The anticipated applications of quantum computers span across science and industry, ranging from quantum chemistry and many-body physics to optimization, finance, and machine learning. Proposed quantum solutions in these areas typically combine multiple quantum algorithmic primitives into an overall quantum algorithm, which must then incorporate the methods of quantum error correction and fault tolerance to be implemented correctly on quantum hardware. As such, it can be difficult to assess how much a particular application benefits from quantum computing, as the various approaches are often sensitive to intricate technical details about the underlying primitives and their complexities. Here we present a survey of several potential application areas of quantum algorithms and their underlying algorithmic primitives, carefully considering technical caveats and subtleties. We outline the challenges and opportunities in each area in an "end-to-end" fashion by clearly defining the problem being solved alongside the input-output model, instantiating all "oracles," and spelling out all hidden costs. We also compare quantum solutions against state-of-the-art classical methods and complexity-theoretic limitations to evaluate possible quantum speedups. The survey is written in a modular, wiki-like fashion to facilitate navigation of the content. Each primitive and application area is discussed in a standalone section, with its own bibliography of references and embedded hyperlinks that direct to other relevant sections. This structure mirrors that of complex quantum algorithms that involve several layers of abstraction, and it enables rapid evaluation of how end-to-end complexities are impacted when subroutines are altered.
Quantum state tomography (QST) is a fundamental task in quantum information science that aims to reconstruct unknown quantum states from measurement data. However, the exponential growth of Hilbert-space dimension with system size makes full tomography of general quantum states statistically and computationally prohibitive. This challenge has motivated extensive research on structured quantum state tomography, where prior structure, such as low-rankness, tensor-network representations, shallow quantum circuits, and neural quantum states, can substantially reduce the effective degrees of freedom and enable scalable recovery. In this review, we provide a unified perspective on QST for structured quantum states through three closely related themes: compact state representations, measurement design, and computational algorithms. After reviewing common models for structured quantum states, we survey existing work on geometric preservation properties of measurement frameworks, ranging from informationally complete POVMs to randomized measurements, and their implications for sample complexity. On the algorithmic side, we review optimization methods for reconstructing structured quantum states from empirical measurements. By connecting QST with broader principles from compressive sensing, matrix sensing, and structured inverse problems, this survey highlights common theoretical foundations underlying sample complexity, measurement efficiency, and scalable recovery.
The complexity class QMA(2), featuring two unentangled quantum provers, occupies a central yet poorly understood position in quantum complexity theory. While superficially a modest extension of QMA, unentanglement fundamentally changes both the algorithmic and complexitytheoretic landscape, connecting QMA(2) to deep questions in separability testing, polynomial optimization, and hardness of approximation. Over the past two decades, QMA(2) has inspired a rich body of work spanning restricted verification models, semidefinite and Sum-of-Squares hierarchies, and surprising links to classical problems such as the Unique Games Conjecture. This survey provides an overview of these developments, emphasizing structural insights, algorithmic techniques, and known barriers, while highlighting how progress on QMA(2) continues to illuminate the interplay between quantum information, convex optimization, and classical complexity theory.
The emergence of quantum computing threatens classical cryptographic systems, motivating the development of cryptographic solutions based on Post-Quantum Cryptography (PQC) and Quantum Key Distribution (QKD). This paper surveys hybrid key-establishment techniques combining PQC and QKD, including XOR/concatenation, KEM combiners, split usage, and cascade approaches, and evaluates their applicability to protocols such as TLS and IPsec. It further compares positions of major standardization bodies (ANSSI, BSI, NCSC, NIST, ETSI) and identifies key challenges in standardization, performance, scalability, and complexity. While not a universal solution, hybridization remains relevant for high-assurance environments.