We propose a general-purpose quantum algorithm for preparing ground states of quantum Hamiltonians from a given trial state. The algorithm is based on techniques recently developed in the context of solving the quantum linear system problem. We show that, compared to algorithms based on phase estimation, the runtime of our algorithm is exponentially better as a function of the allowed error, and at least quadratically better as a function of the overlap with the trial state. We also show that our algorithm requires fewer ancilla qubits than existing algorithms, making it attractive for early applications of small quantum computers. Additionally, it can be used to determine an unknown ground energy faster than with phase estimation if a very high precision is required.

1.
R.
Feynman
,
Int. J. Theor. Phys.
21
,
467
(
1982
).
3.
D. W.
Berry
,
A. M.
Childs
,
R.
Cleve
,
R.
Kothari
, and
R. D.
Somma
,
Phys. Rev. Lett.
114
,
090502
(
2015
).
4.
D. W.
Berry
,
A. M.
Childs
, and
R.
Kothari
, in
2015 IEEE 56th Annual Symposium on Foundations of Computer Science
(
IEEE
,
2015
), pp.
792
809
.
5.
G. H.
Low
and
I. L.
Chuang
,
Phys. Rev. Lett.
118
,
010501
(
2017
).
6.
G. H.
Low
and
I. L.
Chuang
, preprint arXiv:1610.06546 [quant-ph] (
2016
).
7.
S.
Gharibian
,
Y.
Huang
,
Z.
Landau
, and
S. W.
Shin
,
Found. Trends Theor. Comput. Sci.
10
,
159
(
2015
).
8.
A. Y.
Kitaev
, preprint arXiv:quant-ph/9511026 [quant-ph] (
1995
).
9.
D.
Poulin
and
P.
Wocjan
,
Phys. Rev. Lett.
102
,
130503
(
2009
).
10.
D. S.
Abrams
and
S.
Lloyd
,
Phys. Rev. Lett.
83
,
5162
(
1999
).
11.
D. A.
Abanin
and
Z.
Papi
,
Ann. Phys.
529
,
1700169
(
2017
).
12.
A.
Polkovnikov
,
K.
Sengupta
,
A.
Silva
, and
M.
Vengalattore
,
Rev. Mod. Phys.
83
,
863
(
2011
).
L.
D’Alessio
,
Y.
Kafri
,
A.
Polkovnikov
, and
M.
Rigol
,
Adv. Phys.
65
,
239
(
2016
).
13.
E.
Farhi
,
D.
Gosset
,
A.
Hassidim
,
A.
Lutomirski
,
D.
Nagaj
, and
P.
Shor
,
Phys. Rev. Lett.
105
,
190503
(
2010
).
14.
M.-H.
Yung
,
J. D.
Whitfield
,
S.
Boixo
,
D. G.
Tempel
, and
A.
Aspuru-Guzik
, “
Introduction to quantum algorithms for physics and chemistry
,” in
Quantum Information and Computation for Chemistry
(
John Wiley & Sons, Inc.
,
2014
), pp.
67
106
.
15.
J.
Biamonte
,
P.
Wittek
,
N.
Pancotti
,
P.
Rebentrost
,
N.
Wiebe
, and
S.
Lloyd
,
Nature
549
,
195
(
2017
).
16.
E.
Farhi
,
J.
Goldstone
,
S.
Gutmann
, and
M.
Sipser
, preprint arXiv:quant-ph/0001106 [quant-ph] (
2000
).
17.
S.
Jansen
,
M.-B.
Ruskai
, and
R.
Seiler
,
J. Math. Phys.
48
,
102111
(
2007
).
18.
S.
Oh
,
Phys. Rev. A
77
,
012326
(
2008
).
19.
A.
Childs
,
R.
Kothari
, and
R.
Somma
,
SIAM J. Comput.
46
,
1920
(
2017
).
20.

Examples of such algorithms include Refs. 3–6. Notice that algorithms based on Trotter product formulas such as Ref. 2 do not meet this requirement.

21.
G.
Brassard
,
P.
Høyer
,
M.
Mosca
, and
A.
Tapp
,
Quantum Computation and Information
, AMS Contemporary Mathematics Series (
AMS
,
2002
), Vol. 305.
22.

In fact, the Hamiltonian simulation algorithms4–6 only require few ancilla qubits and leave the O(·) expressions in Tables I and II unchanged. Reference 3 requires Olog(d) log(βΔ1log(χ1)/ϵ)log log(βΔ1log(χ1)/ϵ) additional qubits, where H̃=j=1dβjUj with unitaries Uj costing O(Λ) elementary gates and β=j|βj| (see Table 1 of Ref. 6 for an overview).

23.
R. V.
Mises
and
H.
Pollaczek-Geiringer
,
Z. Angew. Math. Mech.
9
,
152
(
1929
).
24.
J.
van Apeldoorn
,
A.
Gilyén
,
S.
Gribling
, and
R.
de Wolf
,
IEEE 58th Annual Symposium on Foundations of Computer Science
(
IEEE
,
2017
), pp.
403
414
.
25.
T. J.
Yoder
,
G. H.
Low
, and
I. L.
Chuang
,
Phys. Rev. Lett.
113
,
210501
(
2014
).
26.
V. V.
Shende
,
S. S.
Bullock
, and
I. L.
Markov
,
IEEE Trans. Comput. -Aided Des. Integr. Circuits Syst.
25
,
1000
(
2006
).
27.

Note that unlike Ref. 26, where λ denotes the overlap, here we write the overlap as λ.

28.
R.
Kothari
, private communication (
2017
).
29.
M. A.
Nielsen
and
I. L.
Chuang
,
Quantum Computation and Quantum Information: 10th Anniversary Edition
, 10th ed. (
Cambridge University Press
,
New York, NY, USA
,
2011
).
30.

Notice that this includes many-body Hamiltonians, as Hamiltonians consisting of n terms acting on at most k qubits are sparse with d = 2kn.

31.

In fact, it is sufficient to assume that the spectrum of H̃ is contained in [0, 1 − τ], where τ is defined in (D9). This ensures that H, as defined below, has entries with modulus at most 1.

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