Quantum Filter Diagonalization: Quantum Eigendecomposition without Full Quantum Phase Estimation (1909.08925v1)

Abstract: We develop a quantum filter diagonalization method (QFD) that lies somewhere between the variational quantum eigensolver (VQE) and the phase estimation algorithm (PEA) in terms of required quantum circuit resources and conceptual simplicity. QFD uses a set of of time-propagated guess states as a variational basis for approximate diagonalization of a sparse Pauli Hamiltonian. The variational coefficients of the basis functions are determined by the Rayleigh-Ritz procedure by classically solving a generalized eigenvalue problem in the space of time-propagated guess states. The matrix elements of the subspace Hamiltonian and subspace metric matrix are each determined in quantum circuits by a one-ancilla extended swap test, i.e., statistical convergence of a one-ancilla PEA circuit. These matrix elements can be determined by many parallel quantum circuit evaluations, and the final Ritz estimates for the eigenvectors can conceptually be prepared as a linear combination over separate quantum state preparation circuits. The QFD method naturally provides for the computation of ground-state, excited-state, and transition expectation values. We numerically demonstrate the potential of the method by classical simulations of the QFD algorithm for an N=8 octamer of BChl-a chromophores represented by an 8-qubit ab initio exciton model (AIEM) Hamiltonian. Using only a handful of time-displacement points and a coarse, variational Trotter expansion of the time propagation operators, the QFD method recovers an accurate prediction of the absorption spectrum.

• The paper introduces a novel hybrid approach that filters eigenstates by evolving time-propagated guess states to approximate spectral properties.
• It utilizes the Rayleigh-Ritz method and an extended swap test with a single ancilla qubit to accurately predict absorption spectra in an 8-qubit model.
• The algorithm balances the benefits of VQE and PEA, reducing circuit complexity while delivering reliable eigenvalue estimates on near-term quantum devices.

Quantum Filter Diagonalization: An Efficient Hybrid Quantum Algorithm

The research presented in this paper introduces a novel hybrid quantum-classical algorithm termed Quantum Filter Diagonalization (QFD). This approach is developed as a middle ground between two established quantum algorithms: the Variational Quantum Eigensolver (VQE) and Quantum Phase Estimation (PEA). The primary focus of QFD is to efficiently approximate the eigendecomposition of a large Hermitian matrix described in sparse Pauli form, which is a common requirement in quantum mechanics and classical optimization problems.

Methodology

QFD employs a variational basis constructed from a set of time-propagated guess states. These guess states, which serve as approximations of the targeted eigenstates, are updated based on time propagation using Trotterized time evolution operators. The variational coefficients associated with these basis states are determined using the Rayleigh-Ritz procedure, which involves solving a generalized eigenvalue problem. This solution is achieved via classical methods that make use of matrix elements obtained from quantum circuits, specifically through an extension of the swap test, which employs a single ancilla qubit.

The QFD method is capable of estimating both ground-state and excited-state energies along with transition properties. The researchers demonstrate the effectiveness of QFD through classical simulations involving an 8-qubit model Hamiltonian constructed for a linear stack of BChl-a chromophores. The results show that using a handful of time-displacement points and a coarse variational Trotter expansion, QFD can accurately predict the absorption spectrum.

Numerical Results

The results section of the paper provides numerical evidence supporting the efficacy of QFD. The researchers tested the method on an ab initio exciton model Hamiltonian representing a stack of chromophores. They found that the QFD approach, employing a limited number of time steps and Trotter expansions, was able to bring about results that significantly reduce errors in predicted excitation energies and oscillator strengths as compared to basic guess states.

Implications and Future Work

The QFD method offers a promising avenue for leveraging quantum resources without the full complexity of PEA, making it more viable for near-term quantum devices characterized by limited coherence times and noise. The approach paves the way for quantum algorithms that balance circuit depth with computational accuracy — a critical consideration in the NISQ era.

While this paper has focused on quantum Hamiltonian diagonalization, future research could explore the application of QFD in broader contexts, such as optimization problems represented by Ising models. Additionally, integrating the boosting techniques from quantum inverse iteration methods could potentially enhance the convergence and accuracy of the algorithm further.

In summary, Quantum Filter Diagonalization represents a valuable contribution to the quantum algorithm toolkit, particularly suitable for situations where VQE is too heuristic or PEA is too resource-intensive. This method's capacity to adapt to the limitations of current quantum hardware while achieving notable accuracy in its predictions positions it as a meaningful step forward in quantum computational approaches.