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Pauli Center for Theoretical Studies

The Swiss National Science Foundation

ETH Zurich (Computer Science and Physics Department)

Quantum Science and Technology

CQT Singapore

QAP European Project

Sandia National Laboratories

Institute for Quantum Computing

id Quantique

Laying the quantum and classical embedding problems to rest

Toby Cubitt, Bristol

joint work with Jens Eisert and Michael Wolf

Quantum channels and master equations are both widely used to describe the dynamics of quantum systems that are subject to noise. Quantum channels, also known as completely-positive maps, are commonly used in quantum information theory, where abstracting away the underlying physics allows one to focus on the information-theoretic aspects of noise. Master equations are frequently used in physics, where the underlying physical processes are the main focus of attention. The connection between these two descriptions is a classic topic in mathematical physics. One direction was solved by the now famous result due to Lindblad, Kossakowski and Gorini, who gave a complete characterisation of master equations that generate completely positive evolutions. However, the other direction has remained open: given a quantum channel, is there a master equation that generates it (and if so, can we deduce it's form)? Physically, this is asking how one can deduce underlying physical processes from experimental observations.

The analogous question can equally well be posed in classical dynamics: given a stochastic map, does there exist a continuous-time Markov chain that generates it? This is known in probability theory as the embedding problem, and it is even older than the quantum version; it was first studied at least as far back as 1937. It too has remained an open problem to this day. In this work, we give complexity theoretic solutions to both the quantum and classical embedding problems: both problems are NP-hard. Moreover, we give an explicit algorithm that reduces the problem to integer semi-definite programming, completing the proof that solving the quantum or classical embedding problem is equivalent to solving P=NP, thus finally laying the embedding problem to rest after more than 70 years.