We study the computational power of shallow quantum circuits
with O(log n) initialized and n^{O(1)} uninitialized ancillary
qubits, where n is the input length and the initial state of
the uninitialized ancillary qubits is arbitrary. First, we show
that such a circuit can compute any symmetric function on n bits
that is classically computable in polynomial time. Then, we
regard such a circuit as an oracle and show that a
polynomial-time classical algorithm with the oracle can estimate
the elements of any unitary matrix corresponding to a
constant-depth quantum circuit on n qubits. Since it seems unlikely
that these tasks can be done with only O(log n) initialized
ancillary qubits, our results give evidences that adding
uninitialized ancillary qubits increases the computational power
of shallow quantum circuits with only O(log n) initialized
ancillary qubits. Lastly, to understand the limitations of
uninitialized ancillary qubits, we focus on
near-logarithmic-depth quantum circuits with them and show
the impossibility of computing the parity function on n bits
