The Learnability of Unknown Quantum Measurements
- Publication Type:
- Working Paper
- Citation:
- 2015
- Issue Date:
- 2015
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Quantum machine learning has received significant attention in recent years,
and promising progress has been made in the development of quantum algorithms
to speed up traditional machine learning tasks. In this work, however, we focus
on investigating the information-theoretic upper bounds of sample complexity -
how many training samples are sufficient to predict the future behaviour of an
unknown target function. This kind of problem is, arguably, one of the most
fundamental problems in statistical learning theory and the bounds for
practical settings can be completely characterised by a simple measure of
complexity.
Our main result in the paper is that, for learning an unknown quantum
measurement, the upper bound, given by the fat-shattering dimension, is
linearly proportional to the dimension of the underlying Hilbert space.
Learning an unknown quantum state becomes a dual problem to ours, and as a
byproduct, we can recover Aaronson's famous result [Proc. R. Soc. A
463:3089-3144 (2007)] solely using a classical machine learning technique. In
addition, other famous complexity measures like covering numbers and Rademacher
complexities are derived explicitly. We are able to connect measures of sample
complexity with various areas in quantum information science, e.g. quantum
state/measurement tomography, quantum state discrimination and quantum random
access codes, which may be of independent interest. Lastly, with the assistance
of general Bloch-sphere representation, we show that learning quantum
measurements/states can be mathematically formulated as a neural network.
Consequently, classical ML algorithms can be applied to efficiently accomplish
the two quantum learning tasks.
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