Pauli I in eoh procedure doesn't become a global phase gate in qiskit evolution_instruction method. Possible consequences

Question

If one has two qubits in arbitrary stats and wishes to apply a controlled version of some global phase gate $P(\varphi) = \begin{pmatrix} e^{i\varphi} & 0 \\ 0 & e^{i\varphi} \end{pmatrix}$ one should obtain:

$$ \left| \psi_0 \right\rangle \left| \psi_1 \right\rangle = \left(c_{00} \left| 0 \right\rangle + c_{01} \left| 1 \right\rangle \right)\left(c_{10} \left| 0 \right\rangle + c_{11} \left| 1 \right\rangle \right) \xrightarrow{\text{CP} }$$

$$ \xrightarrow{\text{CP}} c_{00} \left| 0 \right\rangle \left(c_{10} \left| 0 \right\rangle + c_{11} \left| 1 \right\rangle \right) + c_{01} \left| 1 \right\rangle \left(c_{10} e^{i\varphi} \left| 0 \right\rangle + c_{11} e^{i\varphi} \left| 1 \right\rangle \right), $$

This example shows that in contrast to global phase gate $P$ which doesn't produce any measurable changes in a qubit state, the controlled version of it ($CP$) makes measurable changes in the state of a system of qubits. So it is ok if $P$ is ignored, but it is not ok if $CP$ is ignored (what is happening in the presented code).

Here is a code where I use qiskit.aqua.operators.common.evolution_instruction method, where because of (how I understand) ignoring the global phase gate, the $CP$ gate wasn't obtained (the print(qc.qasm()) command shows no gate is added to the circuit):

import numpy as np
from qiskit import *
from qiskit.aqua.operators import WeightedPauliOperator
from qiskit.aqua.operators.common import evolution_instruction

phase = np.pi
pauli_dict = {'paulis': [{"coeff": {"imag": 0.0, "real": phase}, "label": "I"}]}
identity = WeightedPauliOperator.from_dict(pauli_dict)

pauli_list = identity.reorder_paulis()
instruction = evolution_instruction(pauli_list, evo_time=1, num_time_slices=1, controlled=True)

q = QuantumRegister(2)
qc_temp = QuantumCircuit(q)
qc_temp.append(instruction, q)
qc = qc_temp.decompose()

print(qc.qasm())

Possible consequences: when I tried to implement IPEA algorithm for $H = \begin{pmatrix} E_1 & 0 \\ 0 & E_2 \end{pmatrix}$ Hamiltonian I was estimating $E_2 - E_1$ instead of estimating $E_2$ eigenvalue. This problem was reported here. Problems may arise also for other quantum algos that use PEA as a subroutine (like HHL algo).

Is this a problem/bug? Are there alternatives to qiskit evolution_instruction method that don't ignore phase gates (or Pauli I operator)?

Answer 1

It seems that this is not a problem/bug. In case the global phase gate $P$ is applied in non-controlled form, final matrix for two qubits ($P$ is acting on second qubit) is \begin{equation} I \otimes P = \begin{pmatrix} P & O \\ O & P \\ \end{pmatrix} =\mathrm{e}^{i\phi}I \end{equation} So, $P$ acts as global phase gate and the phase $\mathrm{e}^{i\phi}$ can be ignored.

However, matrix of controlled $P$ is as follows \begin{equation} CP = \begin{pmatrix} I & O \\ O & P \\ \end{pmatrix} \end{equation} Hence, $CP$ is no longer global phase gate but it prepares entangled state.

Answer 2

What does the print(qc.qasm()) yield for you?

Here is what I get:

OPENQASM 2.0;
include "qelib1.inc";
qreg q2[2];
Controlled-Evolution^1 q2[0],q2[1];

Answer 3

I just copied and pasted your code above, and that is the output I get from the print() statement. What version of qiskit do you have?

from qiskit import qiskit_version
print(qiskit_version)