If the claims of D-Wave are correct, the first general purpose quantum computers are already here. How do they work? In this talk we will follow a pedestrian approach to quantum computation. We will travel back in time, to revisit the old fashioned analog computers, such as the spaghetti-sort or the soapy solution to the Steiner tree problem. Quantum adiabatic computation, the basic principle of the D-Wave device, is just an analog computer, but it profits from a spooky property of quantum mechanics called entanglement. In the last part of the talk we will address the (unanswered yet) question: what will be the real power of quantum computers, compared to classical ones?

1. Analog computers and Schr¨odinger’s cats
A pedestrian introduction to Quantum Computers
Javier Rodr´ıguez Laguna
Dto. F´ısica Fundamental, UNED, Madrid Instituto de F´ısica Te´orica (CSIC), Madrid
Facultad de Inform´atica, UCM, Madrid. May 7, 2015.

2. What shall we talk about?
Simulation is the sincerest form of flattery
We’re surrounded by computers, we only have to look
Always bet so that losing is the best outcome
The best path is to take all paths
Solving problems requires some untidiness
Can we learn physics studying computer science?

3. Physicists love to simulate
MthK protein (PDB: 1LNQ),
Bacterial Ca2+-gated binding K channel
Courtesy Agata Kranjc, SISSA
Foam configuration simulations,
[J. Phys. C: Cond. Matt. 16, 4165 (2004)]
Courtesy Dolores Alonso, Trinity College
Pt(111) surface reconstruction,
[see also PRB 67, 205418 (2003)]
Courtesy Raghani Pushpa, SISSA

4. It’s hard to simulate!
• Very often, Nature optimizes.
• Simulating optimization can be pretty hard.
• Target functions can have very complicated landscapes.

5. What if we can’t?
• What if simulation is out of our reach?
• We can profit from that to devise analog computers!
Spaghetti computer Stringy computer

6. Classical Analog Computers
Experiment by Dutta and coworkers, ArXiv: 0806.1340.

7. Nature need not be a Turing machine
• Can we simulate efficiently Nature in a Turing machine?
• Classical mechanics: maybe.
• Quantum mechanics: no way.

8. Schr¨odinger’s cat

9. Schr¨odinger’s cat

10. Entanglement

11. EPR experiment
• Prepare an entangled state |+− − |−+ .
• Measure any component on one spin, you get:
• Entanglement entropy: how much information you lose forgetting one part.

12. Many-Cat Physics
• Also known as many-body physics.
• A quantum pure state is a mapping
ψ : {0, 1}N → C
• Example: α |010101 + β |101010 .
• 2N components... a lot!
• They are not epistemological, they are ontological!!!

13. Qubistic view
• How to represent graphically a pure state?
• QUBISM, developed by us (2012).
0 1
01
00 11
10
AF 0101 . . .
FM 0000 . . .
AF 1010 . . .
FM 1111 . . .

14. Qubistic view

15. Solving all your life problems
• Nodes: life aims, Links: constraints.
Also known as Spin-Glass Problem

16. Solving all your life problems
• Minimize frustration

17. Adiabatic Quantum Computation
HF =
i,j
JijSz
i Sz
j +
i
hiSz
i
H0 =
i
Sx
i
H(t) = (1 − t)HF + tHO
• For t = 0, |Ψ = |→ ⊗N
.
• For t = 1, |Ψ is the solution to our problem!!!
What happens in the middle?

18. Complexity of the State
1e-20
1e-18
1e-16
1e-14
1e-12
1e-10
1e-08
1e-06
0.0001
0.01
1
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
|Ψ|
2
Adiabatic parameter
-0.4
-0.2
0
0.2
0.4
〈Sz〉
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
MaximalvonNeumannentropy

19. The Unfolding Hypothesis
Moreover: complexity typically explodes, a Quantum Phase Transition.

20. Problems studied with AQC
• Exact cover.
• Ramsey numbers.
• Factoring.
• Unsorted search.
— Main experimental problem: the GAP.
— Speed limit, to success probability larger than 1 − ǫ,
dH
dt
/(∆E)2 < ǫ

21. Simulating AQC
• What makes Quantum Computation Special?
• Simulated Thermal Annealing.
Energy target function, E, finite temperature: p(X) ∼ exp(−E/T).
• Simulated Quantum Annealing.
The same, but with P replicas, joined by springs.

22. Matrix Product States
Efficient way to store pure states, ψ : {0, 1}N → C.
ψ(s1, · · · , sN) = Tr A
s1
1 A
s2
2 A
s3
3 · · · A
sN
N
• A±
k are 2N matrices of dimension m × m.
• Somehow, A are similar to finite-automata transition matrices.
• The dimension m is related to entanglement.
• Quantum Wavefunction Annealing: simulate AQC using MPS!
• Bottleneck of QWA: entanglement.

23. Quantum Mythology

24. Physical Predictions of P= NP?
• Can we learn physics by doing computer science?
• Nature need not be a Turing machine... but our computers are!
• Thus, no simulation of AQC will solve NP-complete problems in polynomial
time.
• Thus, our QWA simulation time must scale fast for them.
• Thus, we must encounter a quantum phase transition in our path!!

25. Will Quantum Computation Succeed?
• AQC depends on the existence of a gap.
• Gaps are typically found at Quantum Phase Transitions.
• Computer Science predicts Quantum Phase Transitions...
• YET, Nature is hard to simulate!!!!!!!!
• So... we don’t know.

26. Thank you for your Attention!
• Visit our bar: http://mononoke.fisfun.uned.es/jrlaguna
Thanks to I. Rodr´ıguez-Laguna, S.N. Santalla, G. Sierra, G. Santoro, P. Raghani,
A. Degenhard, M. Lewenstein, A. Celi, E. Koroutcheva, M.A. Mart´ın-Delgado and
R. Cuerno.