Shallow Quantum Circuits with Uninitialized Ancillary Qubits.

We study the computational power of shallow quantum circuits with $n$ input qubits, one output qubit, and two types of ancillary qubits: $O(log n)$ initialized and $n^{O(1)}$ uninitialized qubits. The initial state of the uninitialized ancillary qubits is arbitrary, but we have to return their state into the initial one at the end of the computation. First, we show that such circuits can compute various symmetric functions on $n$ bits, such as threshold functions. Then, we consider a polynomial-time probabilistic classical algorithm with an oracle that can perform such a circuit. We show that it can estimate the elements of any unitary matrix that can be implemented by a constant-depth quantum circuit on $n$ qubits. Since it is not known whether these tasks can be done with only $O(log n)$ initialized ancillary qubits, our results show the possibility that augmenting uninitialized ancillary qubits increases the computational power of shallow quantum circuits with only $O(log n)$ initialized ancillary qubits. On the other hand, we give the cases where augmenting uninitialized ancillary qubits does not. More concretely, we consider near-logarithmic-depth quantum circuits with only $O(log n)$ initialized ancillary qubits such that they include unbounded fan-out gates on a small number of qubits and unbounded Toffoli gates. We show that they cannot compute the parity function on $n$ bits, even when they are augmented by $n^{O(1)}$ uninitialized ancillary qubits.