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 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.

1.
R. P.
Feynman
,
Opt. Photonics News
11
,
11
(
1985
).
3.
A.
Aspuru-Guzik
,
A. D.
Dutoi
,
P. J.
Love
, and
M.
Head-Gordon
,
Science
309
,
1704
(
2005
).
4.
D. A.
Lidar
and
H.
Wang
,
Phys. Rev. E
59
,
2429
(
1999
).
5.
I.
Kassal
,
S. P.
Jordan
,
P. J.
Love
,
M.
Mohseni
, and
A.
Aspuru-Guzik
,
Proc. Natl. Acad. Sci. U.S.A.
105
,
18681
(
2008
).
6.
G.
Ortiz
,
J.
Gubernatis
,
E.
Knill
, and
R.
Laflamme
,
Phys. Rev. A
64
,
022319
(
2001
).
7.
I.
Kassal
and
A.
Aspuru-Guzik
,
J. Chem. Phys.
131
,
4102
(
2009
).
8.
D. S.
Abrams
and
S.
Lloyd
,
Phys. Rev. Lett.
79
,
2586
(
1997
).
9.
R.
Somma
,
G.
Ortiz
,
J. E.
Gubernatis
,
E.
Knill
, and
R.
Laflamme
,
Phys. Rev. A
65
,
42323
(
2002
).
10.
P.
Jordan
and
E.
Wigner
,
Z. Phys.
47
,
631
(
1928
).
11.
J. D.
Whitfield
,
J.
Biamonte
, and
A.
Aspuru-Guzik
,
Mol. Phys.
109
,
735
(
2011
).
12.
F.
Verstraete
and
J. I.
Cirac
,
J. Stat. Mech.: Theory Exp.
09
,
012
(
2005
).
13.
S.
Bravyi
and
A.
Kitaev
,
Ann. Phys.
298
,
210
(
2002
);
14.
R. C.
Ball
,
Phys. Rev. Lett.
95
,
176407
(
2005
).
15.
B. P.
Lanyon
,
J. D.
Whitfield
,
G. G.
Gillett
,
M. E.
Goggin
,
M. P.
Almeida
,
I.
Kassal
,
J. D.
Biamonte
,
M.
Mohseni
,
B. J.
Powell
,
M.
Barbieri
,
A.
Aspuru-Guzik
, and
A. G.
White
,
Nat. Chem.
2
,
106
(
2010
).
16.
A.
Aspuru-Guzik
and
P.
Walther
,
Nat. Phys.
8
,
285
(
2012
).
17.
J.
Du
,
N.
Xu
,
X.
Peng
,
P.
Wang
,
S.
Wu
, and
D.
Lu
,
Phys. Rev. Lett.
104
,
030502
(
2010
).
18.
B. P.
Lanyon
,
C.
Hempel
,
D.
Nigg
,
M.
Müller
,
R.
Gerritsma
,
F.
Zähringer
,
P.
Schindler
,
J. T.
Barreiro
,
M.
Rambach
,
G.
Kirchmair
,
M.
Hennrich
,
P.
Zoller
,
R.
Blatt
, and
C. F.
Roos
,
Science
334
,
57
(
2011
).
19.
R.
Blatt
and
C. F.
Roos
,
Nat. Phys.
8
,
277
(
2012
).
20.
L.
Veis
and
J.
Pittner
,
J. Chem. Phys.
133
,
4106
(
2010
).
21.
L.
Veis
,
J.
Višňák
,
T.
Fleig
,
S.
Knecht
,
T.
Saue
,
L.
Visscher
, and
J.
Pittner
,
Phys. Rev. A
85
,
030304
(
2012
).
22.
P. J.
Love
, “
Back to the future: A roadmap for quantum simulation from vintage quantum chemistry
,” Adv. Chem. Phys. (to be published), e-print arXiv:1208.5524.
23.
N. C.
Jones
,
J. D.
Whitfield
,
P. L.
McMahon
,
M.-H.
Yung
,
R. V.
Meter
,
A.
Aspuru-Guzik
, and
Y.
Yamamoto
, “
Simulating chemistry efficiently on fault-tolerant quantum computers
,” e-print arXiv:1204.0567v1 (unpublished).
24.
P.
Zanardi
,
Phys. Rev. A
65
,
042101
(
2002
).
25.
H. F.
Trotter
,
Proc. Am. Math. Soc.
10
,
545
(
1959
).
26.
M.
Suzuki
,
Phys. Lett. A
165
,
387
(
1992
).
27.
N.
Hatano
, and
M.
Suzuki
, in
Quantum Annealing and Other Optimization Methods
,
Lecture Notes in Physics
Vol.
679
, edited by
A.
Das
, and
B. K.
Chakrabarti
(
Springer
,
2005
), pp.
36
68
.
28.
M. A.
Nielsen
and
I. L.
Chuang
,
Quantum Computation and Quantum Information
(
Cambridge University Press
,
Cambridge
,
2000
).
29.
R.
McWeeny
,
Methods of Molecular Quantum Mechanics
(
Academic
,
1992
).
30.
A.
Szabo
and
N.
Ostlund
,
Modern Quantum Chemistry: Introduction to Advanced Electronic Structure Theory
(
Dover
,
1996
).
31.
A.
Dutta
,
U.
Divakaran
,
D.
Sen
,
B. K.
Chakrabarti
,
T. F.
Rosenbaum
, and
G.
Aeppli
, “
Transverse field spin models: From statistical physics to quantum information
,” e-print arXiv:1012.0653v1 (unpublished).
32.
R. P.
Muller
, “
Python quantum chemistry (pyquante) program
” (
2007
); online at http://pyquante.sourceforge.net/.
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