User:Harold Foppele/Draft:Open-system quantum computing

Open-system formulations
Field	Quantum computing
Applications	Noise modeling, Decoherence, Error correction
Related topics	Open quantum system, Quantum decoherence

Open-system formulations in quantum computing are theoretical methods used to describe how quantum computers interact with their environment, including noise and decoherence. These approaches provide tools to model environmental effects, predict errors, and support error correction strategies using frameworks such as the Lindblad equation and the Redfield equation equations.

Background

In isolated systems, the quantum state \(\psi\) evolves according to a time-dependent Hamiltonian \(H(t)\):

$U(t)={\mathcal {T}}\exp \!\left[-{\frac {i}{\hbar }}\int _{0}^{t}H(t')\,dt'\right]$

where \(\mathcal{T}\) is the time-ordering operator, \(\hbar\) is the reduced Planck constant, and \(U(t)\) is the time evolution.

Real devices are never perfectly isolated; interactions with particles, phonons, photons, and control electronics affect qubit. Open-system methods capture these effects via quantum channels or environment-dependent modifications to the generator of the time evolution.

Formulations

Open-system evolution can be modeled in several ways depending on the type of noise and system-environment coupling.

• Environment-dependent generators:* Some models explicitly include external conditions such as particle density $n$ and relative velocity $v$:

$U(t;n,v)={\mathcal {T}}\exp \!\left[-{\frac {i}{\hbar }}\int _{0}^{t}{\big (}H_{S}+\alpha \,n(t')+\mathbf {v} (t')\cdot \mathbf {P} {\big )}\,dt'\right]$

where $H_{S}$ is the system Hamiltonian, $\alpha$ is a coupling constant, and $\mathbf {P}$ is the momentum operator.

• Lindblad equation (Markovian noise):* For weak, memoryless noise processes, the evolution of the density matrix $\rho$ is described by:

${\frac {d\rho }{dt}}=-{\frac {i}{\hbar }}[H,\rho ]+\sum _{k}\left(L_{k}\rho L_{k}^{\dagger }-{\frac {1}{2}}\{L_{k}^{\dagger }L_{k},\rho \}\right)$

where $L_{k}$ represent specific noise processes such as dephasing. The commutator is $[A,B]=AB-BA$ and the anticommutator is $\{A,B\}=AB+BA$.

• Redfield equation (non-Markovian noise):* For systems with memory effects, the Redfield equation is used:

${\frac {d\rho (t)}{dt}}=-{\frac {i}{\hbar }}[H_{S},\rho (t)]-\int _{0}^{t}d\tau \,\operatorname {Tr} _{E}\left[H_{I}(t),[H_{I}(\tau ),\rho (t)\otimes \rho _{E}]\right]$

where $H_{I}$ is the interaction Hamiltonian and $\rho _{E}$ is the environment density matrix.

• Collisional decoherence:* Collisions with environmental particles lead to loss of spatial coherence:

$C(t)\approx \exp[-\Gamma t],\quad \Gamma \propto n\,v\,\sigma _{\rm {decoh}}$

where $\sigma _{\rm {decoh}}$ is the effective cross-section for decoherence.

Relevance to quantum computing

Open-system models are used to predict errors such as dephasing and relaxation in qubit devices. The dominant noise processes depend on the type of quantum computing platform (e.g., trapped ions, neutral atom qubits, superconducting circuits), but the open-system framework provides a unified language to analyze experimental conditions.

, superconducting circuits), but the open-system framework provides a unified language to analyze experimental conditions.

See also

• Open quantum system
• Quantum decoherence
• Lindblad equation
• Quantum master equation
• Quantum noise
• Density matrix
• Hamiltonian (quantum mechanics)
• Time evolution operator

Further reading

• Breuer, H.-P.; Laine, E.-M.; Piilo, J.; Vacchini, B. (2016). "Colloquium: Non-Markovian dynamics in open quantum systems". Reviews of Modern Physics. 88 (2): 021002. doi:10.1103/RevModPhys.88.021002.{{cite journal}}: CS1 maint: article number as page number (link)