Algorithms · Eigenvalue estimation
Quantum phase estimation
The eigenphase-estimation primitive behind chemistry, simulation, and period-finding workflows.
Strong empiricalUpdated
Open on the mapView frozen graph
A small worked example, computed in your browser. It is not a result from this record's papers.
Phase estimation of P(3*pi/4)
3 counting qubits estimating the phase of P(3π/4), which is 2π·3/8 exactly: a phase 3-bit QPE can represent exactly.
Step 1 of 4 · X
Probability of each outcome after this step
- 1000100%
Puts the target qubit into |1⟩, which is an eigenstate of P(3*pi/4): applying the gate to it multiplies it by a phase and changes nothing else. That is the whole requirement QPE places on its input.
The whole state moves from 0000 to 1000 — one certain outcome before, one certain outcome after, no superposition either side.
Spreads the 3 counting qubits over all 8 values they can hold at once. Each one will later ask about a different power of the phase.
Probability spreads from 1 outcome to 8. All 8 are equally likely.
Counting qubit k applies P(3*pi/4) to the target 2^k times, controlled on itself. The target never changes — it is an eigenstate — so the phase it would have picked up is kicked back onto the control instead. This is the step that writes the answer into the register.
The outcome probabilities do not move at all. What this step changes is phase — 8 distinct relative phases across the state now — which no measurement can see until a later step turns it back into probability.
The inverse Fourier transform reads that phase pattern back out as a binary number. Phase in, bits out.
Probability gathers from 8 outcomes onto 1. 1011 goes from 12.5% to 100.0%.
Readout The counting register reads 011 = 3 with probability 1: angle/(2π) = 3/8, recovered exactly.
Open in Studio- Steps3
- Wires2
- RegistersPhase + eigenstate
- PrecisionBits of phase
- CoreControlled-U
- Papers2
A part this method uses, worked through
This page says inverse QFT
— here is a concrete, runnable instance of it.
QFT then its inverse, on |0101⟩
4 qubits, starting at |0101⟩, applying the QFT then immediately undoing it.
Where it ends up
- 0101100%
The register reads 0101 again, exactly: the QFT and its inverse cancel, the way any unitary and its adjoint do.
Open this part in StudioQPE converts an eigenvalue phase into a classical bit string using controlled powers of a unitary and an inverse QFT. It is powerful, but its coherent-depth and state-overlap requirements are much stronger than a small demo suggests.
TakesNothingWhat joins here
No input port at this edge: the record publishes no gate sequence and no register, so there is nothing here to read one off — and unlike a declared hole, nothing has been recorded about what belongs here.
Nothing in the Atlas meets this end.
ReturnsNothingWhat joins here
No output port at this edge: the record publishes no gate sequence and no register, so there is nothing here to read one off — and unlike a declared hole, nothing has been recorded about what belongs here.
Nothing in the Atlas meets this end.
This record publishes no gate sequence and no register, so there is nothing here to read an interface off. Absent rather than empty. See all 152 →
Quantum phase estimation extracts the phase of an eigenvalue for a unitary acting on an eigenstate . With phase-register qubits prepared in , applying controlled powers for and an inverse QFT concentrates the register's amplitude near the -bit binary expansion of , with success probability for the nearest -bit estimate and error shrinking as — precision scales with the coherent circuit depth needed to apply . This record checks a single-qubit phase rotation with a known eigenstate against the inverse-QFT readout — a controlled setting where is known in advance — matching the entry's caveat: applying QPE to an unknown Hamiltonian's eigenstate additionally requires state overlap and enough coherent evolution time to reach the desired precision.
TakesNothingWhat joins here
No input port at this edge: the record publishes no gate sequence and no register, so there is nothing here to read one off — and unlike a declared hole, nothing has been recorded about what belongs here.
Nothing in the Atlas meets this end.
ReturnsNothingWhat joins here
No output port at this edge: the record publishes no gate sequence and no register, so there is nothing here to read one off — and unlike a declared hole, nothing has been recorded about what belongs here.
Nothing in the Atlas meets this end.
This record publishes no gate sequence and no register, so there is nothing here to read an interface off. Absent rather than empty. See all 152 →
- Registers
- Phase + eigenstate
- Precision
- Bits of phase
- Core
- Controlled-U
- Input
- Approximate eigenstate
- Output
- Eigenphase bits
- Classical post
- Phase interpretation
Hon phase[0..m]c-U²ʲon phase[0..m], eigenstate |ψ⟩QFT†on phase[0..m]
Quantum vs classical
Classical baseline
Classical diagonalization, Lanczos/Krylov methods, or Fourier analysis when the operator has exploitable structure.
Quantum claim
Estimates eigenphases with precision tied to controlled-U evolution time in the coherent access model.
How to compare
Report eigenstate overlap, evolution cost, precision, repetitions, and the classical method used to interpret the result.
Nobody has reviewed this record for gaps yet.
from qiskit import QuantumCircuit
from qiskit.circuit.library import QFT
qc = QuantumCircuit(3)
qc.h([0, 1])
# Controlled powers of U occupy the problem-specific block.
qc.cp(0.5, 0, 2)
qc.cp(1.0, 1, 2)
qc.append(QFT(2, inverse=True), [0, 1])
FINAL_CIRCUIT = qcThe controlled-U block is a phase-rotation toy, not a Hamiltonian simulation implementation.
Where this sits
This record is named by the layer graph at:
- Phase estimation into an ancilla register Method
Takes A circuit for U that can be applied as controlled U^(2^j), a preparation routine for a state whose overlap with the target eigenvector is not negligible, the number of bits of the phase wanted, and the failure probability that may be tolerated. Returns An estimate of the eigenphase to the requested number of bits, with the failure probability it was obtained at, plus the two costs that actually differ between routes: how many ancillas were held at once, and how many sequential rounds were run.
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References
- Quantum measurements and the Abelian Stabilizer Problem
A. Yu. Kitaev · 1995
Foundational quantum measurement setting associated with phase estimation and hidden subgroup structure.
- Tight Bounds for Quantum Phase Estimation and Related Problems
Nikhil S. Mande, Ronald de Wolf · 2023
Provides modern precision and error-cost bounds for phase estimation.