Practical Period Finding on IBM Q -- Quantum Speedups in the Presence of Errors

We implemented Simon's quantum period finding circuit for functions $\mathbb{F}_2^n \rightarrow \mathbb{F}_2^n$ with period $\vec s \in \mathbb{F}_2^n$ up to $n=7$ on the 14-qubit quantum device IBM Q 16 Melbourne. Our experiments show that with a certain probability $\tau(n)$ we measure erroneous vectors that are not orthogonal to $\vec s$. While Simon's algorithm for extracting $\vec s$ runs in polynomial time in the error-free case $\tau(n)=0$, we show that the problem of extracting $\vec s \in \mathbb{F}_2^n$ in the general setting $0 \leq \tau(n) \leq \frac 1 2$ is as hard as solving LPN (Learning Parity with Noise) with parameters $n$ and $\tau(n)$. Hence, in the error-prone case we may not hope to find periods in time polynomial in $n$. However, we also demonstrate theoretically and experimentally that erroneous quantum measurements are still useful to find periods faster than with purely classical algorithms, even for large errors $\tau(n)$ close to $\frac 1 2$.