When you hear ‘Essher,’ images filled with impossible geometries might come to your mind. In our next Research Spotlight, however, we won’t be discussing the (correctly spelled) artist, Escher—yet the same sense of wonder might still arise. Read what Yixian Qiu, a PhD candidate at Centre for Quantum Technologies, has to say about her latest preprint, available at (https://lnkd.in/egWi2c8P):
"Consider the problem of inferring a probability distribution from a prior distribution and a set of measurement data. The available data play the role of constraints-they put restrictions on what the inferred distribution could be-but these data may not suffice to uniquely determine the distribution. In such situations, a common approach is to invoke the minimum relative entropy principle. This principle advocates that the selected distribution be one that simultaneously minimizes the distinguishability (characterized by the relative entropy) from the prior distribution while also respecting the imposed constraints. When the measurement data is presented in the form of expectation values of random variables, the solution to the corresponding relative entropy minimization problem takes the form known as an ‘’Esscher Transform’’. Also known as `exponential tilting' in statistics, the Esscher Transform and its various extensions have since then found many applications, such as option pricing in mathematical finance, importance sampling for rare-event simulation, Lévy processes in financial economics, and tilted empirical risk minimization in machine learning.
In our work, we explore the extension of the classical Esscher Transform to the quantum setting, motivated by solving a (quantum) relative entropy minimization problem. First, we consider the quantized version of the aforementioned constrained relative entropy minimization problem, and characterize the formal solution of this problem. The form of the solution then inspires us to define the 'Quantum Esscher Transform'. Second, we discuss the implementation of the Quantum Esscher Transform on fault-tolerant quantum computers. Our work thus offers new tools and insights for analyzing quantum systems and optimizing quantum processes, with the potential to significantly impact quantum information processing tasks.
With this new tool in hand, we envision various potential applications thereof, including quantum analogous of option pricing, quantum modeling of Lévy processes, and quantum tilted empirical risk minimization under consideration. Our work has been well-received by the community, and in particular was accepted as a talk at a top conference, the Theory of Quantum Computation, Communication, and Cryptography 2024. We will continue to develop the line of research spawned from the Quantum Esscher Transform, and engage with other researchers with different expertises to realize the full potential and impact of this tool."
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