Quantum information and the simulation of quantum systems
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Quantum Information and the 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

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Quantum information and the simulation of quantum systems

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


Quantum information and the simulation of quantum systems

Physics

Theory 1

Theory 2

Exact solution

Approximated

methods

Simulation

Classical

Simulation

Quantum

Simulation


Quantum information and the simulation of quantum systems

  • 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)


Quantum information and the simulation of quantum systems

Is it possible to classically simulate faithfully a quantum system?

represent

Heisenberg model

evolve

read


Quantum information and the simulation of quantum systems

Misconception: NO

Exponential growth of Hilbert space

n

Classical representation requires dn complex coefficients

A random state carries maximum entropy


Quantum information and the simulation of quantum systems

  • 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


Quantum information and the simulation of quantum systems

e.g: efficient description for slightly entangled states

Schmidt decomposition

A B

 = min(dim HA, dim HB)

Schmidt number

A product state will have


Quantum information and the simulation of quantum systems

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


Quantum information and the simulation of quantum systems

Matrix Product States

i

α

canonical form PVWC06

Approximate physical states with a finite  MPS


Quantum information and the simulation of quantum systems

Graphic representation of a MPS

Efficient computation of scalar products

operations


Quantum information and the simulation of quantum systems

Local action on MPS

U


Quantum information and the simulation of quantum systems

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


Quantum information and the simulation of quantum systems

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


Quantum information and the simulation of quantum systems

  • 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


Quantum information and the simulation of quantum systems

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!


Quantum information and the simulation of quantum systems

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


Quantum information and the simulation of quantum systems

Exact entropy for a reduced block in spin chains

At Quantum Phase Transition

Away from Quantum Phase Transition


Quantum information and the simulation of quantum systems

Maximum entropy support for MPS

Maximum supported entanglement


Quantum information and the simulation of quantum systems

Faithfullness = Entanglement support

MPS

Spin chains

Spin networks

PEPS

Area law

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


Quantum information and the simulation of quantum systems

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


Quantum information and the simulation of quantum systems

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

AGK07


Quantum information and the simulation of quantum systems

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?

  • e.g.: Faithfull numbers for entropy? Exact solutions? Smaller errors?

  • Can we simulate better than lattice Monte Carlo?

  • Are MPS and PEPS the best simulation solution?


  • Quantum information and the simulation of quantum systems

    Simulation of the Laughlin wave function

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

    Dimension of the Hilbert space

    Analytic expression for the reduced entropy

    ILO06


    Quantum information and the simulation of quantum systems

    Exact MPS representation of Laughlin wave function

    Clifford algebra

    Optimal solution!

    (all matrices equal but the last!)


    Quantum information and the simulation of quantum systems

    m=2


    Quantum information and the simulation of quantum systems

    Example:

    Normalization of wave function for m=2

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


    Quantum information and the simulation of quantum systems

    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


    Quantum information and the simulation of quantum systems

    are isometries

    =

    Energy


    Quantum information and the simulation of quantum systems

    Trotter 2nd order

    Heisenberg model

    Trotter 2 order, =.001

    Exponential distribution λ

    Poorness of DMRG


    Quantum information and the simulation of quantum systems

    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


    Quantum information and the simulation of quantum systems

    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


    Quantum information and the simulation of quantum systems

    Exact solution for =2

    min

    =

    S= .485704202


    Quantum information and the simulation of quantum systems

    Numerics

    Precision for entropy requires some extra effort

    Trotter higher order

    Random seeds

    (avoiding hysteresis cycles associated to the minimization procedure)

    Boost


    Quantum information and the simulation of quantum systems

    S

    Perfect alignement

    M


    Quantum information and the simulation of quantum systems

    MPS support of entropy obeys scaling law!!

    S

    χ

    ??


    Quantum information and the simulation of quantum systems

    So far

    • Simulation technique

      • representation

      • evolution

      • observables

    Physics

    Entanglement

    Entanglement support

    Exploit MPS, PEPS, MERA

    NEXT


    Quantum information and the simulation of quantum systems

    Contraction of PEPS is #P

    Yet, for translational invariant systems, it comes to iTEBD

    JOVVC07

    Beats quantum Montecarlo!!


    Quantum information and the simulation of quantum systems

    VIDAL Beyond MPS: Entanglement RG

    MERA Unitary networks

    Building the program: detailed check vs MPS


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