Why are S wires the only ones not measured ?

[mspronesti]

Hello lambeq dev team,
after a while using lambeq I started digging a little more into the middle level details of the framework and I noticed that in the produced discopy circuit (and even quantum circuit) the S wires are the only ones not measured.

For instance,

which maps to the following quantum circuit (here I allocated 2 qubits instead of 1 for the S wire)

From what I see, S is the only wire without an effect (first image) / measurement (second image). However, in my mind, S wires contained all the information and even the size of the measured vector depended on it. I don't get why those are, on the contrary, the only one not measured.

Also, I see we do here 16 - 2 measurements, how can the output vector be of size $2^s$ ? Is this because of post selection ? And how you perform it then ?

[Thommy257]

Hi there,

yes, the effects that you can see in the first image are post-selections. Imagine a 3-qubit system where we post-select on the |0> state of the second and third wire. If the first wire is supposed to carry information of our sentence encoding, we basically say that it is only a valid encoding if we record |x00> in our measurment (not measuring the first qubit).

If the post-selection was successful, we are left with a one-qubit system, hence the open wire. It is now up to the user to decide what to do with it. You could for example plug it into a another box that takes an s-wire, or you can read out the state vector of the remaining qubit by performing quantum tomography or other read-out methods.

You can also imagine post-selection as deleting all entries in the state-vector of the system that do not correspond to the desired outcome. By default, the .eval() method of DisCoPy perfoms a state-vector simulation and does precisely that. Hence, if you have n qubits and p post-selections, you are left with a state vector of size 2**(n-p).

I hope that helps.

[mspronesti]

Hi there, thanks for your reply.
From what I understand, the S-wire is not measured in the circuits I showed to allow the user to decide what to do with it but the eval() method of DisCoPy eventually measures it. Is this it ?

I also have another question: from the above pictures, it seems to me that the S wire (which contains the relevant information) only accounts from the gates on the left branches but not from those on the right, even though S controls them via CNOTs and controlled rotation. I miss the intuition behind this. I mean, I think the reason why you do post-selection is to make the result consistent (so, in a sense, to condition it) but intuitively I expected these circuit to "flow" to the other direction.

[akatief]

Hello, thank you for your response. It was very helpful in understanding how post-selection happens.
In the case of long sentences, post-selection drastically reduces the number of measurements we can actually use (in an n qubit system only 1/2^n measurements should result in all zeros, on average). Is this an issue in practice? Shouldn't the number of repeated measurements become prohibitively high?

[Thommy257]

Hi @akatief,
Yes, this is indeed an issue with DisCoCat models. In general, the number of circuit executions required to gather meaningful statistics grows exponentially with the number of post-selections. In other words, cups are expensive! In order to mitigate the problem, one can reduce the number of post-selections per sentence by using the helper method remove_cups or design a model that requires less or zero post-selections. One grammar-based model that is similar to DisCoCat and does not require post-selections is the TreeReader model in conjunction with a CircuitAnsatz where the discard variable is set to True. This will produce circuits containing discards rather than post-selections and will run efficiently on quantum hardware.
I hope this helps.

[akatief]

Very interesting, thank you very much!

[akatief]

It's been a while, but I have a follow-up question: would it be possible to use the measurement results of non-S wires to correct the S wire? This would avoid discarding runs, am I correct? Something similar is done in one-way QC.

[Thommy257]

The .eval() evaluates the state vector after post-selection by default, hence, it's not a measurement:

>> from discopy import Ket, Bra, qubit, Id
>> circ = Ket(0,0).H(0).CX(0,1) >> Id(qubit) @ Bra(0)
>> print(circ.eval())
Tensor(dom=Dim(1), cod=Dim(2), array=[0.70710678+0.j, 0.    +0.j])

However, if you provide a backend (e.g. the qiskit AER simulator), it will run a shot-based simulation and return the measurement probabilities.

>> from pytket.extensions.qiskit import AerBackend
>> backend = AerBackend()
>> circ.eval(backend, n_shots=2**10)
Tensor(dom=Dim(1), cod=Dim(1), array=[0.51464844+0.j])

(it's a bit weird that the return type of the shot-based simulation is complex, I'll look into this)

[Thommy257]

Regarding your second question. This ansatz entangles all qubits in a uniqe way, so the order of application of the controlled gates does not neccessarily mean that the "right-most" gates don't have an effect on the s-wire, they certainly do especially given the post-selection criteria. However, it would be a good idea to try out other ansätze to find one that might be more suitable for your specific discocat model.