Emergent Mind

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

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


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.


  • 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.


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.


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.


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.

Get summaries of trending AI papers delivered straight to your inbox

Unsubscribe anytime.

  1. CaDiCaL, Kissat, Paracooba, Plingeling and Treengeling entering the SAT Competition 2020. In Tomas Balyo, Nils Froleyks, Marijn Heule, Markus Iser, Matti Järvisalo, and Martin Suda, editors, Proc. of SAT Competition 2020 – Solver and Benchmark Descriptions, volume B-2020-1 of Department of Computer Science Report Series B, pages 51–53. University of Helsinki
  2. Gimsatul, IsaSAT and Kissat entering the SAT Competition 2022. In Tomas Balyo, Marijn Heule, Markus Iser, Matti Järvisalo, and Martin Suda, editors, Proc. of SAT Competition 2022 – Solver and Benchmark Descriptions, volume B-2022-1 of Department of Computer Science Series of Publications B, pages 10–11. University of Helsinki
  3. Tapering off qubits to simulate fermionic hamiltonians
  4. Fermionic quantum computation. Annals of Physics, 298(1):210–226, May 2002. arXiv:quant-ph/0003137.
  5. Analysis of superfast encoding performance for electronic structure simulations. Physical Review A, 100(3), sep 2019.
  6. Phase gadget synthesis for shallow circuits. Electronic Proceedings in Theoretical Computer Science, 318:213–228, may 2020.
  7. A generic compilation strategy for the unitary coupled cluster ansatz
  8. Z3: An efficient smt solver. In C. R. Ramakrishnan and Jakob Rehof, editors, Tools and Algorithms for the Construction and Analysis of Systems, pages 337–340, Berlin, Heidelberg, 2008. Springer Berlin Heidelberg.
  9. Cirq Developers. Cirq, July 2023.
  10. P. A. M. Dirac. The quantum theory of the emission and absorption of radiation. Proceedings of the Royal Society of London. Series A, Containing Papers of a Mathematical and Physical Character, 114(767):243–265
  11. Exact multiple-control toffoli network synthesis with sat techniques. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems, 28(5):703–715, May 2009.
  12. Term grouping and travelling salesperson for digital quantum simulation
  13. Improving quantum algorithms for quantum chemistry. Quantum Info. Comput., 15(1–2):1–21, jan 2015.
  14. J. Hubbard and Brian Hilton Flowers. Electron correlations in narrow energy bands. Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences, 276(1365):238–257
  15. Optimal fermion-to-qubit mapping via ternary trees with applications to reduced quantum states learning. Quantum, 4:276, June 2020.
  16. P. Jordan and E. Wigner. Über das paulische äquivalenzverbot. Zeitschrift für Physik, 47(9):631–651, Sep 1928.
  17. P. Jordan and E. Wigner. Über das Paulische Äquivalenzverbot. Zeitschrift für Physik, 47(9):631–651, September 1928.
  18. Paulihedral: A generalized block-wise compiler optimization framework for quantum simulation kernels. In Proceedings of the 27th ACM International Conference on Architectural Support for Programming Languages and Operating Systems, ASPLOS ’22, page 554–569, New York, NY, USA, 2022. Association for Computing Machinery.
  19. Quantum computational chemistry. Rev. Mod. Phys., 92:015003, Mar 2020.
  20. Openfermion: the electronic structure package for quantum computers. Quantum Science and Technology, 5(3):034014, jun 2020.
  21. Microsoft. Q# Language Specification
  22. The Bonsai algorithm: grow your own fermion-to-qubit mapping
  23. Qubit Mapping and Routing via MaxSAT
  24. Noise-adaptive compiler mappings for noisy intermediate-scale quantum computers
  25. Formal constraint-based compilation for noisy intermediate-scale quantum systems. Microprocessors and Microsystems, 66:102–112, April 2019.
  26. Automated optimization of large quantum circuits with continuous parameters. npj Quantum Information, 4(1), may 2018.
  27. Quantum Computation and Quantum Information: 10th Anniversary Edition. Cambridge University Press
  28. Oak Ridge National Lab. ALCC program awards nearly 6 million summit node hours across 31 projects. https://www.olcf.ornl.gov/2020/08/05/alcc-program-awards-nearly-6-million-summit-node-hours-across-31-projects/, 2020. Accessed: 2020-08-16.

  29. Qiskit contributors. Qiskit: An open-source framework for quantum computing
  30. Venkat K. Raman. Handbook of Computational Quantum Chemistry By David B. Cook. Oxford University Press: New York. 1998. 743 pp. ISBN 0-19-850114-5. $140.00. Journal of Chemical Information and Computer Sciences, 40(3):882–882, May 2000. Publisher: American Chemical Society.
  31. Gapless spin-fluid ground state in a random quantum heisenberg magnet. Physical Review Letters, 70(21):3339–3342, May 1993.
  32. Superfast encodings for fermionic quantum simulation. Phys. Rev. Res., 1:033033, Oct 2019.
  33. White dots do matter: Rewriting reversible logic circuits. In Proceedings of the 5th International Conference on Reversible Computation, RC’13, page 196–208, Berlin, Heidelberg, 2013. Springer-Verlag.
  34. Optimal layout synthesis for quantum computing. In Proceedings of the 39th International Conference on Computer-Aided Design, ICCAD ’20. ACM, November 2020.
  35. Optimal qubit mapping with simultaneous gate absorption. In 2021 IEEE/ACM International Conference On Computer Aided Design (ICCAD). IEEE, November 2021.
  36. A comparison of the bravyi–kitaev and jordan–wigner transformations for the quantum simulation of quantum chemistry. Journal of Chemical Theory and Computation, 14(11):5617–5630, 2018. PMID: 30189144.
  37. H. F. Trotter. On the product of semi-groups of operators. Proceedings of the American Mathematical Society, 10(4):545–551
  38. G. S. Tseitin. On the Complexity of Derivation in Propositional Calculus, pages 466–483. Springer Berlin Heidelberg, Berlin, Heidelberg
  39. Arianne Meijer van de Griend and Ross Duncan. Architecture-aware synthesis of phase polynomials for nisq devices
  40. Ewout van den Berg and Kristan Temme. Circuit optimization of hamiltonian simulation by simultaneous diagonalization of pauli clusters. Quantum, 4:322, sep 2020.
  41. Mapping quantum circuits to ibm qx architectures using the minimal number of swap and h operations. In Proceedings of the 56th Annual Design Automation Conference 2019, DAC ’19, New York, NY, USA, 2019. Association for Computing Machinery.

Show All 41

Test Your Knowledge

You answered out of questions correctly.

Well done!