Algorithms · Quantum differential equations · linear
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.
Attested & literatureUpdated
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- Record typeLiterature reference
- Reference implementationmaxwell_equation
- Reported costImproved 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.
- Papers1
A part this method uses, worked through
This page says using Hamiltonian simulation and quantum linear system algorithms as subroutines
— here is a concrete, runnable instance of it.
2 Trotter steps of a transverse-field Ising chain
4 qubits starting at |0000⟩, J=1, h=0.5, dt=0.3, 2 first-order Trotter steps.
Where it ends up
- 000075%
- 00016%
- 10006%
- 00105%
- 12 more outcomes8%
The state spreads from |0000⟩ across several nearby basis states. The exact split is a regression value from this file's own simulator, not a claim about a real material.
Open this part in StudioSimulate 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.
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 →
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..
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
- 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.
- 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 demonstration this record covers states no bound. 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 Quantum differential equations · linear with the strongest classical method for the same instance, input budget, and output metric.
Quantum claim
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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 demonstration this record covers states no bound. Those are the only sources read for this field.
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.
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 with a normalization , preparation unitaries for and , the evolution time , and an error tolerance . Returns A state proportional to , or a history state, together with separately stated matrix-query and initial-state-query complexity.
Related entries
References
- Quantum Algorithm for Simulating the Wave Equation
Pedro C.S. Costa, Stephen Jordan, Aaron Ostrander · 2017
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.