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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

Quantum and classical

AF Heisenberg model for 2 spins

Eigen state

which we get handling the Schrödinger equation

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 ?

- 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

1 spin problem

Wave function :

(2 DoF)

2×2matrix.

…

2 spin problem

4×4matrix.

(22=4 DoF)

Nspin problem

:2N×2Nmatrix

:2NDoF

Computational Cost

- Exponential growth of the dimension

…

…

…

…

…

…

…

…

…

Reduce as

finite

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

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)

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

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 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

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

- 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

We search the ground state energy by optimizing the tensors.

- My research

Iteration

(Down-hill Simplex Method)

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]

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

- 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 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

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

- 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

- 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

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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