Quantum Phase Estimation
The canonical algorithm for reading an eigenvalue of a unitary, and thus of a Hamiltonian, from the phase it applies to an eigenstate.
The task
Given a unitary U and one of its eigenstates |u> with U|u> = e^(2 pi i phi)|u>, quantum phase estimation (QPE, Kitaev) estimates the phase phi to a chosen number of bits. Applied to U = e^(-iHt), the phase directly encodes an energy eigenvalue E of the Hamiltonian, so QPE is the standard route to spectra.
How it works
- Prepare a register of t counting qubits in uniform superposition.
- Apply controlled-U^(2^k) from each counting qubit k to the eigenstate register.
- The counting register accumulates phases proportional to phi across a binary ladder.
- Apply the inverse quantum Fourier transform to the counting register.
- Measure: the outcome is the binary expansion of phi to t bits.
Precision and cost
To resolve phi to precision epsilon requires O(1/epsilon) applications of controlled-U and O(log(1/epsilon)) counting qubits. This Heisenberg-limited scaling, error proportional to one over the total evolution time, is quadratically better than the standard-quantum-limit scaling of sampling-based methods like VQE.
The state-preparation caveat
QPE needs an input state with substantial overlap on the target eigenstate. If the prepared state is a superposition of many eigenstates, QPE returns one of their eigenvalues at random, weighted by overlap. So a good approximate ground state (from Hartree-Fock, VQE, or adiabatic preparation) is a prerequisite for reliably estimating the ground-state energy.
Why it needs fault tolerance
QPE requires long coherent controlled evolution, far deeper than near-term devices tolerate. It is the workhorse of the fault-tolerant era: combined with qubitization it gives the lowest known resource estimates for chemistry and materials eigenvalue problems. Variants such as iterative and Bayesian phase estimation reduce qubit count and adapt to noise, trading a single deep circuit for many shorter adaptive runs.