Quantum Filter Diagonalization with Double-Factorized Hamiltonians

Abstract

We demonstrate a method that merges the quantum filter diagonalization (QFD) approach for hybrid quantum/classical solution of the time-independent electronic Schrödinger equation with a low-rank double factorization (DF) approach for the representation of the electronic Hamiltonian. In particular, we explore the use of sparse "compressed" double factorization (C-DF) truncation of the Hamiltonian within the time-propagation elements of QFD, while retaining a similarly compressed but numerically converged double-factorized representation of the Hamiltonian for the operator expectation values needed in the QFD quantum matrix elements. Together with significant circuit reduction optimizations and number-preserving post-selection/echo-sequencing error mitigation strategies, the method is found to provide accurate predictions for low-lying eigenspectra in a number of representative molecular systems, while requiring reasonably short circuit depths and modest measurement costs. The method is demonstrated by experiments on noise-free simulators, decoherence-and shot-noise including simulators, and real quantum hardware.

Analytical Ground- and Excited-State Gradients for Molecular Electronic Structure Theory from Hybrid Quantum/Classical Methods

“…This leads directly to the rather verbose representation of the Hamiltonian and quantum number operators as specific linear combinations of Pauli operators in the qubit basis, as detailed in Appendix B. Many techniques have been developed to reduce the verbosity of the representation, notably including the double factorization approach [45][46][47][48][49][50][51][52][53], composing the Hamiltonian as a sum of groups of simultaneously observable Pauli operators, with each leaf in the sum corresponding to a specific spin-adapted spatial orbital rotation obtained by tensor factorization of the ERIs. For our purposes herein, it is conceptually sufficient to be able to compute the density matrix in the natural representation of the qubit-basis operators, e.g., to be able to compute the expectation value of each Pauli word for a representation of the operators in terms of a linear combination of Pauli operators.…”

Compressing Many-Body Fermion Operators Under Unitary Constraints

“…This justifies a numerical optimization of this doubles term, as is performed in the unitary cluster Jastrow factor ansatz [10] (k-uCJ) and its variants [9,11]. Reference [12] proposed an approach based on a gradient descent least-squares fitting of µ(l) and J(l) for à under the constraint that the coefficients have the proper symmetry to represent a spin-free chemical Hamiltonian. Although Reference [12] introduces a clever optimization strategy that alternates between one particle basis µ(l) optimization and J(l) coefficient optimization, the overall convergence of the the least-squares fitting is unknown and seems to be limited to six tensor factors before numerical difficulties make optimization challenging.…”

“…Many recently proposed simulation strategies for fermions leverage an analytical sum-of-squares many-body operator decomposition or use a sum-of-squares type ansatz for approximate ground states. Some examples for both near-term quantum computers and fault tolerant quantum computers are the double factorized Trotter steps for chemical Hamiltonians [7], tensor-hypercontraction based hamiltonian dynamics [8], restricted models of generalized coupled-cluster [9][10][11], and compressed density fitting [12]. The sum-of-squares picture unifies these ansätze and suggests a numerical compilation strategy for determining a sum-of-squares operator decomposition with few terms.…”

“…A numerical optimization strategy for a non-orthogonal single particle basis representation of many-body operators has already been used in determining one-particle bases to measure chemical Hamiltonians [13] and compressing manybody operators through numerical density fitting [12]. A classical analog of these methods is matching pursuit where the dictionary is a set of non-orthogonal single particle bases which are obtained as the algorithm progresses [14,15].…”

“…We propose an efficient local search (i.e., greedy) algorithm for recursively decomposing a fermionic two-body operator into a sum-of-squares terms with an iteration cost that scales no worse than a one-particle basis rotation of the two-body operator-O(n 5 ) where n is the number of spin-orbitals describing the problem. A numerical study on decomposition of the unitary coupled cluster operator indicates that a greedy search provides substantial improvements over analytical decompositions such as the singular value decomposition [7] or Takagi [10] decomposition and provides similar performance to least-squares tensor fitting [12] at substantially reduced computational complexity.…”