MPO decomposition into circuits

[ACE07-Sev]

Greetings Johnnie,

Hope you are doing well good sir. I have been wondering about this for a while now. Given that one can compile a MPS to a quantum circuit to perform state preparation, could one analogously compile a MPO to a quantum circuit to perform unitary (operator) preparation in the same manner?

[jcmgray]

Yes sure, one way to think about it is if you just fuse the two legs of each MPO site you get an MPS.

[jcmgray]

I appreciate you expanding on your questions, but again these are basically theoretical questions that might be better asked e.g. on https://quantumcomputing.stackexchange.com/ or by reviewing the tensor network literature. To briefly mention some considerations:

1. Representing an arbitrary dense unitary and converting it to an MPO scales like O(2^N), so this cost dominates regardless.
2. All operators (up to normalization) can be thought of as pure states in a space twice as large, whether it is useful in this form depends very much on what you want to do with it.

[ACE07-Sev]

Understood. Thank you very much for the time and kind guidance!

[ACE07-Sev]

Just to provide some concluding thoughts.

All operators (up to normalization) can be thought of as pure states in a space twice as large, whether it is useful in this form depends very much on what you want to do with it.

What we want to do with this is as before to perform $T$. With (2) I think it's not really feasible to perform $T$ as the number of qubits wouldn't match.

Representing an arbitrary dense unitary and converting it to an MPO scales like O(2^N), so this cost dominates regardless.

I don't quite follow this. Why would unitary preparation differ from state preparation in terms of the benefit they'd receive from MPS/MPO approach? Wouldn't it also be O(N)?

I'd deeply appreciate any resources relevant to $T$ using MPOs and how we would prepare the resulting MPO as a circuit. I won't expand further as you requested, and again would like to extend many thanks for the discussion.

[jcmgray]

What we want to do with this is as before to perform T.

Here my point was, perform T i.e. prepare or perform A and then do what with it? What form you can have it in will depend on this.

I don't quite follow this. Why would unitary preparation differ from state preparation in terms of the benefit they'd receive from MPS/MPO approach?

Here I was just making the point that if you are starting by constructing the MPO (or MPS too) by creating it from a dense array then this classical step scales exponentially in N already. It might still be a useful thing to do (depending on the overall task, see above) but if you want a linearly scaling routine then you will need to start with something already in MPS/MPO form.

[ACE07-Sev]

Here my point was, perform T i.e. prepare or perform A and then do what with it? What form you can have it in will depend on this.

I see. So, given the circuit qc with N qubits (representing a state $\ket{\psi}$) and an operator $A$ applied to N qubits, we want to prepare the circuit op representing $A$ using an MPO. With this, when we append op to qc we would effectively be doing $A\ket{\psi}$.

I'd like to know how this would be done, and whether it would be similar to the MPS approach (basically being able to prepare the MPO with O(N) circuit depth).

Here I was just making the point that if you are starting by constructing the MPO (or MPS too) by creating it from a dense array then this classical step scales exponentially in N already.

Ohh I see your point. No no, my statement on $O(N)$ is solely regarding the circuit depth to prepare the state or operator (going from arbitrary MPS/MPO to circuit). I understand constructing MPS/MPO (going from state/operator to MPS/MPO) is classically expensive.

It might still be a useful thing to do (depending on the overall task, see above) but if you want a linearly scaling routine then you will need to start with something already in MPS/MPO form.

Right, the motivation is to enable the implementation of deep circuits on current hardware. We achieve this by first approximating them using MPS/MPO, and then subsequently preparing the resulting MPS/MPO to achieve $O(N)$ circuit depth as opposed to $O(2^N)$.

For example, with the state preparation we had to construct the MPS from the state vector (using mps.from_dense()) which is classically expensive. However, with this MPS we were able to subsequently prepare a circuit which would not have been feasible on any current hardware before (where we went from 8 million depth to 12 thousand, a massive reduction in the circuit depth).

So, the goal is to compress these deep circuits using this approximate preparation approach. It would allow us to answer "can vs can't". Right now, we can't do any practical state preparation/unitary preparation for states with more than 10 qubits or so. The circuits become too deep. With the MPS/MPO approach we will actually be able to do these, but of course the gotcha will be the classical cost to construct the MPS/MPO. When better alternatives present themselves, we can then evaluate "should vs shouldn't".