Quantum unary approach to option pricing

We present a novel 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 amplification is used to gain quantum advantage. Unary representation offers several advantages over the usual binary representation of asset values, and one disadvantage. On the positive side, unary representation remarkably simplifies the structure and depth of the quantum circuit. Amplitude distributions use quantum superposition to bypass the role of the 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 offers lower resolution given a fixed number of qubits, as compared to 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 practise.