A curated library of research on quantum, quantum-inspired, and classical methods for quantitative finance, assembled and maintained by Applied Quantum Software. 477 entries and growing.
Gustav J L Jager, Martin B Plenio, Hans-Martin Rieser · arXiv:2608.18901 (quant-ph) · 2026 (arXiv, Aug)Tensor-Network / Matrix Product State Machine Learning
Studies matrix product states (MPS, also known as tensor trains) as trainable machine-learning models and how best to optimize them. The authors add a global normalization condition so the trained MPS represents a valid (normalized) quantum state, then compare two ways of finding the locally optimal tensors: standard gradient descent versus an adaptation of the density-matrix renormalization group (DMRG). Relevant to AQS because MPS with DMRG-style optimization is exactly the quantum-inspired machinery behind our own tensor-network models: we productionized a DMRG-based fix for our MPS generator (which had failed to capture cross-asset correlation), and this paper speaks directly to the same design choices, normalization to keep the MPS a proper quantum state and DMRG-versus-gradient-descent training, that determine whether a tensor-network model faithfully learns a target distribution.
Mackenson Polche, Varun Puram, Aditi Lal, Weronika Golletz, Joan Etude Arrow, Vardaan Sahgal, Kumar Ghosh, Giorgio Cortiana, Corey O'Meara · arXiv:2605.24252 (quant-ph); industrial quantum-ML collaboration (energy / smart-metering domain) · 2026 (arXiv, May)Hybrid Quantum-Classical Machine Learning for Time-Series Forecasting
Benchmarks two hybrid quantum-classical models for multi-output (multi-stream) time-series forecasting on a real smart-meter dataset of 103 household electricity-consumption series. The first, KQRC-RM, combines a quantum reservoir with measurement and a kernelized readout on 114 qubits; the second is a Projected Quantum Kernel Gaussian Process with a topology-aware circuit design on 100 qubits that forecasts many series simultaneously. Roughly half the outputs reach high accuracy, with average error metrics reported below the classical baselines on both simulators and real quantum hardware, which the authors read as evidence that utility-scale forecasting is feasible on current devices; they also document a sizeable simulation-versus-hardware performance gap. Relevant to AQS as a direct point of comparison for quantum machine learning on real hardware at 100+ qubit scale for time-series problems: it shares our exact rigor questions (does the quantum model actually beat the best classical baseline, and does any advantage survive the move from simulator to noisy hardware), and its candid reporting of the sim-to-hardware gap mirrors what we see running our own generative time-series model on NISQ devices.
Lucas Arenstein, Michael Kastoryano · arXiv:2608.17636 (q-fin.CP, Computational Finance) · 2026 (arXiv, Aug)Tensor-Train (Tensor Network) Compression for Multi-Asset Option Pricing
Prices European multi-asset options (basket and min/max payoffs) under Levy and affine characteristic-function models by extending the Fourier-cosine (COS) method with tensor-train (TT) compression. Rather than forming the full multi-dimensional grid of characteristic-function values, which grows exponentially in the number of assets, COS-TT-CHF samples the characteristic-function tensor and compresses it into low-rank tensor-train coefficients, so that strike-grid revaluations and Greeks become cheap once the TT representation is built. The authors benchmark against adaptive-quadrature Fourier, direct COS, tensor-Fourier, and quasi-Monte Carlo, showing favorable accuracy-versus-cost scaling from d=2 to d=4 assets and demonstrating feasibility up to d=30 for geometric Brownian motion and d=20 for Variance Gamma, NIG, and Heston, with diagnostics on accuracy, TT rank, runtime, and sensitivities. Relevant to AQS as a directly comparable quantum-inspired (tensor-network) baseline for our Derivative Pricing use case: it is exactly the class of best-in-fleet classical / tensor-train method that a quantum amplitude-estimation approach must beat to claim advantage on high-dimensional option pricing, and it quantifies how far tensor-train compression alone pushes back the curse of dimensionality.
Sam Reifenstein, Satoshi Kako, Farad Khoyratee, Timothee Leleu, Yoshihisa Yamamoto · arXiv:2108.07369 (quant-ph); NTT Research Physics and Informatics (PHI) Laboratories and Stanford University · 2021 (arXiv, Aug 16)Coherent Ising Machines / Optical Computing for Combinatorial Optimization
Extends the Coherent Ising Machine with optical error-correction circuits that use squeezed and anti-squeezed vacuum states to build quantum correlations and steer the machine's symmetry-breaking dynamics, combining an initial search phase with chaotic exploration of candidate solutions. The design targets programmable all-to-all coupling and low-energy, chip-scale optical implementation (for example on thin-film lithium niobate), improving solution quality on hard Ising instances over the standard CIM. Relevant to AQS as the research frontier for analog Ising optimization: it sharpens the question of when a CIM's advantage is genuinely quantum versus classically reproducible, which is exactly the rigor lens we apply to quantum and quantum-inspired optimization methods for finance.
Takahiro Inagaki, Yoshitaka Haribara, Koji Igarashi, Tomohiro Sonobe, Shuhei Tamate, Toshimori Honjo, Alireza Marandi, Peter L. McMahon, Takeshi Umeki, Koji Enbutsu, Osamu Tadanaga, Hirokazu Takenouchi, Kazuyuki Aihara, Ken-ichi Kawarabayashi, Kyo Inoue, Shoko Utsunomiya, Hiroki Takesue · Science 354, 603-606 (2016); DOI 10.1126/science.aah4243; NTT Basic Research Laboratories, NTT Corporation (with Stanford, NII, Osaka University, University of Tokyo) · 2016 (Science, Nov 4)Coherent Ising Machines / Optical Computing for Combinatorial Optimization
Demonstrates a measurement-feedback Coherent Ising Machine (CIM) built from a network of degenerate optical parametric oscillators in a long fiber ring, solving Ising and MAX-CUT problems on fully connected graphs of up to 2000 nodes. The optimization objective is encoded as an Ising Hamiltonian and the optical system relaxes toward a low-energy spin configuration, returning solutions to dense 2000-node instances in about a millisecond. Relevant to AQS as a landmark physics-based approach to the same combinatorial optimization problems (portfolio construction, index tracking, clustering) that our Portfolio Optimization use case targets with QAOA and classical branch-and-bound, and as a benchmark for whether analog Ising solvers beat the best classical heuristics out of sample.
Paul Wilmott · Wiley · 2006 (2nd edition, Wiley; 3-volume set)Quantitative Finance (reference)
The comprehensive classical reference on quantitative finance across three volumes: derivatives pricing and hedging (Black-Scholes and well beyond), stochastic calculus, interest-rate and credit models, volatility modeling, portfolio and risk management (VaR/CVaR), and the numerical methods that trading desks actually use (finite-difference PDE solvers, Monte Carlo, and lattice/tree methods). Relevant to AQS as the canonical statement of the best-classical baselines our quantum and quantum-inspired methods must beat: it defines the pricing, risk, and scenario-generation techniques (Monte Carlo, GARCH-style volatility, VaR backtesting) that the FabriQ estimator and the Generative Model Validation workbench are benchmarked against.
Jielun Chen, E. M. Stoudenmire, Steven R. White · arXiv:2210.08468 (quant-ph); PRX Quantum 4, 040318 (2023) · 2022 (arXiv, Oct 16)Classical / Tensor-Network Simulation of Quantum Circuits
Shows that the core of the Quantum Fourier Transform, the primitive behind Shor's algorithm and quantum phase estimation, generates only a small, qubit-count-independent amount of entanglement: its Schmidt coefficients decay exponentially, and the maximal operator entanglement reported in prior work is due entirely to the bit-reversal step. A matrix-product-state (MPS) analysis therefore simulates the QFT classically in time linear in the number of qubits, and the authors report speedups of several orders of magnitude over the classical FFT for structured data vectors of 10^6 to 10^8 elements. Relevant to AQS as a sharp instance of the quantum-inspired principle behind our tensor-network methods: a quantum primitive with low entanglement is efficiently classical, which both bounds where QFT-based speedups can be genuine and reinforces MPS as a competitive classical tool.
Albert Saiapin, Kim Batselier · arXiv:2608.07043 (cs.MS, preprint) · 2026 (arXiv, Aug 7)Tensor-Network Machine Learning
Introduces tnkm, an open-source JAX library for building and training Tensor Network Kernel Machines (TNKM), which pair nonlinear feature representations with compact low-rank tensor-network parameterizations. The library gives a unified interface across feature maps, tensor-network architectures, and optimizers (alternating least squares and gradient-based methods), and reports competitive prediction accuracy with compact models on nonlinear benchmark problems while keeping training efficient. Relevant to AQS as a modern, reproducible software foundation for the tensor-network (MPS-style) methods that underpin our quantum-inspired risk and generative modeling.
Giovanni Barone-Adesi, Kostas Giannopoulos, Les Vosper · Journal of Futures Markets (reference) · 1999Value-at-Risk Methodology
A semiparametric VaR method that combines a volatility model with historical simulation. Returns are first standardized by a fitted conditional volatility (EWMA or GARCH), whole days of standardized residuals are resampled (which preserves the empirical cross-asset dependence and tail co-movement), and each scenario is then re-inflated to the current volatility level. It is a widely used state-of-the-art classical VaR baseline, and the primary benchmark AQS FabriQ scores its MPS engine against.
Paul Kupiec (1995); Peter Christoffersen (1998) · Journal of Derivatives / International Economic Review (reference) · 1995 (Kupiec), 1998 (Christoffersen)Risk Model Validation
The two standard likelihood-ratio tests for validating a Value-at-Risk model. Kupiec's proportion-of-failures (POF) test checks whether the observed VaR breach rate matches the promised confidence level (unconditional coverage). Christoffersen's test adds an independence check, since breaches should not cluster in time, and combines the two into a conditional-coverage test. Both are computed at each confidence level in AQS FabriQ's VaR gating backtest, and the regime split reports Kupiec separately for calm and stress days.
Robert F. Engle (ARCH, 1982); Tim Bollerslev (GARCH, 1986) · Econometrica / Journal of Econometrics (reference) · 1982 (ARCH), 1986 (GARCH)Time-Series Volatility Modeling
A family of models that capture volatility clustering, the empirical fact that large market moves tend to be followed by more large moves. GARCH(1,1) makes today's variance a weighted sum of a long-run level, yesterday's squared return, and yesterday's variance, and is fit by (quasi-)maximum likelihood. AQS FabriQ uses a GARCH(1,1) volatility filter inside its Filtered Historical Simulation baseline (de-volatilize the returns, resample, then re-inflate by the one-step-ahead volatility forecast) as the state-of-the-art classical benchmark for VaR.
Ulrich Schollwock · arXiv:1008.3477 (Annals of Physics, 2011) · 2011Tensor Networks
The standard reference for matrix-product states (MPS) and the density-matrix renormalization group (DMRG) algorithm that optimizes them. It explains canonical (left / right isometric) gauge forms, two-site variational updates, and singular-value-decomposition truncation to a fixed bond dimension, which is the exact machinery AQS FabriQ uses to train its MPS from market data. Re-canonicalizing the chain each sweep, so every local update happens in a well-normalized gauge, is a key ingredient of a stable trainer.
Zhao-Yu Han, Jun Wang, Heng Fan, Lei Wang, Pan Zhang · arXiv:1709.01662 (Physical Review X, 2018) · 2017 (arXiv), 2018 (Phys. Rev. X)Tensor-Network Machine Learning
Introduces the Born machine: representing a probability distribution as the squared amplitude |psi(x)|^2 of a matrix-product state (MPS), a one-dimensional tensor network borrowed from quantum physics, and training it with a two-site density-matrix renormalization-group (DMRG) sweep that minimizes the negative log-likelihood. Because the model is an MPS it supports exact, independent sampling and its expressiveness is tuned by the bond dimension chi. This is the foundational method behind AQS FabriQ's quantum-inspired MPS scenario generator for multi-asset VaR.
Larry Armijo (1966); Philip Wolfe (Wolfe conditions) · Pacific Journal of Mathematics (reference) · 1966Numerical Optimization
A method for deciding how far to step along a descent direction in iterative optimization. Instead of a fixed step size (which can overshoot and make the objective diverge), the algorithm starts from a trial step and repeatedly shrinks it until the objective falls by at least a fixed fraction of what the gradient predicts, the Armijo sufficient-decrease condition (one of the Wolfe conditions). This guarantees the objective decreases at every iteration. AQS FabriQ's tensor-network (MPS) VaR trainer uses an Armijo backtracking line search on the unit sphere to optimize each block, which is what makes its training numerically stable where an earlier fixed-step trainer could diverge.
Carlo Acerbi, Balazs Szekely · MSCI / Risk Magazine · 2014Risk Model Validation
Expected Shortfall (also called CVaR, the average loss in the tail beyond the VaR quantile) was long thought to be impossible to backtest. Acerbi and Szekely (2014) introduced simple, distribution-free test statistics, most usefully their Test 2 (Z2), that check whether a model's predicted ES is consistent with the tail losses actually realized. In loss convention Z2 is about 0 when the ES is well calibrated, negative when the model under-reserves (ES too low for the losses seen), and positive when it over-reserves. AQS FabriQ uses the Z2 statistic in its VaR/ES gating backtest to score expected-shortfall accuracy alongside the Kupiec and Christoffersen VaR tests.
Robert C. Merton (originator) · Wikipedia (reference article) · 1969Portfolio Optimization
A foundational continuous-time finance problem, formulated and solved by Robert C. Merton in 1969, on how an investor should optimally choose consumption and the fraction of wealth held in a risky asset versus a risk-free asset over time. Cast as a stochastic optimal-control problem (Hamilton-Jacobi-Bellman equation) under constant-relative-risk-aversion (CRRA) utility, it admits a closed-form solution: the Merton fraction pi = (mu - r) / (gamma * sigma^2), a constant proportion of wealth in the risky asset that is independent of both wealth and time. A cornerstone of quantitative portfolio theory and a natural classical benchmark for portfolio-optimization methods.
Mitchell Bordelon, Maurice Garfinkel, Vivek Dixit, Thomas Whitehead, Jenny Holzbauer, Guillermo Mijares Vilarino, Alberto Maldonado Romo, Abhijit Mitra, Vaibhaw Kumar, Jean Utke · Allstate; IBM Quantum · 2026-05-02Quantum Optimization
A quantum-classical hybrid scheme for chance-constrained knapsack problems, a class of stochastic combinatorial optimization in which item weights follow probability distributions and constraints may be violated within a specified risk tolerance, motivated by insurance underwriting and related risk-reward decisions. It uses knapsack-specific QAOA circuits to generate candidate samples, combined with a new self-consistent classical recovery scheme, and is demonstrated on IBM Heron processors with circuits up to 150 qubits and depths up to 177 (3443 gates). The reported solutions are comparable to the classical optimization schemes tested, with the stated potential to outperform purely classical approaches. An Allstate and IBM Quantum collaboration.
Kevin W. Aoun, Florian J. Kiwit, Carlos A. Riofrio, Samer Saab Jr., Charbel Al Bateh, Joe Tekli, Andre Luckow · Lebanese American University (LAU); BMW Group; Ludwig Maximilian University of Munich (LMU) · 2026-05-29Quantum State Preparation
Trains a classical neural network to map input data directly to quantum-circuit parameters, amortizing the per-instance state-preparation cost in quantum machine learning. Reports high-fidelity image-state generation on unseen MNIST / Fashion-MNIST data (fidelities up to 0.992) with large runtime speedups over instance-by-instance optimization. Addresses the data-loading bottleneck for QML.
Jake McNaughton, Mohamed Hibat-Allah · University of Waterloo & Vector Institute; Perimeter Institute for Theoretical Physics; Charles Sturt University · 2025-07-24Neural Quantum States
Uses recurrent neural networks as a neural-quantum-state ansatz for quantum many-body ground states, with an adaptive scheme that trains a smaller RNN and uses it to initialize a larger one, reducing training fluctuations and cost. A recent take on neural quantum states as a classical variational method for simulating quantum systems.
Zhih-Ahn Jia, Biao Yi, Rui Zhai, Yu-Chun Wu, Guang-Can Guo, Guo-Ping Guo · University of Science and Technology of China (USTC) · 2018-08-31Neural Quantum States
Reviews neural-network representations of quantum many-body states (the neural analog of tensor networks): Boltzmann-machine ansatze, their entanglement and representational power, and applications to many-body calculations, state tomography, and classical simulation of quantum computing. Useful background for neural quantum states as a classical, quantum-inspired state representation.
Yilun Zhao, Bingmeng Wang, Wenle Jiang, Xiwei Pan, Bing Li, Yinhe Han, Ying Wang · Institute of Computing Technology, Chinese Academy of Sciences (ICT, CAS); Capital Normal University; Beijing University of Posts and Telecommunications; University of Electronic Science and Technology of China · 2024-08-10Quantum State Preparation
Proposes a pre-trained neural network that directly generates an approximate state-preparation circuit for an arbitrary target state, avoiding the per-state iterative optimization that parameterized-circuit methods require. Aims at a universal, amortized route to approximate quantum state preparation with reduced runtime overhead.
Juan Miguel Arrazola, Thomas R. Bromley, Josh Izaac, Casey R. Myers, Kamil Bradler, Nathan Killoran · Xanadu · 2018-07-27Quantum State Preparation
Trains a continuous-variable quantum neural network (optimized by automatic differentiation in Strawberry Fields) to prepare target photonic states and synthesize gates, reaching over 99 percent fidelity for states such as single photons and cubic-phase / cross-Kerr operations with short-depth circuits. An early machine-learning approach to the state-preparation and gate-synthesis problem on photonic hardware.
Dongwoo Kim, Zhenyu Cui, Daniel K. Park, Chihoon Lee · Stevens Institute of Technology; Yonsei University · 2026-07-16Quantum State Preparation
Uses tensor-train (matrix product state) rank information to build shallow variational circuits that load a basket's joint distribution for amplitude-estimation option pricing. Prepares the asset-wise marginals plus a latent block trained on the basket CDF, targeting linear rather than exponential circuit depth while keeping pricing accuracy. Relevant to the distribution-loading step of quantum derivative pricing.
Juntong Ni, Shiyu Wang, Zewen Liu, Xiaoming Shi, Xinyue Zhong, Zhou Ye, Wei Jin · 2025 (arXiv, Jul; rev. Sep)Time-Series Forecasting
A forecasting architecture for very high-dimensional time series (thousands of channels) that learns hierarchical channel structures via query-based attention with full-rank regularization. Introduces Time-HD, a benchmark of large, diverse, high-dimensional datasets, and reports improved accuracy and efficiency over prior methods. Relevant to forecasting many-asset / many-risk-factor financial series.
Jinsung Yoon, Daniel Jarrett, Mihaela van der Schaar · UCLA / University of Cambridge (van der Schaar Lab) · 2019 (NeurIPS)Generative Models
A GAN framework for generating realistic synthetic time-series data. It combines the flexibility of an unsupervised GAN with the control of supervised autoregressive training in a learned embedding space, using a supervised loss so the generator preserves temporal dynamics (conditional distributions across time) as well as the marginal feature distributions. A key classical baseline for generative modeling of financial return series. The second link is the authors' official TensorFlow implementation.
Kenji Yamanishi, Jun-ichi Takeuchi · NEC / ACM SIGKDD · 2002Changepoint Detection
Online model that fits an autoregressive model with exponential discounting of older observations so it adapts to non-stationarity, producing change/anomaly scores — the engine behind the ChangeFinder algorithm.
Ryan P. Adams, David J. C. MacKay · University of Cambridge · 2007Changepoint Detection
Online Bayesian method that maintains a posterior distribution over the run length (time since the last changepoint), updated recursively as each observation arrives — giving real-time detection with calibrated uncertainty for non-stationary series.
E. S. Page · Biometrika · 1954Changepoint Detection
Classical sequential (control-chart) changepoint method: accumulates the deviation of each observation from a reference level and signals a change when the running sum crosses a threshold. Detects shifts in the mean/variance of a monitored series — regime shifts, structural breaks and volatility changes in financial time series.
Romina Yalovetzky, Martin J. A. Schuetz, Zichang He, Jiayu Shen, Yue Sun, Rudy Raymond, Shauna Sahay, Kishore Perla, Ruben S. Andrist, Grant Salton, Helmut G. Katzgraber, Roger Bongiovanni, Niraj Kumar, Rob Otter · JPMorgan Chase; Amazon Advanced Solutions Lab / AWS; Quantinuum; 55 North Management · July 2026Portfolio Optimization
qReduMIS: a recursive hybrid quantum-classical algorithm for portfolio diversification formulated as a Maximum Independent Set (MIS) problem on an asset-correlation graph (assets = nodes; highly correlated pairs = edges). QAOA measurements identify 'frozen' vertices likely in the optimal set, guiding classical graph reductions before re-solving iteratively. Validated end-to-end on real market data across four indices (up to 225 assets) on Quantinuum's 98-qubit trapped-ion Helios system, with QAOA kernels up to 78 qubits / 1016 two-qubit gates. Reports success rates 0.40 (S&P 100) and 0.95 (Nikkei 225), approximation ratios >= 0.96, and no more than five QPU calls; the optimal time-to-solution scaling exponent is ~3.2x smaller than standalone QAOA (p=2). The authors explicitly do NOT claim a practical quantum advantage over classical methods.
Dániel Léber · Corvinus University of Budapest · October 2025Derivative Pricing
Applies the Feynman path-integral formalism of quantum mechanics to the pricing of multi-asset, path-dependent options — payoffs that depend on the full trajectory of several correlated underliers (e.g. Asian, barrier, and basket-style options). A quantum-inspired analytical technique in the econophysics tradition (Physica A): it borrows quantum-mechanical machinery (path integrals, Hamiltonian/operator methods) but runs on classical hardware. Directly relevant to AQS's derivative-pricing use cases, which target exactly these multi-asset path-dependent payoffs, offering an analytical lens alongside Monte-Carlo and quantum amplitude-estimation approaches. Physica A: Statistical Mechanics and its Applications, Vol. 675, 130644 (Oct 2025). DOI: 10.1016/j.physa.2025.130644.
IBM · 2025Quantum Hardware
IBM Quantum Development Conference 2025 update: fleet-wide EPLG comparison across Heron and Eagle, Heron r3 coherence gains. Context source for AQS IBM catalog specs.
The Quantum Insider · 2026-01-13Quantum Hardware
ibm_boston, IBM's most performant Heron r3 to date: ~2.15e-3 two-qubit error across 100 qubits, dozens of pairs below 1e-3. Source for AQS Heron r3 specs.
Metriq · 2025-07Quantum Hardware
First Heron r3 beta QPU ibm_pittsburgh — best Heron T1/T2 and readout fidelity to date (July 2025). Source for AQS Heron r3 specs.
IBM · 2025Quantum Hardware
IBM's official processor-type reference: Heron r1 (Dec 2023, 133q), r2 (Jul 2024, 156q, TLS mitigation), r3 (Jul 2025, coherence/fidelity/readout gains); native CZ gate. Source for AQS Heron r2/r3 specs and revision dates.
2023-09Quantum Hardware
Experiments on ibm_sherbrooke (127-qubit Eagle r3): median ECR error ~0.0077, SX error ~2e-4, readout ~0.010, 2Q gate time 533 ns, T1 ~293 us, T2 ~173 us. Source for AQS Eagle r3 specs.
2025-09Quantum Hardware
Reports median two-qubit (ECR) gate error, isolated 2Q fidelities, EPLG, and T1/T2 coherence for the 127-qubit Eagle r3 processors ibm_sherbrooke and ibm_brisbane (Apr 2025). Source for AQS's Eagle r3 catalog specs.
IBM · 2023-11Quantum Hardware
IBM's EPLG (error per layered gate) methodology and fleet metrics for Heron-class processors; basis for Heron r2 two-qubit error figures. Source for AQS Heron r2 specs.
2024-10Quantum Hardware
Survey of IBM QPU generations (Eagle, Heron r1/r2) with gate errors, coherence, and CLOPS. Source for AQS Eagle r3 and Heron r2 catalog specs.
CFA Institute (with EDHEC) · 2026Quantum Finance
CFA Institute practitioner webinar (part of its quantum computing series, supported by EDHEC) examining quantum computing's near-term financial applications — optimization, scenario generation, and derivative-pricing potential — alongside the security risks quantum poses to existing blockchain and custody systems, with insights from early implementations.
Vincent van Wingerden, Esperanza Cuenca Gomez · Classiq / NVIDIA · 2026-06-22Quantum Finance
Shows how Classiq's high-level quantum modeling language combined with NVIDIA's CUDA-Q platform streamlines quantum algorithm development for finance. Walks through practical implementations of QAOA for portfolio optimization and IQAE for derivative pricing, and how GPU-accelerated simulation enables faster iterative workflows while approaching quantum advantage in financial applications.