Quantum Phase Estimation (QPE)

qc/algorithms qc/gates

Given a state $|{\psi }_0⟩={\sum }_i{\gamma }_i|E_i⟩$ decomposed in $H$‘s eigenbasis, QPE measures out an eigenvalue $E_i$ with probability $|{\gamma }_i{|}^2$ — in particular, $E_0$ (the ground-state energy) whenever $|{\psi }_0⟩$ has overlap with the true ground state. It’s called “phase estimation” because it extracts $E_i$ from the phase accumulated by $U=e^{-iH\Delta t}$ (see Trotterization for how to actually build $U$).

Circuit (3-bit example)

1. Apply $H$ (Hadamard) to each of 3 ancilla qubits, preparing an equal superposition.
2. Apply controlled-$U$, controlled-$U^2$, controlled-$U^4$ from each ancilla onto the $|{\psi }_0⟩$ register — each ancilla controls a different power of $U$, writing binary digits of the phase into the ancilla register via phase kickback.
3. Apply the inverse Quantum Fourier Transform ($QF{T}^{†}$) to the ancilla register — see The Quantum Fourier Transform for the full circuit derivation.
4. Measure the ancillas — the result is a 3-bit binary approximation of an energy eigenvalue, sampled with probability $|{\gamma }_i{|}^2$.

QPE is the QFT’s one genuinely load-bearing use case: it never loads classical data into amplitude form and its output (an eigenvalue) is exactly the kind of peaked distribution the QFT reads out cheaply — see The Quantum Fourier Transform for why that combination matters.

Worked example — spin precession

A single qubit under $R_z(\pi /2)$ has eigenphase $\phi =1/8$ (since $R_z(\theta )$ contributes phase $\theta /2\pi$ per application). Running QPE with enough ancilla/phase qubits to represent $1/8=0.001$ in binary recovers $\phi$ exactly; with fewer ancillas, the estimate rounds to the nearest representable binary fraction — a concrete illustration of why circuit depth (ancilla count) trades directly against estimation accuracy $\epsilon$.

Pros and cons

Pros: high accuracy, asymptotically optimal query complexity, the natural target algorithm for future fault-tolerant hardware (see Error Correction (EC)).

Cons: needs circuit depth $O(1/\epsilon )$ for error $\epsilon$ — deep circuits with today’s noise levels — and the controlled time-evolutions are nonlocal, adding significant overhead even for small problems. This is exactly why Variational Quantum Eigensolver (VQE) and Quantum Krylov Methods exist as near-term alternatives.

Related

• The Ground-State Problem
• Trotterization — how $U$ is actually implemented
• The Quantum Fourier Transform — the circuit this note’s step 3 uses, now derived in full
• Variational Quantum Eigensolver (VQE)
• Quantum Krylov Methods
• Quantum Chemistry — QPE on H2 (Worked Example) — this circuit run end-to-end on a real molecule, with concrete accuracy-vs-circuit-cost numbers

Self-Check

• Why is QPE called “phase estimation” — what phase, and where does it come from?
• Why does deeper circuit depth trade directly against QPE’s accuracy $\epsilon$?
• Why is QPE considered better suited to future fault-tolerant hardware than to today’s noisy devices?