Quantum block Krylov subspace projector algorithm for computing low-lying eigenenergies

TL;DR

The paper introduces the quantum block Krylov subspace projector (QBKSP), a multireference quantum Krylov method that uses real-time evolution to extract low-lying and degenerate eigenenergies of quantum many-body systems. By building a block Krylov subspace with multiple initial states and a linear time-grid, QBKSP achieves linear scaling in the number of Krylov iterations and relies on structured, compact quantum circuits to evaluate a minimal set of expectation values, from which a generalized eigenproblem is solved classically after regularization. The authors demonstrate that multiple reference states improve convergence and enable resolving degenerate states across model and molecular Hamiltonians, with robustness against finite sampling and hardware noise, while longer evolution times enhance spectral resolution at increased resource cost. The work provides practical guidance on reference-state selection, regularization thresholds, and evolution-time choices, and shows promising applicability to near-term quantum devices for both ground and excited-state calculations in quantum chemistry and condensed-matter models.

Abstract

Computing eigenvalues is a computationally intensive task central to numerous applications in the natural sciences. Toward this end, we investigate the quantum block Krylov subspace projector (QBKSP) algorithm - a multireference quantum Lanczos method designed to accurately compute low-lying eigenenergies, including degenerate ones, of quantum systems. We present three compact quantum circuits tailored to different problem settings for evaluating the required expectation values. To assess the impact of the number and fidelity of initial reference states, as well as time evolution duration, we perform error-free and limited-precision numerical simulations and quantum circuit simulations. Our results show that using multiple reference states significantly enhances convergence, particularly in precision-limited scenarios and in cases where a single reference state fails to capture all target eigenvalues. Additionally, the QBKSP algorithm allows for the determination of degenerate eigenstates and their multiplicities through appropriate convergence conditions.

Figures (18)

• Figure 1: Visualisation of the block-Toeplitz structure of the overlap matrix $S$ and the expectation values in $T$. Each block is of size $B \times B$, where $B$ is the number of reference states, and the different colors represent overlaps of the block Krylov reference states evolved with a given time difference $\Delta t = k \tau$, where $\tau$ is the chosen time step and $k \in {0, 1, ..., K+1}$, with $K$ being the total number of Krylov iterations. The sub-block structure is shown for the case where $\{|\psi^{(b)}_0\rangle\}_{b=1}^B$ and $\hat{H}$ are real, which results in the individual blocks being symmetric. The grey elements denote unit diagonals due to the conservation of the norm under unitary evolution. The different patterns represent transformations of the solid areas of the same color.
• Figure 2: Three quantum circuits for retrieving $\bra{\psi^\beta}U(t)\ket{\psi^\alpha}$, with $n_s$ being the number of qubits representing the system of interest. By replacing the gate $S^\dagger \slash I$ by $I$, the real part of the expectation value is determined, and by setting it to $S^\dagger$, the imaginary part is retrieved. In the first circuit $\ket{\psi^\beta}=V_{\beta\alpha} \ket{\psi^\alpha}$, in the other two $\ket{\psi^\beta}=W_\beta \ket{\mathbf{0}}$, and in all of them $\ket{\psi^\alpha}=W_\alpha \ket{\mathbf{0}}$. For each problem instance, initial reference states, and device characteristics, the quantum circuit version that leads to more efficient quantum circuits should be chosen. For further details on these three quantum circuits, see Appendix \ref{['app:quantumc']}.
• Figure 3: QBKSP convergence for the 10-site Heisenberg model, evaluated for different numbers of initial states and several overlap values. The top panel shows the convergence of the seven lowest eigenvalues as a function of both the Krylov iteration and the number of different quantum circuits required, whereas the bottom panel shows the absolute error with regard to the exact eigenvalues. The model contains three 2-fold degeneracies in the seven lowest-lying distinct eigenvalues. Note that we define chemical accuracy as errors below $1.6$ mHartree.
• Figure 4: Convergence behavior of the QBKSP algorithm for the Heisenberg model as a function of system size. To reflect a practical application of the method to a previously unsolved use-case, no knowledge about the exact eigenenergies is assumed. To terminate the algorithm, convergence is defined as two consecutive iterations where the calculated energy changes less then $0.1$ mHartree. The top panel displays the number of Krylov iterations required to reach convergence, while the bottom panel shows the error of the retrieved energy when compared to the exact energy. The convergence of the five lowest (non-degenerate) eigenvalues is shown for one, two, and three initial reference states, each with a fixed overlap of $0.5$. Notably, all the retrieved eigenvalues have errors below chemical accuracy.
• Figure 5: QBKSP algorithm applied to hydrogen fluoride using both one and four initial states. To compare the QBKSP performance at the same computational cost, the total number of quantum circuits was fixed to 192, i.e., 96 iterations for the single-reference case, and 8 for the four-reference one. The top panel shows the convergence of the three highest and three lowest eigenvalues as a function of the interatomic distance and the bottom panel displays the absolute error with respect to the exact energies.