Assessment of various Hamiltonian partitionings for the electronic structure problem on a quantum computer using the Trotter approximation

Luis A. Martínez-Martínez, Tzu-Ching Yen, and Artur F. Izmaylov

Department of Physical and Environmental Sciences, University of Toronto Scarborough, Toronto, Ontario M1C 1A4, Canada
Chemical Physics Theory Group, Department of Chemistry, University of Toronto, Toronto, Ontario M5S 3H6, Canada

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Solving the electronic structure problem via unitary evolution of the electronic Hamiltonian is one of the promising applications of digital quantum computers. One of the practical strategies to implement the unitary evolution is via Trotterization, where a sequence of short-time evolutions of fast-forwardable (i.e. efficiently diagonalizable) Hamiltonian fragments is used. Given multiple choices of possible Hamiltonian decompositions to fast-forwardable fragments, the accuracy of the Hamiltonian evolution depends on the choice of the fragments. We assess efficiency of multiple Hamiltonian partitioning techniques using fermionic and qubit algebras for the Trotterization. Use of symmetries of the electronic Hamiltonian and its fragments significantly reduces the Trotter error. This reduction makes fermionic-based partitioning Trotter errors lower compared to those in qubit-based techniques. However, from the simulation-cost standpoint, fermionic methods tend to introduce quantum circuits with a greater number of T-gates at each Trotter step and thus are more computationally expensive compared to their qubit counterparts.

Estimation of energy eigenvalues of molecular Hamiltonians with the aid of quantum computers of the future is one of the most envisioned applications for these devices. Algorithms for this task in error-corrected quantum computer platforms are based on notions closely related to the calculation of molecular spectra based on the dynamical evolution of vibronic wavepackets. Thus, central to these algorithms, is the implementation of time propagation of an encoded wavefunction in a quantum computer. This task can be accomplished only approximately for a system evolving under an arbitrary Hamiltonian, as exact propagation for arbitrary time scales is tantamount to knowledge of eigenfunctions and eigenvalues of the simulated Hamiltonian.

One popular strategy to implement time propagation consists on dividing-up the simulated Hamiltonian in a sum of easy-to-diagonalize sub-Hamiltonians such that each of the latter can be translated into quantum-computer circuitry. Then, time propagation can be approximated as a sequential application of time-propagators generated by each of these sub-Hamiltonians, in the so-called Trotter approximation.

The decomposition of molecular Hamiltonians into easy-to-simulate sub-Hamiltonians is not unique, and in fact there exists a myriad of methods that carry out this task. The accuracy of the ensuing Trotter-approximated time-propagator is dependent of the chosen method. In this work, we perform an analysis of several Hamiltonian decomposition methods and get insight on the ideal traits of the individual Hamiltonian fragments that increase the accuracy of time propagation, as well as the cost of their corresponding implementation. Understanding of these characteristics is central for the design of Hamiltonian decomposition methods that aid more accurate time propagation implementations with an optimal balance in their concomitant cost in quantum computers.

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Cited by

[1] Guang Hao Low, Yuan Su, Yu Tong, and Minh C. Tran, "On the complexity of implementing Trotter steps", arXiv:2211.09133, (2022).

[2] Guang Hao Low, Yuan Su, Yu Tong, and Minh C. Tran, "Complexity of Implementing Trotter Steps", PRX Quantum 4 2, 020323 (2023).

[3] Seonghoon Choi, Ignacio Loaiza, and Artur F. Izmaylov, "Fluid fermionic fragments for optimizing quantum measurements of electronic Hamiltonians in the variational quantum eigensolver", Quantum 7, 889 (2023).

[4] Smik Patel, Tzu-Ching Yen, and Artur F. Izmaylov, "Extension of exactly-solvable Hamiltonians using symmetries of Lie algebras", arXiv:2305.18251, (2023).

[5] Oriel Kiss, Michele Grossi, and Alessandro Roggero, "Importance sampling for stochastic quantum simulations", Quantum 7, 977 (2023).

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