Quantum algorithms have been proposed for most computational tasks in trading, but the resulting literature is not a single body of work with a single standard of evidence. This survey partitions it into strands with distinct foundations and assesses each against the strongest available evidence. Pricing and market-risk computation rest on quantum amplitude estimation and carry a proven quadratic advantage in inverse precision; circuit-level resource estimates place the requirement at roughly 8000 logical qubits and a logical clock rate of tens of megahertz, several orders of magnitude beyond demonstrated hardware, with joint estimation of prices and sensitivities the main lever that lowers it. Combinatorial formulations of portfolio selection, multi-period trading trajectories and arbitrage detection map onto quadratic unconstrained binary optimization and have the field’s most mature hardware demonstrations, but an extensive benchmark on 250 real-stock instances finds classical mixed-integer programming and tailored heuristics dominant. Quantum machine learning for return prediction has been substantially dequantized, and a controlled evaluation on cross-sectional equity returns attributes reported advantages to evaluation protocol rather than to the model. A fourth strand exploits entanglement for decision coordination under latency constraints, where the advantage follows from Bell’s theorem rather than a complexity assumption. We identify the classical-quantum interface, superquadratic structure, resource-estimate methodology, nonlocal-game design and adversarial benchmarking as the productive open directions.
Quantum algorithms have been proposed for most computational tasks in trading, but the resulting literature is not a single body of work with a single standard of evidence. This survey partitions it into strands with distinct foundations and assesses each against the strongest available evidence. Pricing and market-risk computation rest on quantum amplitude estimation and carry a proven quadratic advantage in inverse precision; circuit-level resource estimates place the requirement at roughly 8000 logical qubits and a logical clock rate of tens of megahertz, several orders of magnitude beyond demonstrated hardware, with joint estimation of prices and sensitivities the main lever that lowers it. Combinatorial formulations of portfolio selection, multi-period trading trajectories and arbitrage detection map onto quadratic unconstrained binary optimization and have the field’s most mature hardware demonstrations, but an extensive benchmark on 250 real-stock instances finds classical mixed-integer programming and tailored heuristics dominant. Quantum machine learning for return prediction has been substantially dequantized, and a controlled evaluation on cross-sectional equity returns attributes reported advantages to evaluation protocol rather than to the model. A fourth strand exploits entanglement for decision coordination under latency constraints, where the advantage follows from Bell’s theorem rather than a complexity assumption. We identify the classical-quantum interface, superquadratic structure, resource-estimate methodology, nonlocal-game design and adversarial benchmarking as the productive open directions.
Quantum computing is becoming strategically relevant to finance because several core financial bottlenecks are already defined by combinatorial search, expectation estimation, rare-event analysis, representation learning, and long-horizon cryptographic resilience. This review examines that landscape across five connected domains: constrained portfolio optimisation, derivative pricing, tail-risk and scenario estimation, quantum machine learning, and post-quantum security. Rather than treating these topics as isolated demonstrations, the article studies them as linked layers of a financial-computation stack. Across all five domains, the review applies a common evaluative logic: identify the financial bottleneck, specify the relevant quantum primitive, compare it with an explicit classical benchmark, and assess the result under realistic implementation and governance constraints. The main conclusion is measured but consequential. The strongest near-term case for quantum finance lies in carefully designed hybrid workflows rather than blanket claims of universal advantage. Quantum optimisation is most credible when constrained search dominates; amplitude-estimation methods matter most when repeated expectation evaluation is the binding cost; quantum machine learning remains task dependent; and post-quantum cryptography is already strategically necessary because financial infrastructures must migrate before fault-tolerant attacks arrive. By combining system-level synthesis with locally reproducible small-scale case studies on simulated qubit registers, the article is intended both as a review of the field and as a handbook-style entry point for future work.
Recently, several researchers proposed portfolio optimization as a potential use case for quantum optimization. However, the literature is lacking an extensive benchmark quantifying the potential of quantum computers for portfolio optimization. In this work, we contribute to closing this gap. We provide a computational study, comparing quantum approaches against state-of-the-art classical methods on a meaningful, real-world instance set. In particular, we compare quantum annealing and the quantum approximate optimization algorithm against classical mixed-integer programming, simulated annealing, steepest descent local search, tabu search and a problem-tailored heuristic. We consider a variance-minimizing variant of portfolio optimization which we show to be more difficult to solve for classical optimizers than return-maximizing or multi-objective formulations. Our benchmark data set comprises 270 instances with up to 1,000 assets from actual stock data. Due to hardware limitation, quantum methods could only be tested for instances with at most 30 assets. The results show that all instances can be solved to proven optimality by mixed-integer programming in the order of seconds. Moreover, the problem-tailored heuristic consistently outperforms quantum approaches in terms of solution quality for fixed runtime. Thus, we conclude that there is only very limited room for a potential quantum advantage for the considered variant of portfolio optimization.
Abstract Quantum computing is increasingly discussed as a possible source of new computational methods for economics, but its relevance depends on whether economic bottlenecks match quantum algorithmic primitives. This critical review evaluates that match for computational economics. It synthesizes work on quantum amplitude estimation and quantum Monte Carlo for simulation and risk measurement; quadratic unconstrained binary optimization (QUBO), Ising formulations, quantum annealing, and the quantum approximate optimization algorithm for discrete optimization; and quantum linear-algebra and machine-learning primitives for econometrics and model solution. The review argues that there is no general-purpose quantum algorithm for economics analogous to Shor's algorithm for factoring. Credible claims are problem-class specific and depend on state preparation, output requirements, conditioning, hardware noise, embedding overhead, and comparison with strong classical and quantum-inspired baselines. The most mature economic applications are derivative pricing and financial risk, payment-system liquidity optimization, quantum Monte Carlo for stress testing, and early quantum-annealing formulations of dynamic programming. Transportation and infrastructure applications are relevant mainly as networked economic optimization problems. The article proposes an evidence ladder for quantum computational economics and a research agenda centered on reproducible benchmarks, economically meaningful objective functions, end-to-end resource accounting, and evaluation by economic value rather than hardware novelty. JEL Classification: C02; C15; C61; C63; C88; E37; G11; G17; R41
This note separates two things often labelled “quantum finance”: the quantum formalism, which applies quantum-mechanical mathematics on classical computers and is judged by empirical fit, and quantum computers, which are judged by whether they complete tasks classical machines cannot. Classifying the evidence into four types (unconditional; beyond-classical under complexity assumptions; empirical claims on specific problems; asymptotic speedups requiring fault tolerance), it finds that, among the public evidence identified and verified here (cut-off 17 October 2026), no economic or financial task has demonstrated broadly accepted practical quantum advantage against strong classical baselines and end-to-end costs. The strongest unconditional results concern information resources rather than computational speed; the formalism has already produced methods usable on classical hardware and testable empirical results; the most mature provable speedup directly relevant to finance is the quadratic speedup of amplitude estimation, which requires fault-tolerant hardware; and because published estimates place the logical-qubit threshold for elliptic-curve discrete logarithms below that for derivative-pricing advantage, cryptographic infrastructure is a channel through which material impact may arrive earlier — a scenario, not a timing prediction, since the two workloads differ in gate counts, runtimes and fault-tolerance assumptions. The note is a structured synthesis of published evidence and reports no new experiments or measurements.
Quantum machine learning sits at the intersection of two paradigm-shifting technologies: quantum computing and deep learning. By exploiting quantum mechanical phenomena – superposition, entanglement, and interference – QML promises exponential or quadratic speedups over classical algorithms for certain learning tasks, as well as the ability to model quantum data natively. However, the path to practical quantum advantage remains contested, constrained by noisy intermediate-scale quantum (NISQ) hardware, limited qubit counts (50–1,000), coherence times (50–500 μs), and high gate error rates (10⁻³–10⁻⁴). This paper provides a rigorous, technically deep survey of QML: mathematical foundations (Hilbert spaces, unitary evolution, measurement, quantum circuits), canonical algorithms (quantum kernel methods, variational quantum circuits (VQCs), quantum neural networks (QNNs), QSVMs, quantum generative adversarial networks (QGANs), encoding strategies (angle, amplitude, basis, and Hamiltonian encoding), optimization landscapes (barren plateaus, parameter shift rules, ansatz design), hardware aware implementation on superconducting (IBM, Google), trapped ion (IonQ), and neutral atom (Pasqal) platforms, empirical benchmarks on near term devices for classification, generative modeling, and quantum chemistry, and pathways to fault tolerant quantum machine learning (FTQML) with error correction (surface codes, logical qubits). Researchers critically assess claims of quantum advantage, identify scenarios where QML may outperform classical ML (small data, high-dimensional feature spaces from quantum systems, kernel estimation), and propose concrete evaluation metrics (quantum-classical cross-validation, resource estimation).
Quantum computing represents one of the most disruptive technological paradigms of the 21st century, with profound implications for economic modelling, financial optimization, and computational decision-making. Unlike classical computing, quantum systems leverage superposition, entanglement, and interference to solve problems of exponentially higher complexity, a capability with transformative potential for economics and finance. This study examines the intersection of quantum computing and economics, systematically evaluating both the promises and the barriers defining this emerging field. Regarding opportunities, quantum algorithms, such as the Quantum Approximate Optimization Algorithm (QAOA) and Grover's algorithm, demonstrate significant theoretical advantages in portfolio optimization, risk assessment, option pricing, and the resolution of large-scale combinatorial problems that are intractable for classical machines. Quantum machine learning extends these prospects, offering accelerated capabilities for pattern recognition and predictive modelling relevant to financial markets. However, substantial barriers temper these expectations. Current quantum hardware operates in the NISQ (Noisy Intermediate-Scale Quantum) era, where decoherence, gate errors, and a limited number of qubits constrain practical applicability. The gap between theoretical quantum advantage and real-world economic implementation remains significant, compounded by a shortage of specialists, high infrastructure costs, and the absence of standardized programming frameworks tailored to economic applications. Through a bibliometric analysis of existing literature and a critical review of recent experimental results, the study maps the current state of quantum computing adoption in economics, identifies the most active research frontiers—quantum game theory and quantum portfolio optimization—and outlines the conditions under which quantum advantage could become economically relevant. The findings suggest that, although quantum computing will not transform economic practice in the short term, its long-term disruptive potential warrants proactive engagement from economists, financial institutions, and policymakers.