Physics:Quantum Noisy Qubits - Revision history

WikiHarold: Remove imported red links from Quantum page

2026-05-23T23:46:31Z

Remove imported red links from Quantum page

← Older revision		Revision as of 23:46, 23 May 2026
Line 87:		Line 87:
	* Hybrid approaches combining Lindblad and non-Markovian models are under investigation.		* 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.		* Active error suppression techniques (e.g. dynamical decoupling, error mitigation) complement open-system modeling.
	~~{{Quantum mechanics}}~~

	== Further reading ==		== Further reading ==
	* {{cite book \|last=Weiss \|first=Ulrich \|title=Quantum Dissipative Systems \|publisher=World Scientific \|year=2012 \|isbn=978-9814374910}}		* {{cite book \|last=Weiss \|first=Ulrich \|title=Quantum Dissipative Systems \|publisher=World Scientific \|year=2012 \|isbn=978-9814374910}}
Line 105:		Line 103:
	<references />		<references />
	{{Author\|Harold Foppele}}		{{Author\|Harold Foppele}}
	~~[[Category:Quantum mechanics]]~~
	~~[[Category:Physics]]~~

	{{Sourceattribution\|Physics:Quantum Noisy Qubits\|1}}		{{Sourceattribution\|Physics:Quantum Noisy Qubits\|1}}

Maintenance script: Update quantum article navigation for Error correction

2026-05-22T17:19:40Z

Update quantum article navigation for Error correction

← Older revision		Revision as of 17:19, 22 May 2026
Line 1:		Line 1:
	{{Short description\|Frameworks for modeling qubit–environment interactions}}		{{Short description\|Frameworks for modeling qubit–environment interactions}}
	{{Quantum book backlink\|Quantum information and computing}}		{{Quantum book backlink\|Quantum information and computing}}
	{{Quantum article nav\|previous=Physics:Quantum Computing Algorithms in the NISQ Era\|previous label=Computing Algorithms in the NISQ Era\|next=Physics:Quantum ~~Boson sampling~~\|next label=~~Boson sampling~~}}		{{Quantum article nav\|previous=Physics:Quantum Computing Algorithms in the NISQ Era\|previous label=Computing Algorithms in the NISQ Era\|next=Physics:Quantum error correction\|next label=Error correction}}
	<div style="display:flex; gap:24px; align-items:flex-start; max-width:1200px;">		<div style="display:flex; gap:24px; align-items:flex-start; max-width:1200px;">

Maintenance script: Normalize quantum page header order

2026-05-22T11:32:49Z

Normalize quantum page header order

← Older revision		Revision as of 11:32, 22 May 2026
Line 2:		Line 2:
	{{Quantum book backlink\|Quantum information and computing}}		{{Quantum book backlink\|Quantum information and computing}}
	{{Quantum article nav\|previous=Physics:Quantum Computing Algorithms in the NISQ Era\|previous label=Computing Algorithms in the NISQ Era\|next=Physics:Quantum Boson sampling\|next label=Boson sampling}}		{{Quantum article nav\|previous=Physics:Quantum Computing Algorithms in the NISQ Era\|previous label=Computing Algorithms in the NISQ Era\|next=Physics:Quantum Boson sampling\|next label=Boson sampling}}

	<div style="display:flex; gap:24px; align-items:flex-start; max-width:1200px;">		<div style="display:flex; gap:24px; align-items:flex-start; max-width:1200px;">

Maintenance script: Clean Book label remnants and backlink spacing

2026-05-22T11:11:09Z

Clean Book label remnants and backlink spacing

← Older revision		Revision as of 11:11, 22 May 2026
Line 1:		Line 1:
	{{Short description\|Frameworks for modeling qubit–environment interactions}}		{{Short description\|Frameworks for modeling qubit–environment interactions}}

	{{Quantum book backlink\|Quantum information and computing}}		{{Quantum book backlink\|Quantum information and computing}}
	{{Quantum article nav\|previous=Physics:Quantum Computing Algorithms in the NISQ Era\|previous label=Computing Algorithms in the NISQ Era\|next=Physics:Quantum Boson sampling\|next label=Boson sampling}}		{{Quantum article nav\|previous=Physics:Quantum Computing Algorithms in the NISQ Era\|previous label=Computing Algorithms in the NISQ Era\|next=Physics:Quantum Boson sampling\|next label=Boson sampling}}

Maintenance script: Remove hidden BOM characters and direct Book label after Short description

2026-05-22T11:03:21Z

Remove hidden BOM characters and direct Book label after Short description

← Older revision		Revision as of 11:03, 22 May 2026
Line 1:		Line 1:
	{{Short description\|Frameworks for modeling qubit–environment interactions}}~~Book I~~		{{Short description\|Frameworks for modeling qubit–environment interactions}}

	{{Quantum book backlink\|Quantum information and computing}}		{{Quantum book backlink\|Quantum information and computing}}

Maintenance script: Clean Short description prefix and add Book I label

2026-05-22T10:48:23Z

Clean Short description prefix and add Book I label

← Older revision		Revision as of 10:48, 22 May 2026
Line 1:		Line 1:
	~~~~{{Short description\|Frameworks for modeling qubit–environment interactions}}		{{Short description\|Frameworks for modeling qubit–environment interactions}}Book I

	{{Quantum book backlink\|Quantum information and computing}}		{{Quantum book backlink\|Quantum information and computing}}

Maintenance script: Update Quantum information navigation links

2026-05-21T22:21:55Z

Update Quantum information navigation links

← Older revision		Revision as of 22:21, 21 May 2026
Line 2:		Line 2:

	{{Quantum book backlink\|Quantum information and computing}}		{{Quantum book backlink\|Quantum information and computing}}
	{{Quantum article nav\|previous=Physics:Quantum Computing Algorithms in the NISQ Era\|previous label=Computing Algorithms in the NISQ Era\|next=Physics:Quantum ~~random access code~~\|next label=~~Random access code~~}}		{{Quantum article nav\|previous=Physics:Quantum Computing Algorithms in the NISQ Era\|previous label=Computing Algorithms in the NISQ Era\|next=Physics:Quantum Boson sampling\|next label=Boson sampling}}

	<div style="display:flex; gap:24px; align-items:flex-start; max-width:1200px;">		<div style="display:flex; gap:24px; align-items:flex-start; max-width:1200px;">

Maintenance script: Apply Quantum previous-next navigation

2026-05-20T12:25:16Z

Apply Quantum previous-next navigation

← Older revision		Revision as of 12:25, 20 May 2026
Line 1:		Line 1:
	~~~~{{Short description\|Frameworks for modeling qubit–environment interactions}}		{{Short description\|Frameworks for modeling qubit–environment interactions}}

	{{Quantum book backlink\|Quantum information and computing}}		{{Quantum book backlink\|Quantum information and computing}}
			{{Quantum article nav\|previous=Physics:Quantum Computing Algorithms in the NISQ Era\|previous label=Computing Algorithms in the NISQ Era\|next=Physics:Quantum random access code\|next label=Random access code}}

	<div style="display:flex; gap:24px; align-items:flex-start; max-width:1200px;">		<div style="display:flex; gap:24px; align-items:flex-start; max-width:1200px;">

Maintenance script: Improve Book I intro quality

2026-05-20T09:04:19Z

Improve Book I intro quality

← Older revision		Revision as of 09:04, 20 May 2026
Line 1:		Line 1:
	{{Short description\|Frameworks for modeling qubit–environment interactions}}		{{Short description\|Frameworks for modeling qubit–environment interactions}}

	{{Quantum book backlink\|Quantum information and computing}}		{{Quantum book backlink\|Quantum information and computing}}
Line 10:		Line 10:

	<div style="flex:1; line-height:1.45; color:#006b45; column-count:2; column-gap:32px; column-rule:1px solid #b8d8c8;">		<div style="flex:1; line-height:1.45; color:#006b45; column-count:2; column-gap:32px; column-rule:1px solid #b8d8c8;">
	'''Noisy Qubits''' ~~is a Book I topic in the Quantum Collection. 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. 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. Noisy qubits are a fundamental challenge in current Noisy Intermediate-Scale Quantum (NISQ) computers, where physical qubits are susceptible to errors from decoherence, are sensitive to their environment (noisy), imperfect gate operations, and measurement noise.		'''Noisy Qubits''' 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. 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. Noisy qubits are a fundamental challenge in current Noisy Intermediate-Scale Quantum (NISQ) computers, where physical qubits are susceptible to errors from 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.
	~~</div>~~
	~~</div>~~
	</div>		</div>

Maintenance script: Clean missing Book I links and template output

2026-05-20T08:47:31Z

Clean missing Book I links and template output

← Older revision		Revision as of 11:11, 22 May 2026
Line 1:		Line 1:
	{{Short description\|Frameworks for modeling qubit–environment interactions}}		{{Short description\|Frameworks for modeling qubit–environment interactions}}

	{{Quantum book backlink\|Quantum information and computing}}		{{Quantum book backlink\|Quantum information and computing}}
	{{Quantum article nav\|previous=Physics:Quantum Computing Algorithms in the NISQ Era\|previous label=Computing Algorithms in the NISQ Era\|next=Physics:Quantum Boson sampling\|next label=Boson sampling}}		{{Quantum article nav\|previous=Physics:Quantum Computing Algorithms in the NISQ Era\|previous label=Computing Algorithms in the NISQ Era\|next=Physics:Quantum Boson sampling\|next label=Boson sampling}}

← Older revision		Revision as of 08:47, 20 May 2026
Line 1:		Line 1:
	{{Short description\|Frameworks for modeling qubit–environment interactions}}		{{Short description\|Frameworks for modeling qubit–environment interactions}}

	{{Quantum book backlink\|Quantum information and computing}}		{{Quantum book backlink\|Quantum information and computing}}
Line 22:		Line 22:

	==Formulas for Noisy Qubits==		==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:Quantum_book1_noisy_qubits_yellow.png\|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><br>IBM's 50-qubit quantum computer prototype, as exhibited at CES 2018 in Las Vegas -----><br>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]]		Noisy qubits are a fundamental challenge in current Noisy Intermediate-Scale Quantum (NISQ) computers, where physical qubits are susceptible to errors from 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. 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 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 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 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:Quantum_book1_noisy_qubits_yellow.png\|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><br>IBM's 50-qubit quantum computer prototype, as exhibited at CES 2018 in Las Vegas -----><br>They describe how methods from the theory of Open quantum system are applied to qubits and quantum hardware. In practice, qubits are never perfectly isolated: they interact with their environments, leading to decoherence, relaxation, and noise that limit computation. This has made open-system tools—such as Kraus operators, Lindblad master equations, and non-Markovian models—fundamental to modern quantum computing research.[[File:SPDC figure.png \|thumb\|upright=0.6\|Conceptual illustration of entanglement]]

	==Textbooks and surveys==		==Textbooks and surveys==
Line 31:		Line 31:
	== Background formulas that govern noisy qubits ==		== Background formulas that govern noisy qubits ==
	[[File:Quantum_book1_noisy_qubits_yellow.png\|thumb\|Schrödinger-equation example]]		[[File:Quantum_book1_noisy_qubits_yellow.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		Equation used in wave mechanics (see 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>		<math>\nabla^{2}\psi + \frac{8\pi^{2} m}{h^{2}}(E - U)\psi = 0</math>
Line 39:		Line 39:
	<math>H \psi = E \psi</math>		<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.<br>		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 Erwin Schrödinger, who was mainly responsible for wave mechanics.<br>
	[[File:Schrödinger equation wave packet.gif\|thumb\|Schrödinger equation wave packet]]		[[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:		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 Dyson series) for an isolated system is:
	<math>i\hbar\frac{\partial}{\partial t}\|\psi(t)\rangle = H \|\psi(t)\rangle.</math><br>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.		<math>i\hbar\frac{\partial}{\partial t}\|\psi(t)\rangle = H \|\psi(t)\rangle.</math><br>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.

Line 48:		Line 48:
	== From microscopic models to master equations ==		== From microscopic models to master equations ==
	[[File:Klein-graph-7-valent Hamiltonian.svg\|Hamiltonian]]		[[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.		Consider a system 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 ===		=== Kraus representation ===
	[[File:Kraus representation.jpg\|thumb\|110px\|]]		[[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>		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 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.		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.

Line 76:		Line 76:
	== Applications in quantum mechanics ==		== Applications in quantum mechanics ==
	Open-system formulations are essential in quantum hardware design and analysis:		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>		* '''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]].		* '''Error correction:''' Providing physical noise models for the design of error-correcting codes.
	* '''~~[[Wikipedia:Consistent histories\|~~Control techniques:]]''' Informing pulse-shaping and ~~[[Wikipedia:~~dynamical decoupling~~\|dynamical decoupling]]~~ sequences to suppress decoherence.		* '''Control techniques:''' Informing pulse-shaping and dynamical decoupling sequences to suppress decoherence.
	* '''~~[[Wikipedia:Threshold theorem\|~~Fault tolerance:]]''' Guiding thresholds for quantum error correction using Lindblad-type noise models.		* '''Fault tolerance:''' Guiding thresholds for quantum error correction using Lindblad-type noise models.

	== Articles of interest ==		== Articles of interest ==