Improve TwoLocal and RealAmplitudes synthesis when entanglement='full'

What should we add?

When choosing a TwoLocal or RealAmplitude or EfficientSU2 gate with entanglement=full on n-qubits (for VQE ansatz), then we get a circuit with a CX sub-circuit that contains n*(n-1)/2 CX gates. However, we can re-write this circuit with only n-1 CX gates with a linear connectivity (see a code example below).

from qiskit.circuit.library import *
from qiskit.quantum_info import Operator

n = 5
# print (RealAmplitudes(n, reps=1, insert_barriers=True).decompose())
# now note that the CX-subcircuit is in fact the same as this qc below

qc = QuantumCircuit(n)
for i in range(n):
    for j in range(i+1, n):
        qc.cx(i,j)
# print ("qc", qc)
mat = LinearFunction(qc).linear
op = Operator(qc)

qc1 = QuantumCircuit(n)
for i in range(n-1):
    qc1.cx(n-i-2, n-i-1)
# print ("qc1", qc1)
mat1 = LinearFunction(qc).linear
op1 = Operator(qc1)

print((mat == mat1).all())
print (op == op1)

outputs:

True
True

It will be good to update either NLocal or TwoLocal to consider this specific enhancement, as it seems that this is used in many applications.

[ShellyGarion]

We have agreed upon the following API:

• the new method will be called reverse_linear

• the previous method will be called full (as it originally was)

• the documentation will say that for a TwoLocal circuit, if entanglement_blocks='cx' then full and reverse_linear actually provide the same circuit

• for RealAmplitudes and EfficientSU2 circuits, where always entanglement_blocks='cx', the default value for entanglement will be changed from full to reverse_linear

[Cryoris]

Yeah it seems these names were chosen by how they visually look like on the circuit, not by their effect. But at the moment I think these names are mostly just an identifier, without any rigour behind them. If you other suggestions I think we can change them 🙂