  Until fault-tolerance becomes implementable at scale, quantum computing will
heavily rely on noise mitigation techniques. While methods such as zero noise
extrapolation with probabilistic error amplification (ZNE-PEA) and
probabilistic error cancellation (PEC) have been successfully tested on
hardware recently, their scalability to larger circuits may be limited. Here,
we introduce the tensor-network error mitigation (TEM) algorithm, which acts in
post-processing to correct the noise-induced errors in estimations of physical
observables. The method consists of the construction of a tensor network
representing the inverse of the global noise channel affecting the state of the
quantum processor, and the consequent application of the map to informationally
complete measurement outcomes obtained from the noisy state. TEM does therefore
not require additional quantum operations other than the implementation of
informationally complete POVMs, which can be achieved through randomised local
measurements. The key advantage of TEM is that the measurement overhead is
quadratically smaller than in PEC. We test TEM extensively in numerical
simulations in different regimes. We find that TEM can be applied to circuits
of twice the depth compared to what is achievable with PEC under realistic
conditions with sparse Pauli-Lindblad noise, such as those in [E. van den Berg
et al., Nat. Phys. (2023)]. By using Clifford circuits, we explore the
capabilities of the method in wider and deeper circuits with lower noise
levels. We find that in the case of 100 qubits and depth 100, both PEC and ZNE
fail to produce accurate results by using $\sim 10^5$ shots, while TEM
succeeds.
