1. Computational approaches for quantum many-body systems
HGSFP Graduate Days SS2019
Martin Gärttner

2. Course overview Lecture 1: Introduction to many-body spin systems
Quantum Ising model, Bloch sphere, tensor structure, exact diagonalization
Lecture 2: Collective spin models
LMG model, symmetry, semi-classical methods, Monte Carlo
Lecture 3: Entanglement
Mixed states, partial trace, Schmidt decomposition
Lecture 4: Tensor network states
Area laws, matrix product states, tensor contraction, AKLT model
Lecture 5: DMRG and other variational approaches
Energy minimization, PEPS and MERA, neural quantum states

3. Learning goals After today you will be able to …
… explain the matrix product state ansatz.
… translate between tensors and their diagrammatic representation.
… derive that the bond dimension bounds the Schmidt rank.
… evaluate observables for MPS, especially with periodic boundaries.

4. Unreasonably large Hilbert space
To see this, consider the following: This is the Hilbert space of a many-body quantum system. As you can see, it is really big. However, if we randomly pick a state out of this exponentially large space, then it will most likely look like this: So, really scary and weird and not like something that exists in reality.
In fact the physical states, and I will define what I mean by that in a second, occupy only an exponentially small corner of Hilbert space.
Physical
corner
Martin Gärttner - BDC 2018

5. Unreasonably large Hilbert space
Physical states:
Can be reached through local Hamiltonians in polynomial time.
Ground states of local gapped Hamiltonians.
Locally entangled.
So what do we mean by physical states? We could say that only states are physical which can be reached from some given, sufficiently simple, initial state through unitary evolution under physical, i.e. local Hamiltonian in polynomial time. Here, by physical Hamiltonians I mean Hamiltonians that consist of at most pairwise interaction terms. We can also allow for three or four-body terms but the number should be of order 1. In practice, if we simulated quenches, we will start in some simple phase, where the ground state is for example close to a product state. What I am saying is that in such a case by time evolving the state under a physical Hamiltonian will only explore a tiny corner of Hilbert space.
The question arises whether one can use this smallness to make simulations efficient.
A special class of physical states are the low energy eigenstates of local gapped Hamiltonians. The reason why they are special is that they are only locally entangled, meaning that if we pick out a region of space and ask how entangled it is with the rest, then we will find that this entanglement will not grow as we increase the subsystem size but will be proportional to the boundary area between the subsystem and the rest. This is called area law of entanglement and is a very useful property since it allows for an efficient representation of these states.
Area law states
Martin Gärttner - BDC 2018

6. Area law vs. volume law 𝑆 𝑣𝑁 𝜌 𝑋 ~𝑋 volume law 𝑆 𝑣𝑁 𝜌 𝑋 ~𝜕𝑋 area law
Ground states of local gapped Hamiltonians
Only k-body terms, k~1
Excitation gap does not go to zero as 𝑁→∞ 1D
Area law → half-chain entropy doesn’t grow with system size

7. Representing quantum states: MPS
Consider N spins in 1D
| 𝜓 = 𝑖 1 … 𝑖 𝑁 𝑐 𝑖 1 … 𝑖 𝑁 | 𝑖 1 … 𝑖 𝑁
𝑐 𝑖 1 … 𝑖 𝑁 = 𝐴 𝛼 1 [1] 𝑖 1 𝐴 𝛼 1 𝛼 2 [2] 𝑖 2 … 𝐴 𝛼 𝑁−1 [𝑁] 𝑖 𝑁
2 𝑁 parameters
I will here focus on the simples version which is used for 1D systems, which is called matrix product states for reasons that will become obvious in a second.
Lets consider a 1D array of spin ½ for concreteness. In general 2^N parameter.
MPS ansatz: Write the coefficients in terms of tensors, which get contracted in order to evaluate the coeffs. To understand how this works, consider the i’s as fixed and see what we have: row vector times matrix times … times column vector.
Dimension of the matrices = bond dimension.
Compare the number of parameters. -> for small D efficient!
This representation is also useful for other reasons: parameters can be optimized to minimize the energy expectation value of some Hamiltonian, i.e. find the ground state. Also time evolution can be calculated (tDMRG).
The class of states that can be represented by MPS is of course restricted. It can only represent states with low entanglement. To understand this, imagine that we divide the system… entanglement can be measured by entanglement entropy which is the von Neumann entropy of a subsystem after we have traced over the rest. For a product state it is zero since the reduced state is still pure. The more entangled the system is, , i.e. the more information about system B is contained in A, the larger S gets.
It turns out that for a state with given bond dimension D one can at most …. This means that if we want to restrict the bond dimension to something small, then also the entanglement is small. This property is exactly fulfilled for area law entangled state, in 1D this means constant S.

8. Representing quantum states: MPS
Consider N spins in 1D
| 𝜓 = 𝑖 1 … 𝑖 𝑁 𝑐 𝑖 1 … 𝑖 𝑁 | 𝑖 1 … 𝑖 𝑁
𝑐 𝑖 1 … 𝑖 𝑁 = 𝐴 𝛼 1 [1] 𝑖 1 𝐴 𝛼 1 𝛼 2 [2] 𝑖 2 … 𝐴 𝛼 𝑁−1 [𝑁] 𝑖 𝑁
| ↑
| ↓
. . .
| ↑
| ↓
. . .
This efficient representation is known as tensor network states. I will here focus on the simples version which is used for 1D systems, which is called matrix product states for reasons that will become obvious in a second.
Lets consider a 1D array of spin ½ for concreteness. In general 2^N parameter.
MPS ansatz: Write the coefficients in terms of tensors, which get contracted in order to evaluate the coeffs. To understand how this works, consider the i’s as fixed and see what we have: row vector times matrix times … times column vector.
Dimension of the matrices = bond dimension.
Compare the number of parameters. -> for small D efficient!
This representation is also useful for other reasons: parameters can be optimized to minimize the energy expectation value of some Hamiltonian, i.e. find the ground state. Also time evolution can be calculated (tDMRG).
The class of states that can be represented by MPS is of course restricted. It can only represent states with low entanglement. To understand this, imagine that we divide the system… entanglement can be measured by entanglement entropy which is the von Neumann entropy of a subsystem after we have traced over the rest. For a product state it is zero since the reduced state is still pure. The more entangled the system is, , i.e. the more information about system B is contained in A, the larger S gets.
It turns out that for a state with given bond dimension D one can at most …. This means that if we want to restrict the bond dimension to something small, then also the entanglement is small. This property is exactly fulfilled for area law entangled state, in 1D this means constant S.
𝑐 ↑↑…↓↑ =

9. Representing quantum states: MPSB
Consider N spins in 1D
| 𝜓 = 𝑖 1 … 𝑖 𝑁 𝑐 𝑖 1 … 𝑖 𝑁 | 𝑖 1 … 𝑖 𝑁
𝑐 𝑖 1 … 𝑖 𝑁 = 𝐴 𝛼 1 [1] 𝑖 1 𝐴 𝛼 1 𝛼 2 [2] 𝑖 2 … 𝐴 𝛼 𝑁−1 [𝑁] 𝑖 𝑁
2 𝑁 parameters
2𝑁 𝐷 2 parameters
This efficient representation is known as tensor network states. I will here focus on the simples version which is used for 1D systems, which is called matrix product states for reasons that will become obvious in a second.
Lets consider a 1D array of spin ½ for concreteness. In general 2^N parameter.
MPS ansatz: Write the coefficients in terms of tensors, which get contracted in order to evaluate the coeffs. To understand how this works, consider the i’s as fixed and see what we have: row vector times matrix times … times column vector.
Dimension of the matrices = bond dimension.
Compare the number of parameters. -> for small D efficient!
This representation is also useful for other reasons: parameters can be optimized to minimize the energy expectation value of some Hamiltonian, i.e. find the ground state. Also time evolution can be calculated (tDMRG).
The class of states that can be represented by MPS is of course restricted. It can only represent states with low entanglement. To understand this, imagine that we divide the system… entanglement can be measured by entanglement entropy which is the von Neumann entropy of a subsystem after we have traced over the rest. For a product state it is zero since the reduced state is still pure. The more entangled the system is, , i.e. the more information about system B is contained in A, the larger S gets.
It turns out that for a state with given bond dimension D one can at most …. This means that if we want to restrict the bond dimension to something small, then also the entanglement is small. This property is exactly fulfilled for area law entangled state, in 1D this means constant S.
𝑆 𝐴 =−Tr 𝜌 𝐴 log⁡( 𝜌 𝐴 )
Finite entanglement capacity:
𝑆 𝐴 ≤ log 𝐷
𝜌 𝐴 = Tr 𝐵 | 𝜓 𝜓 |

10. References Roman Orus: A practical introduction to tensor networks
Very nice tutorial!
AKLT model: English Wikipedia
Original AKLT paper: Phys. Rev. Lett. 59, 799 (1987)
This efficient representation is known as tensor network states. I will here focus on the simples version which is used for 1D systems, which is called matrix product states for reasons that will become obvious in a second.
Lets consider a 1D array of spin ½ for concreteness. In general 2^N parameter.
MPS ansatz: Write the coefficients in terms of tensors, which get contracted in order to evaluate the coeffs. To understand how this works, consider the i’s as fixed and see what we have: row vector times matrix times … times column vector.
Dimension of the matrices = bond dimension.
Compare the number of parameters. -> for small D efficient!
This representation is also useful for other reasons: parameters can be optimized to minimize the energy expectation value of some Hamiltonian, i.e. find the ground state. Also time evolution can be calculated (tDMRG).
The class of states that can be represented by MPS is of course restricted. It can only represent states with low entanglement. To understand this, imagine that we divide the system… entanglement can be measured by entanglement entropy which is the von Neumann entropy of a subsystem after we have traced over the rest. For a product state it is zero since the reduced state is still pure. The more entangled the system is, , i.e. the more information about system B is contained in A, the larger S gets.
It turns out that for a state with given bond dimension D one can at most …. This means that if we want to restrict the bond dimension to something small, then also the entanglement is small. This property is exactly fulfilled for area law entangled state, in 1D this means constant S.