Amplitude estimation and quantum Monte Carlo integration
Algorithms and benchmarks / Estimation and Monte Carlo
Estimates an expected value, such as the price of a financial derivative or a risk measure, with fewer samples than classical Monte Carlo simulation. Here quantum Monte Carlo means this speedup of Monte Carlo integration, not the classical quantum Monte Carlo methods of physics.
- Era
- Fault-tolerant
- Problem class
- Estimation and Monte Carlo
- Speedup
- Near-quadratic over the best possible classical algorithm for estimating the mean of a bounded-variance subroutine (Montanaro 2015).
How it works
The quantity of interest is encoded in the probability of measuring a marked state. Amplitude estimation combines amplitude amplification with phase estimation to read that probability to precision epsilon with about 1/epsilon uses of the circuit, instead of about 1/epsilon squared samples.
Typical use cases
- Derivative pricing and risk measures in finance.
- Expected values in engineering and scientific simulation.
Documented use cases on related problems
From the use case directory; the sources of these cases may use other methods than this one.
Resource estimates
Figures exactly as stated by their authors, with their assumptions. They are estimates under those assumptions, not forecasts, and they tend to fall as methods improve.
| Quantity | Value as stated |
|---|---|
| Logical qubits | 8k |
| T-depth | 54 million |
| Time needed for advantage | execution in the order of a second |
Assumptions: Benchmark derivatives chosen by the authors (autocallable and Target Accrual Redemption Forward), their re-parameterisation method combining pre-trained variational circuits with fault-tolerant computation; the authors state that these requirements are out of reach of current systems.
Source: S. Chakrabarti et al. 2021, Quantum 5, 463 (2021); arXiv:2012.03819
Demonstrations
- 2020: Option pricing with amplitude estimation on the IBM Q Tokyo device, with a simple error mitigation scheme for noisy two-qubit gates. (N. Stamatopoulos et al. 2020)
Hardware platforms that fit
- Superconducting qubits: Option pricing with amplitude estimation was run on the IBM Q Tokyo superconducting device with error mitigation (Stamatopoulos et al. 2020); the derivative-pricing threshold estimate targets fault-tolerant gate-based machines.
What you need to run it
- Classical simulation
- A state-vector simulator holds 2^n complex amplitudes, so memory doubles with every qubit: about 30 qubits fit in 16 GiB at double precision. Circuits with little entanglement can go further with tensor-network methods. See the guide for the calculator.
- Noisy hardware (NISQ)
- Small proofs of concept with error mitigation; the circuit depth of full amplitude estimation is too large for noisy devices.
- Fault-tolerant machine
- For the derivative benchmarks of Chakrabarti et al. (2021): about 8,000 logical qubits, a T-depth of 54 million, and a run in about a second to beat classical Monte Carlo.
- Annealer or analogue simulator
- Not applicable for this family.
Calculate the memory of a simulation
Where to run it in Europe
Superconducting qubits
- VLQ (Czechia; Inaugurated, per the EuroHPC JU): Superconducting; digital quantum computer. Access: EuroHPC JU call for quantum access in pilot mode. Deployment milestones VLQ
- Euro-Q-Exa (Germany; Inaugurated, per the EuroHPC JU): Superconducting; digital quantum computer. Access: EuroHPC JU call for quantum access in pilot mode. Deployment milestones Euro-Q-Exa
- EuroQCS-Spain (Spain; Inaugurated, per the EuroHPC JU): Superconducting annealing; analogue quantum computer. Access: EuroHPC JU call for quantum access in pilot mode. Deployment milestones EuroQCS-Spain
- Quantum Inspire (Netherlands; In operation, per the source): Online platform with superconducting transmon QPUs, a neutral-atom emulator and simulators; a Spin-2+ electron spin QPU is listed as archived. Access: Quantum Inspire registration.
- LRZ Quantum Integration Centre (QIC) (Germany; In operation, per the source): Superconducting systems from IQM (54 qubits, up to 150 in future), a 20-qubit AQT ion-trap system, and a neutral-atom system being developed by planqc (MAQCS project). Access: LRZ quantum computing consulting.
- Finnish Quantum-Computing Infrastructure (FiQCI) (Finland; In operation, per the source): Hybrid HPC and quantum computing platform: VTT Q50 (53 qubits) and Aalto Q20 (20 qubits), superconducting systems made by IQM, used through the LUMI supercomputer environment. Access: FiQCI calls for academic projects.
- Quantum Spain quantum computer at BSC-CNS (Spain; In operation, per the source): Superconducting; digital quantum computer with a 35-qubit processor, integrated into the MareNostrum 5 supercomputer. Access: Spanish Supercomputing Network (RES) quantum access. Deployment milestones Quantum Spain quantum computer at BSC-CNS
The full infrastructure catalogue · The infrastructure deployment tracker
Software
European tools in the software directory
Other open-source frameworks
- Qiskit (IBM (United States); non-European)
A listing is not an endorsement; check each tool for the features you need.
Limitations and open questions
- A quadratic speedup only: Babbush et al. (2021) argue that such speedups are unlikely to pay off on early fault-tolerant machines.
- Loading the probability distribution into the quantum computer adds a cost that must be counted against the speedup.
In the Quantum Algorithm Zoo
- Searching (Quantum Algorithm Zoo, external site): Oracular Algorithms; speedup as the Zoo states it: Polynomial
The Zoo covers amplitude amplification and estimation within its Searching entry.
Index: the Quantum Algorithm Zoo by Stephen P. Jordan, last updated 2026-09-10, checked 2026-10-23. The portal links to it and copies no description from it. Browse the full Quantum Algorithm Zoo (external site)
Named in European procurements
- Used in: EuroHPC procurement LUMI-Q (now VLQ) (Required in the benchmark suite (scored under the award criteria))
As the tender documents state; how a benchmark is weighted against others is described in each procurement.
Sources
- G. Brassard, P. Høyer, M. Mosca, A. Tapp (2002). Quantum Amplitude Amplification and Estimation. AMS Contemporary Mathematics 305, 53 (2002); arXiv:quant-ph/0005055. Read: abstract page, retrieved 2026-10-23.
- A. Montanaro (2015). Quantum speedup of Monte Carlo methods. Proc. R. Soc. A 471, 20150301 (2015); arXiv:1504.06987. Read: abstract page, retrieved 2026-10-23.
- R. Babbush, J. McClean, M. Newman, C. Gidney, S. Boixo, H. Neven (2021). Focus beyond quadratic speedups for error-corrected quantum advantage. PRX Quantum 2, 010103 (2021); arXiv:2011.04149. Read: abstract page, retrieved 2026-10-23.
- N. Stamatopoulos, D. J. Egger, Y. Sun, C. Zoufal, R. Iten, N. Shen, S. Woerner (2020). Option Pricing using Quantum Computers. Quantum 4, 291 (2020); arXiv:1905.02666. Read: abstract page, retrieved 2026-10-23.
- S. Chakrabarti, R. Krishnakumar, G. Mazzola, N. Stamatopoulos, S. Woerner, W. J. Zeng (2021). A Threshold for Quantum Advantage in Derivative Pricing. Quantum 5, 463 (2021); arXiv:2012.03819. Read: abstract page, retrieved 2026-10-23.