Noisy Simon Period Finding
This addresses the challenge of using current noisy quantum hardware for practical algorithms, offering incremental progress toward quantum advantage before full error correction.
The paper tackles the problem of performing Simon's period-finding algorithm on noisy quantum devices, showing that even with errors, these devices can provide speedups over classical methods by transforming noisy outputs into an LPN instance, with experiments on a 15-qubit IBM device demonstrating this potential advantage.
Let $f: \mathbb{F}_2^n \rightarrow \mathbb{F}_2^n$ be a Boolean function with period $\vec s$. It is well-known that Simon's algorithm finds $\vec s$ in time polynomial in $n$ on quantum devices that are capable of performing error-correction. However, today's quantum devices are inherently noisy, too limited for error correction, and Simon's algorithm is not error-tolerant. We show that even noisy quantum period finding computations may lead to speedups in comparison to purely classical computations. To this end, we implemented Simon's quantum period finding circuit on the $15$-qubit quantum device IBM Q 16 Melbourne. Our experiments show that with a certain probability $τ(n)$ we measure erroneous vectors that are not orthogonal to $\vec s$. We propose new, simple, but very effective smoothing techniques to classically mitigate physical noise effects such as e.g. IBM Q's bias towards the $0$-qubit. After smoothing, our noisy quantum device provides us a statistical distribution that we can easily transform into an LPN instance with parameters $n$ and $τ(n)$. Hence, in the noisy case we may not hope to find periods in time polynomial in $n$. However, we may still obtain a quantum advantage if the error $τ(n)$ does not grow too large. This demonstrates that quantum devices may be useful for period finding, even before achieving the level of full error correction capability.