Quantum unary approach to option pricing

Abstract

We present a quantum algorithm for European option pricing in finance, where the key idea is to work in the unary representation of the asset value. The algorithm needs novel circuitry and is divided in three parts: first, the amplitude distribution corresponding to the asset value at maturity is generated using a low depth circuit; second, the computation of the expected return is computed with simple controlled gates; and third, standard Amplitude Estimation is used to gain quantum advantage. On the positive side, unary representation remarkably simplifies the structure and depth of the quantum circuit. Amplitude distributions uses quantum superposition to bypass the role of classical Monte Carlo simulation. The unary representation also provides a post-selection consistency check that allows for a substantial mitigation in the error of the computation. On the negative side, unary representation requires linearly many qubits to represent a target probability distribution, as compared to the logarithmic scaling of binary algorithms. We compare the performance of both unary vs. binary option pricing algorithms using error maps, and find that unary representation may bring a relevant advantage in practice for near-term devices.

14 (main) + 10 (appendix) pages, 22 figures. Final peer-reviewed version, published in PRA. All suggestions from the referees have been considered. We thank the referees and the journal for all the work

References in corpus (56)

• Quantum Computing in the NISQ era and beyond
• Supplementary information for "Quantum supremacy using a programmable superconducting processor"
• Polynomial-Time Algorithms for Prime Factorization and Discrete Logarithms on a Quantum Computer
• A variational eigenvalue solver on a quantum processor
• Quantum Mechanics helps in searching for a needle in a haystack
• Elementary gates for quantum computation
• Three qubits can be entangled in two inequivalent ways
• A Quantum Approximate Optimization Algorithm
• Quantum Circuit Learning
• Error mitigation for short-depth quantum circuits
• From the Quantum Approximate Optimization Algorithm to a Quantum Alternating Operator Ansatz
• Quantum computing for finance: overview and prospects
• Efficient variational quantum simulator incorporating active error minimisation
• Quantum optimization using variational algorithms on near-term quantum devices
• Data re-uploading for a universal quantum classifier
• Self-Verifying Variational Quantum Simulation of the Lattice Schwinger Model
• Quantum generative adversarial learning
• Practical Quantum Error Mitigation for Near-Future Applications
• Quantum generative adversarial networks
• Quantum Generative Adversarial Networks for Learning and Loading Random Distributions
• Variational Quantum Computation of Excited States
• Quantum advantage with shallow circuits
• Encoding Electronic Spectra in Quantum Circuits with Linear T Complexity
• Quantum algorithms for electronic structure calculations: particle/hole Hamiltonian and optimized wavefunction expansions
• Quantum speedup of Monte Carlo methods
• Quantum Risk Analysis
• Quantum computational finance: Monte Carlo pricing of financial derivatives
• Quantum Process Tomography of a Universal Entangling Gate Implemented with Josephson Phase Qubits
• Variational quantum algorithms for discovering Hamiltonian spectra
• Creating superpositions that correspond to efficiently integrable probability distributions
• Amplitude estimation without phase estimation
• Natural two-qubit gate for quantum computation using the XY interaction
• Efficient Symmetry-Preserving State Preparation Circuits for the Variational Quantum Eigensolver Algorithm
• Variational Fast Forwarding for Quantum Simulation Beyond the Coherence Time
• Option Pricing using Quantum Computers
• Training of Quantum Circuits on a Hybrid Quantum Computer
• Variational Quantum State Diagonalization
• Quantum algorithms and the finite element method
• $XY$-mixers: analytical and numerical results for QAOA
• Variational algorithms for linear algebra
• Solving the Optimal Trading Trajectory Problem Using a Quantum Annealer
• Quantum circuits for strongly correlated quantum systems
• Quantum advantage with noisy shallow circuits in 3D
• Exact Ising model simulation on a quantum computer
• Quantum Approximate Counting, Simplified
• Quantum Algorithm for Spectral Measurement with Lower Gate Count
• Forecasting financial crashes with quantum computing
• Quantum-inspired algorithms for multivariate analysis: from interpolation to partial differential equations
• Quantum Singular Value Decomposer
• Towards Pricing Financial Derivatives with an IBM Quantum Computer
• Near-term quantum algorithms for linear systems of equations
• Entanglement spectroscopy with a depth-two quantum circuit
• Compressed quantum computation using the IBM Quantum Experience
• Matrix Product States Algorithms and Continuous Systems
• Quantum computational finance: quantum algorithm for portfolio optimization
• Quantum simulation by qubitization without Toffoli gates

Cited by in corpus (23)

• Quantum computing for finance
• Qibo: a framework for quantum simulation with hardware acceleration
• Approximate amplitude encoding in shallow parameterized quantum circuits and its application to financial market indicator
• Quantum Methods for Neural Networks and Application to Medical Image Classification
• Prospects and challenges of quantum finance
• Medical image classification via quantum neural networks
• A variational quantum algorithm for the Feynman-Kac formula
• Efficient Hamiltonian Simulation for Solving Option Price Dynamics
• Encoding-Independent Optimization Problem Formulation for Quantum Computing
• Efficient option pricing with unary-based photonic computing chip and generative adversarial learning
• Performance of Superconducting Quantum Computing Chips under Different Architecture Design
• Encoding trade-offs and design toolkits in quantum algorithms for discrete optimization: coloring, routing, scheduling, and other problems
• Application of deep quantum neural networks to finance
• Efficient Quantum Circuits for Accurate State Preparation of Smooth, Differentiable Functions
• Efficient quantum interpolation of natural data
• Quantum encoder for fixed Hamming-weight subspaces
• Pricing multi-asset derivatives by finite difference method on a quantum computer
• Reducing Circuit Depth in Quantum State Preparation for Quantum Simulation Using Measurements and Feedforward
• Efficient Gaussian State Preparation in Quantum Circuits
• Polynomial T-depth Quantum Solvability of Noisy Binary Linear Problem: From Quantum-Sample Preparation to Main Computation
• Low bit-flip rate probabilistic error cancellation
• Multiclass Portfolio Optimization via Variational Quantum Eigensolver with Dicke State Ansatz
• Quantum Speedup of Monte Carlo Integration with respect to the Number of Dimensions and its Application to Finance