Quantum Meets the Minimum Circuit Size Problem

TL;DR

This paper initiates the study of the quantum version of the Minimum Circuit Size Problem, exploring its properties, relationships, and implications in quantum complexity theory and related fields.

Contribution

It defines quantum MCSPs for functions, unitaries, and states, analyzes their complexity, and establishes new reductions and connections to other quantum computational problems.

Findings

Quantum MCSPs are in QCMA, not trivially in NP.

Existence of search-to-decision reductions for quantum MCSPs.

Connections to quantum cryptography, learning, and quantum gravity.

Abstract

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.