Quantum-Centric Algorithm for Sample-Based Krylov Diagonalization

TL;DR

The paper introduces SKQD, a near-term quantum-classical algorithm that combines Krylov subspace methods with sample-based diagonalization to estimate ground-state energies. By time-evolving a reference state and sampling the resulting Krylov states, SKQD forms a manageable subspace that is diagonized classically, with convergence guaranteed under ground-state sparsity and nonzero overlap with the reference. The authors demonstrate scalability on large impurity-models (SIAM with 41 bath sites and a 4-impurity system) using heterogeneous quantum-classical resources, achieving results in close agreement with DMRG. They also highlight SKQD’s noise resilience and reduced circuit-depth advantages, outlining its potential for lattice problems on pre-fault-tolerant hardware while noting current limitations for ab initio applications.

Abstract

Approximating the ground state of many-body systems is a key computational bottleneck underlying important applications in physics and chemistry. The most widely known quantum algorithm for ground state approximation, quantum phase estimation, is out of reach of current quantum processors due to its high circuit-depths. Subspace-based quantum diagonalization methods offer a viable alternative for pre- and early-fault-tolerant quantum computers. Here, we introduce a quantum diagonalization algorithm which combines two key ideas on quantum subspaces: a classical diagonalization based on quantum samples, and subspaces constructed with quantum Krylov states. We prove that our algorithm converges in polynomial time under the working assumptions of Krylov quantum diagonalization and sparseness of the ground state. We then demonstrate the scalability of our approach by performing the largest ground-state quantum simulation of impurity models using a Heron quantum processors and the Frontier supercomputer. We consider both the single-impurity Anderson model with 41 bath sites, and a system with 4 impurities and 7 bath sites per impurity. Our results are in excellent agreement with Density Matrix Renormalization Group calculations.

Figures (9)

• Figure 1: SQKD algorithm. A Krylov subspace is constructed by the time evolution of a reference state $|\psi_0\rangle$ to $d$ different times. At the end of each circuit, the state is measured in the computational basis, yielding a set of measurement outcomes $\mathcal{X}$. The computational basis states sampled in $\mathcal{X}$ are used to span an approximation to the ground state of the system $|\Psi\rangle = \sum_j \mathbf{\Psi}_j |b_j\rangle$ for $b_j \in \mathcal{X}$. The components $\mathbf{\Psi}_j$ are obtained in closed form by the diagonalization of the projection of $H$ in the subspace spanned by the sampled bitstrings.
• Figure 2: SQKD experimental workflow for the ground state of four-impurity model. From left to right: the 4-impurity model in the basis where the baths are diagonal, with the corresponding one-body matrix elements of the Hamiltonian $h_{pq}$. The brown box shows the impurity modes. SKQD is first run in this basis. The first step is the compilation of the free-fermion time evolution into a shallow circuit of Givens rotations. Then, measurement realizations are collected from the quantum device at each Trotter step, followed by an SQD ground-state estimation that uses the configuration recovery procedure, as introduced in Ref. ibm2024chemistry. The one-body reduced density matrix $\Gamma_{\mathbf{kk}'}$ is used to identify $\mathbf{k}$-adjacent natural orbitals, where the impurity mode is only allowed to be mixed with the bath modes corresponding to $\mathbf{k}_f$ and $\mathbf{k}_f- 1$. The resulting Hamiltonian is one where the one-body matrix elements $h_{pq}$ are close to diagonal deep in the Fermi sea and for large values of $\mathbf{k}$, and with off-diagonal two-body matrix elements. SKQD is run in this new basis, requiring the approximate compilation of the free-fermion evolution, and the efficient compilation into a constant-depth circuit of the off-diagonal two-body terms.
• Figure 3: Experiments on quantum processors, and comparison against DMRG.(a)-(c) SIAM with 41 bath sites (85-qubit experiment). (a) Shows the qubit layout, where red and blue qubits correspond to spin-up and spin-down degrees of freedom respectively. The qubit marked with a cross is the qubit representing the impurity. The green qubit is an auxiliary qubit used to implement the time evolution of the Hubbard interaction. (b) Ground state energy error relative to the DMRG estimation as a function of the inverse of the SKQD subspace dimension $D$. Different colors correspond to different values of the onsite repulsion $U$ as indicated in the legend. The Hartree-Fock energies (RHF) and Coupled Cluster with Singles and Doubles (CCSD) errors are shown for reference. (c) Comparison of the two-point spin $\bar{C}_\textrm{S}$ and density $\bar{C}_\textrm{n}$ correlation functions (see Eqs. \ref{['eq: spin correlation']} and \ref{['eq: density correlation']}) obtained with DMRG and SKQD. (d)-(f) 4-impurity model ($L = 4$) with $K = 7$ bath models per impurity (70-qubit experiment). (d) Shows the qubit layout, where red and blue qubits correspond to spin-up and spin-down degrees of freedom respectively. The qubits marked with a cross correspond to the qubits representing the impurity. The green qubits are auxiliary qubits used to implement the time evolution of the Hubbard interaction. (e) SQKD ground state energy estimation in the $\mathbf{k}$-adjacent NOs basis as a function of the configuration recovery step. The HCI and DMRG energy estimations are shown for reference. (f) Effect of orbital optimizations applied to the converged SKQD ground state estimation. The error in the ground-state energy relative to the converged DMRG energy is shown as a function of the self-consistent orbital optimization cycle. The dots connected by the dashed lines correspond to the configuration recovery trajectory shown in panel (e). The solid lines show the improvement of the energy error when the orbitals are optimized to minimize the energy for fixed wave function coefficients. The dots show the error after re-diagonalizing the Hamiltonian in the new basis found by the orbital optimization procedure. The RHF, CISD and HCI errors are shown for reference.
• Figure 4: Dispersion relation of the bath for the 4-impurity model. See Equation \ref{['eq: bath Ham main']}.
• Figure 5: Ground-state energy of the four-impurity model using the classical methods RHF, MP2, CISD, CCSD, and CCSD(T), HCI with truncation threshold $10^{-7}$ , and DMRG with bond dimension $D=4000$.

Theorems & Definitions (17)

• Definition 1: $(\alpha_L, \beta_L)$-sparsity
• Theorem 1
• Theorem 2: Theorem 3.1 in Epperly_2022
• Lemma 1: A state with low energy is close to the ground state
• proof
• proof
• Lemma 3: Each relevant bitstring appears nontrivially in at least one Krylov basis state
• proof
• Lemma 4
• proof