Quantum chemistry with provable convergence via randomized sample-based quantum diagonalization

TL;DR

This paper addresses the challenge of performing reliable quantum-chemistry calculations on near-term quantum devices by extending Krylov-based subspace diagonalization to a hardware-friendly regime. It introduces SqDRIFT, which merges Sample-based Krylov Quantum Diagonalization (SKQD) with the qDRIFT randomized compilation to achieve provable convergence using shallower circuits for molecular Hamiltonians. Theoretical guarantees quantify energy errors and failure probabilities under concentration assumptions, while numerical simulations and IBM hardware experiments demonstrate accurate ground-state energies for polycyclic aromatic hydrocarbons up to coronene. The work also clarifies practical design choices, including orbital basis selection and fermion-to-qubit mappings, and outlines pathways for further extensions to excited states and embedding techniques.

Abstract

Sample-based quantum diagonalization (SQD) is a recently proposed algorithm to approximate the ground-state wave function of many-body quantum systems on near-term and early-fault-tolerant quantum devices. In SQD, the quantum computer acts as a sampling engine that generates the subspace in which the Hamiltonian is classically diagonalized. A recently proposed SQD variant, Sample-based Krylov Quantum Diagonalization (SKQD), uses quantum Krylov states as circuits from which samples are collected. Convergence guarantees can be derived for SKQD under similar assumptions to those of quantum phase estimation, provided that the ground-state wave function is concentrated, i.e., has support on a small subset of the full Hilbert space. Implementations of SKQD on current utility-scale quantum computers are limited by the depth of time-evolution circuits needed to generate Krylov vectors. For many complex many-body Hamiltonians of interest, such as the molecular electronic-structure Hamiltonian, this depth exceeds the capability of state-of-the-art quantum processors. In this work, we introduce a new SQD variant that combines SKQD with the qDRIFT randomized compilation of the Hamiltonian propagator. The resulting algorithm, termed SqDRIFT, enables SQD calculations at the utility scale on chemical Hamiltonians while preserving the convergence guarantees of SKQD. We apply SqDRIFT to calculate the electronic ground-state energy of several polycyclic aromatic hydrocarbons, up to system sizes beyond the reach of exact diagonalization.

Figures (6)

• Figure 1: SqDRIFT quantum-centric supercomputing workflow. The diagram is to be read from top to bottom, left to right. Processes are indicated above the arrows, while results are enclosed in boxes. For details on the various steps, see the main text in the referenced sections.
• Figure 2: F2Q layouting example. Given a set of excitations, exemplified here by two single excitations, no optimization of the routing of fermionic spin orbitals to the qubit register leads to a naive layout with potentially high-weight Paulis. In contrast, an optimized layout leads to lower-weight Paulis (notice the re-shuffled indices), resulting in significant savings in terms of circuit depth. Note that, in principle, there is no reason why indices $1$ and $4$ should not be further placed next to each other. However, since the optimization must balance the weight of all excitations in the currently sampled batch, the optimal solution may not lead to minimal Pauli weight for every single resulting Pauli string and, here, only an exemplary subset is shown for illustration purposes.
• Figure 3: Two-qubit circuit depth improvement due to the fermion-to-qubit layout optimization. We plot the two-qubit gate depth of the transpiled SqDRIFT circuits (for $k=1$; these results are independent of $k$) without and with the fermion-to-qubit layout optimization along the $x$ and $y$ axis, respectively. Thus, if a data point lies below the (red) diagonal, the F2Q layout optimization has decreased the two-qubit gate circuit depth. The two panels show the results obtained for the HF (a) and LO (b) orbitals, respectively. The number of excitations included in each randomized circuit is indicated by the color shade.
• Figure 4: Convergence of the energy error of naphthalene with the length and number of SqDRIFT sequences. In both panels we plot the energy error of noiseless SQD simulations with respect to the exact energy obtained by diagonalization of the full system Hamiltonian. The hollow circles in both panels correspond to repetitive SQD simulations where each one is given a randomly selected $90\%$ of the available unique samples obtained from the noiseless quantum circuit simulations. (a) Along the $x$ axis we vary the number of excitations sampled from the Hamiltonian (i.e. the SqDRIFT length) for each randomized circuit. The green and blue circles indicate the Hatree-Fock (HF) and localized (LO) orbitals, respectively. The HF and Configuration-Interaction including Single and Double Excitations (CISD) reference values are indicated by the dashed dark grey and purple line, respectively. The red line corresponds to the SQD energy obtained from the noiseless samples based collected from the LUCJ ansatz with its parameters fixed to the $t1$ and $t2$ amplitudes of a prior CCSD calculation. (b) Here the $x$ axis indicates the increasing number of SqDRIFT randomizations from which the samples are included in the pool of available bitstrings to sample from. The green and blue lines indicate the convergence of the energy error within the HF and LO orbitals, respectively. The shade of color indicates the number of excitations included in each randomized circuit (as indicated in subfigure a) as well as the overlaid number. In the case of the HF orbitals, the lines of $100$ and $200$ excitations are superimposed.
• Figure 5: Energy error of naphthalene as a function of the diagonalization subspace dimension. Analogously to \ref{['fig:ConvergenceQDriftParams']} we plot the energy error with respect to the exact energy obtained by diagonalization of the full system Hamiltonian. Here, the $x$ axis indicates the varying size of the diagonalization subspace. (a) For the results obtained from noiseless circuit simulations, the distributed data is obtained by taking increasing percentages from $\{10, 20,\dots, 100\%\}$ of the number of unique samples collected. Each such simulation is repeated $10$ times (indicated by the hollow circles) for the data obtained from different SqDRIFT circuit depths (as indicated by the color shade akin to \ref{['fig:ConvergenceQDriftParams']}). Since the subspace dimension cannot be controlled directly, the different simulations place slightly differently along the $x$ axis. A cross with error bars is placed at the center of each set of repetitive simulations. (b) The lowest energy estimates obtained throughout $3$ iterations of SQD post-processing and configuration recovery (CR) applied to the samples obtained from hardware experiments. Circles are indicative of the lowest energy estimate obtained without the use of CR while triangles indicate that CR resulted in an energy decrease. The following classical computations serve as references: Hartree-Fock (dark grey dashed; $-383.384\ \mathrm{Ha}$), CISD (purple dashed; $-383.4866\ \mathrm{Ha}$), CCSD (purple dotted; $-383.4984\ \mathrm{Ha}$).

Theorems & Definitions (3)

• Theorem 1: SqDRIFT convergence guarantees
• proof
• Lemma A.1: Krylov quantum diagonalization with qDRIFT