Quantum Error Suppression with Subgroup Stabilisation

Quantum state purification is the functionality that, given multiple copies of an unknown state, outputs a state with increased purity. This will be an essential building block for near- and middle-term quantum ecosystems before the availability of full fault tolerance, where one may want to suppress errors not only in expectation values but also in quantum states. We propose an effective state purification gadget with a moderate quantum overhead by projecting $M$ noisy quantum inputs to their symmetric subspace defined by a set of projectors forming a symmetric subgroup with order $M$. Our method, applied in every short evolution over $M$ redundant copies of noisy states, can suppress both coherent and stochastic errors by a factor of $1/M$, respectively. This reduces the circuit implementation cost $M$ times smaller than the state projection to the full symmetric subspace proposed by Barenco et al. more than two decades ago. We also show that our gadget purifies the depolarised inputs with probability $p$ to asymptotically $O\left(p^{2}\right)$ with an optimal choice of $M$ when $p$ is small. The sampling cost scales $O\left(p^{-1}\right)$ for small $p$, which is also shown to be asymptotically optimal. Our method provides flexible choices of state purification depending on the hardware restrictions before fully fault-tolerant computation is available.