OBC | Quantum physics out of equilbrium: A new paradigm of computation and information

Quantum physics out of equilibrium:A new paradigm of computation and information

Tomaž Prosen

Department of Physics, Faculty of Mathematics and Physics,University of Ljubljana

OBC, Maribor, May 2012

Tomaž Prosen Quantum physics out of equilibrium

Complexity: Deterministic Chaos

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:On a unit interval x ∈ [0, 1] perform a simple transformation

xt+1 = 2xt (mod 1).



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:On a unit interval x ∈ [0, 1] perform a simple transformation

xt+1 = 2xt (mod 1).

Distance between neighbouring trajectories increases by a factor of 2 in eachstep. Thus, by factor 2t it t steps!


Such map acts simply as a shift of a binary ‘decimal’ point to the right:

[0.b1b2 . . .]2 −→ [0.b2b3 . . .]2



[0.b1b2 . . .]2 −→ [0.b2b3 . . .]2

Butterfly effect demistified: two initial conditions which are less than 1 in amilion appart (2−20), become completely different orbits after only 20 steps!



[0.b1b2 . . .]2 −→ [0.b2b3 . . .]2


[0.100110110111001010110 . . .]2 −→ . . . [0.0 . . .]2

[0.100110110111001010111 . . .]2 −→ . . . [0.1 . . .]2



[0.b1b2 . . .]2 −→ [0.b2b3 . . .]2


[0.100110110111001010110 . . .]2 −→ . . . [0.0 . . .]2

[0.100110110111001010111 . . .]2 −→ . . . [0.1 . . .]2



[0.b1b2 . . .]2 −→ [0.b2b3 . . .]2


[0.100110110111001010110 . . .]2 −→ . . . [0.0 . . .]2

[0.100110110111001010111 . . .]2 −→ . . . [0.1 . . .]2

CHAOS AND INFORMATION:

Baker map creates one bit of information per unit time, exactly the same as astochastic two-outcome event (say a coin toss)


Algorithmic complexity

Complexity(A) = The length of the shortest computer program, or any formalprotocol (in bits), on how to construct A.




A = [11111111111111111111111111111111111111111111111111]

Program: Repeat 1 50 times.




A = [11111111111111111111111111111111111111111111111111]

Program: Repeat 1 50 times.

A = [10010011110101101010101011110001101010101110101101]

Program: Recite A (shortest?)


What about complexity in quantum physics?



What is quantum physics at all?



What is quantum physics at all?Does quantum physics allow classically impossible things?


Three main non-classical properties of quantum mechanics:

Coherent superposition of classically distinct states - possibility ofinterference effects with material particles

Entanglement of states of composite quantum systems - possibility ofteleportation

Quantum measurement and collapse of the wave-function -in the act of quantum measurement God throws dice, and the value of themeasured observable jumps into one of classically allowed values












Quantum superpositions

Schrödinger cat: two possible classical states |alive〉, |dead〉.

Possible quantum state: superposition

|Ψ〉 = |alive〉+ |dead〉


Chaos and quantum superposition

Another paradigm of chaos and dynamical complexity: Billiards


Chaos and quantum superposition

Another paradigm of chaos and dynamical complexity: Billiards

In quantum physics, waves of matter are described by Schrödinger equation.


Chaos and the double slit experiment

A numerical experiment (Casati in Prosen 2005):“Leaking” of quantum particles through two slits in regular and chaotic billiard.

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Composite quantum systems in superpositions

Two Schrödinger cats can be in a superposition of any 4 states:

|alive, alive〉, |alive, dead〉, |dead, alive〉, |dead, dead〉.





The state of a pair of cats is an element of 4-dimensional (Hilbert) space.

|Ψ〉 = α|alive, alive〉+ β|alive, dead〉+ γ|dead, alive〉+ δ|dead, dead〉,

|α|2 + |β|2 + |γ|2 + |δ|2 = 1, α, β, γ, δ ∈ C.







|α|2 + |β|2 + |γ|2 + |δ|2 = 1, α, β, γ, δ ∈ C.

|α|2 is the probability, to see both cats alive when we open the box.







|α|2 + |β|2 + |γ|2 + |δ|2 = 1, α, β, γ, δ ∈ C.

|α|2 is the probability, to see both cats alive when we open the box.

Intriguing observation: Quantum mechanics is the first really fundamentalapplication of complex numbers in nature!


Quantum information - qubits

n two-level quantum systems (e.g. atoms with a single excited state), can bebrought in a superposition of 2n independent many-body states

|Ψ〉 =∑

bj∈{0,1}

αb1,b2,...bn |b1b2 . . . bn〉, bj ∈ {0, 1}.

“Hilbert space is a big place.” (Carlton Caves)




|Ψ〉 =∑

bj∈{0,1}

αb1,b2,...bn |b1b2 . . . bn〉, bj ∈ {0, 1}.


Quantum mechanics naturally generates a linear prescription which transformsstate |Ψ〉 into some other state|Ψ(t)〉 = U|Ψ〉.




|Ψ〉 =∑

bj∈{0,1}

αb1,b2,...bn |b1b2 . . . bn〉, bj ∈ {0, 1}.


Quantum mechanics naturally generates a linear prescription which transformsstate |Ψ〉 into some other state|Ψ(t)〉 = U|Ψ〉.

Feynman (1982): Quantum mechanics of systems in mutual interaction can beused for universal processing of quantum information:quantum computation


Quantum computer: What is this?


It is a UNIVERSAL MACHINE, which can peform arbitrary linear (unitary)transformation of an arbitrary prepared register of n qubits, and at the endperforms the measurement of an arbitrary set of qubits.


It is a UNIVERSAL MACHINE, which can peform arbitrary linear (unitary)transformation of an arbitrary prepared register of n qubits, and at the endperforms the measurement of an arbitrary set of qubits.

The main practical obstacle: decoherence, uncontrollable coupling to theenvironment destroys coherent superpositions.


Quantum Algorithm

A protocol, which reduces a desired transformation U to a sequence of asequence of quantum gates - primitive unitary transformations which act onlyto a single qubit or a pair of qubits at a time.


Quantum Algorithm


Two famous quantum algorithms:

Schor’s algorithm for factoring an integer N:Uses only ∼ (logN)3 quantum operations.The best classical algorithm needs ∼ exp[(logN)1/3] operations.

Grover’s algorithm for searching in an unstructured list of N things:Needs only

√N lookups!


Quantum Algorithm





√N lookups!


Quantum Algorithm





√N lookups!


Why can quantum computer outperform the classicalcomputer?



Because it can use quantum parallelism: it can compute with asuperposition of all 2n states of a quantum register at a time!



Because it can use quantum parallelism: it can compute with asuperposition of all 2n states of a quantum register at a time!

More precisely: Because it can use the entanglement!


Quantum entanglement: Einstein-Podolsky-Rosen (1935) paradox

Entangled pair of two qubits in different places, A and B

A ⇑⇓ · · · B ⇑⇓

|EPAB〉 =1√2

(|0〉A|0〉B + |1〉A|1〉B) =1√2

(|00〉+ |11〉).




A ⇑⇓ · · · B ⇑⇓

|EPAB〉 =1√2

(|0〉A|0〉B + |1〉A|1〉B) =1√2

(|00〉+ |11〉).

NON-LOCALITY of quantum mechanics:The measurement of qubit A triggers an instantaneous transition of the qubitB into the same state (0 or 1).




A ⇑⇓ · · · B ⇑⇓

|EPAB〉 =1√2

(|0〉A|0〉B + |1〉A|1〉B) =1√2

(|00〉+ |11〉).


Is it possible to transport quantum information in this way?




A ⇑⇓ · · · B ⇑⇓

|EPAB〉 =1√2

(|0〉A|0〉B + |1〉A|1〉B) =1√2

(|00〉+ |11〉).


Is it possible to transport quantum information in this way?

Indeed, EPR paradox enables us with two fascinating possibilities:

quantum teleportation and

secure quantum communication (already commercally available viastandard telecom optical cables!).


Already today, they can create in the labs entangled states of nearlymacroscopic objects, say entanglement between mechanical vibrations of asmall metal plate and an electric current (O’Connell et al, Nature 2010):


Chaos and stability of quantum computation

Quantum dynamics - “black box” U(t) - can be chaotic or regular.


Chaos and stability of quantum computation

Quantum dynamics - “black box” U(t) - can be chaotic or regular.

We found that chaotic quantumdynamics is more roboust againstexternal perturbations than regular(Prosen in Žnidarič, 2001 - 2006).We derived a general formula whichdescribes the so-called decay ofquantum fidelity

F (t) = 1−dist(Ψideal(t),Ψperturbed(t))

in a system where ideal dynamicsU(t) is perturbed by a small imper-fection.


Chaos is good.


Chaos is good.

But Quantum is better.


OBC | Quantum physics out of equilbrium: A new paradigm of computation and information

Education

quantum superpositions

act of quantum measurement

map acts

binary decimal point

shortest computer program

dierent orbits

initial conditions

amilion appart