Learning properties of quantum processes is a fundamental task in physics. It is well known that full process tomography scales exponentially in the number of qubits. In this work, we consider online learning quantum processes in a mistake bounded model and prove exponentially improved bounds compared to the stronger notion of diamond norm learning. The problem can be modelled as an interactive game over any given number of rounds, T, between a learner and an adversary. At each timestep, the adversary produces the classical description of an arbitrary n-qubit quantum state and any binary POVM effect. The learner responds with the expected value of the given state evolved under the unkown channel w.r.t. the given effect operator using their guess of the unknown channel.

We show that if the unknown channel is any mixture of (even an exponential number of) known channels, e.g. any Pauli channel, then our proposed online learner’s estimates of the expectation values are epsilon-far from the true expectation at most O(n/epsilon^2) number of times. We complement our result by showing a matching lower bound on the same problem. We further show matching upper and lower bounds on mistakes scaling exponentially in n for online learning arbitrary quantum channels. Finally, we generalize our results for qudit systems having arbitrary memory systems appended with the input state and the POVM effect. Our results open up new directions for efficient online learning of restricted classes of quantum channels, which could potentially lead to efficient algorithms for shadow tomography of special classes of quantum channels.

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