  Small numbers of qubits are one of the primary constraints on the near-term
deployment of advantageous quantum computing. To mitigate this constraint,
techniques have been developed to break up a large quantum computation into
smaller computations. While this work is sometimes called circuit knitting or
divide and quantum we generically refer to it as circuit cutting (CC). Much of
the existing work has focused on the development of more efficient circuit
cutting schemes, leaving open questions on the limits of what theoretically
optimal schemes can achieve. We develop bounds by breaking up possible
approaches into two distinct regimes: the first, where the input state and
measurement are fixed and known, and the second, which requires a given cutting
to work for a complete basis of input states and measurements. For the first
case, it is easy to see that bounds addressing the efficiency of any approaches
to circuit cutting amount to resolving BPP$\stackrel{?}{=}$BQP. We therefore
restrict ourselves to a simpler question, asking what \textit{locally-acting}
circuit cutting schemes can achieve, a technical restriction which still
includes all existing circuit cutting schemes. In our first case we show that
the existence of a locally-acting circuit cutting scheme which could
efficiently partition even a single qubit from the rest of a circuit would
imply BPP$=$BQP. In our second case, we obtain more general results, showing
inefficiency unconditionally. We also show that any (local or otherwise)
circuit cutting scheme cannot function by only applying unital channels.
