Derivative Pricing using Quantum Signal Processing
Nikitas Stamatopoulos, William J. Zeng
Abstract
Pricing financial derivatives on quantum computers typically includes quantum arithmetic components which contribute heavily to the quantum resources required by the corresponding circuits. In this manuscript, we introduce a method based on Quantum Signal Processing (QSP) to encode financial derivative payoffs directly into quantum amplitudes, alleviating the quantum circuits from the burden of costly quantum arithmetic. Compared to current state-of-the-art approaches in the literature, we find that for derivative contracts of practical interest, the application of QSP significantly reduces the required resources across all metrics considered, most notably the total number of T-gates by <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mo>&#x223C;</mml:mo><mml:mn>16</mml:mn></mml:math>x and the number of logical qubits by <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mo>&#x223C;</mml:mo><mml:mn>4</mml:mn></mml:math>x. Additionally, we estimate that the logical clock rate needed for quantum advantage is also reduced by a factor of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mo>&#x223C;</mml:mo><mml:mn>5</mml:mn></mml:math>x. Overall, we find that quantum advantage will require <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mn>4.7</mml:mn></mml:math>k logical qubits, and quantum devices that can execute <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msup><mml:mn>10</mml:mn><mml:mn>9</mml:mn></mml:msup></mml:math> T-gates at a rate of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mn>45</mml:mn></mml:math>MHz. While in this work we focus specifically on the payoff component of the derivative pricing process where the method we present is most readily applicable, similar techniques can be employed to further reduce the resources in other applications, such as state preparation.