The Bravyi-Kitaev transformation for quantum computation of electronic structure (1208.5986v1)

Abstract: Quantum simulation is an important application of future quantum computers with applications in quantum chemistry, condensed matter, and beyond. Quantum simulation of fermionic systems presents a specific challenge. The Jordan-Wigner transformation allows for representation of a fermionic operator by O(n) qubit operations. Here we develop an alternative method of simulating fermions with qubits, first proposed by Bravyi and Kitaev [S. B. Bravyi, A.Yu. Kitaev, Annals of Physics 298, 210-226 (2002)], that reduces the simulation cost to O(log n) qubit operations for one fermionic operation. We apply this new Bravyi-Kitaev transformation to the task of simulating quantum chemical Hamiltonians, and give a detailed example for the simplest possible case of molecular hydrogen in a minimal basis. We show that the quantum circuit for simulating a single Trotter time-step of the Bravyi-Kitaev derived Hamiltonian for H2 requires fewer gate applications than the equivalent circuit derived from the Jordan-Wigner transformation. Since the scaling of the Bravyi-Kitaev method is asymptotically better than the Jordan-Wigner method, this result for molecular hydrogen in a minimal basis demonstrates the superior efficiency of the Bravyi-Kitaev method for all quantum computations of electronic structure.

• The paper introduces the Bravyi-Kitaev transformation that reduces computational overhead from O(n) to O(log n) operations compared to the Jordan-Wigner method.
• The methodology efficiently balances parity and occupation data by defining specific sets, enabling precise and scalable simulation of fermionic systems.
• Practical simulations on molecular hydrogen demonstrate reduced gate counts, highlighting its promise for advancing quantum chemistry and complex molecular modeling.

An Analysis of the Bravyi-Kitaev Transformation for Quantum Computation of Electronic Structure

The paper presents a detailed exploration and application of the Bravyi-Kitaev transformation, advancing its utility in the quantum simulation of electronic structures. Authored by Jacob T. Seeley, Martin J. Richard, and Peter J. Love, the paper details improvements over the traditional Jordan-Wigner transformation by reducing the computational complexity from $O(n)$ to $O(\log n)$ qubit operations. This has significant implications for the efficiency of simulating fermionic systems, such as those encountered in quantum chemistry.

Quantum simulation is posited as a critical application for emerging quantum computing technology, with direct applications in fields such as quantum chemistry and condensed matter physics. Classical computers face exponential time scaling challenges when simulating quantum systems, a problem quantum simulators could overcome with polynomial time scaling advantages as initially suggested by Feynman and others.

Key Contributions

1. Bravyi-Kitaev vs. Jordan-Wigner Transformation: The Jordan-Wigner transformation stands as the conventional method for mapping fermionic operators to qubit operators, requiring $O(n)$ operations for one fermionic operation due to non-locality concerns. The Bravyi-Kitaev method optimally stores parity and occupation number information within sub-logarithmic scaling complexities, promising significantly reduced resource requirements.
2. Efficient Mapping: The paper delineates the construction of the Bravyi-Kitaev transformation, detailing how it balances the trade-offs between the occupation number basis and the parity basis. Important sets, such as the parity set, update set, and flip set, are defined for calculating operator transformations, making it essential for reflecting fermionic creation and annihilation operations on qubits.
3. H$_2$ Molecular Simulation: As an initial application, the transformation is applied to molecular hydrogen (H$_2$) in a minimal basis, demonstrating fewer gate operations for simulating Trotter time-steps than its Jordan-Wigner counterpart. This case paper showcases the Bravyi-Kitaev transformation's computational efficiency for molecular electronic structure simulations.
4. Implications for Future Quantum Chemistry Simulations: The transformation's efficiency suggests its potential utility in facilitating quantum simulations of larger and more complex molecular systems, for which classical simulations are infeasible. The asymptotic performance improvements of Bravyi-Kitaev enable better scaling for increasing molecular complexity.

Numerical Results and Complexity Analysis

The authors present the required number of quantum gates for simulating a single Trotter step in the H$_2$ Hamiltonian. The Bravyi-Kitaev simulation requires 30 single-qubit gates and 44 CNOT gates, compared to 46 and 36, respectively, for the Jordan-Wigner method. Additionally, simulations achieving chemical precision for the ground state eigenvalue require fewer gates, approximately 222 versus 328.

Theoretical and Practical Implications

Theoretically, the work validates the simpler handling of non-local phase information through efficient log-scaled operations. Practically, leveraging the Bravyi-Kitaev transformation could enhance the accessibility of quantum simulation technologies to research in computational chemistry by enabling more complex simulations with fewer resources.

Future Developments

This investigation spurs further research into how these transformations can be extended to more extensive and intricate systems. Future work may delve into developing hybrid methods incorporating Bravyi-Kitaev's principles or integrating machine learning techniques to optimize the selection of non-commutative term ordering in Suzuki-Trotter expansions.

In conclusion, the Bravyi-Kitaev transformation presents a valuable upgrade to how electronic structures are simulated using quantum computers. Its implementation marks a step forward in reducing the computational load of quantum chemistry simulations and sets a foundational block for future work that could see quantum computers tackling previously unsolvable chemical problems.