1. Quantum Spin Systems from the point a view of Quantum Information Theory Frank Verstraete, Ignacio Cirac Max-Planck-Institut für Quantenoptik

2. Overview The quantum repeater, entanglement length and quantum computing with projected entangled pair states (PEPS) MPS/PEPS: basic properties PEPS as variational trial states Illustration: –RG –DMRG Applications: –DMRG with Periodic boundary conditions –Calculating Excitations –Dynamical correlation functions using PEPS –Optimal time evolution algorithm –Finite-T DMRG: Projected entangled pair density operators quantum-TMRG from the perspective of PEPS –Quantum Spin Glasses –Simulation of 2-D systems Ground states Time evolution Conclusion

3. Number of states kept (D)

4. (a) Magnetization vs. transverse magnetic field of the thermal state of the XY model with 60 spins, for temperatures T=0.05,0.5,5 and 50 (bottom to top). (b) Error in the density matrix of the thermal state computed using MPDO, vs. temperature, for a chain with N=8 spins, D=8 (solid) and D=14,20,24 (dashed).

5. 4x4 10x10 20x20

6. Spin systems in Quantum Information Theory Central motto: Entanglement is a resource –Essential ingredient in quantum cryptography, quantum computing, etc. –Basic Unit: Bell-state |00 i +|11 i –Long-range distribution of entanglement can be done via a quantum repeater : –The effective Hamiltonian describing this system is a spin system with nearest neighbour couplings where we have complete local control over spins

7. Given a 1-D state of N spins, what is the amount of entanglement that can be localized between separated spins (qubits) in function of their distance by doing local measurements on the other ones? –Leads to notion of Localizable Entanglement (LE) and associated Entanglement length –The localizable entanglement between two spins is always larger than or equal to the connected 2-point correlation functions –Hence the Correlation length is a lower bound to the Entanglement length: long-range correlations imply long-range entanglement –The LE can be calculated using a combination of DMRG and Monte Carlo Natural question: do there exist ground states with diverging Entanglement length but finite Correlation length? Verstraete, Popp, Cirac 04

8. The spin-1 AKLT-model Singlet Proj. in sym. subspace All correlation functions decay exponentially The symmetric subspace is spanned by 3 Bell states, and hence this ground state can be used as a perfect quantum repeater Diverging entanglement length but finite correlation length LE detects new kind of long range order Antiferro spin-1 chain is a perfect quantum channel Verstraete, Martin-Delgado, Cirac 04

9. Generalizing the AKLT-state: PEPS Every state can be represented as a Projected Entangled Pair State (PEPS) as long as D is large enough Extension to mixed (finite-T) states: take Completely Positive Maps (CPM) instead of projectors : 1-D PEPS reduce to class of finitely correlated states / matrix product states (MPS) in thermodynamic limit (N !1 ) when P 1 =P 2 =  =P 1 –Systematic way of constructing translational invariant states –MPS become dense in space of all states when D !1 PiPi P i+1 P i+2 P i+3 P i+4 P i+5 P i+6 Ex.: 5 qubit state Fannes, Nachtergaele, Werner 92

10. PEPS in higher dimensions: PEPS in QIT: –Quantum Error correction: –Quantum computing Verstraete, Cirac 04

11. Basic properties of PEPS Correlation functions for 1-D PEPS can easily be calculated by multiplying transfer matrices of dimension D 2 : Number of parameters grows linearly in number of particles (NdD c ) with c coordination number of lattice 2-point correlations decay exponentially Holographic principle: entropy of a block of spins is proportional to its surface

12. Spin systems: basic properties Hilbert space scales exponentially in number of spins Universal ground state properties: –Entropy of block of spins / surface of block (holographic principle) –Correlations of spins decay typically exponentially with distance (correlation length) The N-particle states with these properties form a tiny subspace of the exponentially large Hilbert space

13. Ground states are extreme points of a convex set: –Problem of finding ground state energy of all nearest-neighbor transl. invariant Hamiltonians is equivalent to characterizing the convex set of n.n. density operators arising from transl. invariant states Finitely Correlated States / Matrix Product States / Projected Entangled Pair States provide parameterization that only grows linearly with number of particles but captures these desired essential features very well The Hamiltonian defines a hyperplane in (2s+1) 2 dim. space

14. PEPS as variational trial states All expectation values and hence the energy are multi- quadratic in the variables P k Strategy for minimizing energy for N-spin state: –Fix all projectors P i except the j th –Both the energy and the norm are quadratic functions of the variable P j and hence the minimal energy by varying P i can be obtained by a simple generalized eigenvalue problem: H eff and N are function of the Hamiltonian and all other projectors, and can efficiently be calculated by the transfer matrix method –Move on to the (j § 1) th particle and repeat previous steps (sweep) until convergence Compare to MPS-approach of Ostlund and Rommer 95 and Dukelsky et al. 97 : imposing that all P i are equal makes the optimization untractable for large D PiPi P i+1 P i+2 P i+3 P i+4 P i+5 P i+6 Verstraete, Porras, Cirac 04

15. Illustration 1 Wilson’s Renormalization Group (RG) for Kondo-effect: RG approximates ground state and lowest excited states by a family of PEPS –A i are chosen s.t. |   i form orthonormal set, which forces N=identity –Remark that sweeping would give better precision –Excited states are forced to have the same A i k<N, which cannot be true in general; OK for purpose of calculating GS, but better can be done if one wants exact info about spectrum /dynamical correlation functions (see later) P0P0 P1P1 P2P2 P3P3 P4P4 P5P5 P6P6 J 1 2 3 4 5 6 P0P0 P1P1 J P0P0 P1P1 P2P2 J 1 

16. Illustration 2: DMRG The PEPS-method coincides with the B B DMRG-scheme if PEPS with OBC are chosen, and assures that the energy is monotonously decreasing Open boundary conditions enable big speed-up as the calculation of H eff can be done using sparse matrices (computational cost / D 3 ); this follows from the fact that |   i can be chosen orthonormal which implies N=I. White 92

17. Applications –DMRG with Periodic boundary conditions –Calculating Excitations –Dynamical correlation functions / structure factors using PEPS –Optimal time evolution algorithms –Finite-T DMRG: Matrix product density operators Quantum-TMRG from the perspective of PEPS –Quantum Spin Glasses –Simulation of 2-D systems Ground states Time evolution, dynamical correlation functions, …

18. DMRG and periodic boundary conditions Due to boundary effects, periodic boundary conditions (PBC) are preferred above open ones (OBC) in calculating magnetization, (dynamical correlation functions, excitations, … Standard DMRG does not work well in case of PBC: –Underlying PEPS trial state has open boundary conditions –entropy of block in B B -scheme scales as (compare to in B B or B B) Solution: solve generalized eigenvalue problem on PEPS of the form –Remark: sparseness of matrix multiplication partially lost due to impossibility of orthonormalizing basis (simulation time scales as D 5 instead of D 3 ) –Exact translationnally invariant state can be found by using trial PEPS of the form – Verstraete, Porras, Cirac 04 P1P1 P2P2 P3P3 P4P4 P5P5  PNPN

19. Numerical results in case of spin ½ Heisenberg antiferromagnet (N=28): Number of states kept (D) Verstraete, Porras, Cirac 04

20. Excitations Variational formulation: find the PEPS |   i minimizing the energy under the constraint that it is orthogonal to |  0 i –As the constraint is multilinear in the projectors P i, |  1 i can again iteratively be optimized by solving generalized eigenvalue problems; the only difference is that the dD 2 matrices H eff and N are projected onto a dD 2 -1 subspace. –This allows to find excitation energies to same accuracy as for ground states –In case of PBC, trial states with definite momentum can be chosen as –Note that in case of DMRG with multi-state targeting, all states have same A i except at the site one is looking at; the big difference with the method we propose is that here ALL A i are varied at the expense of repeating the algorithm for each excited state again

21. Dynamical Correlation Functions Goal: calculate –Variational formulation within class of PEPS: find (unnormalized) PEPS |  i such that ||A|  0 i -(E 0 -H+  +i  )|  i || 2 is minimized; then G(  )= h  0 |A * |  i –This optimization can again efficiently be done by an iterative method similar to the one already described: fix all projectors but one, then the norm is given by x y H eff x-x y y ; the solution is then H eff x=y/2. –Main difference with correction vector method / D-DMRG: |  0 i and |  i are calculated independently and are not forced to have the same projectors P i everywhere except at the site under consideration This ensures better precision as the variational class of trial states is much larger; note also that the iterative method is efficient as only a set of linear equations has to be solved (scaling in case of OBC: D 3 ) Hallberg 95; Kuhner and White 99; Jeckelmann 02

22. Optimal Time Evolution Goal: given a PEPS |  i (OBC or PBC), describe its time evolution via Trotterization under the action of a local Hamiltonian To describe this 2-step evolution exactly, the dimension of all the bonds has to be multiplied by an interaction-dependent factor (2 for Ising and 4 for Heisenberg) To prevent the dimensions to blow up: variational dimensional reduction of PEPS –Given a PEPS |  D i of dimension D, find the one |  D’ i of dimension D’< D such that || |  D i -|  D’ i || 2 is minimized –This can again be done efficiently in a similar iterative way, yielding a variational and hence optimal way of treating time-evolution within the class of PEPS PiPi P i+1 P i+2 P i+3 P i+4 P i+5 P i+6 Verstraete, Garcia-Ripoll, Cirac 04 Vidal 03; Daley et al. 04; White and Feiguin 04

23. Variational Dimensional Reduction of PEPS Given a PEPS |  D i parameterized by the D £ D matrices A i, find the one |  D’ i parameterized by D’ £ D’matrices B i (D’< D) such as to minimize –Fixing all B i but one to be optimized, this leads to an optimization of the form x y H eff x-x y y, with solution: H eff x=y/2 ; iterating this leads to global optimum –The error of the truncation can exactly be calculated at each step! –In case of OBC: more efficient due to orthonormalization –Note that the algorithm also applies to PBC In the case of OBC, the algorithms of Vidal, Daley et al., White et al. are suboptimal but a factor of 2-3 times faster; a detailed comparison should be made

24. Finite-T DMRG PEPS can represent mixed states by applying completely positive maps (CPM) instead of projectors to the virtual particles –The condition on the M’s ensures positivity; general operators are obtained by relaxing this condition –This PEPS-picture is again complete if dimension of bonds large enough –It turns out to be much more efficient and convenient to work with the purification of  is obtained by tracing over the auxiliary systems a k : Verstraete, Garcia-Ripoll, Cirac 04 See also: Zwolak and Vidal 04

25. –Time-evolution, calculation of dynamical correlation functions, … can now be done on this purification as the physical observables act trivially on the auxiliary systems –Calculation of a thermal state: imaginary time evolution starting from the 1 -T state: The dimension of the bonds of the purification are only a factor d (e.g. 2 in case of qubits) larger than in the zero-T case –In the case of decoherence (simulation of master equations), evolution of  instead of its purification has to be done; the computational cost in case of OBC then scales as D 6 instead of d 3 D 3 and positivity of  is not anymore assured. a good alternative: the quantum jump approach See also: Zwolak and Vidal 04

26. (a) Magnetization vs. transverse magnetic field of the thermal state of the XY model with 60 spins, for temperatures T=0.05,0.5,5 and 50 (bottom to top). (b) Error in the density matrix of the thermal state computed using MPDO, vs. temperature, for a chain with N=8 spins, D=8 (solid) and D=14,20,24 (dashed).

27. quantum-TMRG from the perspective of PEPS Different approach to finite-T DMRG: calculate thermodynamic quantities like the free energy –The operations needed to take this trace can be represented graphically: N i1i1 i1i1 i2i2 i2i2 i3i3 i3i3 i4i4 i4i4 i5i5 i5i5 a1a1 a1a1 a2a2 a2a2 a3a3 a3a3 a4a4 a4a4 a5a5 a5a5 a6a6 a6a6 a7a7 a7a7 a8a8 a8a8   M              Nishino 95; Bursill et al. 96; Shibata 97; Wang et al. 97 i0i0 i0i0  

28. 2 possible methods to sum this: -If N !1 and homogeneous, one can calculate the largest eigenvector of a column transfer matrix (TMRG) using DMRG with PERIODIC boundary condition (note that it is not always possible to get hermitean transfer matrices) -More general approach (also for finite, inhomogeneous and non-hermitean case): propagate a 1-D PEPS from the left to the right by the method of variational dimensional reduction These methods also work for calculating thermodynamic quantities of classical 2-D spin systems (cfr. Nishino); here M=N !1 and an infinite-DMRG like algorithm can be devised that gives very accurate results even at the critical point N i1i1 i1i1 i2i2 i2i2 i3i3 i3i3 i4i4 i4i4 i5i5 i5i5 a1a1 a1a1 a2a2 a2a2 a3a3 a3a3 a4a4 a4a4 a5a5 a5a5 a6a6 a6a6 a7a7 a7a7 a8a8 a8a8   M              i0i0 i0i0  

29. Quantum Spin Glasses We can use the inherent randomness of quantum mechanics to study spin systems with random interactions / fields : encode the randomness in ancilla’s. –This allows to run all possible realizations in 1 run Simple example: –Simulation Hamiltonian :

30. Simulation of 2-D quantum systems Standard DMRG approach: trial state of the form Problems with this approach: dimension of bonds must be exponentially large: - area theorem - only possibility to get large correlations between vertical nearest neighbors We propose trial PEPS states that have bonds between all nearest neighbors, such that the area theorem is fulfilled by construction and all neighbors are treated on equal footing See also Nishino et al.

31. P 11 P 21 P 31 P 41 P 32 P 42 P 22 P 12 P 13 P 23 P 33 P 31 P 44 P 34 P 24 P 14 P 15 P 25 P 35 P 45 The energy of such a state is still a multi-quadratic function of all variable, and hence the same iterative variational principle can be used The big difference: the determination of H eff and N is not obtained by multiplying matrices, but contracting tensors –This can be done using the variational dimensional reduction discussed in the context of TMRG; note that the error in the truncation is completely controlled No (sign) problem with frustrated systems! Possible to devise an infinite dimensional variant

32. Alternatively, the ground state can be found by imaginary time evolution on a pure 2-D PEPS –This can be implemented by Trotterization; the crucial ingredient is again the variational dimensional reduction; the computational cost scales linearly in the number of spins: D 10 –The same algorithm can of course be used for real-time evolution and for finding thermal states. –Dynamical correlation functions can be calculated as in the 1-D PEPS case We have done simulations with the Heisenberg antiferromagnetic interaction and a frustrated version of it on 4 £ 4, 10 £ 10 and 20 £ 20 –We used bonds of dimension 2,3,4; the error seems to decay exponentially in D –Note that we get mean field if D=1 –The number of variational parameters scales as ND 4 and we expect the same accuracy as 1-D DMRG with dimension of bonds D 2

33. 4x4: 36.623 10x10: 2.353 (D=2); 2.473 (D=3) 20x20: 2.440 (D=2); 2.560 (D=3) 20x20

34. Conclusion Projected Entangled Pair States (PEPS) provide a natural parameterization of multipartite entangled states on lattices from the point a view of Quantum Information Theory PEPS provide a solid theoretical basis underlying DMRG, allowing to generalize DMRG to periodic boundary conditions, finite-T, time evolution, dynamical correlation functions, excitations, quantum spin glasses, and most notably higher dimensions