Add support for various gates in the MPS simulator

[yaelbh]

What is the expected behavior?

There are a bunch of gates that the MPS simulator does not support, although other Aer simulators support them. The task is to complete the support in the MPS simulator. See also related issue #1004.

[yaelbh]

List of gates currently supported by the MPS simulator:

{"id", "x",  "y", "z", "s",  "sdg", "h",  "t",   "tdg",  "p", "u1",
   "u2", "u3", "u", "U", "CX", "cx",  "cz", "cp", "cu1", "swap", "ccx"}

For the statevector simulator:

     "cy",     "cp",      "cu1", "cu2",  "cu3",  "swap", "id",   "p",
     "x",      "y",       "z",   "h",    "s",    "sdg",  "t",    "tdg",
     "r",      "rx",      "ry",  "rz",   "rxx",  "ryy",  "rzz",  "rzx",
     "ccx",    "cswap",   "mcx", "mcy",  "mcz",  "mcu1", "mcu2", "mcu3",
     "mcswap", "mcphase", "mcr", "mcrx", "mcry", "mcry", "sx",   "csx",
     "mcsx",   "delay", "pauli"}

To do:

• [x] cy, cswap, cu2, cu3
• [x] r, rx, ry, rz
• [x] rxx, ryy, rzz, rxz
• [ ] All the gates starting with mc
• [x] sx, csx, mcsx
• [x] Check if delay needs to be done
• [x] As for pauli, best is to let the transpiler convert it to a list of regular Pauli gates (unlike the statevector simulator, in the MPS simulator there is no performance gain from this special Pauli gate)

[yaelbh]

As in the statevector simulator, maybe also here we will implement mc gates, and then other gates (some of them are already supported) will become a special case. For example, x, cx, ccx, etc., can also call mcx. Not sure though about it, since with the MPS we also have to handle swaps.

[yaelbh]

I see that mcx is already implemented in the MPS simulator, but it appears to apply only to 3 qubits. I don't know (will check) what happens if the user tries more qubits. x and cx are each implemented separately. Other mc's are not supported.

[yaelbh]

Although mcx (for 3 qubits) is implemented, it is not in the list of gates. Therefore a user cannot use it directly. It's just ccx, which is in the list of gates.

[yaelbh]

I see that some gates are deprecated. I won't add support to them.

[yaelbh]

Working on it now

[yaelbh]

Repeating https://github.com/Qiskit/qiskit-aer/issues/1008#issuecomment-720289414, but this time taking the list of gates supported by each simulator from qasm_simulator.py, employing a couple of changes from #1076, and excluding deprecated gates and non-existing gates (#1070):

List of gates currently supported by the MPS simulator:

'u', 'p', 'cp', 'cx', 'cz', 'id', 'x', 'y', 'z', 'h', 's', 'sdg', 'sx', 't', 'tdg', 'swap', 'ccx', 'unitary', 'roerror', 'delay', 'r', 'rx', 'ry', 'rz', 'rxx', 'ryy', 'rzz', 'rzx'

For the statevector simulator:

'u', 'p', 'r', 'rx', 'ry', 'rz', 'id', 'x', 'y', 'z', 'h', 's', 'sdg', 'sx', 't', 'tdg', 'swap', 'cx', 'cy', 'cz', 'csx', 'cp', 'rxx', 'ryy', 'rzz', 'rzx', 'ccx', 'cswap', 'mcx', 'mcphase', 'unitary', 'diagonal', 'multiplexer', 'initialize', 'kraus', 'roerror', 'delay', 'pauli', 'mcx_gray'

To do:

• [x] cy, cswap, csx
• [x] r, rx, ry, rz
• [x] rxx, ryy, rzz, rxz
• [ ] mcx, mcx_gray, mcphase
• [x] Check if delay needs to be done
• [x] diagonal and kraus are implemented but need to be added to the list of gates, maybe also roerror
• [ ] initialize will be done in #1058
• [ ] As for pauli, best is to let the transpiler convert it to a list of regular Pauli gates (unlike the statevector simulator, in the MPS simulator there is no performance gain from this special Pauli gate)

[yaelbh]

Multi-controlled gates: since they require contraction with many qubits, it is not clear whether their direct application is more efficient than transpilation to 2-qubit gates. Therefore I don't think we should implement them in the MPS simulator.

[yaelbh]

Closing because done in #1088.