CTQW and CTQWDense

Continuous-Time Quantum Walk

Implementation of the continuous-time quantum walk. The model is defined for an arbitrary undirected graph. Hamiltonian is chosen to be adjacency, normalized Laplacian or Laplacian matrix. After the evolution, the state is measured in the canonical basis. The evolution is defined on a system of size equal to graph order. The precise definition can be found in Spatial search by quantum walk by Childs and Goldstone, where both pure walk and search dynamics are described.

The abstract supertype is AbstractCTQW with its default realization CTQW, utilizing sparse matrices. Alternative realization is CTQWDense, which works on standard matrices. The model includes the following types and methods:

Full docs

QuantumWalk.AbstractCTQW — Type.
AbstractCTQW

Abstract CTQW model. By default evolve according to Schrödinger equation and performs measurmenet by taking square of absolute values of its elements. Default representation of AbstractCTQW is CTQW.

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QuantumWalk.CTQW — Type.
CTQW(graph, matrix)

Default representation of AbstractCTQW. matrix defaults to :adjacency. The Hamiltonian is a sparse matrix.

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QuantumWalk.CTQW — Method.
CTQW(graph)

Constructor for CTQW, taking matrix to be :adjacency.

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QuantumWalk.CTQWDense — Type.
CTQWDense(graph, matrix)

Alternative representation of AbstractCTQW. matrix defaults to :adjacency. The Hamiltonian is a dense matrix.

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QuantumWalk.CTQWDense — Method.
CTQWDense(graph)

Constructor for CTQWDense, taking matrix to be :adjacency.

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QuantumWalk.QWEvolution — Method.
QWEvolution([type, ]ctqw)

Creates QWEvolution according to AbstractCTQW model. By default type equals Complex128. The hamiltonian is SparseMatrixCSC.

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QuantumWalk.QWSearch — Method.
QWSearch([type, ]ctqw, marked[, penalty, jumpingrate])

Creates QWSearch according to AbstractCTQW model. By default type equals Complex128, jumpingrate equals largest eigenvalue of adjacency matrix of graph if matrix(CTQW) outputs :adjacency and error otherwise, and penalty equals 0. The hamiltonian is SparseMatrixCSC.

QWSearch(qws_ctqw; marked, penalty)

Updates quantum walk search to new subset of marked elements and new penalty. By default marked and penalty are the same as in qws.

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QuantumWalk.check_qwdynamics — Method.
check_qwdynamics(QWSearch, ctqw, parameters, marked)

Checks whetver combination of ctqw, marked and parameters produces valid QWSearch object. It checks if parameters consists of key :hamiltonian with corresponding value being SparseMatrixCSC or Matrix. Furthermore the hamiltonian needs to be square of size equals to graph(ctqw) order. the hermiticity is not checked for efficiency issue.

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QuantumWalk.check_qwdynamics — Method.
check_qwdynamics(QWEvolution, ctqw, parameters)

Checks iof combination of ctqw and parameters produces valid QWSearch object. It checks if parameters consists of key :hamiltonian with corresponding value being SparseMatrixCSC or Matrix. Furthermore the hamiltonian needs to be square of size equals to graph(ctqw) order. The hermiticity is not checked for efficiency issues.

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QuantumWalk.evolve — Method.
evolve(qwd_ctqw, state, runtime)

Returnes new state creates by evolving state by parameters(qwd_ctqw)[:hamiltonian] for time runtime according to Schrödinger equation.

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QuantumWalk.initial_state — Method.
initial_state(qws_ctqw)

Returns equal superposition of size size and type of parameters(qws_ctqw)[:hamiltonian].

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QuantumWalk.matrix — Method.
matrix(ctqw)

Returns the matrix symbol defining matrix graph used.

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QuantumWalk.measure — Method.
measure(qwd_ctqw, state[, vertices])

Returns the probability of measuring each vertex from vertices from state according to qwd_abstr_ctqw model. If vertices is not provided, full measurement is made. The measurement is done by taking square of absolute value of all elements of state.

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