Eigenvalues and Phases

Before we can understand quantum phase estimation (QPE), we need a precise answer to a simpler question: what does a unitary operator do to its eigenvectors?

Eigenvalues of unitary operators

Recall that a vector $|u\rangle$ is an eigenvector of an operator $U$ with eigenvalue $\lambda$ if

For a general matrix the eigenvalue can be any complex number. But unitary operators obey $U^\dagger U = I$, which places a tight constraint on $\lambda$.

Claim: every eigenvalue of a unitary operator lies on the unit circle of the complex plane, i.e., $|\lambda| = 1$.

Proof: take the norm of both sides of $U|u\rangle = \lambda|u\rangle$:

Because $U$ is unitary it preserves norms, so $\|U|u\rangle\| = \||u\rangle\|$. Dividing both sides by $\||u\rangle\|^2 \neq 0$ gives $|\lambda|^2 = 1$, hence $|\lambda| = 1$.

Any complex number with unit modulus can be written as $e^{i\phi}$ for some real angle $\phi$. This angle is called the phase of the eigenvalue.

Writing the eigenvalue as a phase

Since every unitary eigenvalue satisfies $|\lambda| = 1$, we write

The factor of $2\pi$ is conventional: it makes $\varphi$ a fraction of a full turn, so $\varphi = 0$ corresponds to $\lambda = 1$ and $\varphi = \tfrac{1}{2}$ corresponds to $\lambda = -1$.

The eigenvector equation becomes

The number $\varphi$ encodes everything about how $U$ acts on its eigenvector: it is a pure rotation of the global phase.

A concrete example: the $T$ gate

The $T$ gate (also called $\pi/8$ gate) is defined by

Its eigenvalues are the diagonal entries: $\lambda_0 = 1$ (eigenvector $|0\rangle$) and $\lambda_1 = e^{i\pi/4}$ (eigenvector $|1\rangle$).

For $\lambda_1 = e^{i\pi/4}$ we identify

So the $T$ gate imparts a phase of $\varphi = 1/8$ (one-eighth of a full turn) to its $|1\rangle$ eigenvector. QPE can recover this fraction $1/8$ as an ordinary binary number.

Why this matters for algorithms

Many quantum algorithms reduce to estimating the phase $\varphi$ of a known unitary applied to one of its eigenvectors. Examples include:

• Shor's algorithm — period-finding is recast as phase estimation of a modular-exponentiation unitary.
• Quantum simulation — simulating a Hamiltonian $H$ by running $U = e^{-iHt}$ and reading off energy eigenvalues as phases.
• HHL algorithm — solving linear systems via eigenvalue inversion, where eigenvalues are first estimated as phases.

In each case the core question is the same: given $U$ and $|u\rangle$, find $\varphi$ such that $U|u\rangle = e^{2\pi i\varphi}|u\rangle$. The next lessons build the circuit that answers it.

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