Generalized Eigenvalue Problem in Subspace-Based Excited-State Methods for Quantum Computers.

Area of Science:

• Quantum computing applications in chemistry
• Development of quantum algorithms for chemical problems

Background:

• Quantum computers offer promising applications for solving complex quantum chemistry problems.
• Subspace-based quantum algorithms like QSE and qEOM are suitable for pre-fault-tolerant quantum devices.
• These algorithms often involve solving generalized eigenvalue equations using quantum computations.

Purpose of the Study:

• To analyze the stability and error propagation in subspace-based quantum algorithms for excited-state quantum chemistry.
• To compare the performance of QSE, qEOM, and quantum self-consistent equation-of-motion (q-sc-EOM) methods under realistic quantum computing error conditions.

Main Methods:

• Analytical and numerical analysis of standard and generalized eigenvalue problems.
• Investigation of error magnification in relation to the condition number of the overlap matrix.
• Evaluation of thresholding techniques for ill-conditioned equations and assessment of resulting excited states.

Main Results:

• Errors in eigenvalues drastically magnify with increased condition numbers in generalized eigenvalue problems, destabilizing QSE and qEOM.
• High condition numbers render QSE's working equation ill-conditioned, requiring thresholding which may omit crucial excited states.
• Excited-state methods using standard eigenvalue equations, like q-sc-EOM, are more stable and less sensitive to condition numbers.

Conclusions:

• QSE and qEOM methods are susceptible to instability in excited-state quantum chemistry calculations due to error magnification.
• Quantum self-consistent equation-of-motion (q-sc-EOM) demonstrates greater stability and is a more suitable candidate for near-term quantum computations in chemistry.
• Careful consideration of eigenvalue problem conditioning is crucial for developing robust quantum algorithms for chemistry.