This resource function is obsolete. Use the paclet Wolfram/QuantumFramework instead.

Function Repository Resource:

QuantumSpectralDecomposition

Source Notebook

Perform a spectral decomposition (diagonalization) on a quantum state or operator

Contributed by: Jonathan Gorard and Ruhi Shah

ResourceFunction["QuantumSpectralDecomposition"][QuantumDiscreteState[…]]

performs a spectral decomposition on the specified QuantumDiscreteState.

ResourceFunction["QuantumSpectralDecomposition"][QuantumDiscreteOperator[…]]

performs a spectral decomposition on the specified QuantumDiscreteOperator.

ResourceFunction["QuantumSpectralDecomposition"][QuantumMeasurementOperator[…]]

performs a spectral decomposition on the specified QuantumMeasurementOperator.

ResourceFunction["QuantumSpectralDecomposition"][QuantumHamiltonianOperator[…]]

performs a spectral decomposition on the specified QuantumHamiltonianOperator.

ResourceFunction["QuantumSpectralDecomposition"][QuantumCircuitOperator[…]]

performs a spectral decomposition on the specified QuantumCircuitOperator, represented as a QuantumDiscreteOperator.

Details

The spectral decomposition in linear algebra and functional analysis represents a Hermitian/normal matrix (or linear operator) as a diagonal matrix/operator in some basis. In the context of quantum information theory, it diagonalizes a density matrix, evolution operator, measurement operator, Hamiltonian operator or circuit operator by introducing an appropriate orthonormal basis.

ResourceFunction["QuantumSpectralDecomposition"] will return a QuantumDiscreteState object if the input was a QuantumDiscreteState object, a QuantumDiscreteOperator object if the input was a QuantumDiscreteOperator object, etc. The only exception to this rule is the QuantumCircuitOperator object, whose spectral decomposition is always a QuantumDiscreteOperator object (i.e. ResourceFunction["QuantumSpectralDecomposition"] always returns the operator representation of the resulting circuit).

ResourceFunction["QuantumSpectralDecomposition"] is defined for projection-valued QuantumMeasurementOperator objects, but not for positive operator-valued QuantumMeasurementOperator objects.

The basis elements in the resulting orthonormal basis are named s₁,s₂,….

Examples

Basic Examples (5)

Create a two-qubit pure discrete quantum state in the computational basis (default):

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Perform a spectral decomposition of the state, resulting in a pure state with a diagonalized density matrix and a new (spectral-decomposed) basis:

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Show the basis elements of the new (spectral-decomposed) basis:

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Perform a spectral decomposition of a single-qubit mixed discrete quantum state in the computational basis instead, resulting in a mixed state with a diagonalized density matrix:

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Show the new (spectral-decomposed) basis:

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Perform a spectral decomposition of an arity-2 QuantumDiscreteOperator object, resulting in a QuantumDiscreteOperator object with a diagonalized matrix representation:

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operator = ResourceFunction["QuantumSpectralDecomposition"][
ResourceFunction["QuantumDiscreteOperator"]["Fourier", {1, 2}]];
operator["Operator"]

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Show the new (spectral-decomposed) basis:

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Perform a spectral decomposition of an arity-3 projection-valued QuantumMeasurementOperator object, resulting in a projection-valued QuantumMeasurementOperator object with a diagonalized matrix representation:

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measurement = ResourceFunction["QuantumSpectralDecomposition"][
ResourceFunction["QuantumMeasurementOperator"][
"FourierBasis", {1, 2, 3}]];
measurement["MatrixRepresentation"]

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Show the new (spectral decomposed) basis:

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Perform a spectral decomposition on an arity-1 QuantumHamiltonianOperator object, resulting in a QuantumHamiltonianOperator object with a diagonalized matrix representation:

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$hamiltonian = ResourceFunction["QuantumSpectralDecomposition"][ ResourceFunction[ "QuantumHamiltonianOperator"][{{1 + \[FormalT]^2, 1 - \[FormalT]^2}, {1 - \[FormalT]^2, 1 + \[FormalT]^2}}]]; hamiltonian["Operator"]$

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Show the new (spectral-decomposed) basis:

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Perform a spectral decomposition of an arity-2 QuantumCircuitOperator object, resulting in a QuantumDiscreteOperator object with a diagonalized matrix representation:

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