1. PEPS, matrix product operators and the algebraic Bethe ansatz
Frank Verstraete
University of Vienna
Valentin Murg, Ignacio Cirac (MPQ)
B. Pirvu (Vienna)

2. Matrix Product States and Projected Entangled Pair States as variational states for simulating strongly correlated quantum systems
Why?
History of Quantum Mechanics is one in which we try to find approximate solutions to Schrödinger equation
Actually, this is the reason why quantum computers would be so exciting
Most relevant breakthroughs in context of many-body physics: guess the right wavefunction (BCS, Laughlin, …)
Is there a way to come up with a systematic way of parameterizing the wavefunctions arising in relevant Hamiltonians?
In case of 1-D quantum spin chains: NRG / DMRG : MPS
In case of 2-D quantum spin systems: PEPS / MERA / ….

3. Many-body Hilbert space is a convenient illusion
Size of Hilbert space of system of N particles / modes / … scales exponentially with N.
What is the fraction of states that are physical, i.e. can be created as low-energy states of local Hamiltonians or by a quantum computer in poly time? Exponentially small !!!
Ground states (and low-energy states …) have very special properties
Amount of entanglement is very small: can be formalized using so-called area laws
Ground states have extremal local correlations: all (quasi-)long range correlations are a consequence of the fact that those local correlations must be made compatible with translational invariance
If we want to simulate a many-body system, we should be smarter

4. Matrix Product States
Valence bond state: translational invariant by construction
Has extremal local correlations
Obeys area law by construction
Theorem: if an area law is satisfied, then the state can be well approximated by a MPS:
In case of local gapped 1-D Hamiltonians: area law is guaranteed
Conclusion: all states in finite 1-D chains can be represented by MPS: breakdown of exponential wall !

5. S.R. White’s DMRG et al. H
DMRG can be understood as variational method within this class of VBS/MPS states: alternating least squares
Extension to periodic BC: trivial once formulated like this
A local Hamiltonian (e.g. Heisenberg model) is a special case of a more general type of operators: matrix product operators

6. Matrix product operators
Slight modification of DMRG allows to approximate extremal eigenvectors (and eigenvalues) of hermitean MPO by MPS:
Where else do MPO appear?
Transfer matrices of classical spin systems!
DMRG is therefore basically a method for finding leading eigenvector and eigenvalue of transfer matrix (cfr. Nishino, Baxter)

7. We can also solve another optimization problem involving MPO: given a MPS and an MPO O, find the MPS that minimizes
It turns out that this is also a multiquadratic optimization problem that is very well conditioned and can be solved using DMRG-like sweeping!
Core method for simulating PEPS
The error in the approximation is known exactly!
Leads to a massively parallel time evolution algorithm that does not break translational invariance:

8. Matrix Product Operators and the Bethe ansatz:
Algebraic Bethe ansatz is all about MPO:
Crucial Property of this family of MPO: they all commute (==Yang-Baxter equation):
Gauge transformation of MPS/MPO leave it invariant!

9. What has this to do with the Heisenberg model?
This can easily be seen because is the shift operator (shifts qubits 1,2,3,…N to 2,3,4,…1); taking the derivative replaces one of those “swaps” with the idenity; logarithmic derivative undoes all the other swaps, leaving the Heisenberg Hamiltonian!
It follows that and hence they have the same eigenvectors
Let’s now define new operators similar to but with OBC:
These will play the role of creation operators and commute for all

10. All eigenstates of the Heisenberg model are of the form
The parameters are found by imposing that these are eigenstates of = Bethe equations (follows simply from working out commutation relations; this leads to coupled equations between the )
In terms of MPS/MPO: all eigenstates can exactly be represented as
Can therefore easily be simulated using MPS algorithms: correlation functions, … with absolute error bars!
How to extend to higher dimensions? PEPS!

11. Generalizing MPS to higher dimensions: PEPS
Area law is satisfied by construction : scalable!
Precursors: AKLT, Nishino; PEPS introduced in context of measurement-based quantum computation

12. How to calculate expectation values?
arXiv:cond-mat/
How to calculate expectation values?
Equivalent to contracting tensor network consisting of MPS and MPO!
Obvious way of doing this: recursively use
Optimization: alternating least squares as in DMRG
Alternatively: imaginary time evolution ; infinite algorithm ; renormalization (Gu et al.)

13. Holographic principle: dimensional reduction
Crucial property of MPS/PEPS: dimensional reduction
Start from quantum system in 2 dimensions (2+1)
The PEPS ansatz maps the quantum Hamiltonian to a state corresponding to a partition function in 2 dimensions (2+0)
The properties of such a state are described by a (1+1) dimensional theory (eigenvectors of transfer matrices)
Those eigenvectors are well described by MPS
Properties of MPS are trivial to calculate: reduction to a partition function of a 1-D system (1+0)

14. From here on: all work and slides by Valentin Murg
The Method
Time Evolution
Goal:
Simulation of the Operation
Procedure:
For , the state remains a PEP-state, but with increased virtual Dimension:
Problem:
Virtual Dimension increases exponentially with the number of steps.
Workaround:
Approximate at each step the PEP-state by a PEP-state with reduced virtual Dimension.

15. The Method Time Evolution Goal: Algorithm:8
The Method
Time Evolution
Goal:
Algorithm:
Optimization of the distance site by site:
Optimal scaling of the Algorithm:

16. J1-J2-J3 Heisenberg model
arXiv:
J1-J2-J3 Heisenberg model J3 J1 J2
Frustrated system
We did calculation for systems with open boundary conditions of size up to 14x14 and D=4

17. Classical Phase Diagram
J1J2J3-Model
Classical Phase Diagram
IV. Helicoidal J1 J2
III. Helicoidal J1 J2
I. Néel J1 J2
II. Independent Sublattices J1 J2

18. J1J2J3-Model Quantum Phase Diagram II. Collinear J1 J2
(Order-by-Disorder)

21. J1J3-Model Long Range Order Structure Factor Néel Order:
No long range order!
Néel Order on
four Sublattices:

23. Plaquette order parameter, 8x8 lattice, J3/J1=1/2
Pure Plaquette state: in Pl Q=1, in between Pl Q=1/4

24. Comparison with short range resonating valence bond ground state24
J1J3-Model
Intermediate Phase
Comparison with short range resonating valence bond ground state

25. 34
J1J2-Model
Long Range Order
Structure Factor

26. J1J2-Model Long Range Order Structure Factor Néel Order:
No long range order!
Columnar Order:

27. J1J2-Model
Long Range Order
Local Spin Directions

28. General features of PEPS
Pretty reliable and well-conditioned method with absolute error bars for expectation values of variables
In principle unbiased, like DMRG: we can make the system completely translational invariant
Especially suited for describing spin liquids et al., gapped systems (cfr. MERA for critical systems!)
In case of fermions: make use of classical gauge theories to get right statistics
Everything that can be done with MPS can be done with PEPS (but at a much higher cost)
Cost of algorithm still scales as ND10 , which is very good as a function of systems size, but bad with respect to the bond dimension
Not too bad if compared with DMRG: D5 versus D3
Note also: way less entanglement in 2-D than in 1-D: frustration!
Work in progress for finding alternative methods for contracting tensor networks
Renormalization methods of Gu et al.
Calculating expectation values by Monte Carlo sampling (see talk of Schuch)
Infinite methods (cfr. Talk of Orus)

29. Conclusion
MPS/MPO/PEPS might be a smarter tool to study new states of quantum matter
MPS/MPO/PEPS formalism is very natural way of representing wave functions of strongly correlated quantum systems
How does it compare to MERA (Cfr. Guifre)???
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