Unitary from pattern

[FlorianFuerrutter]

Describe the feature you'd like

Hey all, I'm interested if there is any (efficient) way to get the unitary implemented from a given pattern. I did not find any direct function but I think this could be done easily if one can set the input state arbitrarily (issue #53) and check for flow of a pattern.

Additional context

I would propose a (pseudo) code in that direction:

def get_unitary_from_pattern(pattern):
		
	# check if given pattern has flow, i.e. it implements indeed a unitary
	assert has_flow(pattern)

	# num_qubits .. defined via input/ouput nodes
	assert pattern.num_input_nodes == pattern.num_output_nodes
	
	num_qubits = pattern.num_input_nodes
	N          = 2**num_qubits

	#----- get unitary -----
	# do <i|U|j>=U_ij, where |n>=(0, .., 1, .., 0) std-basis
	# maybe the lib (tensornetwork) allows a more efficient way?

	U = np.zeros((N, N), dtype=np.complex128)
	
	for j in range(N): #can be async

		# set |j> input
		state_j    = np.zeros((N), dtype=np.complex128) 
		state_j[j] = 1
		pattern.set_input(state_j)

		# get matrix elements
		tn        = pattern.simulate_pattern(backend='tensornetwork')
		state_U_j = tn.to_statevector()

		U[:, j] = state_U_j

		#same as
		#for i in range(N): U[i, j] = state_U_j[i]

	return U

Maybe there is a better way to do this? Maybe even with the current code base?

[shinich1]

Hey, thanks for the suggestion. Direct unitary extraction from a pattern would certainly be a good feature (we don't have that yet)!

For pattens with flow (only XY-plane measurements), it might be easier to directly translate into a gate network with CZ (edges connecting different flow paths) and

$$J(\alpha) = \begin{pmatrix} 1 & e^{i\alpha} \\ 1 & -e^{i\alpha} \end{pmatrix}$$

(for XY-plane measurements with angle $\alpha$ along flow paths) gates and work out the unitary directly. For pattens with gflow, as you suggest, it's likely best to do simulations. TN simulation without inputs may give you the unitary (if memory allows).

cc: @masa10-f - would you agree or not? lmk what you think.

[masa10-f]

Thanks for the good comments! @shinich1 @FlorianFuerrutter

As far as I understand, direct construction from tensornetwork seems to be the most efficient method, no matter the type of flow(causal flow / generalize flow / pauli flow). First off, we create a tensornetwork by turning each measurement command into its tensor equivalent(e.g. N -> |+>, E -> 4-rank CZ tensor, M -> projection vector). Then, we contract all the inner edges. After that, we end up with a 2n-rank tensor, which can be reshaped into an n-qubit Unitary matrix. In this approach, we don't have to compute each matrix element one by one; we get them all in one go.

Regarding the implementation, we'll need to modify tensornet.py. It is not hard but may need some tensornetwork know-how.

If you're just looking at patterns with causal flow, @shinich1 's idea is pretty straightforward. We can apply the same techniques used in various quantum circuit libraries.

Just a reminder, the existence of Pauli flow is the weakest condition of Unitary embedding for a given pattern. So what we have to check here is pauli flow :)

[mgarnier59]

@FlorianFuerrutter thanks for the suggestion and sorry for the delay.

• your proposition is exponential so we might try out on small examples but it surely won't scale.
• As we would get the numerical version of the unitary, maybe @masa10-f solution is worth investigating.
• From a MBQC-theory/ZX perspective there are dedicated unitary extraction algorithms in https://arxiv.org/abs/2003.01664 and https://arxiv.org/abs/2109.05654v1 and we are working on those.

We'll update this issue when work has advanced but feel free to reach out if you're still interested.

[thierry-martinez]

@matulni implemented circuit extraction, that will be merged soon: #445