Expressivity of Variational Quantum Machine Learning on the Boolean Cube
Categorical data plays an important part in machine learning research and
appears in a variety of applications. Models that can express large classes of
real-valued functions on the Boolean cube are useful for problems involving
discrete-valued data types, including those which are not Boolean. To this
date, the commonly used schemes for embedding classical data into variational
quantum machine learning models encode continuous values. Here we investigate
quantum embeddings for encoding Boolean-valued data into parameterized quantum
circuits used for machine learning tasks. We narrow down representability
conditions for functions on the $n$-dimensional Boolean cube with respect to
previously known results, using two quantum embeddings: a phase embedding and
an embedding based on quantum random access codes. We show that for any
real-valued function on the $n$-dimensional Boolean cube, there exists a
variational linear quantum model based on a phase embedding using $n$ qubits
that can represent it and an ensemble of such models using $d < n$ qubits that
can express any function with degree at most $d$. Additionally, we prove that
variational linear quantum models that use the quantum random access code
embedding can express functions on the Boolean cube with degree $ d\leq
\lceil\frac{n}{3}\rceil$ using $\lceil\frac{n}{3}\rceil$ qubits, and that an
ensemble of such models can represent any function on the Boolean cube with
degree $ d\leq \lceil\frac{n}{3}\rceil$. Furthermore, we discuss the potential
benefits of each embedding and the impact of serial repetitions. Lastly, we
demonstrate the use of the embeddings presented by performing numerical
simulations and experiments on IBM quantum processors using the Qiskit machine
learning framework.
Dylan Herman Rudy Raymond Muyuan Li Nicolas Robles Antonio Mezzacapo Marco Pistoia