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The Bravyi-Kitaev transformation for quantum computation of electronic structure

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10.1063/1.4768229

### 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)10.1006/aphy.2002.6254; 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.

© 2012 American Institute of Physics

Received 28 August 2012
Accepted 06 November 2012
Published online 12 December 2012

Acknowledgments: The authors thank the Aspuru-Guzik group for their hospitality during the summers of 2011 and 2012, when parts of this work were completed. We are indebted to Jarod Maclean, John Parkhill, Sam Rodriques, Joshua Schrier, Robert Seeley, and James Whitfield for productive. This project is supported by National Science Foundation Centers for Chemical Innovation (NSF CCI) center, “Quantum Information for Quantum Chemistry (QIQC),” Award No. CHE-1037992, and by NSF Award No. PHY-0955518.

Article outline:

I. INTRODUCTION

II. BACKGROUND

A. Fermionic systems and second quantization

B. The Jordan-Wigner transformation

III. ALTERNATIVES TO THE OCCUPATION NUMBER BASIS

A. The parity basis

B. The Bravyi-Kitaev basis

IV. SETS OF QUBITS RELEVANT TO THE BRAVYI-KITAEV BASIS

A. The parity set

B. The update set

C. The flip set

V. THE BRAVYI-KITAEV TRANSFORMATION

A. Representing in the Bravyi-Kitaev basis for *j* even

B. Representing in the Bravyi-Kitaev basis for *j* odd

VI. PAULI REPRESENTATIONS OF SECOND-QUANTIZED OPERATORS IN THE BRAVYI-KITAEV BASIS

A. Number operators:

B. Coulomb and exchange operators:

C. Products of the form

D. Excitation operators:

E. Number-excitation operators:

F. Double-excitation operators:

VII. THE MOLECULAR ELECTRONIC HAMILTONIAN FOR THE HYDROGEN MOLECULE IN THE BRAVYI-KITAEV BASIS

A. The Bravyi-Kitaev Pauli representation of

B. The Bravyi-Kitaev Pauli representation of

VIII. TROTTERIZATION

IX. CONCLUSIONS

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2012-12-12

2014-04-19

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