Power of Uninitialized Qubits in Shallow Quantum Circuits

TL;DR

This paper investigates how uninitialized ancillary qubits enhance the computational capabilities of shallow quantum circuits, demonstrating their power in computing symmetric functions and estimating unitaries, but also their limitations in computing parity.

Contribution

It shows that uninitialized ancillary qubits significantly increase the computational power of shallow quantum circuits, enabling complex tasks beyond the reach of circuits with only initialized qubits.

Findings

Shallow quantum circuits with uninitialized qubits can compute any classically polynomial-time symmetric function.

Such circuits can be used as oracles to estimate unitary matrices of constant-depth quantum circuits.

There are fundamental limitations, such as the inability to compute the parity function with near-logarithmic depth.

Abstract

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.