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**Overview:** 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.

**Authors: **Gili Rosenberg, Poya Haghnegahdar, Phil Goddard, Peter Carr, Kesheng Wu, Marcos López de Prado

**Paper (latest version)**: 1qbit.com/research/solving-the-optimal-trading-trajectory-problem-using-a-quantum-annealer/

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