Observation of Improved Accuracy over Classical Sparse Ground-State Solvers using a Quantum Computer

TL;DR

This work resolves the previously open question of whether a sample-based quantum diagonalization algorithm can outperform standard selected configuration interaction heuristics.

Abstract

We experimentally demonstrate that a hybrid quantum-classical algorithm can outperform purely classical, off-the-shelf selected configuration interaction methods. First, we construct a class of local Hamiltonian problems with sparse ground states, and show that representative classical heuristics fail to find the ground state of a specific 49-qubit instance. Next, we show that the sample-based Krylov quantum diagonalization algorithm, run on an IBM Heron R3 processor, succeeds at the same task. This algorithm uses quantum samples from a grid of time-evolved quantum states, and offers provable convergence guarantees for sparse ground state problems with guiding states. While the problem is also solvable classically using two iterative solvers that we designed specifically to target our Hamiltonian construction, this work resolves the previously open question of whether a sample-based quantum diagonalization algorithm can outperform standard selected configuration interaction heuristics.

Figures (22)

• Figure 1: Sketch of the general Hamiltonian construction. The qubit layout is partitioned into disjoint patches. Within each patch $P_i$, the set $S_0$ of support configurations is coupled to a set $S_1$ of first-order configurations, which are then coupled to the remaining configurations of the qubits in the patch. This defines the patch Hamiltonian $H_{P_i}$. The patches are themselves coupled by $H_\mathrm{coupling}$.
• Figure 2: (a) The sequence of fidelities between all eigenvectors of the $i$th and $(i+1)$th principal submatrices of $H_{S_0}$\ref{['v36_base']}, for each $i=1,2,...,7$. The $j$th eigenvector of the $i$th principal submatrix is labeled $\psi_j^{(i)}$, with $j$ in order of increasing energy. This illustrates that from dimensions 2 to 7, the ground eigenvector remains the same, while from dimension $7$ to $8$ it becomes the first excited state due to the level crossing, which appears visually as displacement of the main nonzero fidelities off of the diagonal in the upper-left of the largest heatmap. The final ground state (whose fidelities are given by the first row in the rightmost heatmap) therefore has no overlap with the previous ground state; it has only $\sim62\%$ fidelity with the entire previous basis (of which $\sim52\%$ is with previous first excited state) so it is significantly different in character from any state that precedes it in iteration. b) The level crossing that is responsible for this new character, made explicit by continuously turning on the final off-diagonal row and column in $H_{S_0}$ via a parameter $\eta$ (i.e., replacing the $1$s in entries $(6,7)$ and $(7,6)$ in \ref{['v36_base']} with $\eta$). Each curve is the evolution of one energy eigenvalue with respect to $\eta$. The level crossing between the lowest two energies is apparent around $\eta=0.8$.
• Figure 3: Layout of the patches defining our Hamiltonian within a heavy-hex graph comprising six hexes. The paths defining each patch are shown by the red arrows. As discussed in \ref{['v57']}, patches are defined by paths with even-indexed qubits corresponding to the support configurations: in this figure, the directed edges point in the direction of increasing index. The edges in the physical qubit layout are shown in light blue wherever they do not overlap with the path edges. Qubit $48$ in the upper right corner is a padding qubit, to complete the sixth hex.
• Figure 4: Weights and maximum distances within the qubit layout of Pauli terms in the Hamiltonian. Weight refers to the number of qubits the Pauli term acts upon. Maximum distance refers to the longest length of the minimum path between any pair of qubits the Pauli term acts upon, with respect to the qubit layout shown in \ref{['fig:v57_layout']}. These distances are given as the number of qubits traversed, including the first and last qubits, so for example a Pauli term acting only on a pair of neighboring qubits would have maximum distance two, while a Pauli acting on three qubits connected along a line would have maximum distance three.
• Figure 5: Improved accuracy of quantum (SKQD) against off-the-shelf classical SCI methods. Comparison of the ground state energy reached for a given subspace dimension for different SCI methods, and the SKQD experiment executed on quantum hardware. Different SCI methods are shown in different panels by the hexagonal markers. The different colors show the number of iterations. The results obtained with SKQD are shown by blue diamond markers in each panel. The sequence of SQKD points is with respect to Krylov dimension.

Theorems & Definitions (14)

• Theorem 1
• proof
• Lemma 2
• proof
• Lemma 3
• proof
• Lemma 4
• proof
• Lemma 5
• proof