# Quantum Algorithms for Portfolio Optimization

arXiv: https://arxiv.org/abs/1908.08040 · HTML: https://mlsanity.com/paper/1908.08040/

## Key facts

- Cited by 8 papers in the map (data through 7 Oct 2026).
- First posted on arXiv 22 Aug 2019.

## Built on by (most cited first, 8 of 8)

- [Quantum Deep Hedging](https://mlsanity.com/paper/2303.16585/) (2023)
- [Improved Financial Forecasting via Quantum Machine Learning](https://mlsanity.com/paper/2306.12965/) (2023)
- [Quantum algorithms for SVD-based data representation and analysis](https://mlsanity.com/paper/2104.08987/) (2021)
- [Photonic Quantum Policy Learning in OpenAI Gym](https://mlsanity.com/paper/2108.12926/) (2021)
- [Classical ensemble of Quantum-classical ML algorithms for Phishing detection in Ethereum transaction networks](https://mlsanity.com/paper/2211.00004/) (2022)
- [A Quantum Computing-based System for Portfolio Optimization using Future Asset Values and Automatic Reduction of the Investment Universe](https://mlsanity.com/paper/2309.12627/) (2023)
- [Quantum Algorithms for Projection-Free Sparse Convex Optimization](https://mlsanity.com/paper/2507.08543/) (2025)
- [A Quantum Model for Constrained Markowitz Modern Portfolio Using Slack Variables to Process Mixed-Binary Optimization under QAOA](https://mlsanity.com/paper/2601.03278/) (2025)

## Abstract (arXiv)

We develop the first quantum algorithm for the constrained portfolio optimization problem. The algorithm has running time widetildeO left( n√r fraczeta kappaδ2 log left(1/εright) right), where r is the number of positivity and budget constraints, n is the number of assets in the portfolio, ε the desired precision, and δ, kappa, zeta are problem-dependent parameters related to the well-conditioning of the intermediate solutions. If only a moderately accurate solution is required, our quantum algorithm can achieve a polynomial speedup over the best classical algorithms with complexity widetildeO left( √rnωlog(1/ε) right), where ω is the matrix multiplication exponent that has a theoretical value of around 2.373, but is closer to 3 in practice. We also provide some experiments to bound the problem-dependent factors arising in the running time of the quantum algorithm, and these experiments suggest that for most instances the quantum algorithm can potentially achieve an O(n) speedup over its classical counterpart.