qml.labs.dla¶
Experimental dynamical Lie algebra (DLA) functionality¶
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Compute the dynamical Lie algebra g from a set of generators using their dense matrix representation. |
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Compute the structure constants that make up the adjoint representation of a Lie algebra. |
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Cartan Decomposition g=k⊕m. |
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Apply a recursive Cartan decomposition specified by a chain of decomposition types. |
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Compute a Cartan subalgebra (CSA) a⊆m. |
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Variational KaK decomposition of Hermitian |
Utility functions¶
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Transform adjoint vector representations back into operator format. |
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Decompose a batch of operators onto a given operator basis. |
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Implementation of the trace inner product ⟨A,B⟩=tr(AB)/dim(A) between two Hermitian operators A and B. |
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Orthonormalize a list of basis vectors. |
Computes the coefficients of one or multiple Hermitian matrices in the Pauli basis. |
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Decomposes a Hermitian matrix or a batch of matrices into a linear combination of Pauli operators. |
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Utility function to check if operators in |
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Helper function to check [ops1,ops2]⊆vspace. |
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Helper function to check if all operators in |
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Helper function to check the validity of a Cartan decomposition g=k⊕m. |
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Apply a |
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Helper function to validate a khk decomposition |
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Boilerplate jax optimization |
Involutions¶
A map θ:g→g from the Lie algebra g to itself is called an involution
when it fulfills θ(θ(g))=g ∀g∈g and is compatible with commutators,
[θ(g),θ(g′)]=θ([g,g′]). Involutions are used to construct a cartan_decomp()
. There are seven canonical
Cartan involutions of real simple Lie algebras (AI, AII, AIII, BDI, CI, CII, DIII
),
see Wikipedia.
In addition, there is a canonical Cartan involution for real semisimple algebras that consist of
two isomorphic simple components (ClassB
), see here.
The Even-Odd involution. |
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The Concurrence Canonical Decomposition Θ(g)=−gT as a Cartan involution function. |
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Khaneja-Glaser involution, which is a type- |
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Canonical Cartan decomposition of type AI, given by θ:x↦x∗. |
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Canonical Cartan decomposition of type AII, given by θ:x↦Y0x∗Y0. |
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Canonical Cartan decomposition of type AIII, given by θ:x↦Ip,qxIp,q. |
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Canonical Cartan decomposition of type BDI, given by θ:x↦Ip,qxIp,q. |
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Canonical Cartan decomposition of type CI, given by θ:x↦x∗. |
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Canonical Cartan decomposition of type CII, given by θ:x↦Kp,qxKp,q. |
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Canonical Cartan decomposition of type DIII, given by θ:x↦Y0xY0. |
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Canonical Cartan decomposition of class B, given by θ:x↦Y0xY0. |