Physics:Quantum Noisy Qubits

Short description: Frameworks for modeling qubit–environment interactions

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Introduction

Artistic impression of an atom 7

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.

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. 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. 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 and relaxation that degrade computational performance.

50-Qubit

Open system models provide mathematical tools for analyzing and mitigating these effects.
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.

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

Schrödinger-equation example

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

${\mathrm{\nabla }}^2\psi +\frac{8{\pi }^{2}m}{h^2}(E-U)\psi =0$

It can also be written in operator form as:

$H\psi =E\psi$

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: $i\hslash \frac{\partial }{\partial t}|\psi (t)⟩=H|\psi (t)⟩.$
The unitary propagator is:$$U(t)={𝒯}\exp (-\frac{i}{\hslash }\underset{0}{\overset{t}{\int }}H(t^{\prime })d{t}^{\prime }),$$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:$${\rho }_S(t)={\mathrm{T}\mathrm{r}}_E[{\rho }_{SE}(t)].$$This partial trace generally produces non-unitary dynamics.

From microscopic models to master equations

Error creating thumbnail: Unable to save thumbnail to destination 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 ${\rho }_{SE}$ evolves unitarily, the reduced density matrix ${\rho }_S$ 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:$${ℰ}(\rho )=\sum _k{E}_k\rho E_k^{†},\sum _k{E}_k^{†}{E}_k=I,$$where the $E_k$ are Kraus operators. This is a general representation of open-system dynamics at discrete times. 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)

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:$$\frac{d\rho }{dt}=-\frac{i}{\hslash }[H,\rho ]+\sum _k(L_k\rho L_k^{†}-\frac{1}{2}\{L_k^{†}{L}_k,\rho \}).$$This generator defines a dynamical semigroup (completely positive, trace-preserving evolution).

For a single qubit, collapse operators commonly model:

• Relaxation (energy decay): $L_{\text{relax}}=\sqrt{{\gamma }_1}{\sigma }_{-}$
• Dephasing: $L_{\text{deph}}=\sqrt{{\gamma }_{\varphi }}{\sigma }_z$

Here ${\gamma }_1$ and ${\gamma }_{\varphi }$ are the relaxation and dephasing rates, respectively.

Redfield equation (non-Markovian)

Redfield equation (non-Markovian)

If the Markov approximation is not applied, the Redfield equation captures memory effects:$$\frac{d{\rho }_S(t)}{dt}=-\frac{i}{\hslash }[H_S,{\rho }_S(t)]-\underset{0}{\overset{t}{\int }}d\tau {\mathrm{T}\mathrm{r}}_E([H_I(t),[H_I(\tau ),{\rho }_S(\tau )\otimes {\rho }_E]]).$$Redfield theory can describe structured environments (e.g. spin baths or photonic reservoirs) but does not guarantee complete positivity without further corrections.

Collisional decoherence

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:$$C(t)\approx \exp (-\mathrm{\Gamma }t),\mathrm{\Gamma }\propto nv{\sigma }_{\text{decoh}},$$with $n$ the particle density, $v$ the relative velocity, and ${\sigma }_{\text{decoh}}$ the scattering cross-section.

Applications in quantum mechanics

Open-system formulations are essential in quantum hardware design and analysis:

• Noise modeling: Estimating dephasing and relaxation times ($T_1$, $T_2$) in superconducting qubits and trapped ions.
• Error correction: Providing physical noise models for the design of error-correcting codes.
• Control techniques: Informing pulse-shaping and dynamical decoupling sequences to suppress decoherence.
• 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.
Open Quantum System Approaches to Superconducting Qubits
Quantum Computer Operating System: The Key to Quantum Power
Building Quantum Computers A Practical Introduction
OpenQASM: The Quantum Programming Language. Assembly Programming for Quantum Computers
What Are Open Quantum Systems? A Complete Guide
Digital Simulation of Single Qubit Markovian Open Quantum Systems: A Tutorial
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.

Further reading

• Weiss, Ulrich (2012). Quantum Dissipative Systems. World Scientific. 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

• Physics:Quantum basics
• Physics:Quantum A Matter Of Size
• Physics:Quantum A Spooky Action at a Distance
• Physics:Quantum A Walk Through the Universe
• Physics:Number of independent spatial modes in a spherical volume
• Physics:Quantum Computing Algorithms in the NISQ Era
• Physics:Quantum Formulas Collection
• Physics:Quantum Matter Elements and Particles
• Physics:Quantum mechanics
• Physics:Quantum mechanics/Timeline
• Physics:Quantum_mechanics/Timeline/Quiz/
• Physics:Quantum mechanics measurements
• Physics:Quantum_Noisy_Qubits
• Physics:Quantum optics beam splitter experiments
• Physics:Quantum: The Secret of Cohesion: How Waves Hold Matter Together
• Physics:Quantum Ultra fast lasers
• Template Quantum optics operators

References

Author: Harold Foppele