Quantum Phase Estimation Algorithm In Quantum Computing

Quantum Computing is one of the revolutionary technologies of the 21st century. With the potential to speed up computations and solve problems that are currently out of reach for classical computers, Quantum Computing is a fascinating field to explore. One of the essential quantum algorithms in the quantum toolbox is the Quantum Phase Estimation (QPE) algorithm. This algorithm is at the core of many other quantum algorithms, such as Shor's factorization algorithm and the quantum-based simulation of physical systems.

Quantum Phase Estimation

In essence, QPE solves the following problem: Given a unitary operator U and a quantum state |u⟩ which is an eigenvector of U, estimate θ in U|u⟩=e2πiθ|u⟩. In other words, it estimates the phase θ of the eigenvalue e2πiθ corresponding to the eigenvector |u⟩ of a unitary operator U.

How it works?

The QPE algorithm starts with two quantum registers initialized to zero. The first register which has n qubits is where we prepare the ancilla state, while the second register is where |u⟩ is held. The algorithm then follows these steps:

1. First, a Qubit Operator is applied to the ancilla register generating a superposition of different frequencies.
2. The controlled unitary stage is then applied, which introduces relative phases between the basis states in our first register.
3. Finally, the inverse Fourier transformation is used to extract the phase information and translated into computational basis outcomes we can read.

Qiskit implementation

Let's illustrate this process with a basic implementation in Qiskit, a Python library for quantum computing programming.

from qiskit import QuantumCircuit, transpile
from qiskit.extensions import UnitaryGate
from qiskit.quantum_info import Statevector
from qiskit.algorithms import PhaseEstimator
from qiskit.tools.visualization import plot_bloch_multivector

# Define our unitary
u = UnitaryGate(data=[[1, 0], [0, np.exp(2 * np.pi * 1.0j / 3)]])

# Define the circuit
qc = QuantumCircuit(1)

# Add the gate to our circuit
qc.append(u, [0])

# Get a statevector simulator
backend = Aer.get_backend('statevector_simulator')
# Run the simulation
result = execute(qc, backend).result()
# Get the statevector
sv = result.get_statevector()

## Do the phase estimation
pe = PhaseEstimator(3, unitary=u)
result = pe.estimate(backend=backend)

print("The estimated phase is ", result.most_likely_phase)

This example leverages Qiskit’s PhaseEstimator class, an implementation of the QPE algorithm, to estimate the phase of a unitary operator. Here, the operator u is applied to a basis state of the second register, and the estimated phase is then printed out.

Keep an eye on the burgeoning field of Quantum Computing, as it promises radical changes in the world of technology.