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Repair Quantum Collection B backlink template

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{{Short description|Frameworks for modeling qubit–environment interactions}}

{{Quantum book backlink|Quantum information and computing}}
 
This resource is intended for advanced undergraduate or graduate learners in physics or quantum information science. It assumes familiarity with linear algebra, quantum mechanics, and density matrix formalism. The page serves as a self-study and research-oriented overview of noise models in quantum systems.

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[[File:Quantum noisy qubits1.jpg]]
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==Formulas for Noisy Qubits==
[[Wikipedia:Noisy intermediate-scale quantum era|Noisy qubits]] are a fundamental challenge in current Noisy Intermediate-Scale Quantum [[Wikipedia:Noisy intermediate-scale quantum computing|(NISQ)]] computers, where physical [[Wikipedia:Qubit|qubits]] are susceptible to errors from [[Wikipedia:decoherence|decoherence]], are sensitive to their environment (noisy), imperfect gate operations, and measurement noise. These errors stem from interactions with the environment and can accumulate during computations, limiting the depth and complexity of algorithms that can be successfully run. [[Wikipedia:quantum advantage|Quantum advantage]] by quantum processors containing up to 1,000 qubits.<ref name="sciencedaily.com">{{Cite web|title=Engineers demonstrate a quantum advantage|url=https://www.sciencedaily.com/releases/2021/06/210601155610.htm|access-date=2021-06-29|website=ScienceDaily|language=en}}</ref> Researchers are developing [[Wikipedia:Cirq|NISQ algorithms]] that leverage limited resources within these noise constraints and exploring new quantum materials and qubit designs to create more robust qubits for the future of fault-tolerant quantum computing.<ref name=":1">{{Cite web|title=What is Quantum Computing?|url=https://www.techspot.com/article/2280-what-is-quantum-computing/|access-date=2021-06-29|website=TechSpot|date=28 June 2021 |language=en-US}}</ref> How [[Wikipedia:qubits|qubits]] interact with their surrounding environment. Unlike isolated quantum systems, real qubits are affected by noise sources such as stray photons, phonons, or control hardware fluctuations. These interactions cause errors including [[Wikipedia:quantum decoherence|decoherence]]<ref>{{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> and relaxation that degrade computational performance.[[File:IBM Q at CES (39660636671).jpg|thumb|100px|50-Qubit]]Open system models provide mathematical tools for analyzing and mitigating these effects.<ref>{{cite book |last1=Rivas |first1=Ángel |last2=Huelga |first2=Susana F. |title=Open Quantum Systems: An Introduction |publisher=Springer |year=2012 |isbn=978-3642233531 |doi=10.1007/978-3-642-23354-8}}</ref> IBM's 50-qubit quantum computer prototype, as exhibited at CES 2018 in Las Vegas -----> They describe how methods from the theory of [[Wikipedia:Open quantum system|Open quantum system]] are applied to qubits and [[Wikipedia:Quantum annealing|quantum hardware]]. In practice, qubits are never perfectly isolated: they interact with their environments, leading to [[Wikipedia:decoherence|decoherence]], [[Wikipedia:relaxation|relaxation]], and noise that limit computation. This has made open-system tools—such as [[Wikipedia:Kraus operators|Kraus operators]], [[Wikipedia:Lindbladian|Lindblad master equations]], and [[Wikipedia:Open quantum system|non-Markovian models]]—fundamental to modern quantum computing research.[[File:SPDC figure.png |thumb|upright=0.6|Conceptual illustration of entanglement]]

==Textbooks and surveys==
treat this intersection as a distinct domain: Breuer & Petruccione’s The Theory of Open Quantum Systems (2002) and Rivas & Huelga’s Open Quantum Systems: An Introduction (2012) present explicit applications to quantum information. Reviews such as Krantz et al., A quantum engineer’s guide to superconducting qubits (2019), and Preskill, Quantum Computing in the NISQ Era and beyond (2018), emphasize that open-system models underpin both noise characterization and the definition of the NISQ regime. Recent tutorials, e.g. Li et al. (2023), treat simulation of open-system dynamics as a computational task in its own right.

As a result, open-system formulations have become central in analyzing qubit performance, setting error-correction thresholds, and guiding fault-tolerant architectures.

== Background formulas that govern noisy qubits ==
[[File:Schrödinger-Gl 04 Lösungsfunktion-Beispiel.png|thumb|Schrödinger-equation example]]
Equation used in wave mechanics (see [[Wikipedia:Quantum mechanics|Quantum mechanics]]) for the wave function of a particle is the time-independent [[Schrödinger equation]]

<math>\nabla^{2}\psi + \frac{8\pi^{2} m}{h^{2}}(E - U)\psi = 0</math>

It can also be written in operator form as:

<math>H \psi = E \psi</math>

where ψ is the wave function, ∇² the Laplace operator, ''h'' the Planck constant, ''m'' the particle's mass, ''E'' its total energy, and ''U'' its potential energy. It was devised by [[Wikipedia:Erwin Schrödinger|Erwin Schrödinger]], who was mainly responsible for wave mechanics. 
[[File:Schrödinger equation wave packet.gif|thumb|Schrödinger equation wave packet]]
The [https://phys.libretexts.org/Bookshelves/Nuclear_and_Particle_Physics/Introduction_to_Applied_Nuclear_Physics_(Cappellaro)/06%3A_Time_Evolution_in_Quantum_Mechanics/6.01%3A_Time-dependent_Schrodinger_equation#:~:text=Unitary%20Evolution,same%20information: time-dependent Schrödinger equation] (see also [[Wikipedia:Dyson series|Dyson series]]) for an isolated system is:
<math>i\hbar\frac{\partial}{\partial t}|\psi(t)\rangle = H |\psi(t)\rangle.</math> The unitary propagator is:<math display="block">U(t) = \mathcal{T}\exp\!\left(-\frac{i}{\hbar}\int_{0}^{t} H(t')\,dt'\right),</math>with <math>\mathcal{T}</math> the time-ordering operator.

For open systems, the state of the system alone is obtained from the full density matrix of system+environment:<math display="block">\rho_S(t) = \mathrm{Tr}_E\,[\rho_{SE}(t)].</math>This partial trace generally produces non-unitary dynamics.

== From microscopic models to master equations ==
[[File:Klein-graph-7-valent Hamiltonian.svg|Hamiltonian]]
Consider a system [[Wikipedia:Hamiltonian (quantum mechanics)|Hamiltonian (quantum mechanics)]] <math>H_S</math>, environment Hamiltonian <math>H_E</math>, and an interaction <math>H_I</math>. The total Hamiltonian is:<math display="block">H = H_S + H_E + H_I.</math>Even if <math>\rho_{SE}</math> evolves unitarily, the reduced density matrix <math>\rho_S</math> typically obeys an integro-differential equation. Approximations lead to different master equations.

=== Kraus representation ===
[[File:Kraus representation.jpg|thumb|110px|]]
Any completely positive trace-preserving (CPTP) map on a quantum state can be written as:<math display="block">\mathcal{E}(\rho)=\sum_{k} E_k\,\rho\,E_k^\dagger,\qquad \sum_k E_k^\dagger E_k = I,</math>where the <math>E_k</math> are [[Wikipedia:Kraus operator|Kraus operators]]. This is a general representation of open-system dynamics at discrete times.<ref>{{cite book |last1=Nielsen |first1=Michael A. |last2=Chuang |first2=Isaac L. |title=Quantum Computation and Quantum Information |publisher=Cambridge University Press |year=2010 |edition=10th anniversary |isbn=978-1107002173}}</ref>
The bit-flip channel is a quantum channel that, with probability <math>p</math>, applies a qubit flip (X-gate), and with probability <math>1 - p </math> does nothing. It is regarded as the quantum analogue of the noise through entanglement with the environment.

=== Lindblad equation (Markovian) ===
[[File:Lindblad equation (Markovian).jpg|thumb|150px|Lindblad equation (Markovian)]]
Under the Born–Markov approximation (weak coupling and short environment correlation times), the system’s density matrix satisfies the Lindblad master equation:<math display="block">\frac{d\rho}{dt} = -\frac{i}{\hbar}[H,\rho] + \sum_{k}\Big(L_k \rho L_k^\dagger - \tfrac{1}{2}\{L_k^\dagger L_k,\rho\}\Big).</math>This generator defines a dynamical semigroup (completely positive, trace-preserving evolution).<ref>{{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>

For a single qubit, collapse operators commonly model:
* '''Relaxation (energy decay):''' <math>L_{\text{relax}}=\sqrt{\gamma_1}\,\sigma_-</math>
* '''Dephasing:''' <math>L_{\text{deph}}=\sqrt{\gamma_\phi}\,\sigma_z</math>

Here <math>\gamma_1</math> and <math>\gamma_\phi</math> are the relaxation and dephasing rates, respectively.

=== Redfield equation (non-Markovian) ===
[[File:Redfield equation (non-Markovian).jpg|thumb|120px|Redfield equation (non-Markovian)]]
If the Markov approximation is not applied, the Redfield equation captures memory effects:<math display="block">\frac{d\rho_S(t)}{dt} = -\frac{i}{\hbar}[H_S,\rho_S(t)] - \int_0^t d\tau\,\mathrm{Tr}_E\big([H_I(t),[H_I(\tau),\rho_S(\tau)\otimes\rho_E]]\big).</math>Redfield theory can describe structured environments (e.g. spin baths or photonic reservoirs) but does not guarantee complete positivity without further corrections.<ref>{{Cite journal|last=Redfield|first=A.G.|date=1965 |title=The Theory of Relaxation Processes |journal=Advances in Magnetic and Optical Resonance |volume=1 |pages=1–32 |doi=10.1016/B978-1-4832-3114-3.50007-6|issn=1057-2732|isbn=978-1-4832-3114-3}}</ref>

=== Collisional decoherence ===
[[File:Cavity Dephasing Simulation.png|thumb|Cavity loses coherence due to dephasing]]
Spin-exchange collisions between alkali metal atoms can change the hyperfine state of the atoms while preserving total angular momentum of the colliding pair. As a result, spin-exchange collisions cause decoherence There has been significant work on correctly identifying the pointer states in the case of a massive particle decohered by collisions with a fluid environment,
A widely used approximation for collisional decoherence assumes exponential suppression of off-diagonal terms:<math display="block">C(t) \approx \exp(-\Gamma t),\qquad \Gamma \propto n\,v\,\sigma_{\text{decoh}},</math>with <math>n</math> the particle density, <math>v</math> the relative velocity, and <math>\sigma_{\text{decoh}}</math> the scattering cross-section.<ref>{{cite journal |last1=Joos |first1=E. |last2=Zeh |first2=H. D. |title=The emergence of classical properties through interaction with the environment |journal=Zeitschrift für Physik B |year=1985 |volume=59 |issue=2 |pages=223–243 |doi=10.1007/BF01725541 |bibcode=1985ZPhyB..59..223J }}</ref>

== Applications in quantum mechanics ==
Open-system formulations are essential in quantum hardware design and analysis:
* '''[[Wikipedia:Quantization (signal processing)|Noise modeling:]]''' Estimating dephasing and relaxation times (<math>T_1</math>, <math>T_2</math>) in superconducting qubits and trapped ions.<ref>{{cite journal |last1=Krantz |first1=Philip |last2=Kjaergaard |first2=Morten |last3=Yan |first3=Fei |last4=Orlando |first4=Terry P. |last5=Gustavsson |first5=Simon |last6=Oliver |first6=William D. |title=A quantum engineer's guide to superconducting qubits |journal=Applied Physics Reviews |year=2019 |volume=6 |issue=2 |article-number=021318 |doi=10.1063/1.5089550 |arxiv=1904.06560 |bibcode=2019ApPRv...6b1318K }}</ref>
* '''[[Wikipedia:Quantum error correction|Error correction:]]''' Providing physical noise models for the design of [[Wikipedia:quantum error correction|error-correcting codes]].
* '''[[Wikipedia:Consistent histories|Control techniques:]]''' Informing pulse-shaping and [[Wikipedia:dynamical decoupling|dynamical decoupling]] sequences to suppress decoherence.
* '''[[Wikipedia:Threshold theorem|Fault tolerance:]]''' Guiding thresholds for quantum error correction using Lindblad-type noise models.

== Articles of interest ==
These articles describe the Quantum system as outlined in this article. 
[https://arxiv.org/html/2402.19241v1 Open Quantum System Approaches to Superconducting Qubits] [https://www.spinquanta.com/news-detail/quantum-computer-operating-system-the-key-to-quantum-power20250116104617 Quantum Computer Operating System: The Key to Quantum Power] [https://www.cambridge.org/highereducation/books/building-quantum-computers/6A73C509D3E0F5F0A566A11F6A566A90#overview Building Quantum Computers A Practical Introduction] [https://quantumzeitgeist.com/openqasm-the-quantum-programming-language/#google_vignette OpenQASM: The Quantum Programming Language. Assembly Programming for Quantum Computers] [https://www.spinquanta.com/news-detail/what-are-open-quantum-systems What Are Open Quantum Systems? A Complete Guide] [https://arxiv.org/abs/2302.02953 Digital Simulation of Single Qubit Markovian Open Quantum Systems: A Tutorial] [https://link.springer.com/chapter/10.1007/978-3-642-23354-8_3 Time Evolution in Open Quantum Systems]

== Challenges ==
* The Lindblad approach assumes memoryless noise and may not capture non-Markovian dynamics in advanced devices.
* Redfield and other non-Markovian models can describe richer environments but are computationally expensive and sometimes unphysical.
* Hybrid approaches combining Lindblad and non-Markovian models are under investigation.
* Active error suppression techniques (e.g. dynamical decoupling, error mitigation) complement open-system modeling.
{{Quantum mechanics}}

== Further reading ==
* {{cite book |last=Weiss |first=Ulrich |title=Quantum Dissipative Systems |publisher=World Scientific |year=2012 |isbn=978-9814374910}}

== Suggested exercises ==
* Derive the Lindblad master equation for a two-level system coupled to a thermal bath.
* Compare Redfield and Lindblad dynamics for weak system–environment coupling.
* Simulate decoherence of a qubit under amplitude damping using Kraus operators.
 
'''This page emphasizes conceptual and mathematical structure rather than step-by-step instruction.'''

=See also=
{{#invoke:PhysicsQC|tocHeadingAndList|Physics:Quantum basics/See also}}

== References ==
<references />
{{Author|Harold Foppele}}
[[Category:Quantum mechanics]]
[[Category:Physics]]

{{Sourceattribution|Quantum Noisy Qubits|1}}

Physics:Quantum Noisy Qubits - Revision history

imported>WikiHarold: Repair Quantum Collection B backlink template

imported>WikiHarold: Repair Quantum Collection B backlink template