On the generalized eigenvalue problem in subspace-based excited state methods for quantum computers

TL;DR

It is shown that excited-state methods that have an eigenvalue equation as the working equation, such as q-sc-EOM, do not have the problems associated with the condition number and could be generally more stable to errors, and therefore, more suitable candidates for excited-state quantum chemistry calculations using quantum computers.

Abstract

Solving challenging problems in quantum chemistry is one of the most promising applications of quantum computers. Within the quantum algorithms proposed for problems in excited state quantum chemistry, subspace-based quantum algorithms, including quantum subspace expansion (QSE), quantum equation of motion (qEOM) and quantum self-consistent equation-of-motion (q-sc-EOM), are promising for pre-fault-tolerant quantum devices. The working equation of QSE and qEOM requires solving a generalized eigenvalue equation with associated matrix elements measured on a quantum computer. Our careful analytical and numerical analysis of the standard and generalized eigenvalue problems, especially in the context of excited-state methods, shows that the errors in eigenvalues magnify drastically with an increase in the condition number of the overlap matrix when a generalized eigenvalue equation is solved in the presence of statistical sampling errors. This makes methods such as QSE unstable to errors that are unavoidable when using quantum computers. Further, at very high condition numbers of the overlap matrix, the QSE's working equation could not be solved without any additional steps in the presence of sampling errors, as it becomes ill-conditioned. It was possible to use the thresholding technique in this case to solve the equation, but the solutions achieved had missing excited states, which may be a problem for future chemical studies. We also show that excited-state methods that have an eigenvalue equation as the working equation, such as q-sc-EOM, do not have the problems associated with the condition number and could be generally more stable to errors, and therefore, more suitable candidates for excited-state quantum chemistry calculations using quantum computers.

Figures (8)

• Figure 1: Eigenstates computed using (a) QSE in the case of the low condition number of the overlap matrix (condition number is 7.1) and (b) q-sc-EOM. The solid lines represent the exact solution (corresponding to an infinite number of shots) while the spheres and bars represent the mean and variance of the data in 10 simulations using the number of shots per matrix element on the x-axis. The geometry of the H$_4$ molecule is shown in the figure. This figure illustrates that when the overlap matrix is well-conditioned ($\kappa(\mathbf{S})=7.1$), QSE and q-sc-EOM show comparable shot-noise convergence and variance, i.e., overlap-conditioning-driven error amplification is not dominant in this regime.
• Figure 2: Eigenstates computed using QSE in the case of increasing condition number of overlap matrix of QSE (condition number indicated on subfigures). The solid lines represent the exact solution (corresponding to an infinite number of shots) while the spheres and bars represent the mean and variance of the data in 10 simulations using the number of shots per matrix element on the x-axis. The geometry of the linear H$_4$ molecule is shown in the subfigures. This figure highlights the rapid amplification of sampling-noise sensitivity in QSE as $\kappa(\mathbf{S})$ increases (from 121.5 to 592.5), consistent with the $1/\lambda_{\min}(\mathbf{S})$ error-amplification factor in Eq. (24) and the shot-scaling trend in Eq. (28).
• Figure 3: Eigenstates computed using q-sc-EOM method for the H$_4$ molecular geometry with condition number of overlap matrix in QSE of 592.5. The solid lines, the spheres and the bars have the same meaning as in previous plots. This figure illustrates that q-sc-EOM remains stable for the same geometry where QSE exhibits a moderately ill-conditioned overlap matrix ($\kappa(\mathbf{S})=592.5$), consistent with the absence of overlap-matrix inversion when $\mathbf{S}=\mathbf{I}$.
• Figure 4: Eigenstates computed using modified QSE method with artificially exact overlap matrix for the H$_4$ molecular geometry with condition number of overlap matrix in QSE of 592.5. The solid lines, the spheres and the bars have the same meaning as in previous plots. This figure shows that using an (artificially) exact overlap matrix substantially reduces QSE errors, supporting the interpretation that noise in $\mathbf{S}$ and its subsequent inversion are key contributors to the observed instability.
• Figure 5: Error in lowest 5 excited states vs the number of shots for H$_{4}$ molecular geometry with condition number 592.5 (geometry shown in the figure). This figure emphasizes the resource implication of overlap conditioning: for $\kappa(\mathbf{S})=592.5$, QSE requires approximately an order-of-magnitude more shots than q-sc-EOM to achieve comparable accuracy for the lowest excited states, consistent with Eq. (28).