# Circuit for pre-factors +-i, -1

### Setting

I have a (6 qubit) circuit which implements a unitary $$U$$.

### Goal

I need the circuits which implement $$-U, iU, -iU$$.

Phase matters, because I later embed a controlled version of $$\pm i U$$ into a bigger circuit.

### Attempt

for 3 qubits, Qiskit decomposes e.g. $$-i$$ into

qnum = 3

unitary = -1j * np.eye(2**qnum)

_qc = QuantumCircuit(qnum)
_qc.unitary(unitary, [*range(qnum)], label='U1')

_qc = transpile(_qc, backend=AerSimulator(), optimization_level=3, basis_gates=['cx', 'id', 'u3'])
_qc.decompose().draw('mpl')


And for four qubits (qnum=4):

Both decompositions are not obvious to me.

Is it known how to find the right gates analytically?

Is this decomposition for 4 qubits really optimal?

• You almost certainly want to get the "global" phase by applying an appropriate Rz(theta) to the later control, instead of by tweaking the circuit being controlled. This will tend to produce a smaller result, since you'll use a a single Z rotation instead of multiple controlled operations to get the effect you're going for. Jun 25, 2022 at 17:14
• You are right, the controlled version of two gates (x, u from Egretta Thula's answer) per qubit quickly becomes a lot. What do you mean by applying 'appropriate Rz(theta) to the later control'? You mean applying Rz to the controlling qubits? How does that work? Jun 26, 2022 at 3:10

To add a global phase $$\phi$$, all what you need is to apply $$X$$-gate followed by $$U(\pi, \phi, \phi + \pi)$$ to any qubit in your circuit.

For example to implement $$iU$$, we have $$\phi = \frac{\pi}{2}$$.

You can use the matrix representation of U-gate to check this. Or, just use the following code to check:

from qiskit import QuantumCircuit
from qiskit.quantum_info.operators import Operator
from qiskit.visualization import array_to_latex
import numpy as np

circ = QuantumCircuit(3)

phase_angle = np.pi / 2

circ.x(0)
circ.u(np.pi, phase_angle, np.pi + phase_angle, 0)

# Check:
op = Operator(circ)
array_to_latex(op)


Similarly, you can implement $$-U$$, and $$-iU$$ by setting phase_angle to $$\pi$$, and $$\frac{3\pi}{2}$$ respectively.

$$iU$$ circuit should look like: