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Spin-Adapted Fermionic Unitaries: From Lie Algebras to Compact Quantum Circuits

Ilias Magoulas, Francesco A. Evangelista

Abstract

Conservation of symmetries plays a crucial role in both classical and quantum simulations of many-body systems, enabling the tracking of states with specific symmetry properties and leading to substantial reductions in the number of optimization parameters. The design of efficient quantum circuits that enforce all symmetries typically encountered in chemistry has remained elusive, mainly due to the interplay of point group and spin symmetries. By exploiting Lie algebraic techniques, we derive exact product formulas representing symmetry-adapted unitaries. These decompositions allow us to design the most efficient symmetry-preserving quantum circuits to date. Finally, we introduce a minimum universal symmetry-adapted operator pool to further reduce the required quantum resources.

Spin-Adapted Fermionic Unitaries: From Lie Algebras to Compact Quantum Circuits

Abstract

Conservation of symmetries plays a crucial role in both classical and quantum simulations of many-body systems, enabling the tracking of states with specific symmetry properties and leading to substantial reductions in the number of optimization parameters. The design of efficient quantum circuits that enforce all symmetries typically encountered in chemistry has remained elusive, mainly due to the interplay of point group and spin symmetries. By exploiting Lie algebraic techniques, we derive exact product formulas representing symmetry-adapted unitaries. These decompositions allow us to design the most efficient symmetry-preserving quantum circuits to date. Finally, we introduce a minimum universal symmetry-adapted operator pool to further reduce the required quantum resources.

Paper Structure

This paper contains 16 sections, 43 equations, 23 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Symmetry Enforcement in Quantum Simulations
  3. Generators of Symmetry-Adapted Fermionic Unitaries
  4. Symmetry-Adapted Circuits: Linear Combination of Unitaries
  5. Symmetry-Adapted Circuits: Wei--Norman Decomposition
  6. Efficient Implementation of Symmetry-Adapted Unitaries
  7. Minimum Universal Symmetry-Adapted Operator Pool
  8. Conclusions
  9. Computational Details
  10. Proof of Closed-Form Similarity-Transformation Expression
  11. Classification of the Lie Algebra $\boldsymbol{\mathop{\mathrm{Lie}}\nolimits\left(A_{P{\mathrel{\uparrow}}P{\mathrel{\downarrow}}}^{Q{\mathrel{\uparrow}}R{\mathrel{\downarrow}}}, A_{P{\mathrel{\uparrow}}P{\mathrel{\downarrow}}}^{Q{\mathrel{\downarrow}}R{\mathrel{\uparrow}}}\right)}$
  12. Effect of Permutations in the Wei--Norman Decompositions of $\boldsymbol{\exp(A_{PP}^{QR})}$
  13. Dynamical Lie Algebras of $\boldsymbol{^{[0]}A_{PQ}^{RS}}$ and $\boldsymbol{^{[1]}A_{PQ}^{RS}}$
  14. CNOT-Efficient Quantum Circuits of Symmetry-Adapted Unitaries
  15. Nuclear Coordinates of the $\text{H}_6$ system
  16. ...and 1 more sections

Figures (23)

  • Figure 1: Quantum circuit implementation of unitary generated by a singlet spin-adapted double excitation operator. The system register is initialized in the state on which the spin-adapted unitary will act. Open and full circles denote controls based on the $\ket{0}$ and $\ket{1}$ states, respectively. The gates $P_0$ through $P_{M-1}$ represent the Pauli strings appearing in the LCU decomposition of the spin-adapted unitary.
  • Figure 2: Numerical values of the $\theta$-dependent parameters defining the Wei--Norman decomposition of $\exp\left( \theta A_{PP}^{QR}\right)$ [\ref{['eq:wn_decomposition']}].
  • Figure 3: Numerical values of the $\theta$-dependent parameters defining the Wei--Norman decomposition of $\exp\left( \theta\,^{[0]}\!A_{PQ}^{RS}\right)$.
  • Figure 4: Numerical values of the $\theta$-dependent parameters defining the Wei--Norman decomposition of $\exp\left( \theta\, ^{[1]}\!A_{PQ}^{RS}\right)$.
  • Figure 5: Qubit excitation-based circuit implementing the unitary $\exp\left( \theta Q_{P{\mathrel{\uparrow}}Q{\mathrel{\downarrow}}}^{R{\mathrel{\uparrow}}S{\mathrel{\downarrow}}}\right)$. In (b), the multiqubit-controlled $R_y$ gate of (a) has been decomposed.
  • ...and 18 more figures