An eigenvalue describes what a quantum circuit transformation does to a special input state called an eigenstate. If U acts on one of its eigenstates, every amplitude is multiplied by the same complex number, the eigenvalue. For quantum circuits, that number has magnitude 1, so its phase rotates every amplitude together rather than changing their relative magnitudes. section 9.2 section 9.7
Figure 9.11 The effect of a quantum circuit U when applied to one of its eigenstates
View the core relationship: applying U to an eigenstate preserves its amplitude pattern while applying the eigenvalue as a shared phase factor. section 9.2
Write the eigenvalue as cis(θ), where θ is its phase. Reapplying U repeatedly accumulates that phase: each application rotates all amplitudes of the eigenstate by θ again. This repeatable phase change turns an otherwise hidden circuit property into a periodic signal that can be measured indirectly. section 9.2 section 9.3
A single-qubit RY eigenvalue
For U = RY(2θ), a prepared single-qubit eigenstate has eigenvalue cis(θ). With θ = π/6, applying RY(π/3) rotates both amplitudes by π/6; applying it twice accumulates twice that phase. State tables and a color-wheel view make the shared rotation visible. section 9.2
Quantum phase estimation (QPE) uses this behavior to estimate an eigenvalue’s phase. Prepare an eigenstate in a target register, place an estimation register in superposition, and apply controlled powers U^(2^k). The accumulated phases encode a frequency in the estimation register; applying the inverse quantum Fourier transform (IQFT) and measuring yields an estimate of θ. section 9.3 section 9.7
Eigenvalues supply the phase information behind several algorithmic uses:
- QPE acts as a reusable building block, including within Shor’s algorithm. section 9.0
- Amplitude estimation examines eigenvalues of a Grover operator; its two nontrivial conjugate eigenvalues relate phase 2θ to the probability of good outcomes through sin²θ. section 9.6.1
- Quantum counting uses amplitude estimation to estimate the number of good outcomes. section 9.7
