Searching new quantum computational methods as the variants of the tree tensor network
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Searching new quantum computational methods as the variants of the Tree Tensor Network. Masashi Orii Kusakabe Lab. outline. Summary and Future Work. Introduction. My research. Algorithms. Numerical Result. Magnetic Order. Microscopic orde r in a m agnet.

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Searching new quantum computational methods as the variants of the tree tensor network

Searching new quantum computational methods as the variants of the Tree Tensor Network

Masashi Orii Kusakabe Lab


Outline

outline

Summary

and

Future

Work

Introduction

  • My research

Algorithms

Numerical

Result


Magnetic order

Magnetic Order

  • Microscopic order in a magnet

  • Ferro magnetic state

  • Antiferro magnetic state


Quantum and classical

Quantum and classical

AF Heisenberg model for 2 spins

Eigen state

which we get handling the Schrödinger equation


Example for real materials

Example for Real materials

  • Example:Shastry-Sutherland model[1] and SrCu2(BO3)2[2]

Singlet (Dimer) state

[3]

or

:O

:B

:Cu

:Sr

:spin 1/2

:Strong AF bond

:Weak AF bond

[1]B. S.Shastry and B. Sutherland Physica108B (1981) 1069-1070

[2] H. Kageyama ,et.al. Phys. Rev. Lett. 82, 3168 (1999)

[3]K. Sparta et. al. Eur. Phys. J. B 19 (2001) 507-516


What s

What’s ?

  • Math(the x-y plane)

    The vector is expressed by 2 basis vectors;

  • Quantum mechanics(spin ½)

  • The wavefunction is likely;

Using

and ,

the vector is rewritten as;

Defining

and ,

the wave function is expresses like a vector;

y

or

r

x


N spin problem

N spin problem

1 spin problem

Wave function :

(2 DoF)

2×2matrix.

2 spin problem

4×4matrix.

(22=4 DoF)

Nspin problem

:2N×2Nmatrix

:2NDoF


Computational c ost

Computational Cost

  • Exponential growth of the dimension

Reduce as

finite

PC has 2^(32)=4GB memory (limitation of PC)


Motivation

Motivation

  • Our motivation is to develop the computational methods to overcome the limitation of PC.

  • Generally, “to improve” means to get better accuracy.

[4]

ΔE

Our Study

MPS[4]

DMRG[1]

Very strong for 1D system

MERA[2]

Promising for 2D system

TTN[3]

・simple

・possibility for 2D system

Degree of freedom

Degree of freedom

[1] Steven R. White , Phys. Rev. Lett. 69, 2863-2866 (1992).

[2] G. Evenbly and G. Vidal, Phys.Lev. B 79, 144108 (2009).

[3] L. Tagliacozzo, G. Evenbly, and G.Vidal , Phys. Rev. B 80, 235127 (2009).

[4] F. Verstraete et al . PRL 93,227205(2004)


My research

My research

Algorithms

Numerical Result


What s ttn

What’s TTN?

We prepare the wave function for the spins using tree-like network.

Tree

  • My research

Tensor

Network

TTN for 4 site system

TTN for 16 site system


How to construct the wave function

How to construct the wave function

An effective Spin of

Χ1DoF (Χ1<4)

Spin ½

4×χ1

DoF

Tree

: Χ1DoF

2

, , or

2 spins → 4 DoF

Tensor

  • My research

Network

4

:2DoF

Reduce!


How to construct the wave function1

How to construct the wave function

  • Exact solution vs TTN for 8site system

TTN

Exact Solution

Χ2=2

4DoF

28=256 DoF

  • My research

Χ1=2

8

8DoF

8

8

8

8 DoF


Searching new quantum computational methods as the variants of the tree tensor network

VP

  • TTN[1] and TTN VP(our study)

  • VP does not change the DoF.

  • The energy is expected to be reduced.

P

  • My research

[1]L. Tagliacozzo, G. Evenbly, and G.Vidal , Phys. Rev. B 80, 235127 (2009).

(a) (b) (c) (d) (e)

Fig:4. Calculated entanglement entropy(EE) for α=1 .


Model hamiltonian

Model: Hamiltonian

  • Hamiltonian for the demonstration

    : 1&2D Heisenberg model with Bond alternation

16site, 2D PBC

  • My research

4site, 1D preodic boundary condition(PBC)

J1=1.0, J2=1.0

For J1=1.0, J2=0.0

→Dimerstate is

The ground state.

(a) (b) (c) (d) (e)

Fig:4. Calculated entanglement entropy(EE) for α=1 .


How to optimize

How to optimize

We search the ground state energy by optimizing the tensors.

  • My research

Iteration

(Down-hill Simplex Method)


How to construct the wave function2

How to construct the wave function

Formula

Program (Fortran)

do i=istart_gs,iend_gs

do j=1,sys_size/2

s1=state_list_a(i,2*j-1)

s2=state_list_a(i,2*j)

v_gs_temp1(i)=v_gs_temp1(i)*w_part(s1,s2,1,1)

end do

s1=state_list_a(i,1)

s2=state_list_a(i,4)

v_gs_temp2(i)=v_gs_temp2(i)*w_part(s1,s2,1,1)

s1=state_list_a(i,2)

s2=state_list_a(i,3)

v_gs_temp2(i)=v_gs_temp2(i)*w_part(s1,s2,1,1)

v_gs_input(i)= cos(linear_coeff(1))* v_gs_temp1(i) &

+ sin(linear_coeff(1)) &

* dcmplx(sin(linear_coeff(2)),cos(linear_coeff(2))) &

* v_gs_temp2(i)

end do

Coding

  • My research


Numerical result for 4 site system 1

Numerical Result for 4 site system[1]

Previous work(TTN[2])

Our work(TTN VP)

  • Χ=2 is enough for VP

  • Next goal is to calculate larger system /other wavefunction

    [1]previous M1 colloquium presentation.

    [2] L. Tagliacozzo, G. Evenbly, and G.Vidal , Phys. Rev. B 80, 235127 (2009).

Diner

→Easy

Uniform

→Difficult

ΧDoF

for any χ

  • My research

4 DoF

P

(a) (b) (c) (d) (e)

Fig:4. Calculated entanglement entropy(EE) for α=1 .


Numerical result 8site

Numerical result 8site

  • 8site 1D PBC(Uniform chain) Energy

χ2

(χ1,χ2)=(1,1)

χ1

(2,1)

(2,2)

(2,3)

(3,1)

(2,4)

(4,1)

(3,2)

Reduced

(3,3)

  • My research

P


Numerical result 8site1

Numerical result 8site

〈Previous Work〉

TTN with Sz conservation[4]

TTN[1]

MERA[3]

MPS[2]

〈Our Work〉

  • My research

[1] L. Tagliacozzo, G. Evenbly, and G.Vidal , Phys. Rev. B 80, 235127 (2009).

[2] F. Verstraete, D. Porras, and J. I. Cirac , Phys. Rev. Lett. 93, 227205 (2004).

[3] G. Evenbly and G. Vidal, Phys. Lev. B 79, 144108 (2009).

[4] S. Singh, R. N. C. Pfeifer, and G. Vidal, arXiv:1008.4774v1.

P

T

T

T


Numerical result 8site2

Numericalresult 8site

  • 8site (1DPBC)

・Among different methods, TTTN+VP gives the best solution with the least degrees of freedom (D=31).MPS(D=32) and also have high accuracy.

P

TTTN

(Translational invariant TTN)

  • My research

= +

T

T

T

T

T

T


Numerical results

Numerical Results

  • 16site 2D PBC

  • 16site 1D PBC

(a)TTTN 1(b)MPS and TTTN2

  • My research

For 1D,MPSis better than TTTN.

For 2D,TTTNis better than MPS.

We must select the optimal network for every Hamiltonian.

T

T

T


Summary

Summary

  • We calculated the ground state of 4,(8,16)spin system.

  • Our result showed that VP(vector projection) reduces the calculated energy for any case.

  • We obtained the exact energy for 8 site system(using TTTN+VP).

  • For 16 site 2D system, TTTN +VP won MPS +VP.

  • We confirmed the fact that MPS +VP is powerful for 1D system but is not for 2D.


Future work

Future Work

Future work

To Calculate …

  • larger system.

    (36site system is our goal.)

  • using other networks.

  • Models for real materials such as Shastry-Sutherland model,2D frustrated system.


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