Paper: arXiv 2311.00825

Abstract

Quantum computers are not yet up to the task of providing computational advantages for practical stochastic diffusion models commonly used by financial analysts. In this paper we introduce a class of stochastic processes that are both realistic in terms of mimicking financial market risks as well as more amenable to potential quantum computational advantages. The type of models we study are based on a regime switching volatility model driven by a Markov chain with observable states. The basic model features a Geometric Brownian Motion with drift and volatility parameters determined by the finite states of a Markov chain. We study algorithms to estimate credit risk and option pricing on a gate-based quantum computer. These models bring us closer to realistic market settings, and therefore quantum computing closer the realm of practical applications.

Complexity vs Empirical Score

  • Math Complexity: 8.5/10
  • Empirical Rigor: 4.0/10
  • Quadrant: Lab Rats — theoretically deep, empirically untested

Why this score: The paper is dense with advanced mathematical formalism including Markov chains, continuous-time stochastic processes, and quantum circuit derivations, but focuses on theoretical algorithm design with no backtested implementations or real data reported.

Research Flowchart

  flowchart TD
  A["Research Goal: Develop quantum algorithms for derivative pricing & credit risk in a regime-switching economy, addressing limitations of existing quantum approaches."] --> B["Methodology: 1. Construct Stochastic Model 2. Derive Quantum Algorithms"]
  B --> C["Key Model Inputs: <br>Regime-Switching Markov Chain<br>Geometric Brownian Motion (GBM)<br>Drift & Volatility Parameters per Regime"]
  C --> D["Computational Process: <br>Gate-based Quantum Algorithms<br>1. Monte Carlo Simulation<br>2. Amplitude Estimation<br>(Estimating Option Prices & Credit Risk)"]
  D --> E["Key Findings & Outcomes: <br>1. Validated model realism for financial markets<br>2. Demonstrated potential for quantum advantage<br>3. Algorithm designed for current NISQ hardware constraints<br>4. Brings quantum finance closer to practical applications"]

Cited References (269)

  1. Quantum computational advantage using photons (2020)
  2. Qiskit: An Open-source Framework for Quantum Computing (2019) — link
  3. Portfolio selection: Efficient diversification of investments (1959)
  4. A comparison of biased simulation schemes for stochastic volatility models (2010)
  5. Markov chains (1998)
  6. Markov process simulation on a real quantum computer (2019)
  7. Quantum-enhanced analysis of discrete stochastic processes (2021)
  8. Prospects and challenges of quantum finance (2020) — link
and 261 more
  1. Quantum computing for Finance: state of the art and future prospects (2020) — link
  2. Benchmarking the Variational Quantum Eigensolver through Simulation of the Ground State Energy of Prebiotic Molecules on High-Performance Computers (2020) — link
  3. Unknotting Unknots (2011) — link
  4. Breaking and Making Quantum Money: Toward a New Quantum Cryptographic Protocol (2010) — link
  5. The Complexity of Quantum States and Transformations: From Quantum Money to Black Holes (2016) — link
  6. Algebraic Cryptanalysis of a Quantum Money Scheme The Noise-Free Case (2015)
  7. Non-quantum cryptanalysis of the noisy version of Aaronson–Christiano’s quantum money scheme (2019)
  8. Franchised Quantum Money (2020) — link
  9. Quantum Tokens for Digital Signatures (2016) — link
  10. Pseudorandom Quantum States (2018)
  11. Toward secure quantum money (2019) — link
  12. Semi-Quantum Money (2020) — link
  13. Quantum Lightning Never Strikes the Same State Twice (2019) — link
  14. One-shot Signatures and Applications to Hybrid Quantum/Classical Authentication (2020)
  15. Almost Public Coins (2020) — link
  16. A Quantum Money Solution to the Blockchain Scalability Problem (2020) — link
  17. On the Power of Quantum Computation (1997) — link
  18. Quantum Complexity Theory (1997) — link
  19. Semi-Quantum Money (2019) — link
  20. On the insecurity of quantum Bitcoin mining (2020) — link
  21. Security and implementation of advanced quantum cryptography: quantum money and quantum weak coin flipping (2019)
  22. Bond Pricing and the Term Structure of Interest Rates: A Discrete Time Approximation (1990) — link
  23. Perspectives of quantum annealing: Methods and implementations (2020) — link
  24. Quantum computational finance: quantum algorithm for portfolio optimization (2018) — link
  25. Forecasting financial crashes with quantum computing (2019)
  26. Towards Prediction of Financial Crashes with a D-Wave Quantum Computer (2019) — link
  27. Quantum Computing for Finance: Overview and Prospects (2019) — link
  28. Towards Pricing Financial Derivatives with an IBM Quantum Computer (2019) — link
  29. What is a quantum simulator? (2014) — link
  30. Quantum sensing (2017) — link
  31. Arbitrary accuracy iterative quantum phase estimation algorithm using a single ancillary qubit: A two-qubit benchmark (2007) — link
  32. Quantum algorithms revisited (1998)
  33. Quantum orbital-optimized unitary coupled cluster methods in the strongly correlated regime: Can quantum algorithms outperform their classical equivalents? (2020)
  34. Toward the first quantum simulation with quantum speedup (2018) — link
  35. Breaking the 49-Qubit Barrier in the Simulation of Quantum Circuits (2017) — link
  36. Quantum Computing in the NISQ era and beyond (2018) — link
  37. A systems perspective of quantum computing (2019) — link
  38. Quantum computing with semiconductor spins (2019) — link
  39. A crossbar network for silicon quantum dot qubits (2018) — link
  40. Toward large-scale fault-tolerant universal photonic quantum computing (2019) — link
  41. Quantum simulations with ultracold atoms in optical lattices (2017) — link
  42. Qiskit Runtime Sampler Primitive — link
  43. Iterative quantum amplitude estimation (2021) — link
  44. Surface codes: Towards practical large-scale quantum computation (2012) — link
  45. Nobel Lecture: Superposition, entanglement, and raising Schr"odinger’s cat (2013) — link
  46. Charge-insensitive qubit design derived from the Cooper pair box (2007) — link
  47. Scalable Quantum Computing with Josephson Charge Qubits (2002) — link
  48. Quantum-state engineering with Josephson-junction devices (2001) — link
  49. Demonstration of a Fundamental Quantum Logic Gate (1995) — link
  50. Elementary Gates for Quantum Computation (1995) — link
  51. Quantum Mechanical Computers (1986) — link
  52. Simulating Physics with Computers (1982) — link
  53. Strengths and Weaknesses of Quantum Computing (1997) — link
  54. A Fast Quantum Mechanical Algorithm for Database Search (1996) — link
  55. Quantum lower bounds by polynomials (2001) — link
  56. The propagation of monetary policy shocks in a heterogeneous production economy (2019) — link
  57. The New Keynesian Transmission Mechanism: A Heterogeneous-Agent Perspective (2019) — link
  58. Monetary Policy with Heterogeneous Agents and Borrowing Constraints (2010) — link
  59. How Investible is Bitcoin? Analyzing the Liquidity and Transaction Costs of Bitcoin Markets (2018) — link
  60. Inflation Targeting (2010)
  61. Equilibrium Traits of Durable Commodity Money (1985) — link
  62. Enabling Deep Negative Rates to Fight Recessions: A Guide (2019) — link
  63. Clearing and Settlement of Interbank Card Transactions: A MasterCard Tutorial for Federal Reserve Payments Analysts (2013) — link
  64. Imperfect Recognizability and Coexistence of Money and Higher-Return Assets (2013) — link
  65. Chapter 1 - The Mechanism-Design Approach to Monetary Theory (2010) — link
  66. A Model of (the Threat of) Counterfeiting (2007) — link
  67. A Simple Search Model of Money with Heterogeneous Agents and Partial Acceptability (2004) — link
  68. Acceptability, Means of Payment, and Media of Exchange (1992) — link
  69. On Ensuring the Acceptability of a New Fiat Money (1994)
  70. Money and the Real World (1972)
  71. Confidence and the Real Value of Money in an Overlapping Generations Economy (1987) — link
  72. Money and Economic Growth (1965)
  73. Assignats and Mandats: A True History, Including an Examination of Dr. Andrew D. White’s “Paper Money Inflation in France” (1877)
  74. Investigating the Impact of Global Stablecoins (2019) — link
  75. Are Stable Coins Stable? (2019) — link
  76. Banking with blockchain-ed big data (2018) — link
  77. Money as a Unit of Account (2017) — link
  78. The Welfare Cost of Inflation and the Role of Money as a Unit of Account (1977)
  79. Competitive Payments Systems and the Unit of Account (1984)
  80. Asset prices and liquidity in an exchange economy (2010)
  81. Barter and Monetary Exchange Under Private Information (1994)
  82. The Overlapping-Generations Model of Fiat Money (1980) — link
  83. A Search-Theoretic Approach to Monetary Economics (1993) — link
  84. On Money as a Medium of Exchange (1989) — link
  85. A theory of a generally acceptable medium of exchange and barter (1989) — link
  86. Experimental evidence on money as a medium of exchange (1996) — link
  87. Of two minds, multiple addresses, and one ledger: Characterizing opinions, knowledge, and perceptions of Bitcoin across users and non-users (2016) — link
  88. The Island of Stone Money (1991)
  89. Operation Bernhard (1962)
  90. Energy dependence of $^ 209 \mathrm Bi $ fragmentation in relativistic nuclear collisions (1981) — link
  91. The Economics of Counterfeiting (2015) — link
  92. A next-generation smart contract and decentralized application platform (2014) — link
  93. Ethereum: a secure decentralised generalised transaction ledger (2014)
  94. Passphrase-protected private key (2012)
  95. Mnemonic code for generating deterministic keys (2013)
  96. Hierarchical deterministic wallets (2013)
  97. Payment Protocol (2013)
  98. Smart Contracts Meet Quantum Cryptography (2019) — link
  99. Anonymity (2011) — link
  100. Privacy as a Public Good: A Case for Electronic Cash (2019)
  101. Designing Central Bank Digital Currencies (2019)
  102. Zcash protocol specification (2016)
  103. Zerocash: Decentralized Anonymous Payments from Bitcoin (2014) — link
  104. Quantitative Analysis of the Full Bitcoin Transaction Graph (2013) — link
  105. Money (1992)
  106. Reducing Shielded Proving Time in Sapling (2018)
  107. Currency and Credit in a Private Information Economy (1989) — link
  108. Micropayments Revisited (2002) — link
  109. Micropayments via Efficient Coin-Flipping (1998) — link
  110. Incentives in BitTorrent Induce Free Riding (2005) — link
  111. Incentives build robustness in BitTorrent (2003)
  112. The Big Problem of Small Change (2002)
  113. Bimetallism: An Economic and Historical Analysis (2000)
  114. Bitcoin and Cryptocurrency Technologies - A Comprehensive Introduction (2016) — link
  115. Price Stability (2009) — link
  116. Scarcity (2016) — link
  117. Fungibility (2016) — link
  118. Designing New Money - The Policy Trilemma of Central Bank Digital Currency (2017)
  119. Why Did the Riksbank Get a Monopoly on Banknotes? (2018)
  120. Digital Currencies (2015)
  121. Payment Systems – History and Challenges (2019)
  122. Proceeding with Caution – A Survey on Central Bank Digital Currency (2019)
  123. Central Bank Cryptocurrencies (2017) — link
  124. Assessing the Impact of Central Bank Digital Currency on Private Banks (2018)
  125. The Digitalization of Money (2019)
  126. Why Pay? An Introduction to Payments Economics (2009)
  127. On the Origin of Money (1892)
  128. On the CNOT-cost of TOFFOLI gates (2009) — link
  129. Quantum Random Number Generators (2017) — link
  130. Infinite Randomness Expansion with a Constant Number of Devices (2014) — link
  131. Robust Protocols for Securely Expanding Randomness and Distributing Keys Using Untrusted Quantum Devices (2016) — link
  132. Certifiable Quantum Dice (2012) — link
  133. Money and the Mechanism of Exchange (1876)
  134. Complexity-Theoretic Foundations of Quantum Supremacy Experiments (2017) — link
  135. Quantum Computing and the Entanglement Frontier (2012)
  136. Algorithms for Quantum Computation: Discrete Logarithms and Factoring (1994) — link
  137. Quantum Phase Estimation with Arbitrary Constant-Precision Phase Shift Operators (2012) — link
  138. Optimal Quantum Algorithm for Polynomial Interpolation (2016) — link
  139. Dynamic General Equilibrium Modeling (2009) — link
  140. Constrained Optimization Approaches to Estimation of Structural Models (2012) — link
  141. Swapping the Nested Fixed Point Algorithm: A Class of Estimators for Discrete Markov Decision Models (2002)
  142. Generalized Autoregressive Conditional Heteroskedasticity (1986) — link
  143. Forecasting with DSGE Models (2010)
  144. Dynamic Discrete Choice Structural Models: A Survey (2010) — link
  145. Introducing Financial Frictions and Unemployment into a Small Open Economy Model (2011) — link
  146. Estimation and Inference in Nonlinear Structural Models (1974)
  147. An Algorithm for Least-Squares Estimation of Nonlinear Parameters (1963) — link
  148. Lecture Notes on Quantum Algorithms (2017)
  149. Quantum Algorithms: An Overview (2016) — link
  150. A Set of Five Independent Postulates for Boolean Algebras, with Application to Logical Constants (1913) — link
  151. A New Notation for Quantum Mechanics (1939) — link
  152. On the Quantum Mechanics of Collisions (1926)
  153. Systems of Logic Based on Ordinals (1939) — link
  154. Quantum Attacks on Bitcoin, and How to Protect Against Them (2018) — link
  155. Quantum Gradient Descent for Linear Systems and Least Squares (2017)
  156. Quantum Algorithms for Gibbs Sampling and Hitting-Time Estimation (2017) — link
  157. Practical Quantum Retrieval Games (2016) — link
  158. Quantum Support Vector Machine for Big Data Classification (2014)
  159. Quantum Attacks on Bitcoin, and How to Protect Against Them (2018) — link
  160. Quantum-Assisted Gaussian Process Regression (2019) — link
  161. Prediction by Linear Regression on a Quantum Computer (2016) — link
  162. Estimating Diagonal Entries of Powers of Sparse Symmetric matrices is BQP-complete (2006)
  163. Quantum Simulated Annealing (2007) — link
  164. Quantum Speedup of Monte Carlo Methods (2015)
  165. Inverting Well Conditioned Matrices in Quantum Logspace (2013) — link
  166. Quantum Algorithm for Data Fitting (2012)
  167. Quantum Algorithms and the Finite Element Method (2016)
  168. Quantum Algorithm for Linear Systems of Equations (2009)
  169. A Complete Characterization of Unitary Quantum Space (2016)
  170. High-Order Quantum Algorithm for Solving Linear Differential Equations (2014) — link
  171. Variable Time Amplitude Amplification and a Faster Quantum Algorithm for Solving Systems of Linear Equations (2010) — link
  172. Numerical Methods in Economics (1998) — link
  173. Quantum Search with Advice (2011) — link
  174. Quantum Algorithms to Solve the Hidden Shift Problem for Quadratics and for Functions of Large Gowers Norm (2009) — link
  175. Quantum Basin Hopping with Gradient-Based Local Optimisation (2005) — link
  176. Fast Quantum Algorithm for Numerical Gradient Estimation (2005) — link
  177. Quantum money with nearly optimal error tolerance (2017)
  178. 24-Hour Relativistic Bit Commitment (2016) — link
  179. Absolutely Covert Quantum Communication (2016) — link
  180. Experimental Realization of Quantum Cheque Using a Five-Qubit Quantum Computer (2017)
  181. Optimal Counterfeiting Attacks and Generalizations for Wiesner’s Quantum Money (2012) — link
  182. Anonymous Quantum Communication
  183. Quantum T-Designs: T-Wise Independence in the Quantum World (2007) — link
  184. Quantum Money with Classical Verification (2012) — link
  185. Component Mixers and a Hardness Result for Counterfeiting Quantum Money (2011)
  186. Quantum State Restoration and Single-Copy Tomography for Ground States of Hamiltonians (2010) — link
  187. Quantum coins (2010)
  188. Breaking and Making Quantum Money: Toward a New Quantum Cryptographic Protocol (2009) — link
  189. Quantum Cryptography: Public Key Distribution and Coin Tossing (1984)
  190. Quantum risk analysis (2019) — link
  191. Quantum unary approach to option pricing (2021) — link
  192. Addition on a Quantum Computer (2000) — link
  193. On a formula for the distribution of the maximum likelihood estimator (1983)
  194. On the pricing of corporate debt: The risk structure of interest rates (1974)
  195. The distribution of loan portfolio value (2002)
  196. A closed-form solution for options with stochastic volatility with applications to bond and currency options (1993)
  197. Efficient state preparation for quantum amplitude estimation (2021)
  198. Quantum pricing with a smile: Implementation of local volatility model on quantum computer (2022)
  199. Pricing with a smile (1994)
  200. Quantum Monte-Carlo Integration: The Full Advantage in Minimal Circuit Depth (2021)
  201. Stochastic volatility (1996)
  202. Alternative models for stock price dynamics (2003)
  203. The Pricing of Options and Corporate Liabilities (1973)
  204. Risk minimizing option pricing in a regime switching market (2008)
  205. American options with regime switching (2002)
  206. Option valuation and hedging strategies with jumps in the volatility of asset returns (1993)
  207. Methods for pricing American options under regime switching (2011)
  208. Valuing options in regime-switching models (1998)
  209. Ergodic control of switching diffusions (1997)
  210. Closed-form solutions for perpetual American put options with regime switching (2004)
  211. Explicit solutions to European options in a regime-switching economy (2005)
  212. A component model for dynamic correlations (2011)
  213. Long-and short-term cryptocurrency volatility components: A GARCH-MIDAS analysis (2018)
  214. Stock market volatility and macroeconomic fundamentals (2013)
  215. Quantum advantage for differential equation analysis (2020)
  216. Hidden quantum Markov models and non-adaptive read-out of many-body states (2010)
  217. Learning hidden quantum Markov models (2018)
  218. Quantum hidden Markov models based on transition operation matrices (2017)
  219. Option pricing with Markov switching (2012)
  220. The econometrics of option pricing (2010)
  221. A new exact solution for pricing European options in a two-state regime-switching economy (2012)
  222. Quantum computing for finance: Overview and prospects (2019)
  223. Long-term risk: An operator approach (2009)
  224. Stock return characteristics, skew laws, and the differential pricing of individual equity options (2003)
  225. Misspecified recovery (2016)
  226. A proposal to extend expected utility in a quantum probabilistic framework (2018)
  227. Quantum random access memory (2008)
  228. From ambiguity aversion to a generalized expected utility. Modeling preferences in a quantum probabilistic framework (2016)
  229. Quantum structure in cognition (2009)
  230. The role of conditioning information in deducing testable restrictions implied by dynamic asset pricing models (1987)
  231. On profile likelihood (2000)
  232. A new notation for quantum mechanics (1939)
  233. A threshold for quantum advantage in derivative pricing (2020)
  234. Option pricing using quantum computers (2020)
  235. Credit risk analysis using quantum computers (2020)
  236. Quantum risk analysis (2019)
  237. Essays on Correlation Modelling (2011)
  238. Probabilistic error cancellation with sparse Pauli-Lindblad models on noisy quantum processors (2022) — link
  239. Confidence intervals for diffusion index forecasts and inference for factor-augmented regressions (2006)
  240. Can small quantum systems learn? (2015)
  241. Quantum algorithm for data fitting (2012)
  242. Quantum support vector machine for big data classification (2014)
  243. Quantum principal component analysis (2014)
  244. Read the fine print (2015)
  245. Quantum inference on Bayesian networks (2014)
  246. Quantum estimation for quantum technology (2009)
  247. Quantum generative adversarial networks (2018)
  248. Generative adversarial networks (2020)
  249. Nonparametric stochastic discount factor decomposition (2017)
  250. An Option Pricing Model with Probability Measure Ambiguity (2020)
  251. Quantum amplitude amplification and estimation (2002)
  252. Quantum speedup of Monte Carlo methods (2015)
  253. Quantum-enhanced machine learning (2016)
  254. Quantum algorithm for linear systems of equations (2009)
  255. Fast quantum algorithms for numerical integrals and stochastic processes (1999)
  256. Quantum measurements and the Abelian stabilizer problem (1995)
  257. Quantum algorithms revisited (1998)
  258. Amplitude estimation without phase estimation (2020)
  259. Creating superpositions that correspond to efficiently integrable probability distributions (2002)
  260. D 'e monstration nouvelle d’un th 'e or `e me concernant les op 'e rations fonctionnelles lin 'e aires (1914)
  261. Quantum Computation and Quantum Information (2010)

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