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Quantum circuit compilation for fermionic excitations using the Jordan-Wigner mapping

Renata Wong

TL;DR

This work provides a practical treatment of the Jordan-Wigner mapping to translate fermionic excitations into qubit operations for the UCCSD ansatz, demonstrated on the hydrogen molecule in a minimal basis. It derives how single and double excitations map to Pauli strings and presents a hardware-aware circuit compilation strategy built from basis-change, parity, rotation, and uncompute steps. The note also discusses handling nonlocal Z-strings, CNOT-direction choices, and the use of common gates like Rx(pi/2) versus sqrt{X} in realistic devices. The results supply explicit circuit templates and decompositions that facilitate implementing fermionic excitations on gate-based quantum hardware, supporting near-term quantum chemistry applications such as VQE and UCCSD.

Abstract

This note bridges the gap between theoretical second quantization and practical quantum hardware by detailing the Jordan-Wigner mapping for the Unitary Coupled Cluster Singles and Doubles (UCCSD) ansatz. Using the hydrogen molecule in a minimal basis as a case study, we explicitly derive the Pauli strings required for single and double excitations. Additionally, we discuss the translation of these operators into quantum circuits, with a focus on implementation nuances such as the difference between mathematical rotations and physical gates like the $\sqrt{X}$ (SX) gate.

Quantum circuit compilation for fermionic excitations using the Jordan-Wigner mapping

TL;DR

This work provides a practical treatment of the Jordan-Wigner mapping to translate fermionic excitations into qubit operations for the UCCSD ansatz, demonstrated on the hydrogen molecule in a minimal basis. It derives how single and double excitations map to Pauli strings and presents a hardware-aware circuit compilation strategy built from basis-change, parity, rotation, and uncompute steps. The note also discusses handling nonlocal Z-strings, CNOT-direction choices, and the use of common gates like Rx(pi/2) versus sqrt{X} in realistic devices. The results supply explicit circuit templates and decompositions that facilitate implementing fermionic excitations on gate-based quantum hardware, supporting near-term quantum chemistry applications such as VQE and UCCSD.

Abstract

This note bridges the gap between theoretical second quantization and practical quantum hardware by detailing the Jordan-Wigner mapping for the Unitary Coupled Cluster Singles and Doubles (UCCSD) ansatz. Using the hydrogen molecule in a minimal basis as a case study, we explicitly derive the Pauli strings required for single and double excitations. Additionally, we discuss the translation of these operators into quantum circuits, with a focus on implementation nuances such as the difference between mathematical rotations and physical gates like the (SX) gate.
Paper Structure (22 sections, 28 equations)

This paper contains 22 sections, 28 equations.

Table of Contents

  1. Introduction
  2. Motivation for the Jordan-Wigner mapping
  3. Coupled cluster theory
  4. Example: excitations in hydrogen molecule
  5. Justification for single excitations
  6. The rule of spin conservation
  7. The Pauli exclusion principle
  8. Example mapping of single and double excitations
  9. Single excitation example
  10. Double excitation example
  11. Quantum circuit implementation for single excitations
  12. Term $e^{-i \frac{\theta}{2} (X_1 Y_0)}$
  13. Term $e^{-i \frac{\theta}{2} (-Y_1 X_0)}$
  14. Handling nonlocal excitations (Z-strings)
  15. Remarks on the implementation
  16. ...and 7 more sections