The Harrow-Hassidim-Lloyd algorithm with qutrits

TL;DR

This paper extends the HHL quantum algorithm from qubits to qutrits, designing new gadgets and implementation schemes, and demonstrates potential resource advantages in quantum chemistry applications.

Contribution

The paper introduces a qutrit version of the HHL algorithm, including Weyl-Heisenberg gadgets and a practical implementation scheme, expanding quantum algorithm capabilities beyond qubits.

Findings

Qutrit HHL requires fewer qudits than qubit HHL for fixed precision.

Qutrit HHL has comparable two-qudit gate complexity to qubit HHL.

Successful testing on simple matrices and application to hydrogen molecule energy calculations.

Abstract

We extend the Harrow-Hassidim-Lloyd (HHL) algorithm, which is well-studied in the qubit framework, to its qutrit counterpart (which we call qutrit HHL, as opposed to qubit HHL, which is HHL using qubits), and develop a program for its implementation. We design Weyl-Heisenberg gadgets, the qutrit equivalents of Pauli gadgets, and come up with a practical implementation scheme for qutrit HHL. We test HHL with qutrits for simple matrices and verify the results against the expected outcomes. We apply the algorithm to quantum chemistry, and in particular, to the potential energy curve calculations of the model problem of the Hydrogen molecule in the split valence basis. We do so for two cases: 1-qutrit and 2-qutrit input states, where the latter makes use of our gadgets. We compare the number of qudits and the number of gates required between qubit and qutrit HHL implementations. In general, we find that for a fixed precision, the qutrit HHL circuit requires fewer number of qudits and comparable number of two-qudit gates than its qubit counterpart.