Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:1208.5986 · The Journal of Chemical Physics

The Bravyi-Kitaev transformation for quantum computation of electronic structure

Jacob T. Seeley, Martin J. Richard, Peter J. Love

Published Dec 12, 2012• 585 citations
arXivHTMLPDF

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 [Ann. Phys. 298, 210 (2002); e-print arXiv:quant-ph/0003137v2], 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 H(2) 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.

DOI
10.1063/1.4768229

Related papers

Bravyi-Kitaev Superfast simulation of electronic structure on a quantum computer

Kanav Setia, James D. Whitfield

Present quantum computers often work with distinguishable qubits as their computational units. In order to simulate indistinguishable fermionic particles, it is first required to map the fermionic state to the state of the qubits. The Bravyi-Kitaev Superfast (BKSF) algorithm can be used to accomplish this mapping. The BKSF mapping has connections to quantum error correction and opens the door to new ways of understanding fermionic simulation in a topological context. Here, we present the first detailed exposition of the BKSF algorithm for molecular simulation. We provide the BKSF transformed qubit operators and report on our implementation of the BKSF fermion-to-qubits transform in OpenFermion. In this initial study of a hydrogen molecule we have compared BKSF, Jordan-Wigner, and Bravyi-Kitaev transforms under the Trotter approximation. The gate count to implement BKSF is lower than Jordan-Wigner but higher than Bravyi-Kitaev. We considered different orderings of the exponentiated terms and found lower Trotter errors than the previously reported for Jordan-Wigner and Bravyi-Kitaev algorithms. These results open the door to the further study of the BKSF algorithm for quantum simulation.

arXiv:1712.00446•The Journal of Chemical Physics•Apr 25, 2018•110 citations

A Comparison of the Bravyi–Kitaev and Jordan–Wigner Transformations for the Quantum Simulation of Quantum Chemistry

Andrew Tranter, Peter J. Love, Florian Mintert +1 more

The ability to perform classically intractable electronic structure calculations is often cited as one of the principal applications of quantum computing. A great deal of theoretical algorithmic development has been performed in support of this goal. Most techniques require a scheme for mapping electronic states and operations to states of and operations upon qubits. The two most commonly used techniques for this are the Jordan-Wigner transformation and the Bravyi-Kitaev transformation. However, comparisons of these schemes have previously been limited to individual small molecules. In this paper, we discuss resource implications for the use of the Bravyi-Kitaev mapping scheme, specifically with regard to the number of quantum gates required for simulation. We consider both small systems, which may be simulatable on near-future quantum devices, and systems sufficiently large for classical simulation to be intractable. We use 86 molecular systems to demonstrate that the use of the Bravyi-Kitaev transformation is typically at least approximately as efficient as the canonical Jordan-Wigner transformation and results in substantially reduced gate count estimates when performing limited circuit optimizations.

arXiv:1812.02233•Journal of Chemical Theory and Computation•Sep 6, 2018•125 citations

The B ravyi– K itaev transformation: Properties and applications

Andrew Tranter, Sarah E. Sofia, Jacob T. Seeley +7 more

Quantum chemistry is an important area of application for quantum computation. In particular, quantum algorithms applied to the electronic structure problem promise exact, efficient methods for determination of the electronic energy of atoms and molecules. The Bravyi–Kitaev transformation is a method of mapping the occupation state of a fermionic system onto qubits. This transformation maps the Hamiltonian ofninteracting fermions to an ‐local Hamiltonian ofnqubits. This is an improvement in locality over the Jordan–Wigner transformation, which results in anO(n)‐local qubit Hamiltonian. We present the Bravyi–Kitaev transformation in detail, introducing the sets of qubits which must be acted on to change occupancy and parity of states in the occupation number basis. We give recursive definitions of these sets and of the transformation and inverse transformation matrices, which relate the occupation number basis and the Bravyi–Kitaev basis. We then compare the use of the Jordan–Wigner and Bravyi–Kitaev Hamiltonians for the quantum simulation of methane using the STO‐6G basis. © 2015 Wiley Periodicals, Inc.

International Journal of Quantum Chemistry•Jul 1, 2015•136 citations

Quantum chemistry and quantum computers -- Testing the Bravyi-Kitaev mapping and Trotter order optimisations

Andrew Tranter

The quantum simulation of quantum chemistry is often cited as one of the most important potential uses for emerging quantum technology. Quantum computing methods are expected to be able to perform full configuration interaction level quantum chemistry calculations with tractable computational resource requirements. However, early quantum computers are likely to be constrained in both the number of qubits available, and the number of gates that can be performed. Implementing chemical algorithms on such quantum devices will require algorithmic improvements. In this thesis, two avenues for optimisation of the canonical method for the simulation of quantum chemistry on quantum computers are discussed. The Bravyi-Kitaev transformation is an alternative to the Jordan-Wigner transformation, the canonical method of mapping electronic states and operators to states and operators of qubits. The resource implications of this transformation scheme are assessed using a variety of chemical examples. Known techniques for optimising quantum circuits for quantum chemistry are applied, and the implications for the use of the Bravyi-Kitaev transformation are considered. Trotterization – the use of Trotter-Suzuki formulae to approximate the evolution operator for the molecular Hamiltonian – is discussed. In particular, ordering schemes designed to minimise the Trotter error are described. Finally, the importance of Trotter ordering when using the Bravyi-Kitaev transformation is considered. The performance of either transformation scheme is found to be highly dependent on the Trotter ordering and level of optimisation chosen. As such, the relative merits of either transformation scheme are found to be not as simple as previously thought, although in most cases the Bravyi-Kitaev transformation is found to be advantageous. The relative merits of both mapping and ordering schemes are considered in the context of performing quantum simulation of quantum chemistry on real quantum devices. Particular focus is paid to molecular systems which are intractable to classical computational simulation. The results emphasise the need for thorough analysis of optimisation schemes, such that quantum chemistry may indeed be a principal use of early quantum computation.

•Jun 1, 2018•2 citations

Geometric Algebra Jordan–Wigner Transformation for Quantum Simulation

Grégoire Veyrac, Zeno Toffano

Quantum simulation qubit models of electronic Hamiltonians rely on specific transformations in order to take into account the fermionic permutation properties of electrons. These transformations (principally the Jordan-Wigner transformation (JWT) and the Bravyi-Kitaev transformation) correspond in a quantum circuit to the introduction of a supplementary circuit level. In order to include the fermionic properties in a more straightforward way in quantum computations, we propose to use methods issued from Geometric Algebra (GA), which, due to its commutation properties, are well adapted for fermionic systems. First, we apply the Witt basis method in GA to reformulate the JWT in this framework and use this formulation to express various quantum gates. We then rewrite the general one and two-electron Hamiltonian and use it for building a quantum simulation circuit for the Hydrogen molecule. Finally, the quantum Ising Hamiltonian, widely used in quantum simulation, is reformulated in this framework.

Entropy•May 8, 2024•3 citations

A unified framework of transformations based on the Jordan–Wigner transformation

Qing-Song Li, Huanyu Liu, Qingchun Wang +2 more

Quantum simulation of chemical Hamiltonians enables the efficient calculation of chemical properties. Mapping is one of the essential steps in simulating fermionic systems on quantum computers. In this work, a unified framework of transformations mapping fermionic systems to qubit systems is presented and many existing transformations-such as Jordan-Wigner, Bravyi-Kitaev, and parity transformations-are included in this framework. Based on this framework, the Multilayer Segmented Parity (MSP) transformation is proposed. The MSP transformation is a general mapping with an adjustable parameter vector, which can be viewed as a generalization of the above-mentioned mappings. Furthermore, the MSP transformation can adjust flexibly when dealing with different systems. Applying these mappings to the electronic structure Hamiltonians of various molecules, the MSP transformation is found to perform better on a number of Pauli operators and gates needed in the circuit of Hamiltonian simulation. The MSP transformation will reduce the qubit gate requirement for Hamiltonian simulation on noisy intermediate-scale quantum devices, and it will provide a much wider choice of mappings for researchers.

arXiv:2108.01725•The Journal of Chemical Physics•Sep 8, 2022•15 citations
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 [Ann. Phys. 298, 210 (2002); e-print arXiv:quant-ph/0003137v2], 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 H(2) 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.
Present quantum computers often work with distinguishable qubits as their computational units. In order to simulate indistinguishable fermionic particles, it is first required to map the fermionic state to the state of the qubits. The Bravyi-Kitaev Superfast (BKSF) algorithm can be used to accomplish this mapping. The BKSF mapping has connections to quantum error correction and opens the door to new ways of understanding fermionic simulation in a topological context. Here, we present the first detailed exposition of the BKSF algorithm for molecular simulation. We provide the BKSF transformed qubit operators and report on our implementation of the BKSF fermion-to-qubits transform in OpenFermion. In this initial study of a hydrogen molecule we have compared BKSF, Jordan-Wigner, and Bravyi-Kitaev transforms under the Trotter approximation. The gate count to implement BKSF is lower than Jordan-Wigner but higher than Bravyi-Kitaev. We considered different orderings of the exponentiated terms and found lower Trotter errors than the previously reported for Jordan-Wigner and Bravyi-Kitaev algorithms. These results open the door to the further study of the BKSF algorithm for quantum simulation.
The ability to perform classically intractable electronic structure calculations is often cited as one of the principal applications of quantum computing. A great deal of theoretical algorithmic development has been performed in support of this goal. Most techniques require a scheme for mapping electronic states and operations to states of and operations upon qubits. The two most commonly used techniques for this are the Jordan-Wigner transformation and the Bravyi-Kitaev transformation. However, comparisons of these schemes have previously been limited to individual small molecules. In this paper, we discuss resource implications for the use of the Bravyi-Kitaev mapping scheme, specifically with regard to the number of quantum gates required for simulation. We consider both small systems, which may be simulatable on near-future quantum devices, and systems sufficiently large for classical simulation to be intractable. We use 86 molecular systems to demonstrate that the use of the Bravyi-Kitaev transformation is typically at least approximately as efficient as the canonical Jordan-Wigner transformation and results in substantially reduced gate count estimates when performing limited circuit optimizations.
Quantum chemistry is an important area of application for quantum computation. In particular, quantum algorithms applied to the electronic structure problem promise exact, efficient methods for determination of the electronic energy of atoms and molecules. The Bravyi–Kitaev transformation is a method of mapping the occupation state of a fermionic system onto qubits. This transformation maps the Hamiltonian ofninteracting fermions to an ‐local Hamiltonian ofnqubits. This is an improvement in locality over the Jordan–Wigner transformation, which results in anO(n)‐local qubit Hamiltonian. We present the Bravyi–Kitaev transformation in detail, introducing the sets of qubits which must be acted on to change occupancy and parity of states in the occupation number basis. We give recursive definitions of these sets and of the transformation and inverse transformation matrices, which relate the occupation number basis and the Bravyi–Kitaev basis. We then compare the use of the Jordan–Wigner and Bravyi–Kitaev Hamiltonians for the quantum simulation of methane using the STO‐6G basis. © 2015 Wiley Periodicals, Inc.
The quantum simulation of quantum chemistry is often cited as one of the most important potential uses for emerging quantum technology. Quantum computing methods are expected to be able to perform full configuration interaction level quantum chemistry calculations with tractable computational resource requirements. However, early quantum computers are likely to be constrained in both the number of qubits available, and the number of gates that can be performed. Implementing chemical algorithms on such quantum devices will require algorithmic improvements. In this thesis, two avenues for optimisation of the canonical method for the simulation of quantum chemistry on quantum computers are discussed. The Bravyi-Kitaev transformation is an alternative to the Jordan-Wigner transformation, the canonical method of mapping electronic states and operators to states and operators of qubits. The resource implications of this transformation scheme are assessed using a variety of chemical examples. Known techniques for optimising quantum circuits for quantum chemistry are applied, and the implications for the use of the Bravyi-Kitaev transformation are considered. Trotterization – the use of Trotter-Suzuki formulae to approximate the evolution operator for the molecular Hamiltonian – is discussed. In particular, ordering schemes designed to minimise the Trotter error are described. Finally, the importance of Trotter ordering when using the Bravyi-Kitaev transformation is considered. The performance of either transformation scheme is found to be highly dependent on the Trotter ordering and level of optimisation chosen. As such, the relative merits of either transformation scheme are found to be not as simple as previously thought, although in most cases the Bravyi-Kitaev transformation is found to be advantageous. The relative merits of both mapping and ordering schemes are considered in the context of performing quantum simulation of quantum chemistry on real quantum devices. Particular focus is paid to molecular systems which are intractable to classical computational simulation. The results emphasise the need for thorough analysis of optimisation schemes, such that quantum chemistry may indeed be a principal use of early quantum computation.
Quantum simulation qubit models of electronic Hamiltonians rely on specific transformations in order to take into account the fermionic permutation properties of electrons. These transformations (principally the Jordan-Wigner transformation (JWT) and the Bravyi-Kitaev transformation) correspond in a quantum circuit to the introduction of a supplementary circuit level. In order to include the fermionic properties in a more straightforward way in quantum computations, we propose to use methods issued from Geometric Algebra (GA), which, due to its commutation properties, are well adapted for fermionic systems. First, we apply the Witt basis method in GA to reformulate the JWT in this framework and use this formulation to express various quantum gates. We then rewrite the general one and two-electron Hamiltonian and use it for building a quantum simulation circuit for the Hydrogen molecule. Finally, the quantum Ising Hamiltonian, widely used in quantum simulation, is reformulated in this framework.
Quantum simulation of chemical Hamiltonians enables the efficient calculation of chemical properties. Mapping is one of the essential steps in simulating fermionic systems on quantum computers. In this work, a unified framework of transformations mapping fermionic systems to qubit systems is presented and many existing transformations-such as Jordan-Wigner, Bravyi-Kitaev, and parity transformations-are included in this framework. Based on this framework, the Multilayer Segmented Parity (MSP) transformation is proposed. The MSP transformation is a general mapping with an adjustable parameter vector, which can be viewed as a generalization of the above-mentioned mappings. Furthermore, the MSP transformation can adjust flexibly when dealing with different systems. Applying these mappings to the electronic structure Hamiltonians of various molecules, the MSP transformation is found to perform better on a number of Pauli operators and gates needed in the circuit of Hamiltonian simulation. The MSP transformation will reduce the qubit gate requirement for Hamiltonian simulation on noisy intermediate-scale quantum devices, and it will provide a much wider choice of mappings for researchers.