TwoQubitWeylChamber

Perfect-entanglers distance measure.

D = D_PE(U; unitarity_weight=0.0, absolute_square=false)

For a given two-qubit gate $Û$, this is defined via the local_invariants $g_1$, $g_2$, $g_3$ as

\[D = g_3 \sqrt{g_1^2 + g_2^2} - g_1\]

This describes the geometric distance of the quantum gate from the polyhedron of perfect entanglers in the Weyl chamber.

This equation is only meaningful under the assumption that $Û$ is unitary. If the two-qubit level are a logical subspace embedded in a larger physical Hilbert space, loss of population from the logical subspace may lead to a non-unitary $Û$. In this case, the unitarity measure can be added to the functional by giving a unitary_weight ∈ [0, 1) that specifies the relative proportion of the $D$ term and the unitarity term.

By specifying absolute_square=true, the functional is modified as $D → |D|²$, optimizing specifically for the boundary of the perfect entanglers polyhedron. This accounts for the fact that $D$ can take negative values inside the polyhedron, as well as the W1 region of the Weyl chamber (the one adjacent to SWAP). This may be especially useful in a system with population loss (unitarity_weight > 0), as it avoids situations where the optimization goes deeper into the perfect entanglers while increasing population loss.

Construct the canonical gate for the given Weyl chamber coordinates.

Û = canonical_gate(c₁, c₂, c₃)

constructs the two qubit gate $Û$ as

\[Û = \exp\left[i\frac{π}{2} (c_1 σ̂_x σ̂_x + c_2 σ̂_y σ̂_y + c_3 σ̂_z σ̂_z)\right]\]

where $σ̂_{x,y,z}$ are the Pauli matrices.

Calculate the maximum gate concurrence.

C = gate_concurrence(U)
C = gate_concurrence(c₁, c₂, c₃)

calculates that maximum concurrence $C ∈ [0, 1]$ that the two two-qubit gate U, respectively the gate described by the Weyl chamber coordinates c₁, c₂, c₃ (see weyl_chamber_coordinates) can generate.

Reference

• [1] Kraus and Cirac, Phys. Rev. A 63, 062309 (2001)

Check whether a given gate is in (a specific region of) the Weyl chamber.

in_weyl_chamber(c₁, c₂, c₃)

checks whether c₁, c₂, c₃ are valid Weyl chamber coordinates.

in_weyl_chamber(U; region="PE")
in_weyl_chamber(c₁, c₂, c₃; region="PE")

checks whether the two-qubit gate U, respectively the gate described by the Weyl chamber coordinates c₁, c₂, c₃ (see weyl_chamber_coordinates) is a perfect entangler. The region can be any other of the regions returned by weyl_chamber_region.

Calculate the local invariants g₁, g₂, g₃ for a two-qubit gate.

g₁, g₂, g₃ = local_invariants(U)

Unitarity of a matrix.

pop_loss = 1 - unitarity(U)

measures the loss of population from the subspace described by U. E.g., for a two-qubit gate, U is a 4×4 matrix. The unitarity is defined as $\Re[\tr(Û^†Û) / N]$ where $N$ is the dimension of $Û$.

Calculate the Weyl chamber coordinates c₁, c₂, c₃ for a two-qubit gate.

c₁, c₂, c₃ = weyl_chamber_coordinates(U)

calculates the Weyl chamber coordinates using the algorithm described in Childs et al. [2].

Reference

• [2] Childs et al.. Phys. Rev. A 68, 052311 (2003)

Identify which region of the Weyl chamber a given gate is located in.

region = weyl_chamber_region(U)
region = weyl_chamber_region(c₁, c₂, c₃)

identifies which region of the Weyl chamber the given two-qubit gate U, respectively the gate identified by the Weyl chamber coordinates c₁, c₂, c₃ (see weyl_chamber_coordinates) is in, as a string. Possible outputs are:

• "PE": gate is in the polyhedron of perfect entanglers.
• "W0": gate is between the identity and the perfect entanglers.
• "W0*": gate is between CPHASE(2π) and the perfect entanglers.
• "W1": gate is between SWAP and the perfect entanglers.

For invalid Weyl chamber coordinates, an empty string is returned.