Latest revision as of 23:46, 23 May 2026

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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.

Formulas for Noisy Qubits

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.^[1] 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.^[2] 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^[3] and relaxation that degrade computational performance.

50-Qubit

Open system models provide mathematical tools for analyzing and mitigating these effects.^[4]
IBM's 50-qubit quantum computer prototype, as exhibited at CES 2018 in Las Vegas ----->
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.

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

Equation used in wave mechanics (see Quantum mechanics) for the wave function of a particle is the time-independent Schrödinger equation

$\nabla^{2} ψ + \frac{8 π^{2} m}{h^{2}} (E - U) ψ = 0$

It can also be written in operator form as:

$H ψ = E ψ$

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.

Schrödinger equation wave packet

The time-dependent Schrödinger equation (see also Dyson series) for an isolated system is: $i ℏ \frac{\partial}{\partial t} | ψ (t) ⟩ = H | ψ (t) ⟩ .$
The unitary propagator is: $U (t) = 𝒯 \exp (- \frac{i}{ℏ} \int_{0}^{t} H (t^{'}) d t^{'}),$ with $𝒯$ the time-ordering operator.

For open systems, the state of the system alone is obtained from the full density matrix of system+environment: $ρ_{S} (t) = {T r}_{E} [ρ_{S E} (t)] .$ This partial trace generally produces non-unitary dynamics.

From microscopic models to master equations

Consider a system Hamiltonian (quantum mechanics) $H_{S}$ , environment Hamiltonian $H_{E}$ , and an interaction $H_{I}$ . The total Hamiltonian is: $H = H_{S} + H_{E} + H_{I} .$ Even if $ρ_{S E}$ evolves unitarily, the reduced density matrix $ρ_{S}$ typically obeys an integro-differential equation. Approximations lead to different master equations.

Kraus representation

Any completely positive trace-preserving (CPTP) map on a quantum state can be written as: $ℰ (ρ) = \sum_{k} E_{k} ρ E_{k}^{†}, \sum_{k} E_{k}^{†} E_{k} = I,$ where the $E_{k}$ are Kraus operators. This is a general representation of open-system dynamics at discrete times.^[5] The bit-flip channel is a quantum channel that, with probability $p$ , applies a qubit flip (X-gate), and with probability $1 - p$ does nothing. It is regarded as the quantum analogue of the noise through entanglement with the environment.

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: $\frac{d ρ}{d t} = - \frac{i}{ℏ} [H, ρ] + \sum_{k} (L_{k} ρ L_{k}^{†} - \frac{1}{2} {L_{k}^{†} L_{k}, ρ}) .$ This generator defines a dynamical semigroup (completely positive, trace-preserving evolution).^[6]

For a single qubit, collapse operators commonly model:

Relaxation (energy decay): $L_{relax} = \sqrt{γ_{1}} σ_{-}$
Dephasing: $L_{deph} = \sqrt{γ_{ϕ}} σ_{z}$

Here $γ_{1}$ and $γ_{ϕ}$ are the relaxation and dephasing rates, respectively.

Redfield equation (non-Markovian)

If the Markov approximation is not applied, the Redfield equation captures memory effects: $\frac{d ρ_{S} (t)}{d t} = - \frac{i}{ℏ} [H_{S}, ρ_{S} (t)] - \int_{0}^{t} d τ {T r}_{E} ([H_{I} (t), [H_{I} (τ), ρ_{S} (τ) \otimes ρ_{E}]]) .$ Redfield theory can describe structured environments (e.g. spin baths or photonic reservoirs) but does not guarantee complete positivity without further corrections.^[7]

Collisional decoherence

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: $C (t) \approx \exp (- Γ t), Γ \propto n v σ_{decoh},$ with $n$ the particle density, $v$ the relative velocity, and $σ_{decoh}$ the scattering cross-section.^[8]

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 {{Short description|Frameworks for modeling qubit–environment interactions}}
+{{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 error correction|next label=Error correction}}
+<div style="display:flex; gap:24px; align-items:flex-start; max-width:1200px;">
-{{Quantum book backlink|Quantum information and computing}}
+<div style="width:280px;">
-<br><br>
+__TOC__
-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.
+</div>
-<div style="float:right; border:1px solid #ccc; padding:4px; background:#f9f9f9; margin:0 0 1em 1em; width:450px;">
+<div style="flex:1; line-height:1.45; color:#006b45; column-count:2; column-gap:32px; column-rule:1px solid #b8d8c8;">
-[[File:Quantum noisy qubits1.jpg]]
+'''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 style="width:300px;">
+[[File:Quantum noisy qubits1.jpg|thumb|280px|Quantum Noisy Qubits.]]
 </div>
 </div>
 ==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><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==
@@ Line 19: / Line 27: @@
 == Background formulas that govern noisy qubits ==
-[[File:Schrödinger-Gl 04 Lösungsfunktion-Beispiel.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>
@@ Line 28: / Line 36: @@
 <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]]
-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.
@@ Line 37: / Line 45: @@
 == 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.
+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 ===
 [[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.
@@ Line 65: / Line 73: @@
 == 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>
+* '''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 ==
@@ Line 79: / Line 87: @@
 * 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}}
@@ Line 97: / Line 103: @@
 <references />
 {{Author|Harold Foppele}}
-[[Category:Quantum mechanics]]
-[[Category:Physics]]
-{{Sourceattribution|Quantum Noisy Qubits|1}}
+{{Sourceattribution|Physics:Quantum Noisy Qubits|1}}

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Physics:Quantum Noisy Qubits: Difference between revisions

Latest revision as of 23:46, 23 May 2026

Formulas for Noisy Qubits

Textbooks and surveys

Background formulas that govern noisy qubits

From microscopic models to master equations

Kraus representation

Lindblad equation (Markovian)

Redfield equation (non-Markovian)

Collisional decoherence

Applications in quantum mechanics

Articles of interest

Challenges

Further reading

Suggested exercises

See also

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