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Fermihedral: On the Optimal Compilation for Fermion-to-Qubit Encoding

(2403.17794)
Published Mar 26, 2024 in quant-ph and cs.ET

Abstract

This paper introduces Fermihedral, a compiler framework focusing on discovering the optimal Fermion-to-qubit encoding for targeted Fermionic Hamiltonians. Fermion-to-qubit encoding is a crucial step in harnessing quantum computing for efficient simulation of Fermionic quantum systems. Utilizing Pauli algebra, Fermihedral redefines complex constraints and objectives of Fermion-to-qubit encoding into a Boolean Satisfiability problem which can then be solved with high-performance solvers. To accommodate larger-scale scenarios, this paper proposed two new strategies that yield approximate optimal solutions mitigating the overhead from the exponentially large number of clauses. Evaluation across diverse Fermionic systems highlights the superiority of Fermihedral, showcasing substantial reductions in implementation costs, gate counts, and circuit depth in the compiled circuits. Real-system experiments on IonQ's device affirm its effectiveness, notably enhancing simulation accuracy.

Simulation of fermionic systems using qubit systems for quantum computing applications.

Overview

  • Fermihedral introduces a novel compiler framework for optimal fermion-to-qubit encoding by converting the encoding task into a SAT problem, enhancing the simulation of fermionic quantum systems on quantum computers.

  • The framework implements a two-step strategy involving conversion to Boolean satisfiability problems solved by SAT solvers and employs clause reduction techniques to manage scalability.

  • The evaluation of Fermihedral shows significant improvements over existing encoding schemes, demonstrating up to 60% reduction in implementation costs and enhanced simulation accuracy on IonQ's quantum device.

  • Fermihedral's approach not only optimizes quantum simulations, particularly for fermionic systems, but also opens new avenues for quantum computing applications and algorithmic developments.

Fermihedral: Achieving Optimal Fermion-to-Qubit Encoding with SAT Solutions

Introduction

The quest for simulating fermionic quantum systems on quantum computers has led to the development of various fermion-to-qubit encoding schemes. A new compiler framework, Fermihedral, revolutionizes this landscape by optimally solving the fermion-to-qubit encoding problem, converting the encoding task into a Boolean Satisfiability (SAT) problem. This approach remolds the encoding process with notable implications for quantum computing, particularly in simulating fermionic systems efficiently.

Core Methodology

Fermihedral introduces a two-step strategy to simplifying and solving the fermion-to-qubit encoding as a SAT problem:

  • Conversion to Boolean Satisfiability Problem: It leverages Pauli algebra to transform the complex constraints and optimization objectives involved in fermion-to-qubit encoding into Boolean expressions. These transformed constraints are then solved using high-performance SAT solvers, outputting the optimal encoding scheme.
  • Clause Reduction Techniques: To address the infeasibility posed by the exponentially large number of clauses, Fermihedral applies two novel techniques:
  1. Ignoring algebraic independence constraints, justified by an exponentially small failure probability analysis.
  2. Employing simulated annealing for larger-scale instances to provide approximate optimal solutions, further reducing the overhead.

Evaluation

The effectiveness of Fermihedral is demonstrated through comprehensive evaluations:

  • Comparison with Existing Encodings: Fermihedral notably outperforms existing encoding schemes like Jordan-Wigner and Bravyi-Kitaev, achieving up to 60% reduction in implementation costs such as gate counts and circuit depth.
  • Real-System Experiments on IonQ's Device: Not only does Fermihedral shine in theory and simulations, but it also excels when tested on real quantum hardware, exhibiting significant improvements in simulation accuracy.

Implications and Future Directions

This pioneering work opens up new vistas for quantum simulation:

  • Optimization of Quantum Simulation: Fermihedral's ability to minimize implementation overhead leapfrogs the efficiency of quantum simulations, particularly for fermionic systems which are central in fields like quantum chemistry and condensed matter physics.
  • Scalable and Practical Quantum Computing: By making fermion-to-qubit encoding more efficient and scalable, Fermihedral paves the way for more practical and extensive applications of quantum computing.
  • Framework for Further Developments: The SAT-based approach presents a versatile framework that could inspire further innovations in quantum computing, ranging from improved encoding schemes to optimized quantum algorithms.

Conclusion

Fermihedral marks a significant advancement in quantum computing, particularly in the simulation of fermionic systems. By solving the fermion-to-qubit encoding problem optimally through SAT solutions and introducing strategies for manageable scalability, it sets a new precedent for the efficiency and practicality of quantum simulations. As we delve deeper into the era of quantum computing, such breakthroughs are pivotal, not just for theoretical exploration but also for harnessing quantum computing's full potential in solving real-world problems.

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