The Quantum Esscher Transform

TL;DR

This paper introduces the quantum Esscher Transform, a quantum generalization of a classical probability tool, and provides an efficient quantum algorithm for its implementation using advanced quantum computing techniques.

Contribution

It defines the quantum Esscher Transform, extending classical concepts to quantum states, and develops a quantum algorithm for its efficient computation on fault-tolerant quantum computers.

Findings

The quantum Esscher Transform generalizes the classical version.

The proposed algorithm achieves accuracy within $ ilde O(rac{ ext{poly}( ext{condition number}, d)}{ ext{poly}(rac{1}{ ext{error}})})$ queries.

Potential applications in quantum information theory and quantum computing implementations.

Abstract

The Esscher Transform is a tool of broad utility in various domains of applied probability. It provides the solution to a constrained minimum relative entropy optimization problem. In this work, we study the generalization of the Esscher Transform to the quantum setting. We examine a relative entropy minimization problem for a quantum density operator, potentially of wide relevance in quantum information theory. The resulting solution form motivates us to define the \textit{quantum} Esscher Transform, which subsumes the classical Esscher Transform as a special case. Envisioning potential applications of the quantum Esscher Transform, we also discuss its implementation on fault-tolerant quantum computers. Our algorithm is based on the modern techniques of block-encoding and quantum singular value transformation (QSVT). We show that given block-encoded inputs, our algorithm outputs a subnormalized block-encoding of the quantum Esscher transform within accuracy $\epsilon$ in $\tilde O(\kappa d \log^2 1/\epsilon)$ queries to the inputs, where $\kappa$ is the condition number of the input density operator and $d$ is the number of constraints.