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Quantum Information and the simulation of quantum systems

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Quantum Informationandthe simulation of quantum systems

José Ignacio Latorre

Universitat de Barcelona

Perugia, July 2007

In collaboration with:

Sofyan Iblisdir, Luca Tagliacozzo

Arnau Riera, Thiago Rodrigues de Oliveira, José María Escartín, Vicent Picó

Román Orús, Artur García-Sáez, Frank Verstraete, Miguel Aguado, Ignacio Cirac

Physics

Theory 1

Theory 2

Exact solution

Approximated

methods

Simulation

Classical

Simulation

Quantum

Simulation

- Classical Theory
- Classical simulation
- Quantum simulation

- Quantum Mechanics
- Classical simulation
- Quantum simulation

Classical computer

?

Quantum computer

Classical simulation of Quantum Mechanics is related to our ability to support

large entanglement

Classical simulation may be enough to handle e.g. ground states

Quantum simulation needed for typical evolution of Quantum systems

(linear entropy growth to maximum)

Is it possible to classically simulate faithfully a quantum system?

represent

Heisenberg model

evolve

read

Misconception: NO

Exponential growth of Hilbert space

n

Classical representation requires dn complex coefficients

A random state carries maximum entropy

- Refutation
- Realistic quantum systems are not random
- symmetries (translational invariance, scale invariance)
- local interactions

- We do not have to work on the computational basis
- use an entangled basis

e.g: efficient description for slightly entangled states

Schmidt decomposition

A B

= min(dim HA, dim HB)

Schmidt number

A product state will have

Vidal 03: Iterate this process

A product state iff

- Slight entanglement iff poly(n)<<dn
- Representation is efficient
- Single qubit gates involve only local update
- Two-qubit gates reduces to local updating

efficient simulation

Matrix Product States

i

α

canonical form PVWC06

Approximate physical states with a finite MPS

Graphic representation of a MPS

Efficient computation of scalar products

operations

Local action on MPS

U

Intelligent way to represent and manipulate

entanglement

Classical analogy:

I want to send 16,24,36,40,54,60,81,90,100,135,150,225,250,375,625

Instruction: take all 4 products of 2,3,5

MPS= compression algorithm

i2=1 i2=2

i2=3 i2=4

Crazy ideas: Image compression

| i2 i1

| i1

105| 2,1

i1=1 i1=2

i1=3 i1=4

RG addressing

level of grey

pixel address

- QPEG
- Read image by blocks
- Fourier transform
- RG address and fill
- Set compression level:
- Find optimal
- gzip (lossless, entropic compression)
- (define discretize Γ’s to improve gzip)
- diagonal organize the frequencies and use 1d RG
- work with diferences to a prefixed table

Max = 81

= 1

PSNR=17

= 4

PSNR=25

= 8

PSNR=31

Crazy ideas: Differential equations

Crazy ideas: Differential equations

Crazy ideas: Shor’s algorithm with MPS

Crazy ideas: Shor’s algorithm with MPS

Constructed: adder, multiplier, multiplier mod(N)

Note: classical problems with a direct product structure!

Back to the central idea: entanglement support

Success of MPS will depend on how much entanglement

is present in the physical state

Physics

Simulation

If

MPS is in very bad shape

Exact entropy for a reduced block in spin chains

At Quantum Phase Transition

Away from Quantum Phase Transition

Maximum entropy support for MPS

Maximum supported entanglement

Faithfullness = Entanglement support

MPS

Spin chains

Spin networks

PEPS

Area law

Computations of entropies are no longer academic exercises but limits on simulations

Physics

Simulation

VLRK02-03

OL04

For 3-SAT

LLRV04

Exact RG on states

VCLRW05

OLRV05

Lipkin model

100-qubit Ex-cover instance

BOLP05

Image compression

L05

OLEC06

RL06

Area law

ILO06

Laughlin

ILO06

Continuous variables

Local (12 levels), nearest neighbor H is QMA-complete!!

AGK07

Keep in mind:

Area law << Volume law

Translational symmetry and locality have reduced dramatically the amount

of entanglement

Worst case (max entropy) remains at phase transition points

- MPS and PEPS are a good representation of QM
- Approach new problems
- Precision
- Can we do any better than DMRG?

Simulation of the Laughlin wave function

Local basis: a=0,..,n-1

Dimension of the Hilbert space

Analytic expression for the reduced entropy

ILO06

Exact MPS representation of Laughlin wave function

Clifford algebra

Optimal solution!

(all matrices equal but the last!)

m=2

Example:

Normalization of wave function for m=2

So far, we have not managed to exploit the product structure

Translational invariant spin chains

Vidal05: iTEBD translationaly invariant infinite system algorithm

commute

commute

All even gates can be performe simultaneously

All odd gates can be performe simultaneously

Use Trotter to combine them

are isometries

=

Energy

Trotter 2nd order

Heisenberg model

Trotter 2 order, =.001

Exponential distribution λ

Poorness of DMRG

Advantage: clean results for infinite half chain entropy

Problem: Poor convergence of entropy

entropy

energy

Maximum half-chain entanglement for Heisenberg model

Consistent with central charge c=1

Attention to spontaneous symmetry breaking

To compute block entropies, use exact coarse graining of MPS

Local basis

Optimal choice!

VCLRW

remains the same and locks the physical index!

After L spins are sequentially blocked

Entropy is bounded

Exact description of non-critical systems

Exact solution for =2

min

=

S= .485704202

Numerics

Precision for entropy requires some extra effort

Trotter higher order

Random seeds

(avoiding hysteresis cycles associated to the minimization procedure)

Boost

S

Perfect alignement

M

MPS support of entropy obeys scaling law!!

S

χ

??

So far

- Simulation technique
- representation
- evolution
- observables

Physics

Entanglement

Entanglement support

Exploit MPS, PEPS, MERA

NEXT

Contraction of PEPS is #P

Yet, for translational invariant systems, it comes to iTEBD

JOVVC07

Beats quantum Montecarlo!!

VIDAL Beyond MPS: Entanglement RG

MERA Unitary networks

Building the program: detailed check vs MPS