Attested & literatureAlgorithmsQuantum differential equations · linear2026-08-12

Quantum algorithm for simulating the wave equation

Simulate the wave equation under Dirichlet and Neumann boundary conditions on a quantum computer, using Hamiltonian simulation and quantum linear system algorithms as subroutines.

Product formula Linear systems

wave equationhamiltonian simulationlinear systemsboundary conditionslaplacian discretization

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

Simulate the wave equation under Dirichlet and Neumann boundary conditions on a quantum computer, using Hamiltonian simulation and quantum linear system algorithms as subroutines. Costa, Jordan and Ostrander present a quantum algorithm for simulating the wave equation under Dirichlet and Neumann boundary conditions. The algorithm uses Hamiltonian simulation and quantum linear system algorithms as subroutines, and relies on factorizations of discretized Laplacian operators. The authors state that this factorization allows for improved scaling in truncation errors and improved scaling for state preparation, relative to general-purpose linear differential-equation algorithms; the abstract gives this comparison in qualitative terms only, without a complexity expression for either side. The authors also state that they consider using the same Hamiltonian-simulation approach for Klein-Gordon equations and Maxwell's equations — the abstract goes no further than naming this as something considered, with no algorithm described and no result reported for either equation.

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:

Graph-Laplacian finite differences Method
Takes The wave equation on a region, a lattice spacing, and Dirichlet or Neumann conditions on the boundary — including the boundary of a scatterer, which is modelled as removed lattice points. Returns A Hermitian generator on the direct sum of the vertex and edge spaces, whose square is the discrete Laplacian, together with the truncation error of the stencil that built it.
Solve a linear ODE du/dt = A(t)u + b(t) Slot
Takes A block-encoding of $A(t)$ with a normalization $\alpha_A \ge \max_t \lVert A(t)\rVert$ , preparation unitaries for $u_0$ and $b$ , the evolution time $T$ , and an error tolerance $\varepsilon$ . Returns A state proportional to $u(T)$ , or a history state, together with separately stated matrix-query and initial-state-query complexity.

How it works

Costa, Jordan and Ostrander present a quantum algorithm for simulating the wave equation under Dirichlet and Neumann boundary conditions. The algorithm uses Hamiltonian simulation and quantum linear system algorithms as subroutines, and relies on factorizations of discretized Laplacian operators. The authors state that this factorization allows for improved scaling in truncation errors and improved scaling for state preparation, relative to general-purpose linear differential-equation algorithms; the abstract gives this comparison in qualitative terms only, without a complexity expression for either side. The authors also state that they consider using the same Hamiltonian-simulation approach for Klein-Gordon equations and Maxwell's equations — the abstract goes no further than naming this as something considered, with no algorithm described and no result reported for either equation. The Classiq library carries this subject under applications · physical_systems. Reported cost: Improved scaling in truncation error and improved scaling for the cost of state preparation, relative to general-purpose linear differential-equation algorithms, for the wave-equation algorithm this abstract presents. The abstract states this comparison only in those qualitative terms — no complexity expression, exponent, or constant is given for the wave-equation algorithm or for the general-purpose algorithms it is compared against, and no bound at all is given for the separately mentioned, merely considered use of the same approach for Klein-Gordon or Maxwell's equations..

Implementation

FrameworkUnsupported

wave-equation-simulation.txt

ALGORITHM: Quantum algorithm for simulating the wave equation
PROBLEM: Simulate the wave equation under Dirichlet and Neumann boundary conditions on a quantum computer, using Hamiltonian simulation and quantum linear system algorithms as subroutines.
IDEA: Costa, Jordan and Ostrander present a quantum algorithm for simulating the wave equation under Dirichlet and Neumann boundary conditions. The algorithm uses Hamiltonian simulation and quantum linear system algorithms as subroutines, and relies on factorizations of discretized Laplacian operators. The authors state that this factorization allows for improved scaling in truncation errors and improved scaling for state preparation, relative to general-purpose linear differential-equation algorithms; the abstract gives this comparison in qualitative terms only, without a complexity expression for either side. The authors also state that they consider using the same Hamiltonian-simulation approach for Klein-Gordon equations and Maxwell's equations — the abstract goes no further than naming this as something considered, with no algorithm described and no result reported for either equation.
REPORTED COST: Improved scaling in truncation error and improved scaling for the cost of state preparation, relative to general-purpose linear differential-equation algorithms, for the wave-equation algorithm this abstract presents. The abstract states this comparison only in those qualitative terms — no complexity expression, exponent, or constant is given for the wave-equation algorithm or for the general-purpose algorithms it is compared against, and no bound at all is given for the separately mentioned, merely considered use of the same approach for Klein-Gordon or Maxwell's equations.
BASIS: The abstract of arXiv:1711.05394 states no big-O expression, qubit count, or gate count. Its complexity claim is comparative and qualitative: "It relies on factorizations of discretized Laplacian operators to allow for improved scaling in truncation errors and improved scaling for state preparation relative to general purpose linear differential equation algorithms." The abstract also names the subroutines the algorithm is built from — "The algorithm uses Hamiltonian simulation and quantum linear system algorithms as subroutines" — without quoting a cost for either. On the further, separately mentioned use of the same approach, the abstract states only "We also consider using Hamiltonian simulation for Klein-Gordon equations" and, in the same sentence, Maxwell's equations — naming both as something considered, with no algorithm described and no result or bound given for either. The Classiq index entry this record covers, applications/physical_systems/maxwell_equation, gives a directory path and a file list (maxwell_2d_simulation.ipynb) and states no bound. Those are the only sources read for this field.
DEMONSTRATED BY: the Classiq library entry applications/physical_systems/maxwell_equation
PRIMARY SOURCE: Pedro C.S. Costa, Stephen Jordan, Aaron Ostrander (2017), Quantum Algorithm for Simulating the Wave Equation — https://arxiv.org/abs/1711.05394

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 Quantum differential equations · linear 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

Quantum Algorithm for Simulating the Wave Equation2017 · Pedro C.S. Costa, Stephen Jordan, Aaron Ostrander

Primary source: it presents the wave-equation algorithm built from Hamiltonian simulation and quantum linear system subroutines, states that factoring the discretized Laplacian improves the scaling of truncation error and of state preparation relative to general-purpose linear differential-equation algorithms, and states that Klein-Gordon equations and Maxwell's equations are considered as further uses of the same Hamiltonian-simulation approach. Consult it for the complexity expressions themselves, for the wave-equation algorithm's qubit and gate counts, and for whatever it says about Klein-Gordon and Maxwell's equations beyond the sentence the abstract gives.

arxiv.org/abs/1711.05394 ↗

Verification

Attested & literature

Peer-reviewed paper

Method: Curation only. The Classiq library publishes this subject as the directory `applications/physical_systems/maxwell_equation`, 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:1711.05394 states no big-O expression, qubit count, or gate count. Its complexity claim is comparative and qualitative: "It relies on factorizations of discretized Laplacian operators to allow for improved scaling in truncation errors and improved scaling for state preparation relative to general purpose linear differential equation algorithms." The abstract also names the subroutines the algorithm is built from — "The algorithm uses Hamiltonian simulation and quantum linear system algorithms as subroutines" — without quoting a cost for either. On the further, separately mentioned use of the same approach, the abstract states only "We also consider using Hamiltonian simulation for Klein-Gordon equations" and, in the same sentence, Maxwell's equations — naming both as something considered, with no algorithm described and no result or bound given for either. The Classiq index entry this record covers, applications/physical_systems/maxwell_equation, gives a directory path and a file list (maxwell_2d_simulation.ipynb) 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, no wave-equation instance was solved, and no timing or resource figure was produced here; the record documents the algorithm the abstract states and does not reproduce or verify any published demonstration of it. The index entry read for this record is at applications/physical_systems/maxwell_equation and gives that directory path and a file list, maxwell_2d_simulation.ipynb, nothing about what the file contains. This Classiq directory pairs with a paper whose subject, on its own abstract, is the wave equation, not Maxwell's equations: the abstract's only sentence naming Maxwell's equations is that the authors also consider using Hamiltonian simulation for Klein-Gordon equations and Maxwell's equations, with no algorithm described, no complexity stated, and no result reported for either equation. This record accordingly documents the wave-equation algorithm and treats the Maxwell connection as exactly that tentative — a stated intention to consider the approach, not a demonstrated algorithm for Maxwell's equations — and it should not be read as a record of a quantum algorithm for Maxwell's equations. The improved-scaling claim that is stated concretely applies to the wave-equation algorithm only, is qualitative rather than quantified — no exponent, no constant, no explicit dependence on the error tolerance or the discretization is given — and is a comparison against general-purpose linear differential-equation algorithms as a class, which is a comparison against a stated point of reference rather than a proven lower bound on that class. The abstract states no experiment, simulation run, or hardware result for the wave-equation algorithm, so none is carried here, and nothing here supports a claim about state preparation cost, readout, or hardware feasibility beyond the qualitative improved scaling for state preparation the abstract itself gives.

Facts

Record type: Literature reference
Classiq library entry: maxwell_equation
Reported cost: Improved scaling in truncation error and improved scaling for the cost of state preparation, relative to general-purpose linear differential-equation algorithms, for the wave-equation algorithm this abstract presents. The abstract states this comparison only in those qualitative terms — no complexity expression, exponent, or constant is given for the wave-equation algorithm or for the general-purpose algorithms it is compared against, and no bound at all is given for the separately mentioned, merely considered use of the same approach for Klein-Gordon or Maxwell's equations.
Classiq publication directory: applications/physical_systems/maxwell_equation
Classiq shelf: applications · physical_systems
Complexity basis: The abstract of arXiv:1711.05394 states no big-O expression, qubit count, or gate count. Its complexity claim is comparative and qualitative: "It relies on factorizations of discretized Laplacian operators to allow for improved scaling in truncation errors and improved scaling for state preparation relative to general purpose linear differential equation algorithms." The abstract also names the subroutines the algorithm is built from — "The algorithm uses Hamiltonian simulation and quantum linear system algorithms as subroutines" — without quoting a cost for either. On the further, separately mentioned use of the same approach, the abstract states only "We also consider using Hamiltonian simulation for Klein-Gordon equations" and, in the same sentence, Maxwell's equations — naming both as something considered, with no algorithm described and no result or bound given for either. The Classiq index entry this record covers, applications/physical_systems/maxwell_equation, gives a directory path and a file list (maxwell_2d_simulation.ipynb) and states no bound. Those are the only sources read for this field.
Circuit: Not supplied

Source

Quantum Algorithm for Simulating the Wave Equation ↗

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