Iterative Phase Estimation¶

IPE example with a 1-qubit gate for ¶

We want to apply the IPE algorithm to estimate the phase for a 1-qubit operator . For example, here we use the -gate.

Let’s apply the IPE algorithm to estimate the phase for -gate. Its matrix is

That is, the -gate adds a phase to the state , leaving unchanged the phase of the state

In the following, we will use the notation and terms used in Section 2 of lab 4 of the Qiskit textbook.

To estimate the phase for the eigenstate , we should find (where ). Therefore to estimate the phase we need exactly two phase bits, i.e. , since . So .

Remember from the theory that for the IPE algorithm, is also the number of iterations, so we need only iterations or steps.

First, we initialize the circuit. IPE works with only 1 auxiliary qubit, instead of counting qubits of the QPE algorithm. Therefore, we need 2 qubits, 1 auxiliary qubit and 1 for the eigen state of -gate, and a classical register of 2 bits, for the phase bits , .

Eigenstate Initialization¶

The initialization consists of application the Hadamard gate to the auxiliary qubit and the preparation of the eigenstate .

def initialize_eigenstate(qc):
    """Initialize the eigenstate and prepare our ancilla qubit"""
    qc.h(0)
    qc.x(1)


initialize_eigenstate(qc_S)
qc_S.draw("mpl")

Application of the Controlled- gates¶

We now have to apply times the Controlled- operators (see also in the docs Two qubit gates), that, in this example, is the Controlled- gate ( for short).

To implement in the circuit, since is a phase gate, we can use the controlled phase gate , with .

theta = 1 * np.pi / 2

cu_circ = QuantumCircuit(2)
cu_circ.cp(theta, 0, 1)
cu_circ.draw("mpl")

Let’s apply times . Since for the first step , and , we have .

for _ in range(2 ** (m - 1)):
    qc_S.cp(theta, 0, 1)
qc_S.draw("mpl")

Measure in x-basis¶

Finally, we perform the measurenment of the auxiliary qubit in x-basis.

In this way we obtain the phase bit and store it in the classical bit .

Note: The eigenvectors of the Hadamard gate are the basis vectors of the x-basis.

def x_measurement(qc, qubit, cbit):
    """Measure 'qubit' in the X-basis, and store the result in 'cbit'"""
    qc.h(qubit)
    qc.measure(qubit, cbit)


x_measurement(qc_S, qr[0], c0)
qc_S.draw("mpl")

Reset the auxiliary qubit¶

As we want to perform an iterative algorithm in the same circuit, we need to reset the auxiliary qubit after the measument gate and initialize it again as before to recycle the qubit.

We use the builtin qc.reset operation to perform a reset of qubit 0 followed by the hadamard operation.

def reset_auxiliary(qc):
    qc.reset(0)
    qc.h(0)


reset_auxiliary(qc_S)
qc_S.draw("mpl")

Phase correction (for step 2)¶

As seen in the theory, in order to extract the phase bit , we perform a phase correction of . Of course, we need to apply the phase correction in the circuit only if the phase bit , i.e. we have to apply the phase correction of only if the classical bit is 1.

So, after the reset we apply the phase gate with phase conditioned by the classical bit ().

with qc_S.if_test((c0, 1)):
    qc_S.p(-np.pi / 2, 0)
qc_S.draw("mpl")

Apply the controlled- gates and x-measurement (for step 2)¶

We apply the operations as we did in the first step. For the second step we have , hence . So we apply once. And then we perform the x-measurment of the qubit , storing the result, the phase bit , in the bit of classical register.

## 2^t c-U operations (with t=m-2)
for _ in range(2 ** (m - 2)):
    qc_S.cp(theta, 0, 1)

x_measurement(qc_S, qr[0], c1)

We now have our final circuit!

qc_S.draw("mpl")

Let’s execute the circuit with the qasm_simulator, the simulator without noise that run locally.

count = execute(qc_S, backend_sim).result().get_counts()
count

{'0 1': 1024}

count0 = execute(qc_S, backend_sim).result().get_counts()
plot_histogram(count0)

count0

{'0 1': 1024}

In the picture we have the same histograms but on the left we have on the x-axis the string with phase bits , and on the right the actual phase in decimal representation.

As we expected we have found with a probability.