Transcript Quantum spin systems from the perspective of quantum

Quantum Spin Systems from the point a view of
Quantum Information Theory
Frank Verstraete, Ignacio Cirac
Max-Planck-Institut für Quantenoptik
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
-4
DMRG (PBC)
DM
10
10
10
-6
RG
Ne
w
-8
< mSi S i+1> / E0 - 1
E 0 /|E0|
10
(O
BC
)
(P
BC
)
mi
-10
0
20
40
Number of states kept (D)
60
(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).

2
2
0
1
Heisenberg
10x10
Frustrated
20x20
-2
0
0
-1
25

50
Frustrated
-2
0
4x4
Heisenberg
25

50
Spin systems in Quantum
Information Theory
• Central motto: Entanglement is a resource
– Essential ingredient in quantum cryptography, quantum computing, etc.
– Basic Unit: Bell-state |00i+|11i
– 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
• 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 Cij   i   j   i  j
– 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
The spin-1 AKLT-model

Singlet

2
1
H   Si .Si 1   Si .Si 1
3 i
i
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
Generalizing the AKLT-state: PEPS
I i i
D
i 1
Pi
Pi+1
Pi+2
Pi+3
Pi+4
Pi+5
Pi+6
Map : H D  H D  H d
• Every state can be represented as a Projected Entangled Pair State (PEPS) as
long as D is large enough
Ex.: 5 qubit state
P ( 2 3  2)
P ( 2  2)
9
P ( 2 3  2)
• 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
Fannes, Nachtergaele, Werner 92
thermodynamic limit (N!1) when P1=P2=L =P1
– Systematic way of constructing translational invariant states
– MPS become dense in space of all states when D!1
• PEPS in higher dimensions:
P  0 00...0  1 11...1
H  00  01  10  11
• PEPS in QIT:
– Quantum Error correction:
H  00  01  10  11
P  0 00  1 11
– Quantum computing
Verstraete, Cirac 04
Basic properties of PEPS
• Correlation functions for 1-D PEPS can easily be calculated
by multiplying transfer matrices of dimension D2 :
• Number of parameters grows linearly in number of particles
c
(NdD ) with c coordination number of lattice
• 2-point correlations decay exponentially
• Holographic principle: entropy of a block of spins is proportional
to its surface
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
• 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
The Hamiltonian defines a hyperplane
in (2s+1)2 dim. space
• 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
PEPS as variational trial states
Pi
Pi+1
Pi+2
Pi+3
Pi+4
Pi+5
• All expectation values and hence the energy E   vbs H  vbs
quadratic in the variables Pk
• Strategy for minimizing energy for N-spin state:
Pi+6
are multi-
– Fix all projectors Pi except the jth
– Both the energy and the norm  vbs  vbs are quadratic functions of the variable Pj and
hence the minimal energy by varying Pi can be obtained by a simple generalized
eigenvalue problem:
Heff 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
Verstraete, Porras, Cirac 04
• Compare to MPS-approach of Ostlund and Rommer 95 and Dukelsky et al. 97 :
imposing that all Pi are equal makes the optimization untractable for large D
Illustration 1
• Wilson’s Renormalization Group (RG) for Kondo-effect:
J
P0
P1
l1
J
P0
l1
J
P1
P2
L
P0
P1
P2
l2
l3
l4
P3
P4
l6
l5
P5
P6
• RG approximates ground state and lowest excited states by a family of PEPS
– Ai 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 Aik<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)
Illustration 2: DMRG
White 92
• 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 Heff can be
done using sparse matrices (computational cost / D3); this follows from the fact
that |i can be chosen orthonormal which implies N=I.
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, …
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 D 2 (compare to D in B
B or B B)
• Solution: solve generalized eigenvalue problem on PEPS of the form
L
P1
P2
P3
P4
P5
PN
– Remark: sparseness of matrix multiplication partially lost due to impossibility of
orthonormalizing basis (simulation time scales as D5 instead of D3 )
– Exact translationnally invariant state can be found by using trial PEPS of the form
 
 Tr A A
i1
1
i2
2

L ANiN  A2i1 A3i2 L ANiN 1 A1iN  L i1i2 LiN
i1i2LiN
Verstraete, Porras, Cirac 04
Numerical results in case of spin ½ Heisenberg antiferromagnet (N=28):
-4
DMRG (PBC)
DM
10
10
10
-6
RG
Ne
w
-8
< mSi S i+1> / E0 - 1
E 0 /|E0|
10
(O
BC
)
(P
BC
)
mi
-10
0
20
40
60
Number of states kept (D)
Verstraete, Porras, Cirac 04
Excitations
• Variational formulation: find the PEPS |1i minimizing the energy under the
constraint that it is orthogonal to |0i
– As the constraint is multilinear in the projectors Pi , |1i can again iteratively be
optimized by solving generalized eigenvalue problems; the only difference is that
the dD2 matrices Heff and N are projected onto a dD2-1 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
2
2
k 1
k 2
 2N k 0 i1 i2

iN
i
i
i
i
k   Tr  e
A1 A2 L AN  e N A21 A32 L ANN 1 A1N  e N A3i1 A4i2 L A2iN  L i1i2 LiN
i1i2 LiN


– Note that in case of DMRG with multi-state targeting, all states have same Ai
except at the site one is looking at; the big difference with the method we propose
is that here ALL Ai are varied at the expense of repeating the algorithm for each
excited state again
Dynamical Correlation Functions
• Goal: calculate
G w  ih   0 A*
1
A 0
E0  H  w  ih
– Variational formulation within class of PEPS: find (unnormalized) PEPS
|ci such that ||A|0i-(E0-H+w+ih)|ci||2 is minimized; then G(w)=h0|A*|ci
– 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 xy Heffx-xy y ; the solution is then Heffx=y/2.
– Main difference with correction vector method / D-DMRG:
Hallberg 95; Kuhner and
White 99; Jeckelmann 02
• |0i and |ci are calculated independently and are not forced to have the same
projectors Pi 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: D3)
Optimal Time Evolution
• Goal: given a PEPS |i (OBC or PBC), describe its time evolution via
Trotterization under the action of a local Hamiltonian
Vidal 03; Daley et al. 04; White and Feiguin 04
Pi+1
Pi
e
e
 iH( i 1,i )t
Pi+2
 iH( i ,i 1)t
Pi+3
e
e
 iH( i 1,i  2 )t
Pi+4
 iH( i  2 ,i  3 )t
Pi+5
e
e
 iH( i  3,i  4 )t
Pi+6
 iH( i  4 ,i 5 )t
e
e
iH( i  6 ,i  7 )t
 iH( i 5,i  6 )t
• 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 |Di of dimension D, find the one |cD’i of dimension D’< D such that
|| |Di-|cD’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
Verstraete, Garcia-Ripoll, Cirac 04
Variational Dimensional Reduction of PEPS
• Given a PEPS |Di parameterized by the D£D matrices Ai, find the one |cD’i
parameterized by D’£D’matrices Bi (D’< D) such as to minimize







  c  Tr   B1i B1i   B2i B2i  L  BNi BNi 

i

i
i



 

 2Tr   B1i A1i   B2i A2i  L  BNi ANi   cst
 i
  i

 i
– Fixing all Bi but one to be optimized, this leads to an optimization of the form
xy Heffx-xy y , with solution: Heffx=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
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
See also: Zwolak and Vidal 04
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 r; r is obtained by tracing over the auxiliary systems ak :
Verstraete, Garcia-Ripoll, Cirac 04
– 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:
M
M
 H 
 H 
exp - H   exp   I exp  
 2M 
 2M 
• 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 r
instead of its purification has to be done; the computational cost in case of
OBC then scales as D6 instead of d3D3 and positivity of r is not anymore
assured.
See also: Zwolak and Vidal 04
• a good alternative: the quantum jump approach
(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).
quantum-TMRG from the perspective
of PEPS
Nishino 95; Bursill et al. 96; Shibata 97; Wang et al. 97
• Different approach to finite-T DMRG: calculate thermodynamic quantities
like the free energy
F(  )  Tr exp - H 

 H q
  H1 
 H 2 
 Tr  exp  exp
L
exp
 


 M 
 M 
 
 M
q
N
H   H
H    X i,i 1
;
 1
;
i 1
X

i ,i 1
– The operations needed to take this trace can be represented graphically:
i0
M
M
a a a a a a a a a a a a a a a a
X
i X X X X X X X X
1 1 2 2
3 3
4 4
5 5
6 6 7 7
8 8
1
M
L L
i2
i2 M
L L
i3
i3 M
L L
i4
i4 M
L L
i5
i5 M
i0L M L
i1
XXXXXXXXX
XXXXXXXXX
XXXXXXXXX
XXXXXXXXX
XXXXXXXXX
N

 

M



, X i1,i  2


0
i0
M
M
a a a a a a a a a a a a a a a a
X
i X X X X X X X X
1 1 2 2
3 3
4 4
5 5
6 6 7 7
8 8
1
M
L L
i2
i2 M
L L
i3
M
i3
L L
i4
M
i4
L L
i5
M
i5
i0L M L
i1
XXXXXXXXX
XXXXXXXXX
XXXXXXXXX
XXXXXXXXX
XXXXXXXXX
N
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
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:
N
N
i 1
i 1
H   S i  S i 1   ( 1) r ( i ) Siz
– Simulation Hamiltonian :
N
N
i 1
i 1
H  I anc   S i  S i 1   Siz,anc  Siz
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.
P11
P12
P13
P14
P15
P21
P22
P23
P24
P25
P31
P32
P33
P34
P35
P41
P42
P31
P44
P45
• 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 Heff 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
X X X X X X X X X
X X X X X X X X X
X X X X X X X X X
X X X X X X X X X
X X X X X X X X X
X X X X X X X X X
No (sign) problem with
frustrated systems!
Possible to devise an infinite
dimensional variant
• 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: D10
– 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 ND4 and we expect the same
accuracy as 1-D DMRG with dimension of bonds D2
2
2
0
1

Heisenberg
Frustrated
-2
0
20x20
0
-1
25

50
Frustrated
Heisenberg
-2
0 4x4: 36.623
25

10x10: 2.353 (D=2); 2.473 (D=3)
20x20: 2.440 (D=2); 2.560 (D=3)
50
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