Download Numerical Renormalization Group methods with Matrix Product States

Quantum Information Theory
and Strongly Correlated Quantum Systems
Frank Verstraete
University of Vienna
Strongly Correlated
Quantum Systems
Quantum Mechanics
Statistical Physics
Quantum Chemistry,
N-representability
Microscopic behavior, quantum
correlations, superposition
principle
Spin systems,
Hubbard models
Topological quantum
order
Quantum Error Correction,
quantum coherence in
macroscopic systems
Frustration,
monogamy of
entanglement,
cryptography
Renormalization Group
(NRG, DMRG)
Simulation
Stochastic non-equilibrium
systems
Entanglement
Quantum
Information
Theory
Macroscopic behavior, order
parameters, phase transitions,
Ground States
PEPS
Area Laws
Entropy
Complexity Theory
Information Theory
Shannon Entropy, channel capacity
Entanglement
Complementary viewpoints on entanglement:
• Quantum information theory: it is a resource that allows for revolutionary
information theoretic tasks
• Quantum many-body physics: entanglement gives rise to exotic phases of
matter
• Numerical simulation of strongly correlated quantum systems: enemy nr. 1!
Of course these viewpoints are mutually compatible:
- Complexity of simulation vs. power of quantum computation
- Topological quantum order vs. quantum error correction
Key question: what kind of superpositions appear in nature?
Hilbert space is a convenient illusion
• Let’s investigate the features of the manifold of states that can be
created under the evolution H(t) for times T polynomial in N: T= Nd
•
•
Solovay-Kitaev: given a standard universal gate set on N spins (cN gates),
then any 2-body unitary can be approximated with log(1/ε) standard gates
such that ║U-Uε║< ε
Given any quantum circuit acting on pairs and of polynomial depth Nd, this
can be reproduced up to error ε by using Nd log(Nd /ε) standard gates. The
total number of states that can hence be created using that many gates
scales as
cN N
•
d
log
Nd

Consider however the DN dimensional hypersphere; the number of points
that are ε-far from each other scales doubly exponential in N:
DN
1
 
 
 
• Conclusion: all physical states live on a tiny submanifold in Hilbert
space; there is no way random states (i.e. following the Haar
measure) can be created in nature
• What about ground states?
Connecting entanglement theory with strongly
correlated quantum systems
•
Strongly correlated quantum systems are at forefront of current experimental
research
– Cfr. Realization of Mott insulator versus superfluid phase transition in optical
lattices (Bloch et al.)
– Building of universal quantum simulators using e.g. ion traps
– No good theoretical understanding yet: main bottleneck is simulation of quantum
Hamiltonians
•
Quantum spin systems form perfect playground for investigating strongly correlated
quantum systems:
– Heisenberg model was put forward by Dirac and Heisenberg already in the ’20s
as candidate Hamiltonian describing magnetism
– Fermi-Hubbard model is believed to be minimal model exhibiting features of high
Tc superconductivity (reduces to Heisenberg in some limit)
– However, still many open questions!
•
Quantum spin models arise naturally in the study of quantum error correcting codes
– Q.E.C led Kitaev to introducing quantum spin model exhibiting new exotic
phases of matter (topological quantum order)
– Intriguing connection between ideas in quantum information and condensed
matter (e.g. cluster states and valence bond states, …)
•
What are the questions we would like to see answered?
– Ground state properties, energy spectrum, correlation length, criticality, connection
between those and entanglement
– Are such systems useful, i.e. do they exhibit the right kind of entanglement and allow for
the right kind of control, for building e.g. quantum repeaters, quantum memory or
quantum computers?
– Finite-T: what kind of quantum properties survive at finite T?
– Connection between amount of entanglement present in system and simulatability on a
classical computer?
– Computational complexity of finding ground states?
– Dynamics: how much entanglement can be created by local Hamiltonian evolution?
•
We already have partial answers to those questions:
– connection between spectrum and correlation length
– criticality in 1-D is accompanied by diverging block entropy.
Not such a signature in 2-D (PEPS)
– Entanglement length in spin systems versus quantum repeaters
– Cluster state quantum computation of Raussendorf and Briegel (cfr. PEPS)
– Kitaev: using Toric Code states as fault-tolerant quantum memory in 4-D
– Finite T: strict area law for mutual information
– MPS/PEPS parameterize manifold of ground states of local Hamiltonians
– Kitaev: finding ground states of disordered local Hamiltonians is QMA-complete (also:
famous N-representability problem)
– Dynamics: Lieb-Robinson bounds
Quantum spin systems
• Provide perfect playground for investigating nature of entanglement
in strongly correlated quantum systems
– Most pronounced quantum effects arise at low temperature as
large quantum fluctuations exist (ground states)
– We assume some geometry and local interactions (cfr. Causality)
such as Heisenberg model
•
Ground states of local spin Hamiltonians are very special:
H   S xi S xj  S yi S yj  S zi S zj
i, j
– Translational invariance implies that energy is completely determined by
n.n. reduced density operator ρ of 2 spins:
E  N . Tr  H ij 
– Finding ground state energy is equivalent to maximizing E over all possible
ρ arising from states with the right symmetry
– The extreme points of the convex set {ρ} therefore correspond to ground
states: ground states are completely determined by their reduced density
operators!
e.g.: H
  S xi S xj  S yi S yj  S zi S zj
i, j
The Hamiltonian defines hyperplanes in
this convex set; convex set is
parameterized as 2 x  z  E ()  0
In infinite dimensions: only unentangled
states are compatible with symmetry,
hence mean-field theory becomes exact
(Based on De-Finetti theorem:
R. Werner ’89)
singlet
•
Difficulty in characterizing this convex set is due to monogamy / frustration
properties of entanglement: a singlet cannot be shared
– Entanglement theory allows to make quantitative statements
•
If local properties of a ground state of a system with N spins and a gap Δ
are well approximated, then also the global ones:
if  i, j
nn
: 

ij

GS
ij
    O    GS O  GS
2 N
 O

fermionic systems vs. spin systems
•
•
Fundamental question: are fermions fundamentally different from
bosons/spins or can local fermionic Hamiltonians be understood as effective
Hamiltonians describing low energy sector of specific local spin systems?
Hilbert space associated to fermions is Fock space, which is obtained via
second quantization:
  a  ... 
   ci1i2 ... a1
i1
 i2
2
i1i2 ...
•
•
•
•
•
What we want to approximate is ci1i2 ...
Effective Hamiltonian for this tensor is obtained by doing the Jordan-Wigner
transformation on the original one (note the ordering of the fermions in
second quantization)
Consider hopping terms in 2-D: J-W induces long-range correlations
Solution: use auxiliary Majorana fermions to turn this Hamiltonian into a
local Hamiltonian of spins (cfr. Kitaev)
Similar but different trick applies to any geometry/dimension and multichannel impurity problems
1
2
3
4
5
10
9
8
7
6
11
12
13
14
15
20
19
18
17
16
Vertical hopping terms become nonlocal by JW-transformation:
9
 9 z x

y
a a  a a      k  10   1    kz  10y
 k 2 
 k 2 
†
1 10
†
10 1
JW
x
1
Solution: add ancillary chains of free fermions bi constructed as follows: define Majorana fermions
ci = bi + bi† , di = i(bi - bi†) and free Hamiltonian
H anc    ick dl
k ,l
As all terms ic k d l are constants of motion (i.e. +1) and commute with each other, we can change
†
†
the original vertical hopping terms a1 a10  a10a1 
 a1† a10  a10† a1 ic1d10 without changing
the physics of the Hamiltonian. Renumbering everything makes everything local after the JW


Entanglement, correlations, area laws
Quantifying the amount of correlations between A and B: mutual information
I AB  S  A   S  B   S  AB 
B
A
All thermal states exhibit an exact area law (as contrasted to volume law)
 AB  exp  H 
F  A   B   Tr H A   B   S  A   B   Tr H AB   S  AB 
 I AB 
1

1


Tr H  A   B   AB   Tr H AB  A   B   AB 

Cirac, Hastings, FV, Wolf
Similar results for ground states (critical systems might get logarithmic corrections)
This is very ungeneric: entanglement is localized around the boundary
This knowledge is being exploited to come up with variational classes of states and associated
simulation methods that capture the physics needed for describing such systems:
* Matrix Product States, Projected Entangled Pair States, MERA
Area laws
•
Main picture: in case of ground states, entanglement is concentrated around the
boundary
cc
Gapped : S 1, 2,, L  
ln    ...
6
cc  1 
Critical : S 1, 2,, L  
1   ln L   ...
12   
Kitaev, Vidal, Cardy, Korepin, …


Gapped : S 1, 2,, L2  a. L  
Critical :


Free fermions : S 1, 2,, L2  a.L ln L  ...


Critical spin : S 1, 2,, L2  a.L  ...
Wolf, Klich
quant-ph/0601075
Topological entropy: detects topological quantum order
locally!
S  ABC   S  AB   S  AC   S  BC   S  A   S  B   S C 
Kitaev, Preskill, Levin, Wen
Ground states of spin Systems
•
Ground states of gapped local Hamiltonians have a finite correlation length:
 l 
OAOB  OA OB  exp   AB 
 C 
A
B
C
l AB  C
•
Let’s analyze this statement from the point of view of quantum information theory,
assuming that
 AB  A  B
– There is a separable purification of ρAB , so there exists a unitary in region C that
disentangles the two parts
– Blocking the spins in blocks of log(ξC) spins, then we can write the state as:
 ABC 
U

 
il ir
l il ir r
, ,il ,ir
– Doing this recursively yields a matrix product state:  i i ... i 
12
N
A
i1
Ai2 ... AiN i1 i2 ... iN
Matrix Product States (MPS)
D
I i i
i 1
Map P : H D  H D  H d
•
•
•
•
•
•
•
•
•
•
State is defined on a d N dimensiona l Hilbert space
Generalizations of AKLT-states (Finitely correlated states, Fannes, Nachtergaele, Werner ‘92)
Gives a LOCAL description of a multipartite state
Translational invariant by construction
Guaranteed to be ground states of gapped local quantum Hamiltonians
The number of parameters scales linearly in N (# spins)
The set of all MPS is complete: Every state can be represented as a MPS as long as D is taken large
enough
The point is: if we consider the set of MPS with fixed D, their reduced density operators already
approximate the ones obtained by all translational invariant ones very well (and hence also all
possible ground states)
MPS have bounded Schmidt rank D
Correlations can be calculated efficiently: contraction of D2x D2 matrices
Numerical renormalization group method of Wilson and Density matrix renormalization group method
of S. White can be reformulated and improved upon as variational methods within class of MPS
Convex set of reduced density operators of
ground states of XXZ-chains approximated with
MPS of D=1,2
•
So how good will MPS approximate ground states? We want find a bound on the scaling of D
as a function of the precision desired and the number of spins N
– We impose
 exN   DN  
with ε independent of N,D
– Because the scaling of the α-entropy of blocks of L spins in spin chains is bounded by
S   
1
cc  1 
ln Tr    
1   ln L 
1
12   
it follows that it is enough to choose
DN 
cst

N f (c )
FV, Cirac
– It shows D only has to grow as a polynomial in the number of particles to obtain a given
precision, even in the critical case!
•
M. Hastings (2007):
– All ground states of gapped Hamiltonians are well represented by MPS because they obey
an area law
– Same proof in principle applies to the higher dimensional generalizations of MPS: PEPS
•
MPS / PEPS are hence the ideal variational class of wavefunctions for simulating strongly
correlated quantum spin systems; in other words: we have identified the right submanifold!
•
What about the complexity of finding this optimal MPS in the worse case?
– Finding ground state of a local 1-D quantum spin chain with a gap that
is bounded below by c||H||/poly(N) is NP-hard
– Proof goes via identifying a family of such Hamiltonians that is NPcomplete and have ground states that are exactly matrix product states
•
Sketch of proof: cfr. Aharonov, Gottesman, Kempe proof of QMA-hardness
of finding GS of 1-D quantum spin systems, but use classical circuit instead
Ground state of corresponding Hamiltonian is of the form
   t t
t

t
ooo...
Nt
t

o o o o...
N (T t 1)
– As all  t  U tU t 1...U1  0
are classical, this ground state has very
few entanglement; in fact it is a MPS with dimension poly(N) and hence
checking energy is in NP; finding MPS is hence NP-complete
– Gap of Hamiltonian comes from random walk
– We can also construct alternative Hamiltonians starting from classical /
quantum reversible cellular automata
Schuch, Cirac, FV
Wilson’s numerical renormalization group
• Consider Kondo-impurity-like problem with Hamiltonian
   k  xk   xk 1   yk   yk 1 
N
k 0
    A i1 i2
2 
i2
i1
i1 ,i2
i
 3   A
 2  i3
3
 ,i3

1
 N  
2
 A  A A ...A
i2
i1
i1 ,i2 ,...
 ,  ,...
i3
i4
iN
3
i1 i2 ... iN 
4
A
i2
5
6
Ai3 Ai4 ... AiN i1 i2 ... iN
i1 ,i2 ,...
• NRG method creates an effective Hamiltonian which is the original
Hamiltonian projected in a basis of matrix product states (MPS)
• Success of NRG follows from the fact that those MPS parameterize well the
low-energy sector of the Hilbert space
• Main new ingredient from DMRG: sweep!
S. White’s DMRG method
Extending DMRG to periodic boundary conditions:
P2
P3
10
P4
-4
DMRG (PBC)
DM
10
10
10

P5
-6
Ne
-8
RG
< mSi S i+1> / E0 - 1
P1
E 0 /|E0|
•
(O
BC
)
w
(P
BC
)
mi
-10
0
20
40
60
PN
Variational dimensional reduction of MPS
•
Given a D-dimensional  D MPS parameterized by the DxD matrices Ai, find  D '
parameterized by D’xD’matrices Bi (D’< D) such as to minimize
  
2


 

 Tr   B1i B1i   B2i B2i    BNi BNi 
 i
  i

 i


 

 2Tr   B1i A1i   B2i A2i    BNi ANi   cst
 i
  i

 i
– Can be minimized variationally by iteratively solving linear systems of equations
•
This can be used to describe both real and imaginary time-evolution
Generalizations of MPS
• PEPS: 2-dimensional generalization of MPS
• MPS and weighted graph states (Briegel, Dur, Eisert, Plenio)
• MERA: multiscale entanglement renormalization ansatz (Vidal)
– Allows to represent critical or scale-invariant wavefunctions
– Can be created using a tree-like quantum circuit of unitaries and
isometries
Generalizations of MPS to higher dimensions
•
The MPS/AKLT picture can be generalized to any geometry : Projected Entangled
Pair States (PEPS)
P maps D4 dimensional
to d dimenional space
•
Properties: Area Law automatically fulfilled; local properties can be approximated very
well ; guaranteed to be ground states of local Hamiltonians; again, every state can be
written as a PEPS
•
A variational approach based on those states provides a solution to the problem of
the numerical renormalization group approach where the number of degrees of
freedom of a block grows exponentially with the size of the block
• How to calculate correlation functions?
– Instead of contracting matrices, we have to contract tensors:
i0
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
M M
i
XX X X X X X X
X
i
M
i
 
X
i X X X X X X X X
M
i
 
X
i X X X X X X X X
M
i
XX X X X X X X
X
i
M
i
X
i MXX X X X X X X
1 1 2 2
1
1
2
2
3
3
4
4
5
5
0
3 3
4 4
5 5
6 6 7 7
8 8
V. Murg, FV, I. Cirac
Examples of PEPS
•
•
•
Cluster states of Briegel and Raussendorf are PEPS with D=2: allow for
universal quantum computation with local measurements only. We can also
construct other states that are universal using PEPS
PEPS with topological quantum order:
– Toric code states of A. Kitaev (D=2): fault-tolerant quantum memory
– Resonating valence bond states (D=3)
PEPS with D=2 can be critical: power law decay of correlations
– Many examples can be constructed by considering coherent versions of
classical statistical models:
 

   exp    i j   1  2   N
 1 2  N
 2
i, j

– Resolves open question about scaling of entanglement in critical 2-D
quantum spin systems: no logarithmic corrections
– PEPS construction shows that for every classical temperature-driven
phase transition there exists a quantum spin model in the same dimension
exhibiting a zero-T quantum phase transition with same features
•
PEPS provide perfect playground for considering open questions like existence
of deconfined criticality: all PEPS are ground states
Conclusion
• Formalism of quantum information theory provides unique perspective on
strongly correlated quantum systems
– MPS/PEPS picture describes low-energy sector of local
Hamiltonians, and opens a whole new toolbox of numerical
renormalization group methods that allows to go where nobody has
gone before
• Similar ideas can be used in context of lattice gauge theories, quantum
chemistry, …
– Frustration and monogamy properties of entanglement
(cryptography), quantum error correction, and the complexity of
simulating quantum systems are basic notions in the fields of
quantum information and statistical physics
– Synergy of quantum information and the theory of strongly correlated
quantum systems opens up many new themes for both fields and
could lead to a much more transparent description of the whole body
of quantum physics
Work described is mainly from: I. Cirac, J. Garcia-Ripoll, M. Martin-Delgado, V. Murg, B. Paredes, D. Perez-Garcia, M. Popp, D.
Porras, C. Shon, E. Solano, M. Wolf (Max Planck Institute for Quantum Optics), J. von Delft, A. Weichselbaum (LMU), U.
Schollwock (RWTH), M. Hastings, G. Ortiz (Los Alamos), T. Osborne (London U)