Attested & literatureAlgorithmsHamiltonian simulation · model systems2026-08-12

Quantum simulation of the kicked rotator model

quantum chaoskicked rotatorlocalizationanderson transitionexponential speedup

Atlas stars stay in the public catalog. Saving this entry to your workspace starts an unstarred private copy.

Simulate the quantum kicked rotator model — used to study quantum chaos, localization and the Anderson transition — with a quantum algorithm that scales better than classical simulation of the same model. Georgeot and Shepelyansky present a quantum algorithm that simulates the quantum kicked rotator model exponentially faster than classical algorithms. They state that this result shows that important physical problems of quantum chaos, localization and the Anderson transition can be modelled efficiently on a quantum computer. They also report a second, related result: a similar algorithm simulates efficiently classical chaos in certain area-preserving maps. The abstract states both results at the level of an asymptotic comparison — exponentially faster, and efficiently — without describing how either algorithm is built, what resource it is measured in, or what base or regime the exponential or the efficient scaling holds over; those specifics, if given at all, are in the paper and not in the abstract read for this record.

Circuit & simulation

problem

encodetransform

algorithm

transformmeasure

readout

measure

What this takes and returns

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 →

Where this sits

This record is named by the layer graph at:

Simulate Hamiltonian evolution Slot
Takes An access model for $H$ — a sum of efficiently exponentiable terms, sparse-access oracles, or a block-encoding — plus an evolution time $t$ and a target error $\varepsilon$ . Returns A circuit approximating $e^{-iHt}$ to within $\varepsilon$ , with a stated query or gate count, an ancilla count, and the norm parameter — sparsity times $\lVert H\rVert_{\max}$ , or the LCU 1-norm — that the cost is measured against.

How it works

Georgeot and Shepelyansky present a quantum algorithm that simulates the quantum kicked rotator model exponentially faster than classical algorithms. They state that this result shows that important physical problems of quantum chaos, localization and the Anderson transition can be modelled efficiently on a quantum computer. They also report a second, related result: a similar algorithm simulates efficiently classical chaos in certain area-preserving maps. The abstract states both results at the level of an asymptotic comparison — exponentially faster, and efficiently — without describing how either algorithm is built, what resource it is measured in, or what base or regime the exponential or the efficient scaling holds over; those specifics, if given at all, are in the paper and not in the abstract read for this record. The Classiq library carries this subject under applications · physical_systems. Reported cost: Exponentially faster than classical algorithms, for simulating the quantum kicked rotator model — a statement about the cost of simulating that one dynamical system, not a bound on solving a decision, search, or optimization problem. The abstract gives no big-O expression, no base for the exponential, no qubit or gate count, and no error dependence; a second, separate claim — that a similar algorithm simulates classical chaos in certain area-preserving maps efficiently — is likewise unquantified..

Implementation

FrameworkUnsupported

quantum-kicked-rotator-simulation.txt

ALGORITHM: Quantum simulation of the kicked rotator model
PROBLEM: Simulate the quantum kicked rotator model — used to study quantum chaos, localization and the Anderson transition — with a quantum algorithm that scales better than classical simulation of the same model.
IDEA: Georgeot and Shepelyansky present a quantum algorithm that simulates the quantum kicked rotator model exponentially faster than classical algorithms. They state that this result shows that important physical problems of quantum chaos, localization and the Anderson transition can be modelled efficiently on a quantum computer. They also report a second, related result: a similar algorithm simulates efficiently classical chaos in certain area-preserving maps. The abstract states both results at the level of an asymptotic comparison — exponentially faster, and efficiently — without describing how either algorithm is built, what resource it is measured in, or what base or regime the exponential or the efficient scaling holds over; those specifics, if given at all, are in the paper and not in the abstract read for this record.
REPORTED COST: Exponentially faster than classical algorithms, for simulating the quantum kicked rotator model — a statement about the cost of simulating that one dynamical system, not a bound on solving a decision, search, or optimization problem. The abstract gives no big-O expression, no base for the exponential, no qubit or gate count, and no error dependence; a second, separate claim — that a similar algorithm simulates classical chaos in certain area-preserving maps efficiently — is likewise unquantified.
BASIS: The abstract of arXiv:quant-ph/0010005 states no big-O expression, qubit count, or gate count. Its cost claim is: "We present a quantum algorithm which simulates the quantum kicked rotator model exponentially faster than classical algorithms." That is a claim about simulating a model, not about solving a decision or optimization problem, and the abstract does not say faster in which resource or with what base. The abstract separately states "important physical problems of quantum chaos, localization and Anderson transition can be modelled efficiently on a quantum computer", which names the problems the simulation bears on without attaching a bound to any of them, and "We also show that a similar algorithm simulates efficiently classical chaos in certain area-preserving maps", whose efficiently is likewise left undefined. The Classiq index entry this record covers, applications/physical_systems/quantum_chaos, gives a directory path and a file list and states no bound. Those are the only sources read for this field.
DEMONSTRATED BY: the Classiq library entry applications/physical_systems/quantum_chaos
PRIMARY SOURCE: B. Georgeot, D. L. Shepelyansky (2000), Exponential Gain in Quantum Computing of Quantum Chaos and Localization — https://arxiv.org/abs/quant-ph/0010005

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.

Quantum vs classical

Classical baseline

Compare Hamiltonian simulation · model systems 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.

Declared gaps

Nobody has reviewed this record for gaps yet.

Literature & references

Exponential Gain in Quantum Computing of Quantum Chaos and Localization2000 · B. Georgeot, D. L. Shepelyansky

Primary source: it presents the quantum algorithm that simulates the quantum kicked rotator model exponentially faster than classical algorithms, states that this makes quantum chaos, localization and the Anderson transition efficiently modellable on a quantum computer, and reports a related algorithm for efficiently simulating classical chaos in certain area-preserving maps. Consult it for how either algorithm is constructed, what resource the speedup is measured in, and any qubit, gate or error bound, none of which the abstract states.

arxiv.org/abs/quant-ph/0010005 ↗

Verification

Attested & literature

Peer-reviewed paper

Method: Curation only. The Classiq library publishes this subject as the directory `applications/physical_systems/quantum_chaos`, pinned in scripts/classiq-parity/classiq-index.json; this record documents the algorithm that entry demonstrates, not the demonstration. The primary reference's title, authors and submission year were read from its arXiv abs page, and the complexity claim taken from a clause of that paper's abstract (The abstract of arXiv:quant-ph/0010005 states no big-O expression, qubit count, or gate count. Its cost claim is: "We present a quantum algorithm which simulates the quantum kicked rotator model exponentially faster than classical algorithms." That is a claim about simulating a model, not about solving a decision or optimization problem, and the abstract does not say faster in which resource or with what base. The abstract separately states "important physical problems of quantum chaos, localization and Anderson transition can be modelled efficiently on a quantum computer", which names the problems the simulation bears on without attaching a bound to any of them, and "We also show that a similar algorithm simulates efficiently classical chaos in certain area-preserving maps", whose efficiently is likewise left undefined. The Classiq index entry this record covers, applications/physical_systems/quantum_chaos, gives a directory path and a file list and states no bound. Those are the only sources read for this field.). No circuit was constructed, simulated or run, and no Qmod model was executed.
Result: Pass · the record's problem statement and primary reference agree with the sources named above.
Caveat: This is a literature record. No circuit was built, compiled, simulated or run for it, and no instance of the kicked rotator model was computed here; the record documents the algorithm as its abstract states it and does not reproduce or verify any published demonstration of it. The index entry read for this record gives only a directory path, applications/physical_systems/quantum_chaos, and a file list, nothing about what those files contain. The exponential speedup the abstract states is for simulating one specific dynamical system, the quantum kicked rotator model, against unnamed classical algorithms for the same task; it is not a claim about solving quantum chaos, localization or the Anderson transition as decision or optimization problems, and the abstract states no base for the exponential, no error dependence, no qubit or gate count, and no regime of validity. What an exponentially faster simulation of a model gives a reader is the ability to compute the model's own dynamics — the same quantities a classical simulation of the kicked rotator would compute, only faster in an unspecified resource — and nothing more: it does not by itself establish a proof about the underlying physics of chaos, localization or the Anderson transition, and the abstract's claim that these can be modelled efficiently is the authors' inference from having an efficient simulator, not a separate, independently bounded result about any of the three phenomena. The second finding, that a similar algorithm efficiently simulates classical chaos in certain area-preserving maps, is stated with the same lack of quantitative detail — no maps are named, no error or resource bound is given, and efficiently is not defined in the abstract. The abstract reports no numerical experiment, no simulation run and no hardware result of the paper's own, so none is carried here, and no comparison against a specific classical algorithm's proven complexity is made — the comparison is against classical algorithms as a class, which is a comparison against a general reference point rather than a proven lower bound for any one classical method.

Facts

Record type: Literature reference
Classiq library entry: quantum_chaos
Reported cost: Exponentially faster than classical algorithms, for simulating the quantum kicked rotator model — a statement about the cost of simulating that one dynamical system, not a bound on solving a decision, search, or optimization problem. The abstract gives no big-O expression, no base for the exponential, no qubit or gate count, and no error dependence; a second, separate claim — that a similar algorithm simulates classical chaos in certain area-preserving maps efficiently — is likewise unquantified.
Classiq publication directory: applications/physical_systems/quantum_chaos
Classiq shelf: applications · physical_systems
Complexity basis: The abstract of arXiv:quant-ph/0010005 states no big-O expression, qubit count, or gate count. Its cost claim is: "We present a quantum algorithm which simulates the quantum kicked rotator model exponentially faster than classical algorithms." That is a claim about simulating a model, not about solving a decision or optimization problem, and the abstract does not say faster in which resource or with what base. The abstract separately states "important physical problems of quantum chaos, localization and Anderson transition can be modelled efficiently on a quantum computer", which names the problems the simulation bears on without attaching a bound to any of them, and "We also show that a similar algorithm simulates efficiently classical chaos in certain area-preserving maps", whose efficiently is likewise left undefined. The Classiq index entry this record covers, applications/physical_systems/quantum_chaos, gives a directory path and a file list and states no bound. Those are the only sources read for this field.
Circuit: Not supplied

Source

Exponential Gain in Quantum Computing of Quantum Chaos and Localization ↗

Kind: curated reference
Reviewed by: Leona Quantum curation pass
License: Citation metadata only; source publication terms apply