_guide-zne-lindblad.rst

Zero-Noise Extrapolation

Zero-Noise Extrapolation: Lindblad dynamics

In an ideal device, the time evolution is unitary, and as is modeled in the intermediate representation of a quantum circuit,

$$\frac{d}{dt}|\psi\rangle=-\frac{i}{\hbar}H(t)|\psi\rangle,$$

where |ψ⟩ is the initial state of the system (e.g., the qubits involved in the operation) and U(t) the unitary time evolution set by a time-dependent Hamiltonian, H(t).

In the simplest scenario for the system-environment interaction, it is still possible to describe the time evolution in terms of operators acting on the system only, at the cost of losing the unitarity of the evolution.

The first required condition to develop such framework, is that the system interacts more weakly with the environment than within its own sub-constituents. This allows to proceed with a perturbative approach to solve the problem, with a coupling constant λ quantifying the magnitude of the first-order expansion terms.

In this case, it is possible to write the time evolution of the density matrix associated to the state, ρ̂ = |ψ⟩⟨ψ|, as

$$\frac{\partial }{ \partial t}\hat{\rho}= \frac{i}{\hbar}\lbrack H(t), \hat{\rho}\rbrack+\lambda \mathcal{L} \lbrack\hat{\rho}\rbrack,$$

where mathcalL is a super-operator acting on the Hilbert space.

The subsequent most straightforward set of sensible approximations includes assuming that at time zero the system and environment are not entangled, that the environment is memoryless, and that there is a dominant scale of times set by the interactions, wich allows to cut off high-frequency perturbations.

These approximations -- called the Born, Markov, and Rotating-Wave approximations, respectively --lead to a so-called Lindblad form of the dissipation, i.e. to a special structure of the system-environment interaction that can be represented with a linear superoperator that always admits the Lindblad form

$$\mathcal{L}\lbrack\hat{\rho}\rbrack=\mathcal{L}\hat{\rho} =\sum_{i=1}^{N^2-1} \gamma_i \left( A_i\hat{\rho} A_i^\dagger - \frac{1}{2}( A_i^\dagger A_i\hat{\rho}+ \hat{\rho}A_i^\dagger A_i )\right) ,$$

where γ_i are constants that set the strenghts of the dissipation mechanisms defined by the jump operators, A_i.