We present a didactic introduction to adiabatic quantum computation (AQC) via the explicit construction of a classical simulator of quantum computers. This constitutes a suitable route to introduce several important concepts for advanced undergraduates in physics: quantum many-body systems, quantum phase transitions, disordered systems, spin-glasses, and computational complexity theory.

1.
M. A.
Nielsen
and
I. L.
Chuang
,
Quantum Computation and Quantum Information
(
Cambridge U.P.
,
Cambridge
,
2001
).
2.
N. D.
Mermin
,
Quantum Computer Science: An Introduction
(
Cambridge U.P.
,
Cambridge
,
2007
).
3.
D.
Castelvecchi
, “
Quantum computers ready to leap out of the lab in 2017
,”
Nature
541
,
9
10
(
2017
).
4.
IBM Quantum Experience
, <https://www.research.ibm.com/ibm-q/>
5.
M.
Hebenstreit
,
D.
Alsina
,
J. I.
Latorre
, and
B.
Kraus
, “
Compressed quantum computation using a remote five-qubit quantum computer
,”
Phys. Rev. A
95
,
052339
(
2017
).
6.
E.
Farhi
,
J.
Goldstone
,
S.
Gutmann
,
J.
Lapan
,
A.
Lundgren
, and
D.
Preda
, “
A quantum adiabatic evolution algorithm applied to random instances of an NP-complete problem
,”
Science
292
,
472
475
(
2001
).
7.
A.
Das
and
B. K.
Charkrabarti
, “
Quantum annealing and analog quantum computation
,”
Rev. Mod. Phys.
80
,
1061
1081
(
2008
).
8.
G. E.
Santoro
,
R.
Martoňák
,
E.
Tosatti
, and
R.
Car
, “
Theory of quantum annealing of an ising spin glass
,”
Science
295
,
2427
2430
(
2002
).
9.
The sample code provided to our students, in C++, is freely available online at <http://github.com/jvrlag/qtoys>. The reader is encouraged to download, understand and modify it, following the instructions given in the appendix.
10.
L.
Susskind
, “
Computational complexity and black hole horizons
,”
Fortsch. Phys.
64
,
24
43
(
2016
).
11.
L.
Susskind
, “
Dear qubitzers, GR=QM
,” e-print arXiv:1708.03040.
12.
V.
Scarani
, “
Quantum computing
,”
Am. J. Phys.
66
,
956
960
(
1998
).
13.
D.
Candela
, “
Undergraduate computational physics projects on quantum computing
,”
Am. J. Phys.
83
,
688
702
(
2015
).
14.
E.
Gerjuoy
, “
Shor's factoring algorithm and modern cryptography. An illustration of the capabilities inherent in quantum computers
,”
Am. J. Phys.
73
,
521
540
(
2005
).
15.
D.
Aharonov
,
W.
van Dam
,
J.
Keme
,
Z.
Landau
,
S.
Lloyd
, and
O.
Regev
, “
Adiabatic quantum computation is equivalent to standard quantum computation
,” in
Proceedings of the 45th Annual IEEE Symposium on Foundations of Computer Science
(
2004
), pp.
42
51
.
16.
R.
Barends
 et al, “
Digitized adiabatic quantum computing with a superconducting circuit
,”
Nature
534
,
222
226
(
2016
).
17.
S.
Boixo
 et al, “
Evidence for quantum annealing with more than one hundred qubits
,”
Nat. Phys.
10
,
218
224
(
2014
).
18.
S. W.
Shin
,
G.
Smith
,
J. A.
Smolin
, and
U.
Vazirani
, “
How ‘quantum’ is the D-wave machine?
,” e-print arXiv:1401.7087.
19.
C.
Papadimitriou
,
Computational Complexity
(
Addison Wesley
,
Reading, MA
,
1994
).
20.

Where is logical or, is logical and and an overline is logical negation.

21.
T.
Weise
, Global Optimization Algorithms—Theory and Application (
2009
), <www.it-weise.de/projects/book.pdf>.
22.
A.
Schrijver
, “
A course on combinatorial optimization
,” <homepages.cwi.nl/~lex/files/dict.pdf> (
2013
).
23.
Time scaling as a power-law of the input size.
24.
S.
Mertens
, “
Computational complexity for physicists
,”
Comput. Sci. Eng.
4
,
31
47
(
2002
).
25.
S.
Aaronson
, “
P=? NP
,” (
2017
), <http://www.scottaaronson.com/papers/pnp.pdf>
26.

A cautionary remark: rigorously, classes P and NP are not defined for optimization problems, but for decision problems, which are solved by stating Yes or No. For example: is 29 a prime number? But optimization problems can be mapped to decision problems, so we will be a bit loose in the terminology.

27.
N.
Blum
, “
A solution to the P vs NP problem
,” e-print arXiv:1708.03486.
28.

Of course, there are also problems which are not NP, e.g., how to win a chess game, but we will not discuss them in this text.

29.
D. E.
Knuth
,
The Art of Computer Programming. Volume 3: Sorting and Searching
(
Addison-Wesley
,
Reading, MA
,
1998
).
30.
H.
Nishimori
,
Statistical Physics of Spin Glasses and Information Processing: An Introduction
(
Clarendon Press
,
Oxford
,
2001
).
31.
M.
Mezard
,
G.
Parisi
, and
M.
Virasoro
,
Spin Glass Theory and Beyond
(
World Scientific
,
Singapore
,
1998
).
32.
F.
Barahona
, “
On the computational complexity of Ising spin glass models
,”
J. Phys. A: Math. Gen.
15
,
3241
3253
(
1982
).
33.
C.
Isenberg
, “
Problem solving with soap films, part I
,”
Phys. Educ.
10
,
452
456
(
1975
).
34.
A. K.
Dewdney
,
The Armchair Universe: An Exploration of Computer Worlds
(
W.H. Freeman & Co.
,
New York
,
1987
).
35.
M. R.
Garey
,
R. L.
Graham
, and
D. S.
Johnson
, “
The rectilinear steiner tree problem is NP-complete
,”
SIAM J. Appl. Math.
32
,
826
834
(
1977
).
36.
D. R.
Lovett
and
J.
Tilley
, “
Illustrating phase transitions with soap films
,”
Am. J. Phys.
59
,
415
421
(
1991
).
37.
S.
Kirkpatrick
,
C. D.
Gelatt
, Jr.
, and
M. P.
Vecchi
, “
Optimization by simulated annealing
,”
Science
220
,
671
680
(
1983
).
38.
D. H.
Wolpert
and
W. G.
Macready
, “
No free lunch theorems for optimization
,”
IEEE Trans. Evol. Comp.
1
,
67
82
(
1997
).
39.
R. P.
Feynman
, “
Simulating physics with computers
,”
Int. J. Theor. Phys.
21
,
467
488
(
1982
).
40.

We would now like to dispel a usual misunderstanding. When quantum computation is developed your computers will still be mostly classical. Usual tasks will be undertaken by a classical CPU. But they will at some moments connect to a “quantum co-processor” devoted to solving certain special problems.

41.
L.
Susskind
and
A.
Friedman
,
Quantum Mechanics: The Theoretical Minimum
(
Basic Books
,
New York
,
2014
).
42.
G.
Vidal
, “
Efficient simulation of slightly entangled quantum computations
,”
Phys. Rev. Lett.
91
,
147902
(
2003
).
43.
J.
Rodriguez-Laguna
, “
Quantum wavefunction annealing of spin glasses on ladders
,”
J. Stat. Mech.
2007
,
P05008
(
2007
).
44.
J.
Rodriguez-Laguna
and
S. N.
Santalla
, “
Physical consequences of P≠NP and the density matrix renormalization group annealing conjecture
,”
J. Stat. Mech.
2014
,
P07006
(
2014
).
45.
L. D.
Landau
and
E. M.
Lifshitz
,
Quantum Mechanics
(
Pergamon Press
,
Oxford
,
1958
).
46.
C.
Wittig
, “
The Landau-Zener formula
,”
J. Phys. Chem. B
109
,
8428
8430
(
2005
).
47.
S.
Sachdev
,
Quantum Phase Transitions
(
Cambridge U.P.
,
Cambridge
,
1999
).
48.
S.
Suzuki
,
J.
Inoue
, and
B. K.
Chakrabarti
,
Quantum Ising Phases and Transitions in Transverse Ising Models
(
Springer
,
Berlin
,
2013
).
49.

Choosing both the Jij and the hi as random uniform deviates in [−1,1].

50.
J.
Rodriguez-Laguna
,
P.
Migdał
,
M.
Ibáñez-Berganza
,
M.
Lewenstein
, and
G.
Sierra
, “
Qubism: self-similar visualization of many-body wavefunctions
,”
New J. Phys.
14
,
053028
(
2012
).
51.
W. H.
Press
,
S. A.
Teukolsky
,
W. T.
Vetterling
, and
B. P.
Flannery
,
Numerical Recipes in C
(
Cambridge U.P.
,
Cambridge
,
1988
).
52.

Notice also that we did not provide a prescription for the division of the system into parts A and B. Different partitions will give rise to different entanglement entropies. Relevant choices usually imply a minimal boundary between the parts.

53.
B.
Altshuler
,
H.
Krovi
, and
J.
Roland
, “
Anderson localization makes adiabatic quantum optimization fail
,”
Proc. Natl. Acad. Sci. U.S.A.
107
,
12446
12450
(
2010
).
54.

The main requirement is that the commutator [Hx,HI]≠0, which is immediately fulfilled for any diagonal HI.

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