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

1. Searching new quantum computational methods as the variants of the Tree Tensor Network Masashi Orii Kusakabe Lab

2. outline Summary and Future Work Introduction • My research Algorithms Numerical Result

3. Magnetic Order • Microscopic order in a magnet • Ferro magnetic state • Antiferro magnetic state

4. Quantum and classical AF Heisenberg model for 2 spins Eigen state which we get handling the Schrödinger equation

5. 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

6. 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

7. 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

8. Computational Cost • Exponential growth of the dimension … … … … … … … … … Reduce as finite PC has 2^(32)=4GB memory (limitation of PC)

9. 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)

10. My research Algorithms Numerical Result

11. 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

12. 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!

13. 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

14. 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 .

15. 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 .

16. How to optimize We search the ground state energy by optimizing the tensors. • My research Iteration (Down-hill Simplex Method)

17. 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

18. 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 .

19. 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

20. 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

21. 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

22. 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

23. 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.

24. 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.