Eigenvector

As explained in Building Quantum Software with Python

An eigenstate, or eigenvector, is a nonzero state vector that a transformation maps to a scalar multiple of itself.

An eigenvector, or eigenstate, of a quantum circuit transformation U is a state vector whose amplitudes are all multiplied by one corresponding eigenvalue when U is applied. The eigenvalue is a complex number with magnitude 1, so this operation can be viewed as rotating every amplitude by the eigenvalue’s phase. A circuit can have multiple eigenstates. section 9.2 section 9.7

Figure 9.11 The effect of a quantum circuit U when applied to one of its eigenstates

This diagram expresses the central invariant: applying U changes an eigenstate only by its eigenvalue factor, rather than changing its amplitude pattern. section 9.2

An eigenstate of RY(2θ)

A one-qubit circuit can prepare the state [i/√2, 1/√2] by applying X and then RX(−π/2). This state is an eigenstate of RY(2θ): applying RY(2θ) rotates both amplitudes by θ, giving eigenvalue cis(θ). With θ = π/6, RY(π/3) supplies the concrete rotation. Applying the gate twice accumulates the same phase rotation again. section 9.2

Eigenstates make a circuit’s otherwise hidden rotation phase usable in phase estimation:

• Prepare an eigenstate in the target register, while an estimation register begins the phase-encoding process. section 9.3
• Apply controlled powers of U; each application contributes the eigenvalue phase to a periodic state in the estimation register. section 9.3
• Apply the inverse QFT and estimate the encoded frequency to recover an estimate of the eigenvalue phase. section 9.3

This eigenstate-phase relationship also supports amplitude estimation. For a Grover operator, the two nontrivial eigenvalues have opposite phases, and their eigenstates place equal total probability on good and bad outcomes. If 2θ is one eigenphase, the probability of good outcomes is sin²θ. section 9.6.1

Figure 9.17 The QPE algorithm where U is an m-qubit circuit and E prepares an eigenstate of the circuit

The circuit diagram shows where eigenstate preparation fits in QPE: a preparation circuit E initializes the target register before controlled applications of U encode its phase. section 9.3

How Building Quantum Software with Python covers it

How deep it goes

Defines eigenstates through amplitude-wide phase rotation and verifies one with a single-qubit RY circuit.

Examples

Demonstrates the state [i/√2, 1/√2] under RY(π/3) and a target-register eigenstate in QPE.

What it assumes

Introduces eigenstates through quantum state vectors, circuit transformations, amplitudes, and phase rotation.

What you can do after

Build it from scratch as an eigenstate-preparation step in a quantum phase estimation circuit.

How it connects

Connects eigenstates to eigenvalues, quantum phase estimation, inverse QFT, amplitude estimation, and Grover operators.

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Also explained in

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  Quantum Programming in DepthAn eigenstate is a quantum state that a quantum gate transforms into the same state multiplied by a scalar phase or eigenvalue.
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  Math and Architectures of Deep LearningA nonzero vector that is scaled without changing direction by a linear transformation.
• 

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  Data Without LabelsAn eigenvector is a nonzero vector whose direction is unchanged when a linear transformation is applied, changing only by a scalar factor.

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Related concepts

• diagonalizationused by · A matrix transformation technique that expresses a square matrix in a basis where it becomes diagonal, simplifying computations such as powers.
• quantum phase estimationused by · Quantum phase estimation is a quantum algorithm that estimates the eigenphase of a unitary operator given a corresponding eigenstate.
• principal component analysisused by · Principal component analysis is a linear dimensionality-reduction technique that transforms data into orthogonal components ordered by the variance they capture.
• eigenvalueuses · An eigenvalue is a scalar by which a linear transformation scales a corresponding nonzero eigenvector without changing its direction.
• orthogonalitycontrasts with · Orthogonality is the relationship between vectors, directions, or states whose inner product is zero, corresponding to perpendicularity in Euclidean space.

This explainer was generated by Manning's knowledge graph from the text of Building Quantum Software with Python. The section links point to the parts of the book it draws on.