Algorithms · Amplitude estimation
Option pricing by amplitude estimation
Price an option, or a portfolio of options — vanilla contracts, multi-asset contracts, and path-dependent contracts such as barrier options — on a gate-based quantum computer, in the setting where the paper takes classical Monte Carlo methods as its point of comparison.
Attested & literatureUpdated
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- Record typeLiterature reference
- Reference implementationoption_pricing
- Reported costA quadratic speedup compared to classical Monte Carlo methods, stated in that form and in no other: the abstract attaches the speedup to amplitude estimation itself, and gives no error scaling, no query count, no qubit count and no gate count for the option-pricing circuits, so nothing here says how many samples or oracle calls pricing a given contract to a given accuracy would take, nor at what accuracy the two methods cross over.
- Papers1
Price an option, or a portfolio of options — vanilla contracts, multi-asset contracts, and path-dependent contracts such as barrier options — on a gate-based quantum computer, in the setting where the paper takes classical Monte Carlo methods as its point of comparison. Stamatopoulos, Egger, Sun, Zoufal, Iten, Shen and Woerner present a methodology to price options and portfolios of options on a gate-based quantum computer using amplitude estimation, which they describe as an algorithm that provides a quadratic speedup compared to classical Monte Carlo methods. The estimator itself is the subject of this catalog's amplitude-estimation record; what this paper adds is the construction around it on the finance side, and the emphasis it puts on the implementation of the quantum circuits required to build the input states and operators that amplitude estimation needs to price the different option types. The contracts it covers are vanilla options, multi-asset options and path-dependent options such as barrier options. The paper reports simulation results that highlight how the circuits it implements price the different option contracts, and it examines the performance of those option-pricing circuits on quantum hardware using the IBM Q Tokyo quantum device. The authors also report employing a simple error-mitigation scheme that they state significantly reduces the errors arising from noisy two-qubit gates.
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 →
Stamatopoulos, Egger, Sun, Zoufal, Iten, Shen and Woerner present a methodology to price options and portfolios of options on a gate-based quantum computer using amplitude estimation, which they describe as an algorithm that provides a quadratic speedup compared to classical Monte Carlo methods. The estimator itself is the subject of this catalog's amplitude-estimation record; what this paper adds is the construction around it on the finance side, and the emphasis it puts on the implementation of the quantum circuits required to build the input states and operators that amplitude estimation needs to price the different option types. The contracts it covers are vanilla options, multi-asset options and path-dependent options such as barrier options. The paper reports simulation results that highlight how the circuits it implements price the different option contracts, and it examines the performance of those option-pricing circuits on quantum hardware using the IBM Q Tokyo quantum device. The authors also report employing a simple error-mitigation scheme that they state significantly reduces the errors arising from noisy two-qubit gates. Reported cost: A quadratic speedup compared to classical Monte Carlo methods, stated in that form and in no other: the abstract attaches the speedup to amplitude estimation itself, and gives no error scaling, no query count, no qubit count and no gate count for the option-pricing circuits, so nothing here says how many samples or oracle calls pricing a given contract to a given accuracy would take, nor at what accuracy the two methods cross over..
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 →
- Record type
- Literature reference
- Reference implementation
- option_pricing
- Reported cost
- A quadratic speedup compared to classical Monte Carlo methods, stated in that form and in no other: the abstract attaches the speedup to amplitude estimation itself, and gives no error scaling, no query count, no qubit count and no gate count for the option-pricing circuits, so nothing here says how many samples or oracle calls pricing a given contract to a given accuracy would take, nor at what accuracy the two methods cross over.
- Complexity basis
- abstract of arXiv:1905.02666: "We present a methodology to price options and portfolios of options on a gate-based quantum computer using amplitude estimation, an algorithm which provides a quadratic speedup compared to classical Monte Carlo methods." That is the only comparative claim the abstract makes; the rest of it states results rather than bounds — simulation results, a run on the IBM Q Tokyo quantum device, and an error-mitigation scheme — and attaches no figure to any of them. The demonstration this record covers states no complexity bound of its own. Those are the only sources read for this field.
- Circuit
- Not supplied
This record is not a concrete circuit.
Quantum vs classical
Classical baseline
Compare Amplitude estimation with the strongest classical method for the same instance, input budget, and output metric.
Quantum claim
This reference exposes a quantum circuit pattern; it does not imply an application-level speedup without a matched benchmark.
How to compare
Report input loading, circuit depth, repetitions, classical preprocessing, post-processing, and wall-clock time together.
Nobody has reviewed this record for gaps yet.
ALGORITHM: Option pricing by amplitude estimation
PROBLEM: Price an option, or a portfolio of options — vanilla contracts, multi-asset contracts, and path-dependent contracts such as barrier options — on a gate-based quantum computer, in the setting where the paper takes classical Monte Carlo methods as its point of comparison.
IDEA: Stamatopoulos, Egger, Sun, Zoufal, Iten, Shen and Woerner present a methodology to price options and portfolios of options on a gate-based quantum computer using amplitude estimation, which they describe as an algorithm that provides a quadratic speedup compared to classical Monte Carlo methods. The estimator itself is the subject of this catalog's amplitude-estimation record; what this paper adds is the construction around it on the finance side, and the emphasis it puts on the implementation of the quantum circuits required to build the input states and operators that amplitude estimation needs to price the different option types. The contracts it covers are vanilla options, multi-asset options and path-dependent options such as barrier options. The paper reports simulation results that highlight how the circuits it implements price the different option contracts, and it examines the performance of those option-pricing circuits on quantum hardware using the IBM Q Tokyo quantum device. The authors also report employing a simple error-mitigation scheme that they state significantly reduces the errors arising from noisy two-qubit gates.
REPORTED COST: A quadratic speedup compared to classical Monte Carlo methods, stated in that form and in no other: the abstract attaches the speedup to amplitude estimation itself, and gives no error scaling, no query count, no qubit count and no gate count for the option-pricing circuits, so nothing here says how many samples or oracle calls pricing a given contract to a given accuracy would take, nor at what accuracy the two methods cross over.
BASIS: abstract of arXiv:1905.02666: "We present a methodology to price options and portfolios of options on a gate-based quantum computer using amplitude estimation, an algorithm which provides a quadratic speedup compared to classical Monte Carlo methods." That is the only comparative claim the abstract makes; the rest of it states results rather than bounds — simulation results, a run on the IBM Q Tokyo quantum device, and an error-mitigation scheme — and attaches no figure to any of them. The demonstration this record covers states no complexity bound of its own. Those are the only sources read for this field.
PRIMARY SOURCE: Nikitas Stamatopoulos, Daniel J. Egger, Yue Sun, Christa Zoufal, Raban Iten, Ning Shen, Stefan Woerner (2019), Option Pricing using Quantum Computers — https://arxiv.org/abs/1905.02666
This is a literature reference record, not an executable circuit.A reference record, not runnable source. Leona cannot execute it, so it cannot be saved to your Library as a circuit.
Related entries
- Quantum amplitude estimation
Algorithms
- Amplitude amplification
Algorithms
- Quantum phase estimation
Algorithms
- Quantum counting
Algorithms
References
- Option Pricing using Quantum Computers
Nikitas Stamatopoulos, Daniel J. Egger, Yue Sun, Christa Zoufal, Raban Iten, Ning Shen, Stefan Woerner · 2019
Primary source: it states the methodology for pricing options and portfolios of options on a gate-based quantum computer with amplitude estimation, the quadratic speedup over classical Monte Carlo it attributes to that estimator, the contract types covered (vanilla, multi-asset, and path-dependent such as barrier options), the simulation results, the run on the IBM Q Tokyo quantum device, and the error-mitigation scheme for noisy two-qubit gates. Consult it for the circuits that build the input states and operators, for the accuracies and qubit counts behind the simulations and the hardware run, and for the mitigation scheme itself — the abstract states none of them.