Quantum Phase Estimation: More Qubits, More Accuracy

Determine Phase of an Eigenvector of a Unitary Operator

Phase Estimation using Qubits [Photo by Jeremy Bezanger on Unsplash]

Quantum Phase Estimation (QPE) is an algorithm that’s used to estimate the phase (or eigenvalue) of an eigenvector of a unitary operator. Since this post will use many important concepts discussed in the Quantum Fourier Transform post, please check it for a review. What you can expect to learn from this post —

1. Concept of Phase Kickback.
2. Controlled Unitary Operator and Phases.
3. Theory and Formulation of QPE Algorithm.
4. Implementation of QPE with Qiskit.
5. How More Qubits Help Precision Phase Estimation.

Without any delay let’s begin.

Concept of Phase Kickback:

Consider a unitary operator (or a gate) acting on a qubit |ψ⟩ as below

Let’s consider a state |ψ⟩=1/√2 (|0⟩+|1⟩) and the outcome of measuring the state will yield 0.5 for state |0⟩ and 0.5 for state |1⟩. if we apply the unitary operator U on this state we will get —

But the outcome of measuring the state will yield the same as before. This is problematic in the sense that we cannot measure the global phase simply this way. It’s also better to…