱼₖ = Σᵢ βᵢ⟨φⱼ|Uᵢ|φₖ⟩.
• The overlap matrix E is computed from the inner products ⟨φⱼ|φₖ⟩, representing the normalization structure of the ansatz.
• The classical optimization step solves a QCQP: minimize ⟨H(α)⟩ = αᵀDα subject to αᵀEα = 1, where α ∈ ℂᵐ.
• Convex relaxation techniques—Lagrangian and semidefinite relaxation—are applied to compute efficiently computable lower bounds on the optimal energy.
• A sufficient condition for a local minimum to be global is derived using results from Bar-On et al. (1994), enabling solvers to terminate when this condition is met.

Experimental results

Research questions

• RQ1 Can a hybrid quantum-classical algorithm approximate the ground state energy of a Hamiltonian without requiring iterative classical-quantum feedback?
• RQ2 Can the use of overlap matrices D and E from quantum hardware enable a classical QCQP to yield accurate ground state approximations?
• RQ3 Can convex relaxation techniques provide efficiently computable lower bounds on the ground state energy for a given ansatz?
• RQ4 Is there a sufficient condition that allows a classical solver to verify global optimality in the non-convex QCQP, thus enabling reliable convergence?
• RQ5 How does the performance of this algorithm compare to VQE in terms of robustness to noise and barren platet avoiding?

Key findings

• For a toy model with a = 0.4 and b = 0.2, the algorithm computes the ground state energy as -0.824621, matching the true ground state energy to six decimal places.
• The optimal ansatz coefficients are found to be α = [-0.87033111, 0.1221769, 0.1221776]ᵀ, demonstrating convergence to a high-accuracy solution.
• The algorithm provides a lower bound on the ground state energy via convex relaxation, which is computationally efficient and valid for any given ansatz.
• A sufficient condition for a local minimum to be global is derived, allowing solvers to use this as a stopping criterion, enhancing reliability.
• The method avoids the barren plateau problem inherent in VQE by eliminating parameter tuning and feedback loops, relying instead on a single-shot quantum computation of matrix elements.
• The success of the algorithm is highly dependent on the expressivity of the ansatz; a poor ansatz may lead to poor approximations, highlighting the importance of ansatz design.