Quantum Computing Algorithms in the NISQ Era

Short description: Overview of quantum algorithms suited for current NISQ hardware, including VQE for chemistry, QAOA for optimization, and classics like Grover and Shor

Quantum computers do not compute by flipping a long string of zeros and ones, but by coaxing tiny quantum objects into behaving like complex waves of possibility. That’s the intuitive leap behind quantum computing: instead of bits that are definitely 0 or 1, quantum computers use qubits that can exist in superpositions of states, become entangled so their states are linked across space, and exploit interference to amplify correct answers while canceling wrong ones. These phenomena: superposition, entanglement, and interference are the conceptual tools that let quantum algorithms explore many possible solutions at once in ways classical algorithms cannot.

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## Contents

* 1 Introduction
* 1.1 Keywords:
* 1.2 Quantum Computing Algorithms
* 1.2.1 Fundamental Concepts
* 1.3 Key Algorithms
* 1.3.1 Shor's Algorithm (1994):
* 1.3.2 Grover's Algorithm (1996):
* 1.3.3 Variational Quantum Algorithms (VQAs):
* 1.3.4 Key Formulas in VQAs
* 1.3.4.1 Additional Visuals and Considerations
* 1.3.4.2 Quantum Machine Learning Algorithms:
* 1.3.4.3 Applications and Impact
* 1.3.4.4 Challenges and Future Directions
* 1.4 Promising Age of Quantum Computing
* 1.4.1 From Classical to Quantum Optimization
* 1.5 Quantum Algorithms
* 1.6 Emerging paradigm
* 1.7 Quantum Advantage: From NISQ to Fault-Tolerance
* 1.7.1 The NISQ Bridge and AI Integration
* 1.7.2 The Transition to Fault-Tolerant Computing (FTQC)
* 1.7.3 Future Outlook and Machine Learning
* 1.8 Key NISQ Algorithms
* 1.8.1 Variational Quantum Eigensolver (VQE) in Quantum Chemistry
* 1.8.2 Ansätze and Architectural Innovations
* 1.8.3 AI-Driven Optimization and Training
* 1.8.4 Recent Advancements (2025–2026)
* 1.9 Quantum Approximate Optimization Algorithm (QAOA)
* 1.9.1 Optimization Landscapes and Training Strategies
* 1.9.2 Digital-Analog Approaches and Hardware Awareness
* 1.9.3 Practical Feasibility and 2025–2026 Milestones
* 1.10 Amplitude Amplification
* 1.10.1 Integration with NISQ and QML
* 1.10.2 Overcoming Noise and Decoherence
* 1.10.3 Application-focused and gap analyses
* 1.10.4 Applications and 2025–2026 Trends
* 1.11 Foundational Classics
* 1.12 Grover's Algorithm and Search Optimization
* 1.12.1 Challenges in the NISQ Era
* 1.12.2 Applications in Machine Learning and Kernels
* 1.12.3 2026 Outlook and Dequantization
* 1.13 Shor's Algorithm and Cryptographic Implications
* 1.13.1 The Cryptographic Threat
* 1.13.2 Modern Cryptography
* 1.13.3 Resource Estimates for 2026
* 1.13.4 Current State and Hybrid Security
* 1.14 Emerging 2025–2026
* 1.14.1 Quantum Echoes
* 1.14.2 Sample-Based Diagonalization
* 1.15 Visuals
* 1.15.1 Circuit Diagrams
* 1.15.2 VQE Energy Landscapes
* 1.15.3 QAOA Parameter Landscapes
* 1.16 Formulas (Gates)
* 1.16.1 Quantum gates
* 1.16.2 Hadamard Gate (H)
* 1.16.3 Measurements (Readout)
* 1.17 Further Reading
* 1.18 Cross Links
* 1.18.1 Noisy Qubits (Error Impact)
* 2 See also
* 3 Table of contents (185 articles)
* 4 Index
* 5 Full contents
* 5.1 Sources
* 6 The Learning Quiz
* 6.1 References

# Introduction

Quantum computing  Field  | Quantum information science

|

Type  | Computing paradigm
Based on  | Quantum mechanics
Key units  | Qubit
Main phenomena  | Superposition, entanglement, interference
Typical models  | Quantum circuit model, adiabatic quantum computing, measurement-based quantum computing
Applications  | Cryptanalysis, quantum simulation, optimization, search
Notable algorithms  | Shor's algorithm, Grover's algorithm

Because of properties, quantum machines have the potential to transform domains where classical approaches struggle. Simulating complex molecules for chemistry and materials science, tackling hard optimization problems in logistics and finance, and accelerating certain kinds of machine-learning and search tasks. Researchers have already demonstrated early milestones where quantum processors performed narrowly defined tasks far faster than classical machines, milestones sometimes called quantum supremacy or quantum advantage. These show the field is progressing from theory toward demonstrable speedups [501][502][503][504][505]. The current era, often referred to as the NISQ (Noisy Intermediate-Scale Quantum) era, is characterized by machines with tens to hundreds of qubits that are inherently noisy and prone to errors. Fully fault-tolerant, error-corrected quantum computing remains an engineering challenge, but progress is rapid. Large technology companies and startups alike continue to publish new processors, algorithms, and roadmaps. Quantum computing is no longer just a theoretical curiosity, it is an active, multidisciplinary race among physicists, engineers, and computer scientists to turn exotic quantum effects into practical advantage.[506]

### Keywords:

Quantum Computing, Artificial intelligence, Machine Learning, Unsupervised learning, Cryptography, Cyber Security, Qubits, Quantum algorithms, Quantum verification, Quantum error correction, Quantum applications, Variational Quantum Algorithms (VQAs), Noisy Intermediate-Scale Quantum (NISQ), Variational Quantum Eigensolver (VQE), Quantum Approximate Optimization Algorithm (QAOA),Quantum Neural Networks (QNNs), Quantum advantage, Barren plateaus, Noise mitigation, Quantum optimization, Quantum chemistry, Quantum machine learning, Hybrid quantum-classical algorithms.

### Quantum Computing Algorithms

Quantum computing algorithms leverage the principles of quantum mechanics, such as superposition, entanglement, and interference, to solve problems more efficiently than classical algorithms in certain domains. Unlike classical bits, which are binary (0 or 1), quantum bits (qubits) can exist in multiple states simultaneously, enabling parallel computation on an exponential scale. This section explores key quantum algorithms, their mechanisms, applications, and current limitations as of 2026.

#### Fundamental Concepts

Quantum algorithms operate on quantum circuits, which consist of quantum gates applied to qubits. The Quantum Fourier Transform (QFT), for instance, is a core building block analogous to the classical Discrete Fourier Transform but exponentially faster for certain tasks. It decomposes periodic functions into their frequency components and underpins many advanced algorithms.

## Key Algorithms

#### Shor's Algorithm (1994):

Developed by Peter Shor, this algorithm efficiently factors large integers and computes discrete logarithms, tasks that are computationally infeasible for classical computers at scale. It exploits QFT to find the period of a function related to the number being factored.

* Mechanism: Initialize qubits in superposition, apply modular exponentiation, use QFT to identify periodicity, and derive factors via continued fractions.
* Applications: Threatens RSA encryption, driving research into post-quantum cryptography. In practice, implementations on noisy intermediate-scale quantum (NISQ) devices have factored small numbers (e.g., 21 using IBM's quantum systems in recent demos). In January 2026, JPMorgan Chase implemented a quantum streaming algorithm achieving exponential space advantage for real-time processing of large datasets, building on Shor's principles for financial applications.
* Complexity: Polynomial time O((log⁡n)3), versus exponential for classical methods.

#### Grover's Algorithm (1996):

Lov Grover's search algorithm provides a quadratic speedup for unstructured search problems, such as finding an item in an unsorted database.

* Mechanism: Uses amplitude amplification to boost the probability of measuring the correct state. Starts with uniform superposition, applies an oracle to mark the target, and reflects amplitudes iteratively.
* Applications: Optimization, database search, and machine learning (e.g., accelerating k-nearest neighbors). Recent variants like Quantum Approximate Optimization Algorithm (QAOA) extend it to combinatorial problems like MaxCut. As of early 2026, Google's Quantum AI lab has advanced QAOA variants for hybrid workflows in optimization tasks.
* Complexity: O(√N) queries, compared to O(N) classically.

#### Variational Quantum Algorithms (VQAs):

* Mechanism: Parameterize a quantum circuit (ansatz), measure expectation values, and optimize classically using gradient descent.
* Applications: Quantum chemistry (simulating molecular energies, e.g., Google's Sycamore processor modeling hydrogen chains) and finance (portfolio optimization). In January 2026, Quandela highlighted VQAs as key to early industrial use cases in hybrid computing for drug discovery and materials science.

VQAs are hybrid quantum-classical methods designed for near-term quantum devices (NISQ era). They leverage a parameterized quantum circuit (ansatz) to prepare a trial quantum state, measure a cost function (often an expectation value), and use a classical optimizer to adjust parameters and minimize the cost. This approach is inspired by the variational principle in quantum mechanics, which states that for a Hermitian operator like a Hamiltonian H, the expectation value in any trial state |ψ⟩ is an upper bound on the ground state energy E0:

⟨ψ|H|ψ⟩≥E0

The goal is to find parameters θ that minimize this expectation value to approximate E0 or solve optimization problems.

A typical VQA workflow involves:

* Preparing a quantum state |ψ(θ)⟩ = U(θ) |0⟩, where U(θ) is the parameterized circuit.
* Computing the cost C(θ) = ⟨ψ(θ)| H |ψ(θ)⟩ via measurements on a quantum computer.
* Optimizing θ classically (e.g., using gradient descent or other methods) to minimize C(θ).

Quamtum Computer vs. Classic computer

This diagram illustrates the standard hybrid loop of a VQA, showing the interplay between quantum state preparation, measurement, and classical optimization.

### Key Formulas in VQAs

* General Cost Function: In many VQAs, the cost is defined as: C(θ)=⟨ψ(θ)|H|ψ(θ)⟩ where H encodes the problem (e.g., a molecular Hamiltonian for quantum chemistry). To compute this, H is often decomposed into Pauli operators: H=∑kckPk with Pk being products of Pauli matrices (I, X, Y, Z), and the expectation value is measured term-by-term.

* Variational Quantum Eigensolver (VQE): VQE aims to find the ground state of H. The ansatz U(θ) generates trial states, and the energy is minimized: E(θ)=⟨ψ(θ)|H|ψ(θ)⟩⟨ψ(θ)|ψ(θ)⟩ (normalized if needed). Assuming ∣ψ(θ)⟩ is normalized; otherwise, include the denominator. For example, in a simple ansatz, θ parameterizes rotations like Ry(θ) = e-iθ Y/2.

The workflow of a typical variational quantum algorithm

This visual depicts the VQE workflow, highlighting the parameter optimization loop for estimating ground state energies.

* Quantum Approximate Optimization Algorithm (QAOA): QAOA solves combinatorial optimization problems by alternating "problem" and "mixer" Hamiltonians over p layers |ψ(γ→,β→)⟩=∏j=1pe−iβjHBe−iγjHC|+⟩⊗n where HC encodes the cost function (e.g., for MaxCut: HC = ∑⟨i,j⟩ Zi Zj), and HB = ∑i Xi is the transverse field mixer. The cost is then: C(γ→,β→)=⟨ψ|HC|ψ⟩ Parameters γj, βj are optimized to approximate the optimal solution.

Variational Quantum Algorithms - From Theory to NISQ-Era

A detailed diagram showing the layered structure of QAOA circuits and the optimization process.

#### Additional Visuals and Considerations

For nonlinear problems (e.g., in fluid dynamics or other simulations), VQAs can extend to cost functions involving higher powers or nonlinear terms. One way to formulate this is through a composite observable, such as a product of expectation values:F=⟨ψ(1)|O1|ψ(1)⟩∏j=2r⟨ψ(j)|Oj|ψ(j)⟩
where |ψ(j)⟩ are copies of trial states, and Oj are operators. The cost C=∑kℜ{Fk} is minimized similarly, often via approximations due to the inherent linearity of quantum mechanics.  This image provides an overview of VQA applications, including circuit representations for optimization tasks.

Challenges like barren plateaus (where gradients vanish) can affect trainability, often mitigated by problem-specific ansatzes. For more on implementations, see experimental setups in photonic or superconducting qubits.

#### Quantum Machine Learning Algorithms:

Algorithms like HHL (Harrow-Hassidim-Lloyd) solve linear systems exponentially faster, aiding tasks in data analysis and AI.

* Mechanism: Encodes matrices into quantum states and uses phase estimation.
* Applications: Solving differential equations in fluid dynamics or recommendation systems. Recent 2026 developments include Google's Quantum Echo algorithm for interpreting NMR spectra in biomedical applications.

#### Applications and Impact

Quantum algorithms promise breakthroughs in cryptography, drug discovery (via molecular simulations), logistics (optimization), and climate modeling. For example, in 2025, IonQ demonstrated a fault-tolerant version of Shor's on trapped-ion qubits, factoring 2048-bit numbers in simulations. In January 2026, D-Wave announced its acquisition of Quantum Circuits Inc., planning to release superconducting gate-model systems later in the year, enabling broader annealing and gate-based applications. Additionally, Microsoft and Atom Computing are set to deliver an error-corrected quantum computer to Denmark's Novo Nordisk Foundation in 2026, focusing on fault-tolerant simulations for pharmaceutical research. QuEra plans to make its error-correction-ready machine available globally this year, advancing neutral atom-based algorithms.

#### Challenges and Future Directions

* Error Correction: Quantum error-correcting codes (e.g., surface codes) are essential but require thousands of physical qubits per logical qubit. In 2026, research intensifies, with QuEra and Atom Computing leading deliveries of error-corrected systems.
* Scalability: As of 2026, systems like IBM's Eagle (127 qubits) and Google's Bristlecone successors are advancing, but full fault-tolerance is projected for the 2030s. D-Wave's January breakthrough in scalable technology aims to address this with hybrid gate-model and annealing approaches.
* Hybrid Approaches: Combining quantum with classical computing mitigates current hardware limitations, as seen in cloud platforms from AWS Braket and Microsoft Azure Quantum. Trends in 2026 emphasize hybrid quantum-classical infrastructures as industry standards.

In summary, quantum algorithms represent a paradigm shift, but their practical realization depends on overcoming decoherence and scaling hardware. Ongoing research focuses on algorithm-hardware co-design to unlock their full potential, with 2026 marking key milestones in error correction and industrial adoption.

## Promising Age of Quantum Computing

Photonic quantum computers are currently prominent contenders in fault-tolerant quantum computation (FTQC). These advanced architectures utilize photons as the medium for qubit encoding and manipulation [586], exhibiting inherent resilience against decoherence and noise, even at room temperature. This makes them exceptionally well-suited for scalable and FTQC. Photonic quantum computing also stands out for enabling the construction of modular, easily networked quantum computers, holding significant potential for practical applications [587][588]. Many scientists believe that the first thoughts about quantum computers emerged with the 1982 lecture by Richard Feynman [416][417]. Feynman had envisioned the possibility of creating a quantum machine that can reproduce quantum physics on the basis of the principles of quantum mechanics. In Feynman’s conception, computers compatible with the basic principles of quantum mechanics may be needed to model natural phenomena because "Nature is fundamentally quantum mechanical" [418].
The development of quantum computers has revealed many possibilities for such thoughts to be translated into reality because they are able to utilize the vast calculation capabilities needed to model quantum systems in a way that takes advantage of the properties offered by quantum mechanics, including superposition, interference, and entanglement [419]. The pace of progress in developing a physical quantum computer was glacial, due in part to difficult technical difficulties that make it difficult to shield and consistently control the dynamics of the quantum mechanical properties manifested at such very basic scales of nature as electron spin or photon polarization [420].

### From Classical to Quantum Optimization

Classical optimization algorithms face limits in speed and scalability, especially for complex problems. Quantum Optimization Algorithms (QOAs) solve this by converting problems into quantum Hamiltonians and finding the lowest energy state as the best solution. Using quantum effects like superposition, they explore many solutions at once. QOAs can also run on NISQ devices, which blend quantum and classical computing for practical use. The table below shows a comparison between classical and quantum optimization:

Feature  | Classical Optimization  | Quantum Optimization

|
|

Basic Unit  | Bit (0 or 1)  | Qubit (0 and 1 simultaneously)
Computation Type  | Sequential  | Parallel (via superposition)
Speed  | Limited for large data  | Potential exponential speedup
Scalability  | Hardware-dependent  | Promising but experimental
Current Status  | Mature  | Emerging and evolving

## Quantum Algorithms

Recent survey work synthesizes how quantum algorithms map onto real-world application areas, such as chemistry, optimization, cryptography, machine learning, and finance carefully weighing theoretical speedups against practical resource costs and engineering constraints. Dalzell [507] provide a useful, application-oriented perspective that emphasizes subtle caveats regarding when quantum advantage actually materializes and the need for end-to-end complexity considerations [507]. Contemporary literature stresses that theoretical asymptotics (e.g., Shor’s exponential speedup) must be evaluated alongside requirements for fault tolerance, qubit counts, and realistic gate/noise budgets[508][506][509]. Beyond these, Huynh et al. [510] explore quantum-inspired machine-learning approaches that bridge classical and quantum paradigms, offering hybrid algorithms deployable on today’s near-term hardware [510]. Grigoryan et al. [511] provides a comprehensive review of quantum-computing models, including gate-based, adiabatic, and measurement-based approaches,analyzing their algorithmic implications and domain specific applications [511]. Industry-oriented analyses published in EPJ Quantum Technology [512] emphasize how algorithmic progress interacts with hardware engineering, highlighting persistent gaps between theoretical quantum advantage and practical scalability [512]. Additionally, the Quantum Algorithm Zoo [513] serves as an evolving catalogue of hundreds of quantum algorithms, classified by domain and computational model, offering researchers a living reference for tracking progress across the field [513]. Together, these surveys illustrate that while theoretical speedups remain intellectually compelling, their translation into practical advantage depends critically on hardware maturity, hybrid algorithm design, and integrated benchmarking frameworks. Quantum computers work by within the rules of quantum mechanics to overcome problems that regular computers struggle with. They've evolved, from early ideas rooted in quantum physics to practical uses in computer science today [417]. Building a full-scale, industrial quantum computer is a big deal; it could shake up fields like cybersecurity and beyond.

The first real quantum algorithm that outpaced classical ones came from Daniel Simon [447]. Then came others like the Deutsch-Jozsa algorithm, which tackles problems needing tons of queries exponentially faster, basically, it cuts down the computing grunt work to check if algorithms are balanced or robust. The Bernstein-Vazirani algorithm solves "black-box" puzzles efficiently, Simon's speeds up certain computations, and Shor's is cracking integer factorization and discrete logarithm problems [419]. All these rely on the quantum Fourier transform.

Grover's algorithm is developed for searching unstructured databases to find specific items, and quantum counting handles broader searches. Both use "amplitude amplification," that boosts quantum computers ability to solve problems way faster than old-school methods. This technique powers other quantum fields, like machine learning, simulations, and advanced searches.

More recently, there's the quantum approximate optimization algorithm, which focuses on graph theory problems [448]. It mixes quantum and classical computing in a hybrid setup.

Basically, quantum software comes down to two main models that shape programs and their use: the quantum gate model [449] and quantum annealing [450].

The gate model is like a quantum version of classical logic gates. It manipulates qubits (quantum bits) using gates that tap into cool quantum effects like superposition (being in multiple states at once) and entanglement (linked particles influencing each other instantly). It's an approach, using algorithms like Shor's or Grover's, so it has many applications. The challenge is decoherence, where quantum states fail quickly, so error correction is crucial.

Quantum annealing, is an approximate adiabatic quantum computing (which is equivalent to the gate model but specialized) used for optimization problems. Like the quantum system naturally settle into its lowest-energy state, water finding the lowest point in a landscape. It uses quantum tunneling to take barriers efficiently and coherent during the process. It is more flexible on errors because leaning on the system's behavior, making it resistent against some glitches.

## Emerging paradigm

Quantum computing is a new paradigm that draws on the following principles of quantum mechanics:
Quantum mechanics to tackle computational difficulties that cannot be addressed by classical computers. This article gives a brief introduction to the basic concepts of qubits, the unique properties of quantum mechanics including superposition, interference, uncertainty relations, superposition and entanglement, and the problem of creating scalable, fault-tolerant systems. It discusses important quantum algorithms and the possibilities.
Applications in areas such as cryptography, optimization, finance, chemistry, among many other including machine learning. It emphasizes the significance of verification frameworks for the verification of quantum programs’ reliability [518]. Literature reviews examples of significant contributions include a presentation on insights derived from recent surveys on quantum algorithms, qubit technologies, and software verification methods. A discussion about challenges that still need to be met, like correcting errors.[503]
Several key issues in modern micro-architecture design, such as overhead, hardware directions for future research.

## Quantum Advantage: From NISQ to Fault-Tolerance

Quantum advantage refers to scenarios where quantum computers perform tasks that classical computers cannot solve efficiently, as demonstrated in supremacy experiments involving random circuit sampling[504][430]. Current devices, known as NISQ systems (Noisy Intermediate-Scale Quantum), utilize hybrid quantum-classical algorithms to mitigate hardware limitations through classical optimization feedback[147][427]. While these systems excel in variational methods for simulation and optimization, they suffer from decoherence that limits computation time and introduces errors. Consequently, techniques such as probabilistic error cancellation and zero-noise extrapolation are essential[556][38].

### The NISQ Bridge and AI Integration

NISQ serves as a vital bridge between theoretical promise and practical utility. Quantum sensors and memories can exponentially enhance our ability to learn about physical systems, a claim recently validated in experimental settings[121][67]. For instance, quantum devices enable the efficient characterization of many-body physics by leveraging AI for noise mitigation and circuit optimization[132][133]. AI is now indispensable across the entire quantum stack: from qubit design and calibration to real-time error correction and the interpretation of complex output[155][159][161]. This hardware-algorithm co-design is crucial for overcoming "barren plateaus" in variational training landscapes[8][9].

### The Transition to Fault-Tolerant Computing (FTQC)

In contrast to the heuristic nature of NISQ, Fault-Tolerant Quantum Computing (FTQC) employs error-correcting codes to create reliable logical qubits from noisy physical ones, enabling scalable computation for high-complexity problems[2][524]. However, this transition requires massive overhead; executing algorithms such as Shor’s for cryptography may necessitate millions of physical qubits[552][518]. The path toward full FTQC involves intermediate partial error correction phases, where users dynamically allocate resources between corrected and uncorrected qubits to optimize performance based on available hardware[145][339].

### Future Outlook and Machine Learning

NISQ research focuses on Quantum Machine Learning (QML) using kernel methods and generative models, while FTQC offers speedups in linear algebra and molecular simulations[25][98]. Advanced techniques like shadow tomography provide insights into the nature of these quantum speedups[7][8]. Hardware remains fragile and error-correction overhead is a significant barrier[31][32], innovations in 2025–2026 provide steady progress toward practical utility[25][26]. Progress will require a multidisciplinary approach where hardware, software, and AI-driven fault tolerance use the potential of quantum mechanics[33][55][501].

## Key NISQ Algorithms

### Variational Quantum Eigensolver (VQE) in Quantum Chemistry

The Variational Quantum Eigensolver (VQE) is a hybrid algorithm designed for estimating the ground state energies of molecular Hamiltonians on NISQ hardware[431][113]. By combining a parameterized quantum circuit (ansatz) with classical optimization loops, VQE minimizes energy expectations in a noise-resistant manner, making it highly suitable for near-term molecular simulations[432][117].

### Ansätze and Architectural Innovations

VQE leverages Parameterized Quantum Circuits (PQCs) with ansätze inspired by the underlying physics of the problem, such as the Unitary Coupled Cluster (UCC) for electronic structure calculations[35][146]. To improve resource efficiency, Adaptive VQE variants dynamically build circuits by adding operators one at a time, which significantly reduces the quantum hardware requirements compared to fixed-depth circuits[121][344]. Furthermore, subspace expansions have extended VQE’s utility beyond ground states to include excited states, broadening its potential for applications in drug discovery and materials science[68][123].

### AI-Driven Optimization and Training

A primary challenge in VQE is the "barren plateau" problem, regions in the optimization landscape where gradients vanish, making training difficult[551][119]. AI integration, particularly through reinforcement learning and surrogate models, has become essential for navigating these landscapes and optimizing parameters effectively[125][97]. These AI-boosted strategies improve trainability and help mitigate the effects of hardware noise, allowing for more accurate approximations of electronic structures[6][95].

### Recent Advancements (2025–2026)

As of 2025, advancements in AI-boosted VQE have enabled more sophisticated molecular dynamics simulations and real-time applications[131][152]. While early experiments focused on small molecules like , the integration of advanced error mitigation and hybrid time-evolution methods by 2026 has allowed for the simulation of increasingly larger and more complex systems[16][119][19]. These evolving hybrid tools continue to transform quantum chemistry, moving the field toward high-precision modeling and practical industrial utility[109][111].

## Quantum Approximate Optimization Algorithm (QAOA)

The Quantum Approximate Optimization Algorithm (QAOA) is a leading variational framework designed to solve combinatorial optimization problems, such as Max-Cut, by mapping them onto Ising Hamiltonians[448][163]. The algorithm operates by applying alternating layers of a problem Hamiltonian (which encodes the cost function) and a mixer Hamiltonian (which drives transitions between states)[522][127]. Because of its relatively shallow circuit depth, QAOA is particularly well-suited for the noisy environments of NISQ hardware[175][176].

### Optimization Landscapes and Training Strategies

A central challenge in QAOA is the high-dimensional parameter landscape, which is often riddled with local minima that can trap classical optimizers[131][167]. To ensure convergence to a global optimum, researchers employ advanced strategies such as:

* Recursive QAOA (RQAOA): This variant improves scalability by iteratively reducing the problem size, effectively eliminating variables until the remaining problem can be solved classically or with minimal quantum resources[574][181].
* Warm-Start and Initialization: Convergence is highly sensitive to initial parameters; "warm-start" strategies and reinforcement learning are increasingly used to provide high-quality starting points[410][191][119].
* AI-Enhanced Tuning: By 2026, AI meta-learning and generative flow networks have become standard tools for exploring parameter spaces and automating circuit synthesis[169][183].

### Digital-Analog Approaches and Hardware Awareness

To maximize efficiency, QAOA has evolved toward hardware-aware designs, such as digital-analog QAOA[149][189]. This approach combines the flexibility of digital gates with the continuous time-evolution of analog simulation, significantly reducing the error rates associated with fully digitized circuits[574][165]. These innovations, alongside robust error mitigation, allow for higher performance ratios on graph-based optimization problems compared to traditional gate-based methods[123][177].

### Practical Feasibility and 2025–2026 Milestones

As of 2025, benchmarks have demonstrated the feasibility of QAOA on 30-qubit systems for real-world applications in logistics and finance, such as portfolio optimization[212][171][185]. While classical heuristics remain competitive, the integration of AI for parameter tuning and the rise of hardware-specific variants are positioning QAOA as a viable tool for complex supply chain modeling and financial risk assessment in the near-term quantum era[173][160][162].

## Amplitude Amplification

Amplitude amplification is a fundamental quantum primitive and a generalization of Grover’s algorithm. It works by iteratively increasing the probability amplitude of "target" states while suppressing undesired ones, effectively providing a quadratic speedup for unstructured searches and sampling tasks[508][195]. In the NISQ era, this technique has evolved from a theoretical search tool into a critical component for data processing and state preparation[50][197].

### Integration with NISQ and QML

In the context of Quantum Machine Learning (QML), amplitude amplification is used to enhance kernels and assist in high-dimensional data encoding[132][199].

* Anomaly Detection: By amplifying outlier states, the algorithm aids in identifying rare patterns within complex datasets[150].
* Hybrid Frameworks: It is frequently integrated with variational circuits to prepare inputs for algorithms like Quantum Principal Component Analysis (QPCA) or to enhance the results of sampling tasks without the immediate need for Quantum RAM (QRAM)[211][221][219].

### Overcoming Noise and Decoherence

The primary limitation of amplitude amplification in the NISQ regime is that each iteration (or "Grover rotation") increases the circuit depth. Dephasing and gate errors accumulate, eventually causing the fidelity of the amplified state to collapse after a certain number of iterations[41][203].

To counter these effects, AI-assisted implementations and "quantum-inspired" variants have emerged. These methods use machine learning to learn robust encodings and optimize the number of amplification steps, ensuring that the process remains productive despite the hardware's inherent noise[66][18][213].

### Application-focused and gap analyses

Domain-specific surveys and studies across finance, chemistry, logistics, and machine learning emphasize persistent gaps between theoretical quantum advantage and experimental feasibility. While algorithmic proposals demonstrate promising asymptotic speedups, end-to-end resource analyses are frequently incomplete, and assumptions about idealized, error-free hardware dominate much of the literature [514][515]. Verification and benchmarking remain at an early stage, with limited experimental validation and inconsistent reporting of quantum resources [507][511][516]. Recent reviews have highlighted that realistic quantum advantage demands hardware-software co-design, integrating insights from algorithm development, quantum control engineering, and compiler optimization [517][512][518][519]. Studies in finance and logistics note that problem encodings and quantum data-loading overheads often offset theoretical speedups, calling for transparent resource estimation frameworks [514][520]. Similarly, in quantum chemistry and materials science, Grigoryan et al. (2025) and related works underscore the necessity of aligning algorithmic complexity with hardware noise and decoherence limits [511][521]. Emerging meta-analyses propose standardized benchmarking and reproducibility protocols for quantum algorithms such as the QBench and QPack initiatives which aim to quantify algorithmic efficiency relative to hardware constraints [522]. Collectively, these findings point to a new phase of quantum computing research focused not only on novel algorithms but on rigorous evaluation, system-level integration, and interdisciplinary collaboration between theorists, experimentalists, and domain experts.

### Applications and 2025–2026 Trends

By 2025, amplitude amplification has become central to Gaussian Boson Sampling for complex statistical modeling and device characterization speedups[132][217].

* Real-time Processing: Emerging hybrids are now capable of real-time applications in high-dimensional data processing, bypassing the traditional "bottleneck" of data loading[129][215].
* Dequantization Risks: Researchers remain cautious of "dequantization", where classical algorithms are discovered that match the quantum speedup, motivating a shift toward applications that offer the most robust theoretical advantages[207][208].

As the field moves toward fault-tolerance, the lessons learned in making amplitude amplification noise-resilient are expected to form the basis for high-fidelity quantum search and sampling in future scalable systems[209][416].

## Foundational Classics

## Grover's Algorithm and Search Optimization

Grover’s algorithm is a cornerstone of quantum computing, providing a mathematically proven quadratic speedup for unstructured search problems. By iteratively applying a quantum oracle and a diffusion operator, the algorithm amplifies the probability amplitudes of "target" states within a database, allowing a search of items in approximately steps[528][486].

### Challenges in the NISQ Era

While theoretically robust, Grover’s algorithm faces significant hurdles on NISQ devices due to the requirement for high-precision gates and long coherence times. Each "Grover iteration" increases the circuit depth, making the algorithm highly susceptible to hardware noise and dephasing[411][412]. To overcome these physical constraints, researchers utilize:

* Quantum-Inspired Variants: These algorithms mimic quantum logic on classical hardware or utilize simplified quantum circuits to achieve near-quantum performance without the full coherence requirements[50][161].
* AI-Assisted Compression: AI techniques are increasingly employed to compress and optimize Grover circuits, reducing the gate count and making the algorithm more resilient to the "noise floor" of current hardware[566][577].

### Applications in Machine Learning and Kernels

Beyond simple database searches, Grover’s algorithm serves as a foundational primitive for Quantum Machine Learning (QML).

* Feature Selection: Grover-type primitives are used to build QML kernels that efficiently identify the most relevant features in high-dimensional datasets[560][174].
* Optimization Subroutines: The algorithm is frequently used as a subroutine within broader hybrid quantum-classical optimization frameworks to speed up the search for global minima[135][139].

### 2026 Outlook and Dequantization

As of 2026, the focus has shifted toward hybrid Grover-based methods that combine quantum search with classical post-processing to maintain a competitive advantage over "dequantized" classical algorithms (classical algorithms inspired by quantum logic that attempt to match their speed)[135][136]. Projections for late 2026 suggest that these hybrid approaches will become standard in advancing QML kernel methods, particularly for complex data processing tasks where high-dimensional feature selection is critical[161][162][137].

## Shor's Algorithm and Cryptographic Implications

Shor’s algorithm is perhaps the most famous quantum algorithm, providing an exponential speedup for integer factorization. By exploiting the Quantum Fourier Transform (QFT) to find the period of a function, it can factorize large integers in polynomial time, a task that is practically impossible for the most powerful classical supercomputers using current methods[457][572].

### The Cryptographic Threat

The primary significance of Shor's algorithm lies in its ability to break RSA cryptography, which secures the majority of modern digital communications. This threat has become the primary driver for the global transition toward Post-Quantum Cryptography (PQC) standards and the development of Quantum Key Distribution (QKD) hybrids to ensure long-term data security[570][145][513].

### Modern Cryptography

The advent of quantum computers heralds a new ground-breaking era within the realm of data integrity and cybersecurity. With improving scalable computing power, quantum computers can effortlessly break the security of traditional cryptosystems, relying on factorization and discrete logarithms, both of which are considered hard problems for classical computers. By contrast, quantum computers have efficient processing capabilities to solve these hard problems within polynomial time [589]. For example, an adversary equipped with a quantum computer may break the RSA(Rivest-Shamir-Adleman) security in polynomial time by exploiting Shor’s algorithm for factoring large numbers. It is clear that such a possibility, despite not yet practical, poses potential threats to the integrity of communication networks [590] that need to be analyzed and mitigated. In fact the potential threat represented by the Shor’s algorithm has led to new developments in classical cryptographic approaches, with the work on post-quantum cryptography and on a completely new paradigm to grant security named quantum cryptography [591], or more precisely Quantum-Key Distribution (QKD). The novelty of QKD is that, instead of adding layers of security based on conventional (i.e. computationally hard to solve) algorithms, it uses fundamental properties of quantum particles to protect information from unauthorized parties. QKD protocols, are themselves composite algorithms where transmission of quantum signals, encryption/decryption, signatures, authentication, and hashing are all combined [592] to achieve (theoretically) unconditional security.

(Illustrative visuals from Wikimedia Commons related to Shor's algorithm and Quantum Key Distribution protocols.)

### Resource Estimates for 2026

Shor’s algorithm has been demonstrated on NISQ hardware for instances (factoring small numbers), it is not yet viable for industrial-scale decryption. As of 2026, the scientific community has the following resource requirements for a fault-tolerant implementation:

* The Qubit Gap: Projections for 2026 estimate that factorizing a standard 2048-bit RSA key would require approximately  (one million) physical qubits when using surface codes for error correction[426][339][518].

* AI-Driven Optimization: To bring these numbers down, AI is now extensively used to perform automated circuit optimization and to refine resource estimations. Machine learning models identify the most efficient gate sequences, potentially reducing the physical qubit overhead required for the modular exponentiation step, the most "intensive" part of the algorithm[505][523][565].

### Current State and Hybrid Security

In the current 2025–2026 landscape, Shor’s algorithm remains a "future-facing" threat. The hardware capable of running a full-scale version, has already forced an evolution in security standards:

* NISQ Limitations: On current devices, only small-scale factorization is possible, serving primarily as a benchmark for qubit quality and gate fidelity[502][513].
* Security Evolution: The focus has shifted to "Harvest Now, Decrypt Later" protection, where communications are increasingly secured using hybrid protocols that combine classical PQC with quantum-resistant hardware layers[426][523].

## Emerging 2025–2026

### Quantum Echoes

A schematic diagram of the present echo-generation process. (a) An electron-hole pair is created by a photo-excitation pulse. (b) After the excitation, the electron and hole move in opposite directions. (c) A driving electric-field pulse reverses the relative velocity of the electron-hole pairs, resulting in the recombination of the pairs and the emission of echo pulses. Credit: Atsushi Ono

Quantum echoes restore coherent states in noisy quantum systems, using AI-driven error mitigation to extend coherence times on NISQ devices[349][287]. Applications include enabling longer computations and supporting deeper algorithms such as time-dependent simulations[269][407]. Advances reported in 2025 include reinforcement learning–based feedback mechanisms for real-time error correction[404][405].

Quantum computers have successfully run a verifiable algorithm that surpasses the ability of supercomputers. Quantum verifiability means the result can be repeated on our quantum computer, or any other of the same caliber, to get the same answer, confirming the result. This repeatable, beyond-classical computation is the basis for scalable verification, bringing quantum computers closer to becoming tools for practical applications.[585]

New technique works like a highly advanced echo. It sends a carefully crafted signal into a quantum system (qubits on Willow chip), perturb one qubit, then precisely reverse the signal’s evolution to listen for the "echo" that comes back.

This quantum echo is special because it gets amplified by constructive interference , a phenomenon where quantum waves add up to become stronger. This makes our measurement incredibly sensitive.

Original expansion: Emerging from quantum error correction (QEC) research, quantum echo techniques integrate with neural quantum states for modeling condensed matter systems[406][150]. Bayesian inference methods are used to optimize open-system dynamics[399][400]. Scalability challenges are addressed through hybrid quantum–classical approaches[401][402]. Preserving coherence is a key requirement for quantum network architectures anticipated by 2026[403][289].

From sources: Quantum echoes restore coherent states in noisy systems[349][287] by using AI-based error mitigation[269][407]. They are applied in NISQ devices to achieve longer coherence times[404][405]. Integration with neural quantum states enables advanced simulations[406][150]. Bayesian methods optimize system dynamics[399][400] and support progress in condensed matter research[401][402]. Scalability remains a challenge[403][289]. Hybrid quantum networks projected for 2026 rely critically on coherence preservation[290][291][292][293].

Sample-based quantum diagonalization (SQD) overview

### Sample-Based Diagonalization

Sample-based diagonalization estimates eigenvalues through sampling techniques, optimized with machine learning for hybrid simulations on NISQ devices[581][344]. It promises improved efficiency in 2025–2026 for applications in quantum chemistry and optimization, employing approaches such as Fourier Neural Operators to model system dynamics[248][2]. The method builds on shadow tomography, with AI surrogate models used to bypass noisy quantum hardware[406][41].

Original perspective: As a potential post-NISQ tool, sample-based diagonalization supports multidisciplinary algorithm discovery[66][579]. Quantum-specific foundation models are being developed to learn reusable primitives[411][412]. Diffusion-based techniques assist circuit synthesis[174][410]. Efficiency gains are expected to grow in hybrid quantum–classical workflows[294][295].

From sources: Sample-based diagonalization estimates eigenvalues via sampling[581][344] and is optimized with machine learning for hybrid simulations[248][2]. It is projected to be efficient for chemistry and optimization tasks in 2025–2026[406][41]. The approach builds on shadow tomography[66][579]. AI surrogate models help bypass noisy hardware[411][412]. Multidisciplinary collaboration supports continued algorithm discovery[174][410], with growing efficiency for chemical applications[294][295][296][297].

## Visuals

### Circuit Diagrams

Circuit diagrams illustrate quantum gates and qubits, for example a VQE circuit with ansatz layers[367][114]. QAOA parameterized mixers and Hamiltonians are also visualized[212][298]. These diagrams help learners understand quantum operations[299][300].

From sources: Circuit diagrams illustrate gates and qubits[367][114]. For example, a VQE circuit with ansatz layers[212][298]. QAOA parameter layers are also depicted[299][300].

### VQE Energy Landscapes

Energy landscapes depict the optimization paths of variational algorithms, with barren plateaus visualized using machine learning tools[98][8]. AI-generated optimization paths highlight convergence behavior[9]. Visualization of these landscapes provides insight into training dynamics and algorithm performance.

From sources: Energy landscapes show optimization paths[98][8], including barren plateaus[9]. (Illustrative visuals from Wikimedia Commons related to quantum potential wells, analogous to energy landscapes in VQE.)

### QAOA Parameter Landscapes

Parameter landscapes visualize cost functions for tuning, enhanced by AI meta-learning[212][229]. Landscapes guide parameter selection[13] Multi-dimensional costs intuitive for learners.

From sources: Parameter-landscapes visualize cost-functions[212][229]. For tuning[13]. (Illustrative visuals from Wikimedia Commons related to quantum annealing landscapes, analogous to parameter landscapes in QAOA.)

## Formulas (Gates)

Basic gates such as Hadamard and CNOT, optimized with RL for NISQ[114][367]. RL optimizes gate sequences[409][310]. NISQ suited gates for efficiency[311][312]. From sources: Basisgates like Hadamard and CNOT[114][367].

### Quantum gates

are the building blocks of quantum circuits, analogous to logic gates in classical computing. Basic gates like the Hadamard (H) and Controlled-NOT (CNOT) are fundamental for creating superposition and entanglement, respectively. In the NISQ era, these gates are often optimized using reinforcement learning (RL) to minimize errors and improve efficiency on noisy hardware [114][367]. RL algorithms can dynamically select and sequence gates to adapt to device-specific noise profiles, reducing circuit depth and enhancing fidelity [409][310]. For NISQ-suited implementations, gates are chosen for their low error rates and compatibility with limited coherence times, prioritizing single-qubit rotations and two-qubit entanglers [311][312].

(Illustrative visuals from Wikimedia Commons showing Hadamard gate, CNOT gate, and a quantum circuit involving both.)

### Hadamard Gate (H)

The Hadamard gate applies a uniform superposition to a single qubit, transforming |0⟩ to (1/√2)(|0⟩ + |1⟩) and |1⟩ to (1/√2)(|0⟩ - |1⟩). Its matrix representation in the computational basis is: H = (1/√2)

1 | 1

|

1 | -1

Action on basis states:
\- H |0⟩ = (1/√2) (|0⟩ + |1⟩) \- H |1⟩ = (1/√2) (|0⟩ - |1⟩)

In RL-optimized NISQ circuits, H gates are often interleaved with error-mitigating sequences to preserve coherence [114][36] Controlled-NOT Gate (CNOT). The CNOT gate is a two-qubit entangling gate that flips the target qubit if the control qubit is in |1\rangle. It is essential for creating multi-qubit correlations. The matrix in the computational basis (|00⟩, |01⟩, |10⟩, |11⟩) is:

CNOT =

1 | 0 | 0 | 0

|
|
|

0 | 1 | 0 | 0
0 | 0 | 0 | 1
0 | 0 | 1 | 0

Action:

Control on first qubit: \text{CNOT} |x\rangle |y\rangle = |x\rangle |y \oplus x\rangle (where \oplus is modulo-2 addition).

RL optimization refines CNOT sequences by learning noise-resilient decompositions, often reducing two-qubit gate counts for NISQ efficiency [40] From sources: Basic gates like Hadamard and CNOT are highlighted for their role in foundational circuits [114][36]. These formulas provide the mathematical core, while RL adaptations address practical NISQ constraints. For implementation, see tools like Qiskit or PennyLane

### Measurements (Readout)

Readout errors reduced with ML, including neural decoders for codes[340][41]. Enhances fidelity via neural networks[2][313]. Readout improves accuracy[314][315].

From sources: Readout errors reducing with ML[340][41]. (Illustrative visuals from Wikimedia Commons related to quantum measurement circuits and parity measurements, analogous to readout error correction processes.)

## Further Reading

See reviews on QML and challenges, including 2025 surveys on NISQ innovations and quantum diplomacy[437][421]. Challenge overviews for future directions[577][518]. 2025 NISQ surveys[316][317]. From sources: See reviews at QML[437][421]. and challenges[577][518].

## Cross Links

### Noisy Qubits (Error Impact)

Noise impacts the performance of quantum algorithms, making error mitigation essential through AI-based decoders and virtual distillation techniques[270][340]. Decoders can effectively reduce errors[145][307], while virtual distillation helps purify quantum states[308][309].

From sources: Noise affects algorithm performance[270][340], making mitigation essential[145][307].

# See also

# Table of contents (185 articles)

# Index

Core theory

1\. Foundations

2\. Conceptual and interpretations

3\. Mathematical structure and systems

4\. Atomic and spectroscopy

5\. Wavefunctions and modes

6\. Quantum dynamics and evolution

7\. Measurement and information

8\. Quantum information and computing

Applications and extensions

9\. Quantum optics and experiments

10\. Open quantum systems

11\. Quantum field theory

12\. Statistical mechanics and kinetic theory

13\. Condensed matter and solid-state physics

14\. Plasma and fusion physics

15\. Timeline

16\. Advanced and frontier topics

Book II

* Matter by scale

Book III

* Methods and tools

Book IV

* Data Analysis Techniques

# Full contents

1\. Foundations (11) ↑ Back to index

1\. Physics:Quantum basics

2\. Physics:Quantum Postulates

3\. Physics:Quantum Hilbert space

4\. Physics:Quantum Observables and operators

5\. Physics:Quantum mechanics

6\. Physics:Quantum mechanics measurements

7\. Physics:Quantum state

8\. Physics:Quantum system

9\. Physics:Quantum superposition

10\. Physics:Quantum probability

11\. Physics:Quantum Mathematical Foundations of Quantum Theory

2\. Conceptual and interpretations (14) ↑ Back to index

12\. Physics:Quantum Interpretations of quantum mechanics

13\. Physics:Quantum Wave–particle duality

14\. Physics:Quantum Complementarity principle

15\. Physics:Quantum Uncertainty principle

16\. Physics:Quantum Measurement problem

17\. Physics:Quantum Bell's theorem

18\. Physics:Quantum Hidden variable theory

19\. Physics:Quantum nonlocality

20\. Physics:Quantum contextuality

21\. Physics:Quantum Darwinism

22\. Physics:Quantum A Spooky Action at a Distance

23\. Physics:Quantum A Walk Through the Universe

24\. Physics:Quantum The Secret of Cohesion and How Waves Hold Matter Together

25\. Physics:Quantum measurement problem

3\. Mathematical structure and systems (13) ↑ Back to index

26\. Physics:Quantum Density matrix

27\. Physics:Quantum Exactly solvable quantum systems

28\. Physics:Quantum Formulas Collection

29\. Physics:Quantum A Matter Of Size

30\. Physics:Quantum Symmetry in quantum mechanics

31\. Physics:Quantum Angular momentum operator

32\. Physics:Quantum Runge–Lenz vector

33\. Physics:Quantum Approximation Methods

34\. Physics:Quantum Matter Elements and Particles

35\. Physics:Quantum Dirac equation

36\. Physics:Quantum Klein–Gordon equation

37\. Physics:Quantum pendulum

38\. Physics:Quantum configuration space

4\. Atomic and spectroscopy (14) ↑ Back to index

Quantum atomic structure and spectroscopy: orbitals, energy levels, and emission and absorption spectra.

39\. Physics:Quantum Atomic structure and spectroscopy

40\. Physics:Quantum Hydrogen atom

41\. Physics:Quantum number

42\. Physics:Quantum Multi-electron atoms

43\. Physics:Quantum Fine structure

44\. Physics:Quantum Hyperfine structure

45\. Physics:Quantum Isotopic shift

46\. Physics:Quantum defect

47\. Physics:Quantum Zeeman effect

48\. Physics:Quantum Stark effect

49\. Physics:Quantum Spectral lines and series

50\. Physics:Quantum Selection rules

51\. Physics:Quantum Fermi's golden rule

52\. Physics:Quantum beats

5\. Wavefunctions and modes (9) ↑ Back to index

A quantum wavefunction showing probability amplitude in space; the square of its magnitude gives the probability density.

53\. Physics:Quantum Wavefunction

54\. Physics:Quantum Superposition principle

55\. Physics:Quantum Eigenstates and eigenvalues

56\. Physics:Quantum Boundary conditions and quantization

57\. Physics:Quantum Standing waves and modes

58\. Physics:Quantum Normal modes and field quantization

59\. Physics:Number of independent spatial modes in a spherical volume

60\. Physics:Quantum Density of states

61\. Physics:Quantum carpet

6\. Quantum dynamics and evolution (17) ↑ Back to index

62\. Physics:Quantum Time evolution

63\. Physics:Quantum Schrödinger equation

64\. Physics:Quantum Time-dependent Schrödinger equation

65\. Physics:Quantum Stationary states

66\. Physics:Quantum Perturbation theory

67\. Physics:Quantum Time-dependent perturbation theory

68\. Physics:Quantum Adiabatic theorem

69\. Physics:Quantum Scattering theory

70\. Physics:Quantum S-matrix

71\. Physics:Quantum tunnelling

72\. Physics:Quantum speed limit

73\. Physics:Quantum revival

74\. Physics:Quantum reflection

75\. Physics:Quantum oscillations

76\. Physics:Quantum jump

77\. Physics:Quantum boomerang effect

78\. Physics:Quantum chaos

7\. Measurement and information (9) ↑ Back to index

79\. Physics:Quantum Measurement theory

80\. Physics:Quantum Measurement operators

81\. Physics:Quantum Projective measurement

82\. Physics:Quantum POVM

83\. Physics:Quantum Weak measurement

84\. Physics:Quantum Measurement collapse

85\. Physics:Quantum entanglement

86\. Physics:Quantum Zeno effect

87\. Physics:Quantum limit

8\. Quantum information and computing (10) ↑ Back to index

88\. Physics:Quantum information theory

89\. Physics:Quantum Qubit

90\. Physics:Quantum Entanglement

91\. Physics:Quantum Gates and circuits

92\. Physics:Quantum Computing Algorithms in the NISQ Era

93\. Physics:Quantum Noisy Qubits

94\. Physics:Quantum random access code

95\. Physics:Quantum pseudo-telepathy

96\. Physics:Quantum network

97\. Physics:Quantum money

9\. Quantum optics and experiments (5) ↑ Back to index

Experimental quantum physics: qubits, dilution refrigerators, quantum communication, and laboratory systems.

98\. Physics:Quantum Nonlinear King plot anomaly in calcium isotope spectroscopy

99\. Physics:Quantum optics beam splitter experiments

100\. Physics:Quantum Ultra fast lasers

101\. Physics:Quantum Experimental quantum physics

102\. Physics:Quantum optics

103\. Template:Quantum optics operators

10\. Open quantum systems (9) ↑ Back to index

104\. Physics:Quantum Open systems

105\. Physics:Quantum Master equation

106\. Physics:Quantum Lindblad equation

107\. Physics:Quantum Decoherence

108\. Physics:Quantum dissipation

109\. Physics:Quantum Markov semigroup

110\. Physics:Quantum Markovian dynamics

111\. Physics:Quantum Non-Markovian dynamics

112\. Physics:Quantum Trajectories

11\. Quantum field theory (20) ↑ Back to index

Structural dependency map of quantum field theory.

113\. Physics:Quantum field theory (QFT) basics

114\. Physics:Quantum field theory (QFT) core

115\. Physics:Quantum Fields and Particles

116\. Physics:Quantum Second quantization

117\. Physics:Quantum Harmonic Oscillator field modes

118\. Physics:Quantum Creation and annihilation operators

119\. Physics:Quantum vacuum fluctuations

120\. Physics:Quantum Propagators in quantum field theory

121\. Physics:Quantum Feynman diagrams

122\. Physics:Quantum Path integral formulation

123\. Physics:Quantum Renormalization in field theory

124\. Physics:Quantum Renormalization group

125\. Physics:Quantum Field Theory Gauge symmetry

126\. Physics:Quantum Non-Abelian gauge theory

127\. Physics:Quantum Electrodynamics (QED)

128\. Physics:Quantum chromodynamics (QCD)

129\. Physics:Quantum Electroweak theory

130\. Physics:Quantum Standard Model

131\. Physics:Quantum triviality

132\. Physics:Quantum confinement problem

12\. Statistical mechanics and kinetic theory (9) ↑ Back to index

133\. Physics:Quantum Statistical mechanics

134\. Physics:Quantum Partition function

135\. Physics:Quantum Distribution functions

136\. Physics:Quantum Liouville equation

137\. Physics:Quantum Kinetic theory

138\. Physics:Quantum Boltzmann equation

139\. Physics:Quantum BBGKY hierarchy

140\. Physics:Quantum Relaxation and thermalization

141\. Physics:Quantum Thermodynamics

13\. Condensed matter and solid-state physics (13) ↑ Back to index

142\. Physics:Quantum Band structure

143\. Physics:Quantum Fermi surfaces

144\. Physics:Quantum Semiconductor physics

145\. Physics:Quantum Phonons

146\. Physics:Quantum Electron-phonon interaction

147\. Physics:Quantum Superconductivity

148\. Physics:Quantum Topological phases of matter

149\. Physics:Quantum well

150\. Physics:Quantum spin liquid

151\. Physics:Quantum spin Hall effect

152\. Physics:Quantum phase transition

153\. Physics:Quantum critical point

154\. Physics:Quantum dot

14\. Plasma and fusion physics (8) ↑ Back to index

Conceptual illustration of plasma physics in a fusion context, showing magnetically confined ionized gas in a tokamak and the collective behavior governed by electromagnetic fields and transport processes.

155\. Physics:Quantum Fusion reactions and Lawson criterion

156\. Physics:Quantum Plasma (fusion context)

157\. Physics:Quantum Magnetic confinement fusion

158\. Physics:Quantum Inertial confinement fusion

159\. Physics:Quantum Plasma instabilities and turbulence

160\. Physics:Quantum Tokamak core plasma

161\. Physics:Quantum Tokamak edge physics and recycling asymmetries

162\. Physics:Quantum Stellarator

15\. Timeline (8) ↑ Back to index

163\. Physics:Quantum mechanics/Timeline

164\. Physics:Quantum mechanics/Timeline/Pre-quantum era

165\. Physics:Quantum mechanics/Timeline/Old quantum theory

166\. Physics:Quantum mechanics/Timeline/Modern quantum mechanics

167\. Physics:Quantum mechanics/Timeline/Quantum field theory era

168\. Physics:Quantum mechanics/Timeline/Quantum information era

169\. Physics:Quantum mechanics/Timeline/Quantum technology era

170\. Physics:Quantum mechanics/Timeline/Quiz

16\. Advanced and frontier topics (16) ↑ Back to index

171\. Physics:Quantum topology

172\. Physics:Quantum battery

173\. Physics:Quantum Supersymmetry

174\. Physics:Quantum Black hole thermodynamics

175\. Physics:Quantum Holographic principle

176\. Physics:Quantum gravity

177\. Physics:Quantum De Sitter invariant special relativity

178\. Physics:Quantum Doubly special relativity

179\. Physics:Quantum arithmetic geometry

180\. Physics:Quantum unsolved problems

181\. Physics:Quantum Yang-Mills mass gap

182\. Physics:Quantum gravity problem

183\. Physics:Quantum black hole information paradox

184\. Physics:Quantum dark matter problem

185\. Physics:Quantum neutrino mass problem

186\. Physics:Quantum matter-antimatter asymmetry problem

* Quantum computing

## Sources

* [1] Yunfei Wang and Junyu Liu 03-2024 QML concepts, NISQ techniques, and fault-tolerant approaches, including fundamental concepts, algorithms, and statistical learning theory.
* [2] Yuri Alexeev et al. 12-2025 How AI advances QC challenges across hardware and software, from device design to applications, highlighting future opportunities and obstacles.
* [3] Sukhpal Singh Gill et al. 04-2025 Examining foundations, visions, hardware advancements, quantum cryptography, software, and scalability, discussing challenges and trends in QC.
* [4] V. Raseena 12-2025 Quantum computing:foundations, algorithms, and emerging applications.
* [5] Variational Quantum Algorithms: From Theory to NISQ-Era Applications Challenges and Opportunities
* [6] A Survey on Quantum Optimization Algorithms. ijrpr, 2025.
* [7] Artificial intelligence for quantum computing. Nature Communications, 2025.
* [8] AI in Quantum Computing: Why NVIDIA-lead Researchers Say It's Key. The Quantum Insider, Dec 2025.
* [9] Quantum Computing Industry Trends 2025: A Year of Breakthrough Milestones and Commercial Transition. SpinQ, Oct 2025.
* [10] Quantum Computing: Vision and Challenges.
* [11] The year to become Quantum-Ready. Microsoft Azure Quantum Blog, Jan 2025.
* [12] The Year of Quantum: From concept to reality in 2025. McKinsey, Jun 2025.
* [13] Quantum computing: foundations, algorithms, and emerging applications. Frontiers in Quantum Science and Technology, 2025.
* [14] Unlocking the Power of Quantum Computing with Practical Benchmarking Tools. cs.lbl.gov, Dec 2025.
* [15] Quantum Echoes algorithm is a big step toward real-world applications for quantum computing. Google Blog, Oct 2025.

#  The Learning Quiz

## References

1. ↑ S. Aaronson. Read the fine print. Nature Physics, 11(4):291–293, 2015. doi:10.1038/nphys3270
2. ↑ 2.0 2.1 2.2 2.3 2.4 S. Aaronson. Shadow tomography of quantum states. In Proceedings of the 50th Annual ACM SIGACT Symposium on Theory of Computing (STOC 2018), pages 325–338, 2018. doi:10.1145/3188745.3188807
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508. ↑ 508.0 508.1 508.2 G. Brassard, P. Høyer, M. Mosca, and A. Tapp. Quantum amplitude amplification and estimation. arXiv:quant-ph/0005055, 2000. (No published DOI).
509. ↑ 509.0 509.1 M. Carter and E. Gheorghiu. QBench and QPack: frameworks for benchmarking quantum algorithms. ACM Trans. Quantum Comput., 6(3), 2024. doi:10.1145/3691041
510. ↑ 510.0 510.1 510.2 CertiQ. CertiQ – compiler/circuit equivalence verification. arXiv, 2025. (No published DOI).
511. ↑ 511.0 511.1 511.2 511.3 511.4 E. Chae, J. Choi, and J. Kim. An elementary review on basic principles and developments of qubits for quantum computing. Nano Converg., 11:11, 2024. doi:10.1186/s40580-024-00418-5
512. ↑ 512.0 512.1 512.2 512.3 S.-J. Chen and Y. Tsai. Quantum-safe networks for 6G. 2:1, 2025. doi:10.69709/caic.2025.102135
513. ↑ 513.0 513.1 513.2 513.3 513.4 J.-S. Chen, E. Nielsen, M. Ebert, V. Inlek, K. Wright, V. Chaplin, et al. Benchmarking a trapped-ion quantum computer with 30 qubits. Quantum, 8:1516, 2024. doi:10.22331/q-2024-11-07-1516
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515. ↑ 515.0 515.1 A. Chohan. A comparative review of quantum bits: superconducting, topological, spin, and emerging qubit technologies. SSRN Preprint, 2024. Available online at: https://papers.ssrn.com/sol3/papers.cfm?abstract_id=4979773. (No DOI).
516. ↑ 516.0 516.1 Colobridge Blog. Quantum computing 2025 — comparison of leading Qubit technologies. Available online at: https://blog.colobridge.net/wp-content/uploads/2025/09/Comparison-of-Leading-Qubit-Technologies_англ-1-1024x536.jpg, 2025. (No DOI, website).
517. ↑ 517.0 517.1 CoqQ. CoqQ – deductive verification framework for quantum programs. arXiv, 2025. (No published DOI).
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527. ↑ A. Grigoryan, S. Kumar, and P. R. Pinheiro. A review on models and applications of quantum computing. Computation, 7(3):39, 2025. doi:10.3390/computation7030039
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529. ↑ A. W. Harrow, A. Hassidim, and S. Lloyd. Quantum algorithm for linear systems of equations. Phys. Rev. Lett., 103:150502, 2009. doi:10.1103/PhysRevLett.103.150502
530. ↑ D. Herman. A survey of quantum computing for finance. arXiv:2201.02773, 2022. (No published DOI).
531. ↑ S. Hiremath, P. Kumar, and R. Singh. A literature survey on quantum computing in next generation challenges in circuit design and applications of future enabling technologies. Int. J. Comput. Appl., 187(1):1–8, 2025. (No DOI).
532. ↑ C. Huot, S. V. Heng, T.-K. Kim, and Y. Han. Quantum autoencoder for enhanced fraud detection in imbalanced credit card dataset. IEEE Access, 12:169671–169681, 2024. doi:10.1109/ACCESS.2024.3496901
533. ↑ L. Huynh, J. Hong, A. Mian, H. Suzuki, Y. Wu, and S. Camtepe. Quantum-inspired machine learning: a survey. arXiv preprint arXiv:2308.11269, 2023. (No published DOI).
534. ↑ IBM Quantum. IBM osprey: 433-qubit quantum processor. IBM Research Blog, 2023. Available online at: https://research.ibm.com/blog/ibm-osprey. (No DOI).
535. ↑ IBM Quantum. IBM quantum roadmap. IBM Research Blog, 2025. Available online at: https://research.ibm.com/blog/ibm-quantum-roadmap. (No DOI).
536. ↑ E. Khabiboulline, J. Romero, and A. Kubica. Bridging theory and implementation in quantum computing: challenges and opportunities. Commun. ACM, 68(7):72–81, 2025. doi:10.1145/3679312
537. ↑ D. Kim, Y. Yamamoto, and S. Gill. Photonic and hybrid quantum processors: integration challenges and opportunities. IEEE Trans. Quantum Eng., 6:5402311, 2025. doi:10.1109/TQE.2025.5402311
538. ↑ A. Kumar. A brief history of quantum computing. Medium, 2022. Available online at: https://miro.medium.com/v2/resize:fit:1400/1*N3CWuTrxr-tg27I2_9Xrmw.png. (No DOI).
539. ↑ R. Laflamme, E. Knill, and W. H. Zurek. Demonstration of a five-qubit quantum computer. IBM Research News Archive, 2001. (No DOI).
540. ↑ S. Lee, D. Patel, and Y. Wang. Quantum algorithms for quantum molecular systems: a survey. WIREs Comput. Mol. Sci., 15(2):e70020, 2024. doi:10.1002/wcms.70020
541. ↑ D. Lewis, S. Soudjani, and P. Zuliani. Formal verification of quantum programs: theory, tools and challenges. arXiv:2110.01320, 2021. (No published DOI).
542. ↑ C. Majdoubi, S. E. Mendili, and Y. Gahi. Quantum cryptology in the big data security era. Int. J. Adv. Comput. Sci. Appl., 15, 2024. doi:10.14569/ijacsa.2024.0150761
543. ↑ A. A. Mamun, A. Abrar, M. Rahman, M. S. Salek, and M. Chowdhury. Enhancing TCPS security: a shift to PQC. arXiv:2411.13023, 2024. (No published DOI).
544. ↑ T. Monroe. Recent advances in trapped-ion quantum processors. Nat. Photonics, 18:210–223, 2024. doi:10.1038/s41566-024-01987-y
545. ↑ J. O. Moral. Cybersecurity in critical infrastructures: a PQC perspective. 2024. (No DOI).
546. ↑ S. Morales, E. Leue, and F. Bauer-Marquart. Hybrid-QEC: integrating classical and quantum verification for fault-tolerant compilation. npj Quantum Inf., 11(33):1–14, 2025. doi:10.1038/s41534-025-00832-y
547. ↑ M. A. Nielsen and I. L. Chuang. Quantum Computation and Quantum Information. 10th Anniversary Edition. Cambridge University Press, 2010. doi:10.1017/CBO9780511976667
548. ↑ M. Ong, J. Kwon, and C. Schilling. Survey of reliability challenges in quantum software stacks: testing, debugging, and verification. arXiv preprint arXiv:2409.06741, 2024. (No published DOI).
549. ↑ D. J. Ott and C. Peikert. Post quantum cryptography migration. arXiv:1909.07353, 2019. (No published DOI).
550. ↑ M. Patel and R. Zurek. Integrated quantum hardware–software co-design: a roadmap toward practical quantum advantage. IEEE Access, 13:78512–78528, 2025. doi:10.1109/ACCESS.2025.3459821
551. ↑ 551.0 551.1 A. Peruzzo et al. A variational eigenvalue solver on a photonic quantum processor. Nat. Commun., 5:4213, 2014. doi:10.1038/ncomms5213
552. ↑ 552.0 552.1 J. Preskill. Quantum computing in the NISQ era and beyond. Quantum, 2:79, 2018. doi:10.22331/q-2018-08-06-79
553. ↑ PRISM. PRISM – probabilistic model checking tool. ACM Digital Library, 2025. (No DOI).
554. ↑ Qafny. Qafny – proof system for quantum programs. arXiv, 2025. (No published DOI).
555. ↑ QEC. Quantum equivalence checker – circuit equivalence tool. dcs.gla.ac.uk, 2025. (No DOI).
556. ↑ 556.0 556.1 QHLProver. QHLProver – Quantum hoare logic prover. arXiv, 2025. (No published DOI).
557. ↑ QPMC. QPMC – quantum program/protocol model checker. opus.lib.uts.edu.au, 2025. (No DOI).
558. ↑ IBM Quantum. Understanding multi-qubit interactions and two-qubit gates. IBM Quantum Learning, 2024. Available online at: https://quantum-computing.ibm.com. (No DOI).
559. ↑ IBM Quantum. IBM quantum system two architecture and modular scaling approach. IBM Research Blog, 2025. Available online at: https://research.ibm.com/blog/ibm-quantum-system-two. (No DOI).
560. ↑ 560.0 560.1 Quantum Algorithm Zoo. Comprehensive catalogue of quantum algorithms. Available online at: https://quantumalgorithmzoo.org/, 2025. (No DOI).
561. ↑ Quantum Computing. Pioneers of quantum computing. SlideShare Presentation, 2021. Available online at: https://image.slidesharecdn.com/quantumcomputing-211118090146/85/quantum-computing-5-320.jpg. (No DOI).
562. ↑ D. Reynolds, T. Bravyi, and S. Lloyd. The next decade of quantum computing: from concept to scalable systems. Nat. Rev. Phys., 7:530–547, 2025. doi:10.1038/s42254-025-00943-y
563. ↑ W. Robert. Cryptographic techniques for IoMT security. 2024. (No DOI).
564. ↑ N. Rodriguez-Alvarez and F. Rodriguez-Merino. Performance and storage analysis of CRYSTALS-Kyber. 2025. (No DOI).
565. ↑ 565.0 565.1 S. K. Sahu and K. Mazumdar. Analysis of quantum cryptography applications. 2024. (No DOI).
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567. ↑ A. Scrivano. Comparative study of classical and post-quantum algorithms. 2025. (No DOI).
568. ↑ N. Sharma, R. Verma, T. Patel, B. G. Wilkes, and B. Panuganti. Applications of quantum computing in clinical care. Front. Med., 12:1573016, 2025. doi:10.3389/fmed.2025.1573016
569. ↑ Y. Shi. CertiQ: a mostly-automated verification of a realistic quantum compiler. arXiv:1908.08963, 2019. (No published DOI).
570. ↑ 570.0 570.1 P. W. Shor. Polynomial-time algorithms for prime factorization and discrete logarithms on a quantum computer. arXiv:quant-ph/9508027, 1995. (No published DOI).
571. ↑ SMT solvers – Z3. Yices. ACM Digital Library, n.d. (No DOI).
572. ↑ 572.0 572.1 S. Sonko. Quantum cryptography and U.S. digital security. 2024. (No DOI).
573. ↑ STORM. STORM – probabilistic model checking tool. ACM Digital Library, 2025. (No DOI).
574. ↑ 574.0 574.1 574.2 V. Uotila, J. Ripatti, and B. Zhao. Higher-order portfolio optimization with quantum approximate optimization algorithm. arXiv preprint arXiv:2509.01496, 2025. (No published DOI).
575. ↑ Veri-QEC. Veri-QEC – verification framework for quantum error-correcting programs. arXiv, 2025. (No published DOI).
576. ↑ T. Vidal, F. Roser, and M. Lewis. Quantum optimization for finance and logistics: benchmarking real-world feasibility. npj Quantum Inf., 11(27), 2025. doi:10.1038/s41534-025-00790-4
577. ↑ 577.0 577.1 577.2 577.3 F. Zaman, S. Ali, M. Hussain, and A. Khan. A survey on quantum machine learning: current trends, challenges, opportunities, and the road ahead. arXiv preprint arXiv:2310.10315, 2023. (No published DOI).
578. ↑ E. Zeydan. Quantum technologies for beyond 5G and 6G networks. 2025. (No DOI).
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580. ↑ Q. Zhao and M. Kumar. Benchmarking and noise characterization for heterogeneous quantum hardware. Phys. Rev. Appl., 21(4):045012, 2024. doi:10.1103/PhysRevApplied.21.045012
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584. ↑ Unlocking the Power of Quantum Computing with Practical Benchmarking Tools. cs.lbl.gov, Dec 2025. URL https://cs.lbl.gov/news-and-events/news/2025/unlocking-the-power-of-quantum-computing-with-practical-benchmarking-tools/.
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586. ↑ O'Brien, Jeremy L. (7 December 2007). "Optical Quantum Computing". Science 318 (5856): 1567–1570. doi:10.1126/science.1142892. https://www.science.org/doi/10.1126/science.1142892.
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592. ↑ S. Pirandola and S. L. Braunstein, “Physics: Unite to build a quantum internet,” Nature, vol. 532, no. 7598, pp. 169–171, 2016.

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