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| 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.[1][2][3]
Background
In isolated systems, the quantum state \(\psi\) evolves according to a time-dependent Hamiltonian \(H(t)\):
![{\displaystyle U(t)={\mathcal {T}}\exp \!\left[-{\frac {i}{\hbar }}\int _{0}^{t}H(t')\,dt'\right]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4af4c945f4e9f0028b34bae5b2fcbecf1ab5bf79)
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.[1][2]
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 and relative velocity :
![{\displaystyle 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]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/cdf92575ad14950f2bb4321d70eec00ceae2d25b)
where is the system Hamiltonian, is a coupling constant, and is the momentum operator.
- Lindblad equation (Markovian noise):* For weak, memoryless noise processes, the evolution of the density matrix is described by:
![{\displaystyle {\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)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ead9a07ecb2cffceb1cf1cbeb42d19d3dded64c1)
where represent specific noise processes such as dephasing. The commutator is and the anticommutator is .[4][5]
![{\displaystyle [A,B]=AB-BA}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4a3b93b316dd0b6b0ab2c71e486c901ddfe6e79a)
- Redfield equation (non-Markovian noise):* For systems with memory effects, the Redfield equation is used:
![{\displaystyle {\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]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/02efaf220c5b5482437a9bfc0bd6edf100bea62d)
where is the interaction Hamiltonian and is the environment density matrix.
- Collisional decoherence:* Collisions with environmental particles lead to loss of spatial coherence:
![{\displaystyle C(t)\approx \exp[-\Gamma t],\quad \Gamma \propto n\,v\,\sigma _{\rm {decoh}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0add0c48bcfda453cd8d6e10a014456e99a72546)
where 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.[1][2][3]
See also
References
- ^ a b c Breuer, Heinz-Peter; Petruccione, Francesco (2002). The Theory of Open Quantum Systems. Oxford University Press. ISBN 978-0199213900.
- ^ a b c Rivas, Ángel; Huelga, Susana F. (2012). Open Quantum Systems: An Introduction. Springer Briefs in Physics. Springer. doi:10.1007/978-3-642-23354-8.
- ^ a b Gneiting, Clemens; Nori, Franco (2017). "Quantum evolution in open systems: Master equations and dynamical maps". Journal of Statistical Physics. 168 (6): 1223–1240. doi:10.1007/s10955-017-1901-0.
- ^ Lindblad, Göran (1976). "On the generators of quantum dynamical semigroups". Communications in Mathematical Physics. 48 (2): 119–130. doi:10.1007/BF01608499.
- ^ Gorini, Vittorio; Kossakowski, Andrzej; Sudarshan, E. C. G. (1976). "Completely positive dynamical semigroups of N-level systems". Journal of Mathematical Physics. 17: 821–825. doi:10.1063/1.522979.
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.
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