Yunchao Liu

Recently, several quantum benchmarking algorithms have been developed to characterize noisy quantum gates on today's quantum devices. A well-known issue in benchmarking is that not everything about quantum noise is learnable due to the existence of gauge freedom, leaving open the question of what information about noise is learnable and what is not, which has been unclear even for a single CNOT gate. Here we give a precise characterization of the learnability of Pauli noise channels attached to Clifford gates, showing that learnable information corresponds to the cycle space of the pattern transfer graph of the gate set, while unlearnable information corresponds to the cut space. This implies the optimality of cycle benchmarking, in the sense that it can learn all learnable information about Pauli noise. We experimentally demonstrate noise characterization of IBM's CNOT gate up to 2 unlearnable degrees of freedom, for which we obtain bounds using physical constraints. In addition, we give an attempt to characterize the unlearnable information by assuming perfect initial state preparation. However, based on the experimental data, we conclude that this assumption is inaccurate as it yields unphysical estimates, and we obtain a lower bound on state preparation noise.

Abstract Understanding the power of random quantum circuit sampling experiments has emerged as one of the most pressing topics in the near-term quantum era. In this work we make progress toward bridging the major remaining gaps between theory and experiment, incorporating the effects of experimental imperfections into the theoretical hardness arguments. We do this first by proving that computing the output probability of an $m$-gate random quantum circuit to within additive imprecision $2^{-O(m^{1+\epsilon})}$ is #P-hard for any $\epsilon>0$, an exponential improvement over the prior hardness results of Bouland et al. and Movassagh which were resistant to imprecision $2^{-O(m^3)}$. This improvement very nearly reaches the threshold ($2^{-O(m)}/\text{poly}(m)$) sufficient to establish the hardness of sampling for constant-depth random quantum circuits. To prove this result we introduce new error reduction techniques for polynomial interpolation, as well as a new robust Berlekamp-Welch argument over the Reals which may be of independent interest. Second we show that these results are still true in the presence of a constant rate of noise, so long as the noise rate is below the error detection threshold. That is, even though random circuits with a constant noise rate converge rapidly to the maximally mixed state, the (exponentially) small deviations in their output probabilities away from uniformity remain difficult to compute. Interestingly, we then show that our two main results are in tension with one another, and the latter result implies the former result is essentially optimal with respect to additive imprecision error, even with substantial generalizations of our techniques.