Scalable Quantum Computation of Highly Excited Eigenstates with Spectral Transforms

Abstract

We propose a natural application of Quantum Linear Systems Problem (QLSP) solvers such as the HHL algorithm to efficiently prepare highly excited interior eigenstates of physical Hamiltonians in a variational manner. This is enabled by the efficient computation of inverse expectation values, taking advantage of the QLSP solvers' exponentially better scaling in problem size without concealing exponentially costly pre/post-processing steps that usually accompanies it. We detail implementations of this scheme for both fault-tolerant and near-term quantum computers, analyse their efficiency and implementability, and discuss applications and simulation results in many-body physics and quantum chemistry that demonstrate its superior effectiveness and scalability over existing approaches.

Many-Body Excited States with a Contracted Quantum Eigensolver

“…Alternative techniques include equation of motion 25,26 or quantum linear response approaches. 27 These can yield entire sets of properties, such as excitation energies, dipole moments, response properties, etc., but they do not always directly provide the resulting states.…”

Measurement Optimization Techniques for Excited Electronic States in Near-Term Quantum Computing Algorithms

“…However, it is unclear how effective these optimized measurement techniques would perform in extensions of VQE developed for excited electronic states. 26−38 In popular excited state VQE algorithms, including quantum subspace expansion (QSE) 26−30 and multistate contraction (MC), 31,32 one has to measure many expectation values in addition to Note that while the quantum measurement problem also exists in quantum Krylov methods 39−43 and methods targeting highly excited states, 44 which are promising alternatives to QSE and MC-VQE, consideration of such methods is outside the scope of the current work. Furthermore, the performance analysis exploring the impact of the nonunit quantum gate fidelities and error mitigation techniques will be the topic of our future work.…”

“…Note that while the quantum measurement problem also exists in quantum Krylov methods − and methods targeting highly excited states, which are promising alternatives to QSE and MC-VQE, consideration of such methods is outside the scope of the current work. Furthermore, the performance analysis exploring the impact of the nonunit quantum gate fidelities and error mitigation techniques will be the topic of our future work.…”