# Geodesic Algorithm for Unitary Gate Design with Time-Independent Hamiltonians

**Authors:** Dylan Lewis, Roeland Wiersema, Juan Carrasquilla, Sougato Bose  
**arXiv:** [2401.05973](https://arxiv.org/abs/2401.05973)  
**Date:** 2024-01-11  
**Summary page:** https://www.summarizepaper.com/en/arxiv-id/2401.05973v1/

## Key points

- Larger multi-qubit quantum gates can improve the efficiency of quantum circuits
- Time-independent Hamiltonians with physical interaction terms can be used to achieve this
- The algorithm proposed in this study uses geodesics on a Riemannian manifold to determine necessary Hamiltonian strengths for a target gate
- The method outperforms traditional gradient descent in finding solutions for standard multi-qubit gates like Toffoli and Fredkin
- New high fidelity parity check gates are discovered with potential applications in various quantum codes
- This research presents a novel algorithm for designing unitary gates using time-independent Hamiltonians by leveraging geodesics and differential programming techniques.

## Summary

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In the field of quantum computing, larger multi-qubit quantum gates have the potential to significantly improve the efficiency of quantum circuits. These gates can help mitigate noise and improve error correction schemes for noisy intermediate-scale quantum (NISQ) devices. One approach is using time-independent Hamiltonians with physical interaction terms. This study proposes an algorithm utilizing geodesics on a Riemannian manifold to determine necessary Hamiltonian strengths for a target gate. By analyzing the geodesic direction, they update the terms accordingly using differential programming. The authors compare their method with traditional gradient descent and show improved efficiency in finding solutions for standard multi-qubit gates like Toffoli and Fredkin. They also discover new high fidelity parity check gates with potential applications in various quantum codes, highlighting its impact on fault-tolerant quantum computing advancement. In summary, this research presents a novel algorithm for designing unitary gates using time-independent Hamiltonians by leveraging geodesics and differential programming techniques.