Tutorial: tensor network basics #1193

EmilianoG-byte · 2024-08-16T04:35:21Z

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github-actions · 2024-08-16T04:35:32Z

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Qottmann

Great skeleton @EmilianoG-byte @Shiro-Raven !

Mainly minor suggestions, and one bigger one towards the last section for quantum circuit applications

demonstrations/tutorial_tn_basics.py

Qottmann · 2024-08-16T07:52:58Z

demonstrations/tutorial_tn_basics.py

+- For this reason there exist heuristics for optimizing contraction path complexity. NP problem -> no perfect solution but great heuristics (https://arxiv.org/pdf/2002.01935).
+    (optional) mention the idea behind some of them 
+    Link to quimb examples.
+- CODE: show this using np.einsum, timeit, and very large dimensions expecting to see a difference.


timeit is a good idea, could also print out dimensions at different steps along a contraction path

Coming back to this, I printed the dimensions locally but I don't quite get what insight we get from this. Since we only have three tensors, the dimensions of intermediate tensors (AB) and (BC) are actually exactly the same. This also by construction to get the expected scaling in the computational cost.

I could come up with an example where the dimensions vary between contraction paths but probably would have to be more complex and not just a "triangle-like" tensor network

Yeah sounds like the example is too simple to show the desired property

I could come up with an example where the dimensions vary between contraction paths

💯

just to be sure, I guess also the timing is the same in the situation you describe rn?

Well the timing varies and that is what I was trying to convey here, as the timing scales exactly as we would expect it from the complexity analysis I discussed some lines before. What would you like to show with the dimensions? That some paths result in tensors of larger intermediate size?

What would you like to show with the dimensions? That some paths result in tensors of larger intermediate size?

exactly :)

demonstrations/tutorial_tn_basics.py

Qottmann · 2024-08-16T08:02:24Z

demonstrations/tutorial_tn_basics.py

+From tensor networks to quantum circuits:
+^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
+- Quantum circuits are a restricted subclass of tensor networks
+- show examples on https://arxiv.org/pdf/1912.10049 page 8 and 9 showing a quantum circuit for a bell state, defining each component as a tensor and show their contraction.


I would suggest to do concrete examples of gates already above, e.g. single qubit gates = matrices, 2 qubit gates like CNOT = 4 leg tensor with 2 "in" and 2 "out" legs etc.

and then focus here on how e.g. an expectation value $\langle \psi_0 | U^\dagger H U |\psi_0\rangle$ is a tensor network contraction of the initial state - unitary circuit - hamiltonian unitary circuit and initial state again. From the above we should be able to understand all the sub-components and here we are fusing them together

there are some subtleties here:

While in priciple a n-qubit state is a n-legged (=2^n sized) tensor, $\psi_0$ is often a product state, so it is just n independent vectors.

H is often the sum of multiple operators. It is beyond the scope of this tutorial to go in depth, but there are ways to efficiently represent such a sum of tensors (e.g. MPOs and generalizations thereof). The "naive" thing to do for a sum of operators $H = \sum_i h_i$ is to do separate evaluations for each $\langle \psi_0 | U^\dagger h_i U |\psi_0\rangle$ and sum them in the end.

$\langle \psi_0 | U^\dagger$ and $U |\psi_0\rangle$ appear twice in the equation, if one can represent their result efficiently we can re-use it (not a given as it may result in a large 2^n tensor)

Great suggestions! Do you think is worth going into the details of boolean tensor networks when talking about the CNOT (for instance):

While in priciple a n-qubit state is a n-legged (=2^n sized) tensor, $\psi_0$
is often a product state, so it is just n independent vectors.

doesn't this go more into the direction of MPS (which is out of the scope of this tutorial)?

The "naive" thing to do for a sum of operators

If I am not mistaken, since we are assuming an exact contraction of the tensor network without approximating it as an MPS, this naive way is the only option, no? Or is it possible to use the Hamiltonian as an MPO and contract it with the state vector even tho this is not in MPS form?

if one can represent their result efficiently we can re-use it (not a given as it may result in a large 2^n tensor)

here again you mean efficiently as an MPS? Or do you mean this thing that Quimb does where it reuses contraction paths and other parts of the computation? (I couldn't find the link but I remember seeing something along these lines)

If I am not mistaken, since we are assuming an exact contraction of the tensor network without approximating it as an MPS, this naive way is the only option, no? Or is it possible to use the Hamiltonian as an MPO and contract it with the state vector even tho this is not in MPS form?

Update: Actually I thought more about it, and I think I was wrong on this. I cannot think of a real constraint of why just contracting a general tensor circuit (no MPS) with an MPO would not work.

doesn't this go more into the direction of MPS (which is out of the scope of this tutorial)?

A product state is something different. You can interpret it as an MPS with trivial virtual bond dimension 1 but that is not the point

If I am not mistaken, since we are assuming an exact contraction of the tensor network without approximating it as an MPS, this naive way is the only option, no? Or is it possible to use the Hamiltonian as an MPO and contract it with the state vector even tho this is not in MPS form?

MPOs are typically exact and of course you can contract them with something that is not an MPS :)

here again you mean efficiently as an MPS? Or do you mean this thing that Quimb does where it reuses contraction paths and other parts of the computation? (I couldn't find the link but I remember seeing something along these lines)

The latter :) though I am actually not sure what https://github.com/PennyLaneAI/pennylane/blob/master/pennylane/devices/default_tensor.py#L799 does

Do you think is worth going into the details of boolean tensor networks when talking about the CNOT (for instance):

Up to you! I personally dont find that too important but if you like it feel free to include it :)

I have now added a section on tensor networks-quantum computing. Lmk what you think!

is the demo overall ready for review?

I would think so @Qottmann :). I can read the whole thing between today and tomorrow to find typos but content-wise, I am happy with what it has.

I just have two TODO's on the script for myself which are to add some details to some figures.

Great! Then I'd suggest to do a self-review first, and when it's ready for a full review please tag me in the PR :)

github-actions · 2024-08-21T23:48:27Z

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EmilianoG-byte · 2024-08-30T19:10:39Z

Initial draft from last week:

DRAFT:

Definition of a tensor as an n-dimensional array. Show notation with n indices, stating that it belongs to C^{d1,...,dn}. Define rank, index, dimension (mention how these terms are sometimes used (wrongly?) interchangeably in literature).
Graphical notation. Mention that there exist certain representations in the literature that allow to represent properties of the tensors (e.g. symmetry, orthogonality). In our case, we can adhere to a general circle.
Specific examples used in a day-to-day: scalars, vectors, matrices. Mention that for quantum states, we can adopt the convention that the legs in one direction mean that the state belongs to one Hilbert space, and the legs to the other side to the dual space.
CODE: include code using numpy creating a >2 dimensional array.

Show the matrix multiplication in terms of summation over indices, then using the diagrammatic representation. This results in another rank 2 tensor (matrix)
Analagously, we can represent matrix-vector multiplication resulting in a rank 1 tensor (vector). Just as we expected!
We can generalize this concept to tensors. This is done by summing over repeated indices (just as in einstein convention - external link for it) resulting in another tensor made up of the open legs of all the tensors together.
In diagrammatic notation, this is simply sticking together legs with same indices! (show nice diagram with >3 tensors). We have just formed a network of tensors, i.e. a Tensor Network!
CODE: Talking about einstein convetion, we can perform this contraction of tensors using np.einsum.

Mention that the resulting tensor network doesn't change but the way we arrive to the final tensor affects how expensive it is to get there.
Show how can we can calculate the complexity of a contraction by means of a simple example using 2 matrices (rank 2 tensors): dimension_contracted x (dimensions_open).
Intuition behind: we perform one operation (contraction) and repeat many times to "populate" the resulting tensor (dimension_open1 x dimension_open2). Show the equation with indices.
Show an example with at least three tensors where they all have different dimensions. Walk through it showing that choosing to contract two indices (the ones with lower dimensions) results in a worst computational complexity than contracting other ones (the ones with higher dimensions).

Very nice source with visual explanations that we can cite: https://www.math3ma.com/blog/matrices-as-tensor-network-diagrams

Qottmann · 2024-10-18T11:42:26Z

hey @EmilianoG-byte @Shiro-Raven how are things coming along? Please let me know when you have a first draft that is ready for review (don't forget to self-review first) :)

… is fixed

modify size of diagrams

EmilianoG-byte · 2024-10-29T09:02:25Z

Hi @Qottmann ! I have finished checking the spelling and grammar so I believe the demo is now ready for a review :).

If anything, I saw my last 3 drawings could use some improvement in the thickness of the lines, but I guess that’s a minor detail I can correct in the following days :D

Qottmann · 2024-10-29T09:04:47Z

@EmilianoG-byte awesome :) you can expect a review at the latest by eow

Qottmann

Great first draft @EmilianoG-byte , congrats!

I left a bunch of nitpicky comments, in particular feel free to ignore those marked as "personal opinion" at your discretion.

The demo starts relatively slow in the beginning (which is great pedagogically) and then very quickly goes very fast (also understandable since that is in the nature of these highly complex topics).

I wonder if you can adjust the pace on either ends to make the experience smoother (I understand this is a very vague and hard-to-implement suggestion, but perhaps you get an idea). Perhaps it is also more a matter of the framing of the scope of the demo, making it clear in the beginning, end and thoughout what this demo is trying to achieve.

Perhaps as a good exercise for you to answer first and then use to translate into the draft: who is the target audience of this demo? what is the intention of writing this demo? and what should a reader take away from it?

I think the content itself it already great, it is just a matter of framing and scoping of the text :)

Qottmann · 2024-11-04T10:22:14Z

demonstrations/tutorial_tn_basics.py

+
+Part of the excitement surrounding tensor networks is due to their ability to represent complex data efficiently, which allows for — among other things — fast classical simulations. In addition, the diagrammatic language accompanying tensor networks makes working with them intuitive and suitable for describing a vast range of mathematical concepts, including quantum circuits.
+
+In this tutorial, we aim to provide an introduction to tensor networks with a focus on their applications in quantum computing. We choose to start by discussing the basic notions and definitions of tensors and tensor networks and work our way up to more advanced topics such as contraction paths and algorithms used to simulate quantum computers using tensor networks.


(personal opinion)
could be reduced or even removed

Qottmann · 2024-11-04T10:22:29Z

demonstrations/tutorial_tn_basics.py

+.. tip::
+    Some authors refer to the indices of the tensors as their dimensions. In this tutorial, these two concepts will have different meanings, although related.
+
+For example, a scalar :math:`s` is a rank-0 tensor, a vector :math:`v_i` is a rank-1 tensor, and a matrix :math:`G_{i,j}` is a rank-2 tensor.


(personal opinion)
move above tip

Qottmann · 2024-11-04T10:23:45Z

demonstrations/tutorial_tn_basics.py

+where each :math:`i_n` is an *index* of dimension :math:`d_n`—it takes integer values such that :math:`i_n \in [1, d_n]`—and the number of indices :math:`r` is known as the *rank* of the tensor. We say :math:`T` is a rank-:math:`r` tensor.
+
+.. tip::
+    Some authors refer to the indices of the tensors as their dimensions. In this tutorial, these two concepts will have different meanings, although related.


what is the meaning of dimension here? I'd either name the meaning of "dimension" directly here or leave the comment alltogether as it doesnt help at this stage

Qottmann · 2024-11-04T10:28:24Z

demonstrations/tutorial_tn_basics.py

+
+Does the last diagram seem familiar? It is because this is the representation of a single-qubit gate! We will see later in this tutorial the relation between quantum circuits and tensor networks. 
+
+When working within the quantum computing notation, we adopt the convention that drawing the leg of a quantum state (i.e., a vector) to the right corresponds to a ket, i.e., a vector living in the Hilbert space, while drawing the legs to the left means they are a bra vector, i.e., living in the dual space.


Is the differentiation between bra and kets via the direction they are pointing a common definition and actually necessary here? Instead of ascribing duals to left-pointing legs, you can also just indicate complex conjugation of $v$ as the concept of transposition becomes irrelevant in tensor diagrams (leaving only complex conjugation)

Qottmann · 2024-11-04T10:30:23Z

demonstrations/tutorial_tn_basics.py

+print("Rank-3 tensor: \n", tensor_rank3)
+
+##############################################################################
+# We can create a tensor of arbitrary rank following a similar procedure. This recursive approach illustrates how a rank-:math:`r` tensor can be seen as consisting of nested rank-:math:`(r-1)` tensors, represented in code by adding another level to the nested bracket structure ``[tensor_rank_r-1]``.


(personal opinion)
the [tensor_rank_r-1] at the end of the sentence reads odd

Qottmann · 2024-11-04T11:03:55Z

demonstrations/tutorial_tn_basics.py

+#     :align: center
+#     :width: 45%
+# 
+# In the right-hand side of the equality we have assumed a specific form for the U tensor in terms of local 2-qubit gates, which is often the case when dealing with real quantum hardware. In addition, it is common for the initial state to be a product state such as :math:`|0\rangle^{\otimes N}`, hence the form of the tensor in the diagram.


Note that in the diagram depicted as it is right now, also the initial state is assumed to be a product state since all tensors are independent

a general input state would be one big rank-#qubits tensor

Qottmann · 2024-11-04T11:06:31Z

demonstrations/tutorial_tn_basics.py

+# 
+# When the observable of interest is *local*, i.e., it acts on a few neighbouring qubits, we can calculate the expectation value by considering only the section of the quantum circuit within the *reverse light cone* (causal cone) of the observable :math:`O_l`.
+# 
+# .. figure:: ../_static/demonstration_assets/tn_basics/12-expectation-local.png


the convention you chose in the beginning with bras and kets is not doing you any favors here, the reversal of bra and ket is unnecessarily confusing the image imo

Qottmann · 2024-11-04T11:06:48Z

demonstrations/tutorial_tn_basics.py

+#     :align: center
+#     :width: 70%
+# 
+# Then, the sections outside of the light cone (grayed-out gates in the figure above) can be ignored since these are contractions resulting in the identity: :math:`G G^\dagger = I`. This helps us decrease the size of the tensor to be contracted, and consequently, the computational expense, by focusing on the section of the circuit with support inside the light cone of the observable - i.e., the gates that affect the calculation of the expectation value.


Qottmann · 2024-11-04T11:10:01Z

demonstrations/tutorial_tn_basics.py

+# 
+# The tensor network corresponding to the computation of this vector is
+# 
+# .. figure:: ../_static/demonstration_assets/tn_basics/13-sample.png


very cool illustration!

Qottmann · 2024-11-04T11:13:56Z

demonstrations/tutorial_tn_basics.py

+#     :align: center
+#     :width: 70%
+# 
+# Analogously as done with the expectation values, these contractions only involve the sections of the circuit within the light cone of **both** the projection with :math:`| \hat{x}_1 \rangle` and the contraction with the COPY tensor (diagonal computation). This procedure can be repeated recursively using the chain rule equation until we obtain the full bitstring :math:`(\hat{x}_1, \hat{x}_2, \hat{x}_3, \ldots, \hat{x}_N)`. To obtain more samples, we repeat the procedure from the beginning - this is what makes every sample memoryless or a perfect sample from the probability distribution.


what do you mean by this?

this is what makes every sample memoryless

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Tutorial: tensor network basics #1193

Tutorial: tensor network basics #1193

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		Part of the excitement surrounding tensor networks is due to their ability to represent complex data efficiently, which allows for — among other things — fast classical simulations. In addition, the diagrammatic language accompanying tensor networks makes working with them intuitive and suitable for describing a vast range of mathematical concepts, including quantum circuits.

		In this tutorial, we aim to provide an introduction to tensor networks with a focus on their applications in quantum computing. We choose to start by discussing the basic notions and definitions of tensors and tensor networks and work our way up to more advanced topics such as contraction paths and algorithms used to simulate quantum computers using tensor networks.


		Does the last diagram seem familiar? It is because this is the representation of a single-qubit gate! We will see later in this tutorial the relation between quantum circuits and tensor networks.

		When working within the quantum computing notation, we adopt the convention that drawing the leg of a quantum state (i.e., a vector) to the right corresponds to a ket, i.e., a vector living in the Hilbert space, while drawing the legs to the left means they are a bra vector, i.e., living in the dual space.

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