In this work, we initiate the study of the Minimum Circuit Size Problem
(MCSP) in the quantum setting. MCSP is a problem to compute the circuit
complexity of Boolean functions. It is a fascinating problem in complexity
theory — its hardness is mysterious, and a better understanding of its
hardness can have surprising implications to many fields in computer science.

We first define and investigate the basic complexity-theoretic properties of
minimum quantum circuit size problems for three natural objects: Boolean
functions, unitaries, and quantum states. We show that these problems are not
trivially in NP but in QCMA (or have QCMA protocols). Next, we explore the
relations between the three quantum MCSPs and their variants. We discover that
some reductions that are not known for classical MCSP exist for quantum MCSPs
for unitaries and states, e.g., search-to-decision reduction and
self-reduction. Finally, we systematically generalize results known for
classical MCSP to the quantum setting (including quantum cryptography, quantum
learning theory, quantum circuit lower bounds, and quantum fine-grained
complexity) and also find new connections to tomography and quantum gravity.
Due to the fundamental differences between classical and quantum circuits, most
of our results require extra care and reveal properties and phenomena unique to
the quantum setting. Our findings could be of interest for future studies, and
we post several open problems for further exploration along this direction.

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