Algorithms · Amplitude estimation
Quantum amplitude estimation
A finance- and Monte-Carlo-facing primitive that makes the oracle and error model visible.
Strong empiricalUpdated
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A small worked example, computed in your browser. It is not a result from this record's papers.
Canonical amplitude estimation
3 counting qubits estimating a = sin²(θ) for the one-qubit state A|0⟩ = RY(π/4)|0⟩, θ = π/8.
Step 1 of 4 · RY(π/4)
Probability of each outcome after this step
- 000085%
- 100015%
Prepares the target qubit as A|0⟩ = cos(θ)|0⟩ + sin(θ)|1⟩; the quantity being estimated is a = sin²(θ), the probability that measuring this qubit alone would give |1⟩.
Probability spreads from 1 outcome to 2.
Spreads the 3 counting qubits into an equal superposition of all 8 values.
Probability spreads from 2 outcomes to 16.
Applies controlled powers of the Grover operator Q = A Z A⁻¹ Z, one power per counting qubit: phase estimation of Q, using A|0⟩ in place of a single eigenstate.
Probability moves between the same 16 outcomes. Down: 0110 10.7% → 1.8%, 0010 10.7% → 1.8%. This is the step that entangles the qubits: from here on no single qubit has a state of its own, only the register as a whole does.
An inverse QFT on the counting register reads the phase into a 3-bit estimate.
Probability gathers from 16 outcomes onto 4. 0001 goes from 1.8% to 25.0%.
Readout The counting register reads 001 or 111, each equally likely: these are θ/π = 1/8 and its mirror 1 − 1/8 = 7/8, both exact in 3 bits by construction. Either one gives a = sin²(π/8) = (2 − √2) / 4 exactly, the standard amplitude-estimation formula a = sin²(πλ) applied to the smaller of the two readings.
Open in Studio- Steps3
- Wires3
- OutputProbability estimate
- TargetError ε
- AccessCoherent oracle
- Papers1
A part this method uses, worked through
This page says phase estimation
— here is a concrete, runnable instance of it.
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.
Where it ends up
- 1011100%
The counting register reads 011 = 3 with probability 1: angle/(2π) = 3/8, recovered exactly.
One of its steps changes only phase: the outcome probabilities stay put while the state does not.
Open this part in StudioAmplitude estimation estimates the success probability embedded in a quantum state. It is attractive for risk, option pricing, and integration models, but only after state preparation and oracle costs are included.
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 →
Amplitude estimation extracts the amplitude hidden in a state prepared by a state-preparation operator , where is typically a success probability or expectation of interest. The original construction runs phase estimation on the Grover-style operator built from and two reflections; 's eigenphases encode , so measuring phase-register qubits yields an estimate with error scaling as queries to , compared with independent classical samples needed for the same error via a Chernoff/Hoeffding bound — a quadratic query reduction assuming coherent access to . This record checks a small Bernoulli-amplitude toy circuit against repeated classical sampling; the query-count advantage, as the entry's caveat states, depends entirely on state-preparation and oracle-access costs a toy circuit cannot certify.
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 →
- Output
- Probability estimate
- Target
- Error ε
- Access
- Coherent oracle
- Classical
- Monte Carlo sampling
- Quantum
- Amplitude / phase
- Readout
- Estimate a scalar
Aon state preparationQon state preparation, good/bad ancillaQPEon good/bad ancilla, phase register
Quantum vs classical
| Metric | Classical | Quantum |
|---|---|---|
| Query complexity (error ε) | O(1/ε²) | O(1/ε) |
| Cost model | Monte Carlo samples | coherent oracle calls |
| Asymptotic speedup | baseline | quadratic |
Classical baseline
Classical Monte Carlo with independent samples and a confidence interval for the same scalar quantity.
Quantum claim
Quadratic reduction in query/sample complexity under coherent state-preparation and oracle access.
How to compare
State the confidence interval, estimator variant, data-loading circuit, and total repetitions; a lower query count alone is not a business speedup.
Nobody has reviewed this record for gaps yet.
from qiskit import QuantumCircuit
qc = QuantumCircuit(3)
# A prepares a Bernoulli amplitude.
qc.ry(0.7, 0)
# Grover iterate and phase-estimation controls follow.
qc.barrier()
FINAL_CIRCUIT = qcThe circuit is a readable access-model sketch; benchmark data must state the estimator variant and oracle cost.
Where this sits
This record is named by the layer graph at:
- Quantum simulation of the KvN representation Method
Takes Access oracles for the components of (for example a linear part , a quadratic part , a forcing term ), a preparation unitary for , the evolution time , and an error tolerance . Returns A normalized state -close to , a history state over , or an estimate of an observable of the solution.
- Coherent amplitude-estimation readout Method
Takes A preparation routine with , or repeated copies of ; a description of ; a target additive error and a confidence . Coherent, controlled access to and is required by some methods here and by none of the sampling-based ones. Returns A scalar estimate with a stated additive-error guarantee, plus the shot or query budget and the maximum circuit depth actually consumed.
Related entries
- Grover unstructured search
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- Quantum phase estimation
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- Variational quantum eigensolver for ground-state energy
Algorithms
References
- Quantum Amplitude Amplification and Estimation
Gilles Brassard, Peter Hoyer, Michele Mosca, Alain Tapp · 2000
Introduces amplitude amplification and the amplitude-estimation scaling used in Monte Carlo applications.