  An ideal system of $n$ qubits has $2^n$ dimensions. This exponential grants
power, but also hinders characterizing the system's state and dynamics. We
study a new problem: the qubits in a physical system might not be independent.
They can "overlap," in the sense that an operation on one qubit slightly
affects the others.
  We show that allowing for slight overlaps, $n$ qubits can fit in just
polynomially many dimensions. (Defined in a natural way, all pairwise overlaps
can be $\leq \epsilon$ in $n^{O(1/\epsilon^2)}$ dimensions.) Thus, even before
considering issues like noise, a real system of $n$ qubits might inherently
lack any potential for exponential power.
  On the other hand, we also provide an efficient test to certify exponential
dimensionality. Unfortunately, the test is sensitive to noise. It is important
to devise more robust tests on the arrangements of qubits in quantum devices.
