# Quantum Many-Body Physics¶

My knowledge of quantum many-body physics is still very sparse. Nevertheless…

This chapter gives a brief overview of tensor networks and how they are used to simulate quantum circuits.

## Tensor Networks¶

- Orus,
**A practical introduction to tensor networks: Matrix product states and projected entangled pair states**, 2014 [101] - chapter 2-4“a partly non-technical introduction to selected topics on tensor network method”

Examples of TN methods: Density Matrix Renormalization Group (DMRG), Matrix product states (MPS), Projected Entangled Pair States (PEPS)

Wide variety of numerical methods for strongly correlated systems, each with their own limitations, e.g. Density Functional Theory (DFT) depends strongly on the modeling of the exchange and correlation interactions

We can think of TN states as quantum states given in some entanglement representation.

The Hilbert space of a quantum many-body system is a really big place with an incredibly large number of quantum states, but low-energy states of realistic Hamiltonians are not just “any” state in the Hilbert space: they are heavily constrained by locality so that they must obey the

*entanglement area-law*.The manifold containing [the candidate low-energy] states is just a tiny, exponentially small, corner of the gigantic Hilbert space. The vast majority of the Hilbert space is reachable only after a time evolution that would take O(exp(N )) time, given some initial quantum state, most of the Hilbert space is unreachable in practice.

A tensor network is the re-formulation of a rank-k tensor in terms of tensors of rank < k, using

*index contraction*i.e. “sum over all the possible values of the repeated indices of a set of tensors”Tensor networks can nicely by visualized as diagrams, such as to easily handle calculations with TN.

The total number of operations that must be done in order to obtain the final result of a TN contraction depends heavily on the order in which indices in the TN are contracted.

- Vidal,
**Efficient classical simulation of slightly entangled quantum computations**, 2003, [128]Quantum states (as the initial state of a quantum circuit) can be expressed in terms of local Schmidt decompositions of each of the partitions based on separating the set of qubits at one of the (sorted) qubits. This allows to write the state as a product of matrices. Applying single and two-qubit gates results in updating only one of two of these matrices. The computation cost of the updates depends on the maximum rank of the Schmidt decompositions i.e. it depends on the degree of entanglement.

“The evolution of a pure state of n qubits can be simulated by using computational resources that grow linearly in n and exponentially in the entanglement.”

“explain tensor networks and some associated methods as quickly and as painlessly as possible”

The connection between tensor networks and quantum circuits is elucidated at the end of Section 2.

…

Describes the MPS in terms of SVD (the Schmidt decompositions) getting only the largest singular values i.e. truncating the Hilbert space and therefore compressing data. “Efficient MPS-generating algorithms are given an indirect, compact description of a state […], and they then iteratively produce an MPS that closely approximates that state”.

- Bridgeman,
**Hand-waving and interpretive dance: an introductory course on tensor networks**, 2017 [23] - to read…“forms the basis for a seven lecture course, introducing the basics of a range of common tensor networks and algorithms”

- Markov et al.,
**Simulating quantum computation by contracting tensor networks**, 2008 [88] - chapter 3-…makes the connection between density (super)operators and tensor contraction - to re-read carefully

A quantum circuit C can be naturally regarded as a tensor network N(C): each gate is regarded as the corresponding tensor.

…

Implementation and algorithms

- NVIDIA’s cuTensorNet
See

*Description of tensor networks*for important computation steps: Find a cost-optimal tensor network contraction path, possibly with slicing and additional constraints. Create and tune a tensor network contraction plan.

- Qiskit’s Matrix product state simulation method, cites Vidal (2003) and Schollwoeck (2011)
- Pennylane’s tutorial Tensor-network quantum circuits
- Tensor Network algorithms on tensornetwork.org

More to read

- Wood et al.,
**Tensor networks and graphical calculus for open quantum systems**, 2015 [136] cited in Quantum channels. - Huang et al.,
**Efficient parallelization of tensor network contraction for simulating quantum computation**, 2021 [64] - Seitz et al.,
**Simulating quantum circuits using tree tensor networks**, 2023 [119]

More references