# Solving the Optimal Trading Trajectory Problem Using a Quantum Annealer

arXiv: https://arxiv.org/abs/1508.06182 · HTML: https://mlsanity.com/paper/1508.06182/

## Key facts

- Cited by 12 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 22 Aug 2015.

## Built on by (most cited first, 12 of 12)

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- [Relaxation-assisted reverse annealing on nonnegative/binary matrix factorization](https://mlsanity.com/paper/2501.02114/) (2025)
- [Quantum-Inspired Machine Learning for Molecular Docking](https://mlsanity.com/paper/2401.12999/) (2024)
- [Large-scale portfolio optimization with variational neural annealing](https://mlsanity.com/paper/2507.07159/) (2025)
- [Variational Quantum Circuit-Based Reinforcement Learning for Dynamic Portfolio Optimization](https://mlsanity.com/paper/2601.18811/) (2026)
- [A Generative AI Framework for Intelligent Utility Billing CO 2 Analytics and Sustainable Resource Optimisation](https://mlsanity.com/paper/2605.16250/) (2026)
- [Benchmarking Quantum Algorithmic Resilience for CVaR Portfolio Optimization: The Expressibility-Coherence Trade-off](https://mlsanity.com/paper/2606.07727/) (2026)
- [A Novel Hybrid Quantum Reservoir Computing (nHQRC) for Phase Transition Detection in Non-Equilibrium Dynamical Systems](https://mlsanity.com/paper/2607.16281/) (2026)

## Abstract (arXiv)

We solve a multi-period portfolio optimization problem using D-Wave Systems' quantum annealer. We derive a formulation of the problem, discuss several possible integer encoding schemes, and present numerical examples that show high success rates. The formulation incorporates transaction costs (including permanent and temporary market impact), and, significantly, the solution does not require the inversion of a covariance matrix. The discrete multi-period portfolio optimization problem we solve is significantly harder than the continuous variable problem. We present insight into how results may be improved using suitable software enhancements, and why current quantum annealing technology limits the size of problem that can be successfully solved today. The formulation presented is specifically designed to be scalable, with the expectation that as quantum annealing technology improves, larger problems will be solvable using the same techniques.