# 6. Tensor Network Design¶

The section contains some notes on the design of quimb’s tensor network functionality that should be helpful for a) understanding how certain functions work better, and b) implementing more advanced functions or contributing to quimb.

[1]:

%config InlineBackend.figure_formats = ['svg']

# the main tensor and tensor network functionality
import quimb.tensor as qtn

# backend agnostic array functions
from autoray import do


## 6.1. Tensor¶

The Tensor object, unsurprisingly, is the core object in quimb.tensor. It has three main attributes.

### 6.1.1. Tensor.data¶

This is any n-dimensional array looking object (i.e. something with a .shape attribute). Whenever array functions are performed on a tensors .data, autoray is used to dispatch to the correct backend functions, so quimb itself does no explicit conversion of array backends.

[2]:

data = do('random.uniform', size=(3, 4, 5), like='cupy')


Having said that, if the supplied data has no .shape attribute numpy.asarray() will be called on it.

### 6.1.2. Tensor.inds¶

This is an ordered tuple of index names - one for each of the dimensions of .data. Functions that would normally require ‘axes’ to be specified instead can just use names, and these are propagated through all operations. For example, operations such as transposing the underyling array can just be done via reordering the inds of the tensor.

[3]:

inds = ('a', 'b', 'c')


Moreover, the names of indices explicitly define the geometry of bonds (edges) in tensor networks at this local tensor level - there are no extra ‘index’ objects and this information is thus completely local.

### 6.1.3. Tensor.tags¶

These are an ordered set (quimb has a quick dict based implementation - oset) of an arbitrary number of extra identifiers for the tensor. The main use of these is that within a tensor network you can use any number of simultenous labelling schemes to identify your tensors with. You probably don’t need to use any if just using raw tensors.

[4]:

tags = qtn.oset(['hello', 'world'])


Note

oset is used in many places across quimb.tensor in order to make all operations deterministic, unlike set. You can supply any (ideally itself ordered) iterable to tags and it will be converted for you.

### 6.1.4. Creating & Contracting Tensors¶

We can use our data, inds and (optional) tags to create a tensor:

[5]:

T = qtn.Tensor(data=data, inds=inds, tags=tags)
T

[5]:

Tensor(shape=(3, 4, 5), inds=('a', 'b', 'c'), tags=oset(['hello', 'world']))

[6]:

T.draw(color='hello')


Just on their own (i.e. not within a tensor network) just the labelled dimension aspect of tensors is already very convenient. Here’s fuse(), for example, which would usually require some non-trivial combination of axis finding, permuting, reshaping etc:

[7]:

T1 = T.fuse({'ac': ('a', 'c')})
T1

[7]:

Tensor(shape=(15, 4), inds=('ac', 'b'), tags=oset(['hello', 'world']))


More importantly, the indices of a tensor also exactly define the subscripts as would appear in a implicit Einstein summation, such that if we create the following tensor:

[8]:

T2 = qtn.Tensor(
# use whatever backend function T1's data is using
data=do('random.uniform', size=(6, 15), like=T1.data),
# 'd' is new, 'ac' is shared with T1
inds=['d', 'ac'],
tags=['T2'])
T2

[8]:

Tensor(shape=(6, 15), inds=('d', 'ac'), tags=oset(['T2']))


The fact that both tensors have a 'ac' index means that when combined the two corresponding dimensions are either implicitly or explicitly summed over (contracted) like so:

$T_{3}[b, d] = \sum_{ac} T_1[ac, b] T_2[d, ac]$
[9]:

T1.add_tag('T1')
(T1 | T2).draw(color=['T1', 'T2'])

[10]:

T3 = T1 @ T2
# == qtn.tensor_contract(T1, T2)
# == T1.contract(T2)
T3

[10]:

Tensor(shape=(4, 6), inds=('b', 'd'), tags=oset(['hello', 'world', 'T1', 'T2']))

[11]:

T3.draw(color=['T1'])


The @ operator eagerly contracts two tensors according to this summation (via tensor_contract()), thereby removing the 'ac' index. As you can see, contracting two tensors also merges their tags.

Hint

In quimb, the names of the indices entirely define the geometry of the tensor network. If you want two tensor dimensions to be contracted just name them the same thing. Alternatively use connect() to name them for you. The actual name is not so important, in as much as its unique and shared, you can generate new names automatically using rand_uuid().

The fact that the specific name of internal indices has no effect on the overall oject represented is made use of by TensorNetwork - see - Index Mangling.

### 6.1.5. Copying & Modifying Tensors¶

You can copy a tensor with copy(), (note by default this doesn’t copy the .data - see below), and update it with modify(). Routing all changes to tensors through modify() allows various checks, and for tensors to automatically inform any tensor networks they belong to, for example.

An example of a typical tensor method that demonstrates the above is the following:

[12]:

def sum_reduce(self, ind, inplace=False):
"""Sum the array axis corresponding to ind.
"""
# easily enables copy vs inplace version
t = self if inplace else self.copy()

# get the axis ind corresponds to
axis = t.inds.index(ind)

# use autoray.do to be backend agnostic
new_data = do('sum', t.data, axis=axis)
new_inds = t.inds[:axis] + t.inds[axis + 1:]

# update tensor using modify
t.modify(data=new_data, inds=new_inds)

# return t even if inplace to allow method chaining
return t

sum_reduce(T1, 'b')

[12]:

Tensor(shape=(15,), inds=('ac',), tags=oset(['hello', 'world', 'T1']))


A convention that quimb adopts is for many common methods to have inplace versions specifically named with a trailing underscore, like:

Warning

All array functions in quimb as assumed to be pure, i.e. return a new array and not modify the original. This allows tensor copies by default to not copy their .data, which saves memory and is more efficient with backends that e.g. construct an explicit computational graph. You should only inplace modify the actual arrays if you know what you are doing, or after calling .copy(deep=True) first.

## 6.2. TensorNetwork¶

As described above, there is already an implicit tensor network geometry introduced by simply matching up tensor index names - and indeed you could just keep track of the tensors yourself.

However, its generally useful to collect tensors into a TensorNetwork object, which enables $$\mathscr{O}(1)$$ access to any tensors based on their inds or tags, as well as encapsulating all sorts of functionality the depends on the network relations of the tensors.

[13]:

ta = qtn.rand_tensor((2, 3, 3), ['a', 'x', 'y'], tags='A')
tb = qtn.rand_tensor((2, 3, 3), ['b', 'y', 'z'], tags='B')
tc = qtn.rand_tensor((2, 3, 3), ['c', 'z', 'x'], tags='C')

tn = qtn.TensorNetwork([ta, tb, tc])
tn

[13]:

<TensorNetwork(tensors=3, indices=6)>

[14]:

tn.draw(color=['A', 'B', 'C'])


Note

By default TNs take copies of the input tensors, (but again, not the actual data array). This can be turned off and it is possible e.g. to create multiple TNs that view the same tensors - see Virtual TNs.

Tensor networks also have three key attributes…

### 6.2.1. TensorNetwork.tensor_map¶

The key storage of a TensorNetwork is the tensor_map attribute. This is a mapping of ‘tids’ - unique integers reprensenting nodes in the network - to the actual tensor at that node, which need not be unique.

[15]:

tn.tensor_map

[15]:

{0: Tensor(shape=(2, 3, 3), inds=('a', 'x', 'y'), tags=oset(['A'])),
1: Tensor(shape=(2, 3, 3), inds=('b', 'y', 'z'), tags=oset(['B'])),
2: Tensor(shape=(2, 3, 3), inds=('c', 'z', 'x'), tags=oset(['C']))}


This means that the same tensor can be in the same or different tensor networks multiple times.

Hint

'tids' are essentially vertex labels in the (hyper) graph of the TN.

### 6.2.2. TensorNetwork.ind_map¶

This is a mapping of every index in the tensor network to the ordered set of ‘tids’ of the tensors which have that index.

[16]:

tn.ind_map

[16]:

{'a': oset([0]),
'x': oset([0, 2]),
'y': oset([0, 1]),
'b': oset([1]),
'z': oset([1, 2]),
'c': oset([2])}


For example if we wanted all tensors with the 'z' index, we could call:

[17]:

[tn.tensor_map[tid] for tid in tn.ind_map['z']]

[17]:

[Tensor(shape=(2, 3, 3), inds=('b', 'y', 'z'), tags=oset(['B'])),
Tensor(shape=(2, 3, 3), inds=('c', 'z', 'x'), tags=oset(['C']))]


Hint

ind_map[ix] is essentially the set of (hyper) graph vertices the (hyper) edge ix is incident to.

### 6.2.3. TensorNetwork.tag_map¶

Similarly, this is a mapping of every tag in the tensor network to the ordered set of ‘tids’ of the tensors which have that tag.

[18]:

tn.tag_map

[18]:

{'A': oset([0]), 'B': oset([1]), 'C': oset([2])}


If we wanted all tensors tagged with 'B' we could call:

[19]:

[tn.tensor_map[tid] for tid in tn.tag_map['B']]

[19]:

[Tensor(shape=(2, 3, 3), inds=('b', 'y', 'z'), tags=oset(['B']))]


#### 6.2.3.1. Selecting Tensors Based on Tags - the which kwarg¶

tags are the main high level method for labelling and accessing tensors. Many functions which take a tags and which arg, e.g. select() internally call _get_tids_from_tags(). When supplied to this, the which kwarg takes four options:

• 'all' return tids for tensors which match all the tags

• '!all' return tids for tensors which don’t match all the tags

• 'any' return tids for tensors which match any of the tags

• '!any' return tids for tensors which don’t match any of the tags

[20]:

tn._get_tids_from_tags(['A', 'B'], which='any')

[20]:

oset([0, 1])


### 6.2.4. Copying & Modifying Tensor Networks¶

Tensor networks also have a copy() method which is used heavily. As with creation, by default this copies both the TN and the tensors it contains (but again, not their data).

In terms of modifying tensor networks, quimb takes the general approach that you directly modify tensors in a TN, which then automatically tell the TN, and possibly other TNs, about any relevant changes.

[21]:

# get the tensor identified by tag 'A'
t = tn['A']

# inplace rename an index
t.reindex_({'a': 'b'})

# changes are automatically reflected in tn
tn.ind_map

[21]:

{'x': oset([0, 2]),
'y': oset([0, 1]),
'b': oset([1, 0]),
'z': oset([1, 2]),
'c': oset([2])}

[22]:

# 'b' should now be traced over
tn.draw(color=['A', 'B', 'C'])


This functionality is enabled by tensors having ‘owners’ - see Tensor ‘Owners’.

The following function demonstrates the anatomy of a typical tensor network function. It performs the (probably not useful task) of:

1. Getting all the tensors corresponding to a particular tag

2. Getting all the tensors neighboring to these

3. Performing a backend-agnostic tanh on the data of all these tensors

[23]:

def tanh_neighbors(self, tag, inplace=False):
"""Example function that demonstrates the anatomy of a
typical tensor network  method, using tensor_map,
ind_map, tag_map, and modify.
"""
# easily handle both copy and non copy methods
tn = self if inplace else self.copy()

# get the unique tensor identifiers from our tag
tids = tn.tag_map[tag]

# collect neighbors in here -> oset for determinism
neighbors = qtn.oset()

# for every tid in our tagged region
for tid in tids:

# get the tensor
t = tn.tensor_map[tid]

# for each of its inds
for ix in t.inds:

# add all tensors with that index to neighbors
neighbors |= tn.ind_map[ix]

# now apply tanh to our expanded region
for tid in neighbors:
# always use modify and autoray to update tensors
t = tn.tensor_map[tid]
t.modify(data=do('tanh', t.data))

# return tn even if self (i.e. inplace) for method chaining
return tn

qtn.TensorNetwork.tanh_neighbors = tanh_neighbors

# create trailing underscore inplace version
import functools
qtn.TensorNetwork.tanh_neighbors_ =  \
functools.partialmethod(tanh_neighbors, inplace=True)


In use:

[24]:

tn['A'].data.item(0)

[24]:

-0.49010884995279475

[25]:

tn.tanh_neighbors_('B')

[25]:

<TensorNetwork(tensors=3, indices=5)>

[26]:

# since A is connected to B, its data should have been tanh'd
tn['A'].data.item(0)

[26]:

-0.45430282125028054


### 6.2.5. Combining TNs¶

#### 6.2.5.1. Index Mangling¶

While it is very convenient generally to have the tensor network geometry defined ‘locally’ just by Tensor.inds, there is one main drawback.

1. Imagine we have two tensor networks, tn1 and tn2, both with the same outer indices - e.g. ('k0', 'k1', 'k2', 'k3', ...).

2. When we combine these these networks we expect the new TN to automatically represent the overlap of these networks - i.e. with the 'k{}' indices contracted.

3. However, if tn2 has the same internal indices as tn1 (e.g. it actually is tn1, or is a copy of or is derived from tn1 etc.), then these indices will now clash and appear four times in the new TN.

The solution quimb adopts is that when you combine two or more tensor networks, inner indices in the latter will be mangled - ‘inner’ being defined as those appearing $$\geq 2$$ times. This is fairly natural since renaming internal indices has no effect on the overall TN object represented, but it does mean you should only rely on outer index names being preserved.

Hint

In general in quimb, you should keep track of the external indices of a TN and the tags describing the internal tensor structure. If and when you need explicit index names you can retrieve them from the tensors with, e.g.:

#### 6.2.5.2. Virtual TNs¶

When you create a TensorNetwork, from some collection of tensors and/or other tensor networks, the tensors are added via:

By default the tensors (but not the data - see above), are copied, so that they only appear in the new TN. This behavior corresponds to the virtual=False option and overloaded operators:

• &

• &=

used to combine tensors and tensor networks.

[27]:

tx = qtn.rand_tensor([3, 4], ['a', 'b'], tags='X')
ty = qtn.rand_tensor([4, 5], ['b', 'c'], tags='Y')

tn = tx & ty
# == qtn.TensorNetwork([tx, ty], virtual=False)

tx is tn['X']

[27]:

False


Any changes to tx won’t affect tn.

If you want to tensor networks to ‘view’ existing tensors, either to have the tensors appear in multiple networks, or simply because you know a copy is not needed, you can use the virtual=True option. This corresponds to the overloaded operators:

• |

• |=

used to combine tensors and tensor networks.

[28]:

tn = tx | ty
# == qtn.TensorNetwork([tx, ty], virtual=True)

tx is tn['X']

[28]:

True


Now any changes to tx will affect tn. The virtual kwarg can also be supplied to copy.

[29]:

tx is tn.copy(virtual=True)['X']

[29]:

True


#### 6.2.5.3. Tensor ‘Owners’¶

The piece of technology that enables tensors to tell, possibly mutiple, tensor networks about changes to their inds and tags is owners. You probably won’t need to interact with this, but it might be useful to understand, what happens behind the scenes.

When a tensor is added to a tensor network, the tensor itself stores a weakref to the tensor network. If the tensor’s indices or tags are then changed using modify(), it can tell each tensor network it has been added to to update their ind_map and tag_map correctly.

[30]:

tx = qtn.rand_tensor([3, 4], ['a', 'b'], tags='X')
ty = qtn.rand_tensor([4, 5], ['b', 'c'], tags='Y')
tz = qtn.rand_tensor([4, 5], ['b', 'c'], tags='Z')

tn_xy = tx | ty
tn_xz = tx | tz

tx.owners

[30]:

{8778386051729: (<weakref at 0x7fbe09536310; to 'TensorNetwork' at 0x7fbe09512910>,
0),
8778386051714: (<weakref at 0x7fbe09536360; to 'TensorNetwork' at 0x7fbe09512820>,
0)}

[ ]:




A weakref is used so as not to prevent garbage collection of TNs, with stale weakrefs being simply cleared out.

Note

Since TensorNetwork by default copies tensors before adding them, and a freshly copied tensor has no owners, most tensors are usually only ‘owned’ by a single tensor network.

## 6.3. Structured (1D, 2D, …) Tensor Networks¶

So far we have only talked about the design of generic tensor networks of any geometry, and noted that the external indices of a TN, plus the tags of the tensors inside it, are the things that essentially define it.

Specific, structured, tensor networks such as MPS or PEPS are implemented as subclasses of TensorNetwork, with extra properties that describe, for example, a ‘format’ for how the physical indices and tensors are tagged as a function of coordinate. Methods that rely on assuming such structure can then be invoked.

As an example consider MatrixProductState, which is a subclass of TensorNetwork but also has methods mixed in from the following:

The MPS class then has the following extra properties that are required:

[31]:

qtn.MatrixProductState._EXTRA_PROPS

[31]:

('_site_tag_id', '_site_ind_id', 'cyclic', '_L')

• '_site_tag_id' being the string formatter that converts site coordinate to site tag

• '_site_ind_id' being the string formatter that converts site coordinate to site ind

• 'cyclic' describing whether the MPS has periodic or open boundary conditions

• '_L' describing the number of sites.

These are usually generated automatically, from defaults or context:

[32]:

mps = qtn.MPS_rand_state(L=100, bond_dim=20, cyclic=True)
mps.site_tag_id, mps.site_ind_id, mps.cyclic, mps.L

[32]:

('I{}', 'k{}', True, 100)

[33]:

mps = qtn.MatrixProductState(
[
do('random.normal', size=(7, 2), like='numpy'),
do('random.normal', size=(7, 7, 2), like='numpy'),
do('random.normal', size=(7, 7, 2), like='numpy'),
do('random.normal', size=(7, 2), like='numpy'),
],
site_ind_id='b{}'
)
mps.site_tag_id, mps.site_ind_id, mps.cyclic, mps.L

[33]:

('I{}', 'b{}', False, 4)


### 6.3.1. Converting Structured Tensor Networks¶

Sometimes you might have created or modified another TensorNetwork such that you know it matches the structure of some more specific geometry and now want to use relevant methods, for example, if you have constructed a 2D tensor network and now want to use contract_boundary(). The following describes how to convert between generic and structured tensor networks without going via the raw constructors, that might be fiddly and inefficient.

Imagine we have the following tensors and generic TN:

[34]:

t00 = qtn.rand_tensor([2, 4, 4], inds=['k0,0', '00-01', '00-10'], tags=['I0,0', 'ROW0', 'COL0'])
t01 = qtn.rand_tensor([2, 4, 4], inds=['k0,1', '00-01', '01-11'], tags=['I0,1', 'ROW0', 'COL0'])
t10 = qtn.rand_tensor([2, 4, 4], inds=['k1,0', '00-10', '10-11'], tags=['I1,0', 'ROW1', 'COL1'])
t11 = qtn.rand_tensor([2, 4, 4], inds=['k1,1', '01-11', '10-11'], tags=['I1,1', 'ROW1', 'COL1'])


These make up a little PEPS-like 2x2 TN:

[35]:

tn = (t00 | t01 | t10 | t11)
tn

[35]:

<TensorNetwork(tensors=4, indices=8)>

[36]:

tn.draw(color=['I0,0', 'I0,1', 'I1,0', 'I1,1'])


However it doesn’t have any of the special PEPS methods yet. Rather than retrieving raw arrays and calling PEPS(arrays), we can use the following functions to directly instantiate a PEPS:

Note

Note the following only work because we have already correctly tagged and labelled the tensors, thus ‘promising’ that the PEPS structure exists.

#### 6.3.1.1. from_TN¶

[37]:

# we need to tell the constructor what values we are using for the following:
qtn.PEPS._EXTRA_PROPS

[37]:

('_site_tag_id', '_row_tag_id', '_col_tag_id', '_Lx', '_Ly', '_site_ind_id')

[38]:

peps = qtn.PEPS.from_TN(
tn,
Lx=2,
Ly=2,
site_tag_id='I{},{}',
site_ind_id='k{},{}',
row_tag_id='ROW{}',
col_tag_id='COL{}',
)
peps

[38]:

<PEPS(tensors=4, indices=8, Lx=2, Ly=2, max_bond=4)>

[39]:

peps.show()

    4
●━━━━●
╱┃4  ╱┃4
┃  4 ┃
●━━━━●
╱    ╱


#### 6.3.1.2. TensorNetwork.view_as¶

We can also use the method version, view_as(), which enables an inplace option:

[40]:

tn.view_as_(
qtn.tensor_2d.TensorNetwork2DFlat,
Lx=2,
Ly=2,
site_tag_id='I{},{}',
row_tag_id='ROW{}',
col_tag_id='COL{}',
)

[40]:

<TensorNetwork2DFlat(tensors=4, indices=8, Lx=2, Ly=2, max_bond=4)>

[41]:

# tn is now a 'FLat 2D TN'
tn.show()

    4
●━━━━●
┃4   ┃4
┃  4 ┃
●━━━━●



#### 6.3.1.3. TensorNetwork.view_like¶

Finally, and often most conveniently, if you have an existing structured TN with extra attributes you want to match, you can call view_like(), which defaults to picking up all the necessary attributes from whatever you supply to the like= kwarg:

[42]:

other_peps = qtn.PEPS.rand(Lx=2, Ly=2, bond_dim=5)

tn.view_like_(other_peps)

[42]:

<PEPS(tensors=4, indices=8, Lx=2, Ly=2, max_bond=4)>

[43]:

# tn is now a PEPS with its special methods
tn.expand_bond_dimension(9).show()

    9
●━━━━●
╱┃9  ╱┃9
┃  9 ┃
●━━━━●
╱    ╱


This is useful if you perform modifications to a specific type of TN, such that it loses its structured identity, but then you want to recast it as original type, knowing that all the tags and indices are still correct.

### 6.3.2. Compatible Subclasses¶

One final feature to note regarding specific tensor network subclasses is that when you combine certain TNs with the & or | operators, if they are both ‘compatible’ versions of an inherited structured TN, they keep that structure:

[44]:

# so if we combine two MPS, which are both TensorNetwork1D
mps_a = qtn.MPS_rand_state(10, 7)
mps_b = qtn.MPS_rand_state(10, 7)

# we get a TensorNetwork1D
mps_a | mps_b

[44]:

<TensorNetwork1D(tensors=20, indices=28, L=10, max_bond=7)>

[45]:

# and if we combine two PEPS, which are both TensorNetwork2D
peps_a = qtn.PEPS.rand(10, 10, 7)
peps_b = qtn.PEPS.rand(10, 10, 7)

# we get a TensorNetwork2D
peps_a | peps_b

[45]:

<TensorNetwork2D(tensors=200, indices=460, Lx=10, Ly=10, max_bond=7)>


One key thing this allows is for structured contraction schemes to remain avaiable once norm- or expectation- TNs have been formed for example, without calling view_like().

## 6.4. Standard vs. Hyper Indices & Tensor Networks¶

• In standard summation expressions and tensor networks, if an index appears twice it is contracted, and if it appears once, it is an ‘outer’ index that will be retained. Such a setup is assumed to be the default when contracting tensors, and an error will be raised if an index is encountered more than twice.

• On the other hand, it is a perfectly valid ‘sum-of-products-of-tensor-entries’ expression to have indices appear an arbitrary number of times. It can be very beneficial to do so in fact. quimb supports this, with the main difference being that you will have to explicitly name the output_inds for contractions, which can no longer be automatically inferred.

[46]:

T1 = qtn.rand_tensor((2, 2), ('a', 'x'))
T2 = qtn.rand_tensor((2, 2), ('b', 'x'))
T3 = qtn.rand_tensor((2, 2), ('c', 'x'))

# we'll get an error due to 'x' unless we specify output_inds
qtn.tensor_contract(T1, T2, T3, output_inds=['a', 'b', 'c'])

[46]:

Tensor(shape=(2, 2, 2), inds=('a', 'b', 'c'), tags=oset([]))


The above has performed the summation:

$T[a, b, c] = \sum_{x} T_1[a, x] T_2[b, x] T_2[c, x]$

I.e. everything that doesn’t appear in the output_inds is summed over and removed. Indices such as ‘x’ can be thought of as hyperedges:

[47]:

(T1 | T2 | T3).draw(highlight_inds=['x'])


Note

Contractions of ‘standard’ tensor networks can be performed just using pairwise ‘matrix-multiplication equivalent’ contractions, i.e. using tensordot(). While ‘hyper’ tensor networks are still contracted using pairwise operations, these might require routines equivalent to ‘batched-matrix-multiplication’ and so einsum().