Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

The Harrow-Hassidim-Lloyd algorithm with qutrits

Tushti Patel, V. S. Prasannaa

TL;DR

This work extends the Harrow-Hassidim-Lloyd algorithm from qubits to qutrits, enabling HHL computations in a ternary quantum architecture and applying it to quantum chemistry benchmarks. The authors design a qutrit-based circuit, define a practical gate set, and demonstrate through toy matrices and an H$_2$ potential energy curve that qutrit HHL can achieve comparable accuracy with fewer clock qudits and similar two-qudit gate costs when hardware supports reliable qutrits. They map molecular linear-response equations to an HHL linear-system framework, use isometry and swap-test techniques for overlap measurements, and benchmark against classical CISD/LCCSD references, observing correlation energies within about $2\times10^{-4}$ of the classical values. The results indicate a favorable resource trade-off for qutrit HHL at fixed precision, suggesting a practical path for early quantum chemistry advantages on higher-dimensional quantum hardware. The work also provides detailed comparisons of qudit resources, highlighting how ternary encoding reduces qudit counts by a factor of $\log_2(3)$ while maintaining competitive gate requirements.

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). We design the circuit for the algorithm and develop a program for its implementation. 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 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.

The Harrow-Hassidim-Lloyd algorithm with qutrits

TL;DR

This work extends the Harrow-Hassidim-Lloyd algorithm from qubits to qutrits, enabling HHL computations in a ternary quantum architecture and applying it to quantum chemistry benchmarks. The authors design a qutrit-based circuit, define a practical gate set, and demonstrate through toy matrices and an H potential energy curve that qutrit HHL can achieve comparable accuracy with fewer clock qudits and similar two-qudit gate costs when hardware supports reliable qutrits. They map molecular linear-response equations to an HHL linear-system framework, use isometry and swap-test techniques for overlap measurements, and benchmark against classical CISD/LCCSD references, observing correlation energies within about of the classical values. The results indicate a favorable resource trade-off for qutrit HHL at fixed precision, suggesting a practical path for early quantum chemistry advantages on higher-dimensional quantum hardware. The work also provides detailed comparisons of qudit resources, highlighting how ternary encoding reduces qudit counts by a factor of while maintaining competitive gate requirements.

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). We design the circuit for the algorithm and develop a program for its implementation. 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 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.

Paper Structure

This paper contains 16 sections, 12 equations, 6 figures, 5 tables.

Table of Contents

  1. Introduction
  2. The HHL algorithm
  3. HHL with qutrits
  4. A brief introduction to ternary logic
  5. Basic gates for qutrits
  6. One-qutrit gates
  7. Two-qutrit gates
  8. Qutrit HHL: working principles and quantum circuit
  9. Application of qutrit HHL for molecular ground-state energy calculations
  10. Results
  11. Testing with toy matrices
  12. Quantum chemistry simulations
  13. Comparison of qubit and qutrit HHL
  14. Analyzing the number of qudits
  15. Analyzing the number of 2-qudit gates
  16. ...and 1 more sections

Figures (6)

  • Figure 1: Schematic representation of the HHL algorithm.
  • Figure 2: Quantum Fourier transform circuit for qutrits.
  • Figure 3: Quantum phase estimation circuit for qutrits.
  • Figure 4: Uniform controlled-rotation circuit for 1 control qutrit.
  • Figure 5: Potential energy curves with energies specified in units of Hartree for the $H_2$ molecule. The HF energies have each been shifted by an amount of $-0.005$ Hartree for ease of visual representation.
  • ...and 1 more figures