Quantum Many-Body Linear Algebra, Hamiltonian Moments, and a Coupled Cluster Inspired Framework

TL;DR

The paper addresses the challenge of computing ground-state properties via linear-algebra primitives by casting Hamiltonian moments $M_n=⟨Φ|H^n|Φ⟩$ into a quantum many-body framework. It introduces a coupled-cluster–inspired construction of Hamiltonian moments (moment CC, or mCC), deriving differential equations and diagrammatic relations to obtain truncated, tractable approximations such as mCCSD. Numerical experiments on systems like $H_{10}$ and $N_2$ show that mCC moments can reproduce exact or near-exact results in some regimes and, in stretched geometries, can improve over standard CC methods while avoiding nonlinear amplitude solutions. The work suggests a broader, non-perturbative path to quantum many-body linear algebra and hints at extensions to Green's-function and multi-reference approaches with potential practical impact for efficient ground-state estimation.

Abstract

We propose a general strategy to develop quantum many-body approximations of primitives in linear algebra algorithms. As a practical example, we introduce a coupled-cluster inspired framework to produce approximate Hamiltonian moments, and demonstrate its application in various linear algebra algorithms for ground state estimation. Through numerical examples, we illustrate the difference between the ground-state energies arising from quantum many-body linear algebra and those from the analogous many-body perturbation theory. Our results support the general idea of designing quantum many-body approximations outside of perturbation theory, providing a route to new algorithms and approximations.

Figures (5)

• Figure 1: Coupled Cluster Singles (CCS) diagrams and their mCCS counterparts for a 1-body Hamiltonian. Single (double) lines correspond to CCS (mCCS) amplitudes $\hat{T}_1 (\hat{W}_1)$. When generating $\hat{T}_1$ by the standard CC iteration, energy denominators appear, but this is not the case for $W_1$. Note also that for (c), translating from the CCS to mCCS diagrams introduces a combinatorial factor arising from the different time-orderings (here, the factor is 2, as seen by the two identical mCC diagrams in (c)).
• Figure 2: Convergence patterns of various quantum many-body linear algebra algorithms (power method, Chebyshev iteration, and Lanczos method) with approximate RmCCSD moments, compared to RCCSD and FCI(DMRG) on $\mathrm{H_{10}}(\mathrm{N_2})$. The symbols denote when the termination criterion has been reached.
• Figure 3: Potential energy curves of $\rm N_{2}$ obtained by the quantum many-body Lanczos algorithm with RmCC moments truncating to singles and doubles (RmCCSD, upper panel), and an approximate triples (RmCCSDT-1, lower panel). The dashed-and-dotted traces from top to bottom show Lanczos convergence with increasing iterations $k$.
• Figure 4: Potential energy curves of $\rm N_{2}$ obtained by the quantum many-body Lanczos algorithm with UmCC moments truncating to singles and doubles (UmCCSD, upper panel), and an approximate triples (UmCCSDT-1, lower panel). The dashed-and-dotted traces from top to bottom show Lanczos convergence with increasing iterations $k$.
• Figure 5: Potential energy curves of $\rm H_{10}$ ring obtained by the quantum many-body Lanczos algorithm with RmCC moments truncating to singles and doubles (RmCCSD). The dashed-and-dotted traces from top to bottom show Lanczos convergence with increasing iterations $k$.