Draft:Open-system formulations in quantum computing.: Difference between revisions

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{{User:Harold_Foppele/Infobox quantum computing

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'''Open-system formulations in [[quantum computing]]''' are theoretical methodsframeworks usedfor todescribing describethe howinteraction of [[quantum computer|quantum computersqubit]] interacts with their surrounding [[Environmentenvironment (systems)|environment]], including [[Quantum noise|noise]] and [[Quantum decoherence|decoherence]]. TheseSuch approachesinteractions provide toolslead to model environmental effects, predictincluding [[Quantum errorquantum correctionnoise|errorsnoise]], and support [[Quantum errorquantum correctiondecoherence|error correctiondecoherence]]. strategiesMaster usingequation frameworksapproaches such as the [[Lindblad equation]] and the [[Redfield equation]] equationsare widely used to represent these processes mathematically.<ref name="BreuerPetruccione">{{cite book |last1=Breuer |first1=Heinz-Peter |last2=Petruccione |first2=Francesco |title=The Theory of Open Quantum Systems |publisher=Oxford University Press |year=2002 |isbn=978-0199213900}}</ref><ref name="RivasHuelga">{{cite book |last1=Rivas |first1=Ángel |last2=Huelga |first2=Susana F. |title=Open Quantum Systems: An Introduction |series=Springer Briefs in Physics |publisher=Springer |year=2012 |doi=10.1007/978-3-642-23354-8}}</ref><ref name="GneitingNori">{{cite journal |last1=Gneiting |first1=Clemens |last2=Nori |first2=Franco |title=Quantum evolution in open systems: Master equations and dynamical maps |journal=Journal of Statistical Physics |year=2017 |volume=168 |issue=6 |pages=1223–1240 |doi=10.1007/s10955-017-1901-0}}</ref>

== Background ==

In an [[isolated system|isolated systems]], the [[Quantum state|quantum state]] \(\psi\) evolves unitarily according to a [[Hamiltonian (quantum mechanics)|time-dependent Hamiltonian]] \(H(t)\):

<math>U(t)=\mathcal{T}\exp\!\left[-\frac{i}{\hbar}\int_0^t H(t')\,dt'\right]</math>

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where {{math|{T}}} is the time-ordering operator, {{math|H}} is the reduced Planck constant, and {{math|U(t)}} is the time evolution operator.

RealRealistic quantum devices are nevernot perfectly isolated;. interactionsInteractions with [[particleexternal physics|particles]],degrees of freedom—such as [[Phonon|phononsphonon]]s, [[Photon|photonsphoton]]s, and [[Control engineering|control electronics]]electronics—introduce affecterrors that alter [[qubit]] states.<ref Open-systemname="RivasHuelga" methods/> capture theseThese effects viacan be described in terms of [[Quantumquantum channel|quantum channels]]s or environment-dependent modifications to the generatorsystem’s ofeffective the time evolutiondynamics.<ref name="BreuerPetruccione" /><ref name="RivasHuelga" />

== Formulations ==

Open-systemSeveral evolutionmathematical canapproaches beexist modeledfor inmodeling severalopen-system waysdynamics, depending on the typenoise ofcharacteristics [[Quantumand noise|noise]]the and system-environmentsystem–environment coupling.

* '''Lindblad equation (Markovian noise)''': For weak and [[Markov process|memoryless]] noise, the evolution of the [[density matrix]] \(\rho\) is given by:

*'''Environment-dependent generators''':* Some models explicitly include external conditions such as [[Number density|particle density]] <math>n</math> and relative velocity <math>v</math>:

<math>U(t;n,v)=\mathcalfrac{Td\rho}\exp\!\left[{dt}=-\frac{i}{\hbar}[H,\int_0^t \big(H_Srho]+\alphasum_k\,nleft(t')+L_k\mathbf{v}(t')rho L_k^\cdotdagger-\mathbftfrac{1}{P2}\big){L_k^\dagger L_k,dt'\rho\}\right])</math>

where <math>L_k</math> represent specific [[Quantum noise|noise processes]] such as [[Quantum decoherence#Dephasing|dephasing]]. The commutator is <math>[A,B]=AB-BA</math> and the anticommutator is <math>\{A,B\}=AB+BA</math>.<ref name="Lindblad">{{cite journal |last=Lindblad |first=Göran |title=On the generators of quantum dynamical semigroups |journal=Communications in Mathematical Physics |year=1976 |volume=48 |issue=2 |pages=119–130 |doi=10.1007/BF01608499 |bibcode=1976CMaPh..48..119L }}</ref><ref name="GKS">{{cite journal |last1=Gorini |first1=Vittorio |last2=Kossakowski |first2=Andrzej |last3=Sudarshan |first3=E. C. G. |title=Completely positive dynamical semigroups of N-level systems |journal=Journal of Mathematical Physics |year=1976 |volume=17 |issue=5 |pages=821–825 |doi=10.1063/1.522979 |bibcode=1976JMP....17..821G }}</ref>

where <math>H_S</math> is the [[Hamiltonian (quantum mechanics)|system Hamiltonian]], <math>\alpha</math> is a coupling constant, and <math>\mathbf{P}</math> is the momentum operator.

* '''LindbladRedfield equation (non-Markovian noise)''':* For weak,environments [[Markovwith process|memoryless]]memory [[Quantum noise|noise processes]]effects, the evolution of the [[DensityRedfield matrix|density matrixequation]] <math>\rho</math> is described byused:

<math>\frac{d\rho}{dt}=-\frac{i}{\hbar}[H,\rho]+\sum_k\left(L_k\rho L_k^\dagger-\frac12\{L_k^\dagger L_k,\rho\}\right)</math>

where <math>L_k</math> represent specific [[Quantum noise|noise processes]] such as [[Quantum decoherence#Dephasing|dephasing]]. The commutator is <math>[A,B]=AB-BA</math> and the anticommutator is <math>\{A,B\}=AB+BA</math>.<ref name="Lindblad">{{cite journal |last=Lindblad |first=Göran |title=On the generators of quantum dynamical semigroups |journal=Communications in Mathematical Physics |year=1976 |volume=48 |issue=2 |pages=119–130 |doi=10.1007/BF01608499}}</ref><ref name="GKS">{{cite journal |last1=Gorini |first1=Vittorio |last2=Kossakowski |first2=Andrzej |last3=Sudarshan |first3=E. C. G. |title=Completely positive dynamical semigroups of N-level systems |journal=Journal of Mathematical Physics |year=1976 |volume=17 |pages=821–825 |doi=10.1063/1.522979}}</ref>

*'''Redfield equation (non-Markovian noise)''':* For systems with [[memory effect|memory effects]], the [[Redfield equation]] is used:

<math>\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]</math>

where <math>H_I</math> is the [[Interaction Hamiltonian|interaction Hamiltonian]] and <math>\rho_E</math> is the [[Density matrix|environment density matrix]].

* '''Collisional decoherence''':* CollisionsEnvironmental with environmental [[particle physics|particles]]collisions leadcan toreduce loss of [[Quantum decoherence|spatial coherence]]:

*'''Collisional decoherence''':* Collisions with environmental [[particle physics|particles]] lead to loss of [[Quantum decoherence|spatial coherence]]:

<math>C(t) \approx \exp[-\Gamma t], \quad \Gamma \propto n\,v\,\sigma_{\rm decoh}</math>

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== Relevance to quantum computing ==

Open-system models are usedcentral to predict [[Quantumunderstanding error correction|errors]]processes such as [[Quantumquantum decoherence#Dephasing|dephasing]] and [[Relaxationrelaxation (physics)|relaxation]] in [[qubit]] devices. The dominant [[Quantumsources of noise|noise processes]] depend on the type of [[Quantum computing|quantum computing]]underlying platform (e.g., [[Trappedtrapped ion|trapped ions]] qubits, [[Quantum computing#Neutral atoms|neutral atom qubits]], or [[Superconductingsuperconducting quantum computing|superconducting circuits]]), but the open-system framework provides a unified language to analyze [[Experimental physics|experimental conditions]].<ref name="BreuerPetruccione" /><ref name="RivasHuelga" /><ref name="GneitingNori" /> By providing a unified framework, open-system methods guide both experimental characterization and the design of [[quantum error correction|error-correction]] strategies.

== See also ==

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* [[Density matrix]]

* [[Hamiltonian (quantum mechanics)]]

* [[Time evolution operator]]

== References ==

<references />

<ref name="BreuerPetruccione">{{cite book |last1=Breuer |first1=Heinz-Peter |last2=Petruccione |first2=Francesco |title=The Theory of Open Quantum Systems |publisher=Oxford University Press |year=2002 |isbn=978-0199213900}}</ref>

<ref name="RivasHuelga">{{cite book |last1=Rivas |first1=Ángel |last2=Huelga |first2=Susana F. |title=Open Quantum Systems: An Introduction |series=Springer Briefs in Physics |publisher=Springer |year=2012 |doi=10.1007/978-3-642-23354-8 |arxiv=1104.5242 |isbn=978-3-642-23353-1 }}</ref>

<ref name="GneitingNori">{{cite journal |last1=Gneiting |first1=Clemens |last2=Nori |first2=Franco |title=Quantum evolution in open systems: Master equations and dynamical maps |journal=Journal of Statistical Physics |year=2017 |volume=168 |issue=6 |pages=1223–1240 |doi=10.1007/s10955-017-1901-0}}</ref>

<ref name="Lindblad">{{cite journal |last=Lindblad |first=Göran |title=On the generators of quantum dynamical semigroups |journal=Communications in Mathematical Physics |year=1976 |volume=48 |issue=2 |pages=119–130 |doi=10.1007/BF01608499 |bibcode=1976CMaPh..48..119L }}</ref>

<ref name="GKS">{{cite journal |last1=Gorini |first1=Vittorio |last2=Kossakowski |first2=Andrzej |last3=Sudarshan |first3=E. C. G. |title=Completely positive dynamical semigroups of N-level systems |journal=Journal of Mathematical Physics |year=1976 |volume=17 |issue=5 |pages=821–825 |doi=10.1063/1.522979 |bibcode=1976JMP....17..821G }}</ref>

</references>

== Further reading ==

* {{cite journal |last1=Breuer |first1=H.-P. |last2=Laine |first2=E.-M. |last3=Piilo |first3=J. |last4=Vacchini |first4=B. |title=Colloquium: Non-Markovian dynamics in open quantum systems |journal=Reviews of Modern Physics |year=2016 |volume=88 |issue=2 |page=021002 |doi=10.1103/RevModPhys.88.021002 |arxiv=1505.01385 |bibcode=2016RvMP...88b1002B |hdl=2434/387123 }}

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