# Design draft: improving tensor-product dimensions in QuTiP

Qobj instantiation and mathematical operations have a large overhead, mostly because of handling the dims parameter in tensor-product spaces. I’m proposing one possible way to speed this up, while also gaining some additional safety and knowledge about mathematical operations on tensor-product spaces.

The steps:

1. rigourously define the “grammar” of dims, and allow all of dimensions.py to assume that this grammar is followed to speed up parsing
2. maintain a private data structure type dimensions._Parsed inside Qobj which is constructed once, and keeps all details of the parsing so they need not be repeated. Determine Qobj.type from this data structure
3. maintain knowledge of the individual type of every subspace in the full Hilbert space (e.g. with a list). There is still a “global” Qobj.type, but this can now be one in the set {'bra', 'ket', 'oper', 'scalar', 'super', 'other'}. 'other' is for when the individual elements do not all match each other. Individual elements cannot be 'other'. 'scalar' is added to operations can keep track of tensor elements which have been contracted, say by a bra-ket product—operations will then broadcast scalar up to the correct dimensions on certain operations.
4. dimension parsing is now sped up by using the operation-specific type knowledge. For example, bra + bra -> bra, and ket.dag() -> bra. Step 3 is necessary to allow matrix multiplication to work. These lookups could be done with enum values instead of string hashing.

Note: this is part of a design discussion for the next major release of QuTiP. I originally wrote this on 2020-07-13, and any further discussion may be found at the corresponding GitHub issue.

As of QuTiP 4.5 (and all previous versions), Qobj instantiation is slow and this permeates through to all operations on Qobj. Matrix multiplication, scalar multiplication, addition and so forth all need to instantiate new Qobj instances, and the time penalty for this is on the order of ~50µs per object. This results in more and more code than needs to bypass Qobj for speed, and in some cases (e.g. qutip.control) makes the use of Qobj prohibitively expensive. This obviously is not ideal, since Qobj is our primary data type.

The majority of this time loss is due to inferring the type of an object from its dimensions, and on unnecessary copying of data at initialisation. This is exacerbated by operations often instantiating an out parameter as out = Qobj(), and then doing things like out.dims = ..., out.data = .... This causes runtime checks to be done at every stage, so the penalty of initialisation can sometimes be paid several times over in simple operations (a particularly notable example is in the implicit promotion of scalars to operators in addition, taking over 500µs to execute 1 + qutip.qeye(2)). A lot of this can be completely avoided, however, simply by instantiating the objects using all known information, not relying on inference.

In particular, various operations know what the type of their outcome is by a simple lookup table: addition is only defined between operations of the same type and maintains that type, whereas the adjoint has the mapping

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6_ADJOINT_TYPE_LOOKUP = {
'ket': 'bra',
'bra': 'ket',
'oper': 'oper',
'super': 'super',
}

If this information is supplied to Qobj.__init__ (and the fact that it need not copy data we’ve created specially for it…), we can hugely slash the overhead of mathematical operations while maintaining their safety.

The issues start to come once we look at matrix multiplication and tensor-product spaces. The tensor allows us to construct objects which are a mixture of several different types, and matrix multiplication wants to be able to contract scalar product spaces so that bra * ket gives a scalar.

## Problems with dimension handling

The current dimension handling in QuTiP is simple and intuitive until tensor-product structures are considered. At this point, it starts to become more complicated. In particular, the type of a Qobj is tied to its dimensions, but it becomes difficult to define this once there is tensor product structure. Some of this is because QuTiP allows us to construct objects which do not have a really rigourous mathematical backing to them, such as I . |g> - the tensor product of an operator and a ket. QuTiP assigns this a type 'oper', though the way it reaches this decision is more like:

1. is it a ket? [no]
2. is it a bra? [no]
3. is it a super-operator? [no]
4. if here, it must be an operator

Such objects do have a use. Let’s say we have a system with two computational qubits and one ancillary qubit, we’ve performed a calculation on it and ended up in some state |x> = |a>.|b>.|c>, and we want to extract the computational subspace when the ancilla bit is projected onto |0>. We can do this in a mathematically rigourous way with

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4>>> projector = qutip.tensor(qutip.qeye([2, 2]), qutip.basis(2, 0).proj())
>>> (projector * x).ptrace([0, 1])
Quantum object: dims = [[2, 2], [2, 2]], shape = (4, 4), type = oper, isherm = True
...

which will always return a density matrix. Alternatively, we can instead define the operator (note proj() to create |g><g| has become dag() to simply make <g|) as

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4>>> projector = qutip.tensor(qutip.qeye([2, 2]), qutip.basis(2, 0).dag())
>>> projector * x
Quantum object: dims = [[2, 2], [1, 1]], shape = (4, 1), type = ket
...

which gets us what we wanted.

This is not necessarily common, but it is useful in some circumstances.

## Other problem discussions

There have been some cases of complaint about the handling of tensor-product spaces in QuTiP in the past (see this discussion in the Google group), but these largely revolve around people not liking the idea that we enforce the tensor structure to be maintained at all. My reading of these issues is that some people would like to see dims removed, or make mathematical operations effectively ignore it.

## Solutions

### Rejections

Personally, I think enforcing the tensor product structure catches a whole lot of potential issues in code when working with objects in different Hilbert spaces, and so far I’ve not actually seen any examples where I think frequent overriding of the dims is necessary. I’d argue that the Qobj constructor taking a dims parameter is sufficient for any use-case which needs to manually set the dimensions because they’re passing in an object constructed outside of QuTiP functions.

Removing dims also makes a lot of operations harder to do. Various places in the code permute the order of the tensored spaces, and dropping dims means that the user has to “remember” the tensor product structure themselves so that they can pass it in, and we can then know what to reorder. Clearly this is undesirable—the dims are a non-computable property of the object, and therefore should be carried around as a data attribute on the class.

The alternative in the Google group that’s sometimes suggested is to keep dims for these use-cases, but make it more of a suggestion, so that any two objects which satisfy left.shape == right.shape should be compatible for addition-like operations, and ones which satisfy left.shape[1] == right.shape[0] should be compatible for matrix-multiplication-like operations. Again, I personally would tend to reject this on the grounds that enforcing the tensor-product structure is respected is a core part of what Qobj does; it ensures that the operations are mathematically possible, and that operations between different Hilbert spaces are not mixed.

### Proposal

First off, we can sidestep some of these issues by improving library code which creates Qobj instances. Operations like Qobj.__add__ already know exactly what the output dimensions are, what the the type must be, and other things like if Hermiticity has been preserved. We move away from the outdated style of

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4out = Qobj()
out.data = left.data + right.data
out.dims = left.dims
...

to one which passes all the information in one go:

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5out = Qobj(left.data + right.data,
copy=False,
dims=left.dims,
type=left.type,
isherm=left._isherm and right._isherm)

This is more verbose, but significantly faster. With no other changes to the code, doing this can save around one-quarter of the overhead on several Qobj operations. Moving to the new data-layer types also gets large improvements in instantiation time.

This is fine, except for matrix multiplication of tensor structures. In these, like in the example above, the matrix multiplication can cause tensor structures to contract, and so they then become incompatible with their previous Hilbert spaces. If we instead maintain a list of 'type' and introduce a 'scalar' type, such objects can sensibly be broadcast back up to the correct size when needed, treating the spaces containing as identities of the correct dimension. I envisage that this may have some nice use-cases within qip, for example a gate on a single qubit could be represented by a two-by-two matrix with all other dimensions scalars, rather than requiring the whole Hilbert space to be represented all the time. Optimisations can be done using only the required elements of the subspace, and only broadcast up to the full representation once at the end.

Further, we can ease the burden of parsing the dimensions in the first place. I haven’t fully attempted this yet so I don’t have full details on this, but I imagine there is some internal information we can keep after a single parsing pass that will make other operations simpler. This is particularly true of super-operators, which often care about the input and output shapes of the spaces, necessitating several more calls to np.prod. Since this information has to be computed at type-inference time, it’s easy to save it and remove the overhead. I’d propose having this type be internal, something like dimensions.Parsed and storing it as a protected attribute on Qobj instances.