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Annealing/Evolution

Single system-bath coupling

This package implements the AbstractFactory pattern for a quantum evolution via a concrete type Annealing. A complete quantum evolution is assembled from the following parts:

  1. Hamiltonian: Any object that implements the AbstractHamiltonian interface
  2. Initial state: A state vector/density matrix
  3. (Optional) System bath coupling – system part
  4. (Optional) System bath coupling – bath part
  5. (Optional) System-bath interaction – InteractionSet
  6. (Optional) Additional control protocols

For example, the following code block

H = DenseHamiltonian([(s)->1-s, (s)->s], -[σx, σz]/2, unit=:ħ)
u0 = PauliVec[1][1]
coupling = ConstantCouplings(["Z"], unit=:ħ)
bath = Ohmic(1e-4, 4, 16)
annealing = Annealing(H, u0; coupling=coupling, bath=bath)

constructs a standard single-qubit annealing process with the total Hamiltonian

\[H(s) = -(1-s)\frac{X}{2} - s\frac{Z}{2} + Z \otimes B + H_B \ ,\]

where $\{B, H_B\}$ forms an Ohmic bath.

Multiple system-bath couplings

If multiple system-bath couplings are needed, they can be merged into an InteractionSet via Interactions before constructing Annealing.

coupling_1 = ConstantCouplings(["X"])
bath_1 = Ohmic(1e-4, 4, 16)
interaction_1 = Interaction(coupling_1, bath_1)

coupling_2 = ConstantCouplings(["Z"])
bath_2 = Ohmic(1e-4, 0.1, 16)
interaction_2 = Interaction(coupling_2, bath_2)

interaction_set = InteractionSet(interaction_1, interaction_2)
annealing = Annealing(H, u0, interactions=interaction_set)

The above code block builds the same single-qubit Hamiltonian coupled to two different Ohmic baths via both $\sigma_z$ and $\sigma_x$ operators

\[H(s) = -(1-s)\frac{X}{2} - s\frac{Z}{2} + X \otimes B_1 + Z \otimes B_2 + H_B \ .\]

Other interaction types

Sometimes it is impractical to directly start from the system-bath interaction Hamiltonian. For example, often the Lindblad equation is the starting point. In such cases, instead of building Interactions from AbstractCouplings and AbstractBath, the user can specify different subtypes of AbstractInteraction. For example,

lind = Lindblad(0.1, σz)

defines the Lindblad superoperator

\[\gamma \Big( L \rho L^\dagger - \frac{1}{2}\big\{L^\dagger L, \rho\big\}\Big) \ ,\]

where $L=Z$ and $\gamma=0.1$. In the above example, Lindblad is a subtype of AbstractInteraction that can be combined into an InteractionSet. To learn more about this example, please see the tutorial.