Motivated by notions of quantum heuristics and by average-case rather than worst-case algorithmic analysis, we define quantum computational advantage in terms of individual problem instances. Inspired by the classical notions of Kolmogorov complexity and instance complexity, we define their quantum versions. This allows us to define queasy instances of computational problems, like e.g. Satisfiability and Factoring, as those whose quantum instance complexity is significantly smaller than their classical instance complexity. These instances indicate quantum advantage: they are easy to solve on a quantum computer, but classical algorithms struggle (they feel queasy). Via a reduction from Factoring, we prove the existence of queasy Satisfiability instances; specifically, these instances are maximally queasy (under reasonable complexity-theoretic assumptions). Further, we show that there is exponential algorithmic utility in the queasiness of a quantum algorithm. This formal framework serves as a beacon that guides the hunt for quantum advantage in practice, and moreover, because its focus lies on single instances, it can lead to new ways of designing quantum algorithms.
Motivated by notions of quantum heuristics and by average-case rather than worst-case algorithmic analysis, we define quantum computational advantage in terms of individual problem instances. Inspired by the classical notions of Kolmogorov complexity and instance complexity, we define their quantum versions. This allows us to define queasy instances of computational problems, like e.g. Satisfiability and Factoring, as those whose quantum instance complexity is significantly smaller than their classical instance complexity. These instances indicate quantum advantage: they are easy to solve on a quantum computer, but classical algorithms struggle (they feel queasy). Via a reduction from Factoring, we prove the existence of queasy Satisfiability instances; specifically, these instances are maximally queasy (under reasonable complexity-theoretic assumptions). Further, we show that there is exponential algorithmic utility in the queasiness of a quantum algorithm. This formal framework serves as a beacon that guides the hunt for quantum advantage in practice, and moreover, because its focus lies on single instances, it can lead to new ways of designing quantum algorithms.
Quantum algorithms with super-polynomial or polynomial speedups have been central to the promise of quantum computing, yet their real-world impact remains unclear because most analyses are asymptotic and worst-case. This work develops a systematic framework for characterizing practical quantum advantage that explicitly separates three components of speedup: (i) oracle/algorithmic advantage (Salg), (ii) data/instance structure advantage (Sdata), and (iii) implementability advantage under realistic noise and resource constraints (S impl). We first formalize quantum speedup as a triple (Salg, Sdata, S impl) that jointly captures theoretical complexity, distribution of input instances, and hardware feasibility. We then analyze seven canonical algorithms-Grover search, amplitude estimation, phase estimation, HHL, VQE, QAOA, and Hamiltonian simulationand extract common structural patterns that lead to genuine advantage over classical algorithms for specific problem families. Building on these patterns, we derive sufficient and necessary conditions for average-case quantum advantage in terms of problem symmetries, spectral properties, and data-access models. For NISQ-era devices, we incorporate realistic noise models and compilation overheads to define a practical advantage region in the space of problem size, error rates, and algorithm depth. The framework yields decision procedures and flowcharts that, given a candidate problem and hardware profile, can certify whether a proposed quantum algorithm is likely to achieve practical advantage or not. Case studies in finance (Monte Carlo risk analysis), chemistry (electronic structure estimation), and optimization (QAOA-type problems) illustrate how our method can guide algorithm and hardware co-design, and also show negative results where no advantage is achievable under realistic constraints. This work turns "quantum advantage" from a vague narrative into a quantitatively testable property of algorithmproblem-hardware triples.
We provide a rigorous theoretical analysis of quantum algorithms for shortest path problems in weighted graphs, establishing complexity-theoretic foundations for quantum acceleration in this fundamental computational domain. Building upon recent quantum adaptations of classical graph algorithms, we develop a unified framework that characterizes when and why quantum approaches offer computational advantages for single-source and all-pairs shortest path computations. Our analysis encompasses both positive-weight and negative-weight scenarios, deriving conditional speedup theorems that depend on graph structure, weight distribution, and oracle access models. We prove that quantum amplitude amplification and quantum walks can provide polynomial to exponential speedups under specific structural assumptions, while establishing hardness results that identify problem instances where quantum advantage is provably absent. The theoretical framework introduces novel complexity measures that capture graph-theoretic properties relevant to quantum algorithm performance, including quantum diameter, entanglement capacity, and interference efficiency. We extend our analysis to dynamic graph scenarios where edge weights evolve over time, establishing bounds on quantum advantage preservation under graph modifications. This work contributes fundamental insights into quantum algorithmic graph theory, clarifying the boundaries of quantum computational advantage and providing theoretical guidance for algorithm design in network optimization, routing, and graph analytics applications.
Query complexity is a common tool for comparing quantum and classical computation, and it has produced many examples of how quantum algorithms differ from classical ones. Here we investigate in detail the role that oracles play for the advantage of quantum algorithms. We do so by using a simulation framework, Quantum Simulation Logic (QSL), to construct oracles and algorithms that solve some problems with the same success probability and number of queries as the quantum algorithms. The framework can be simulated using only classical resources at a constant overhead as compared to the quantum resources used in quantum computation. Our results clarify the assumptions made and the conditions needed when using quantum oracles. Using the same assumptions on oracles within the simulation framework we show that for some specific algorithms, such as the Deutsch-Jozsa and Simon's algorithms, there simply is no advantage in terms of query complexity. This does not detract from the fact that quantum query complexity provides examples of how a quantum computer can be expected to behave, which in turn has proved useful for finding new quantum algorithms outside of the oracle paradigm, where the most prominent example is Shor's algorithm for integer factorization.
Abstract This guide is based on the IBM Quantum course Quantum Algorithmic Foundations and addresses a central question in quantum computation: Do quantum algorithms offer provable advantages beyond the oracle (black-box) query model, in more standard computational settings where inputs are given explicitly? Answering this question requires building a rigorous theoretical foundation. The course begins with a detailed discussion of computational cost in both classical and quantum models, focusing on how complexity scales with input size rather than on absolute execution time. The emphasis is placed on algorithmic efficiency, independent of hardware improvements. To ground the discussion, the material draws heavily from computational number theory, including problems such as: Integer arithmetic Greatest common divisor (GCD) computation Integer factorization Although number theory represents a specific application domain, it provides a clear and powerful framework for analyzing computational complexity and illustrating the principles underlying quantum speedups. A key theme of the lesson is the distinction between algorithmic scalability and hardware performance. The primary concern is how the cost of an algorithm grows as problem instances become larger — a perspective that remains valid across technological generations. The course concludes by addressing a critically important concept: implementing classical computations on quantum computers. The significance of this idea is not to replace classical machines, but to use classical algorithms as subroutines within quantum procedures. This hybrid approach leverages decades of classical algorithmic research while opening new pathways toward quantum computational advantage. Through practical experimentation on IBM Q, this work connects theoretical foundations with hands-on quantum programming, reinforcing the structural principles that underlie advanced quantum algorithm design.
Abstract This guide is based on the IBM Quantum course Fundamentals of Quantum Algorithms and explores the computational advantages enabled by quantum information. The focus is on what can be achieved using a single quantum computer, rather than distributed quantum systems. The primary objective of the course is to analyze how quantum computation can outperform classical computation in specific problem classes. In particular, the material examines quantum query algorithms, which provide a structured framework for understanding quantum speedups. Key themes include: The conceptual foundations of quantum computational advantage. The role of superposition, interference, and entanglement in algorithm design. Query complexity and how quantum algorithms reduce the number of required oracle calls compared to classical approaches. Foundational algorithms that demonstrate provable quantum speedups. The central guiding question is: What advantages can a quantum computer potentially offer over a classical computer? Through hands-on experimentation on IBM Q, the work connects theoretical analysis with practical implementation, enabling deeper insight into how quantum resources translate into measurable computational gains.
Traditional methods for analyzing the computational complexity of algorithms, predominantly rooted in classical computer models, struggle to accurately characterize the potential advantages offered by quantum computation. This work proposes a novel approach utilizing category theory to provide a more fundamental and abstract framework for assessing the complexity of quantum algorithms. Instead of focusing on the specifics of quantum hardware, we examine the underlying categorical structures that define the computational processes inherent in these algorithms. This allows us to identify and analyze complexity properties that are independent of the physical implementation. We define key concepts like quantum states as objects and quantum gates as morphisms, establishing a formal connection between quantum computation and category theory. This framework facilitates a more rigorous and generalized understanding of quantum complexity, potentially revealing new insights and providing a basis for comparing different quantum algorithms. The core claim is that traditional complexity analysis is inadequate, and the mechanism is the application of category theory for a more abstract and generalized framework.