Power of Uninitialized Qubits in Shallow Quantum Circuits

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 O(poly(n)) uninitialized qubits. The initial state of the uninitialized ancillary qubits is arbitrary, and we have to return their state into the initial one at the end of the computation. First, to show the strengths of such circuits, we consider a class of symmetric functions on n bits, including those (such as threshold functions) for which it is not known whether there exist shallow quantum circuits with only O(log n) initialized ancillary qubits. We show that there exists an O(log n)-depth quantum circuit for any function in the class with O(log n) initialized and O(n log n) uninitialized ancillary qubits. Its existence shows the possibility that augmenting uninitialized ancillary qubits increases the computational power of shallow quantum circuits. The depth decreases to O(log n) when we use unbounded fan-out gates and unbounded Toffoli gates. Then, we consider the limitations of shallow quantum circuits with uninitialized ancillary qubits, which include unbounded fan-out gates and unbounded Toffoli gates. We show that, when the number of qubits on which unbounded fan-out gates act is O(poly(log n)), for any constant 0 ≤ δ < 1, there does not exist an O(log n)-depth quantum circuit for the parity function on n bits with O(log n) initialized and O(poly(n)) uninitialized ancillary qubits.