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We introduce a technique to estimate error-mitigated expectation values onnoisy quantum computers. Our technique performs shadow tomography on a logicalstate to produce a memory-efficient classical reconstruction of the noisydensity matrix. Using efficient classical post-processing, one can mitigateerrors by projecting a general nonlinear function of the noisy density matrixinto the codespace. The subspace expansion and virtual distillation can beviewed as special cases of the new framekwork. We show our method is favorablein the quantum and classical resources overhead. Relative to subspace expansionwhich requires $O\left(2^{N} \right)$ samples to estimate a logical Pauliobservable with $[[N, k]]$ error correction code, our technique requires only$O\left(4^{k} \right)$ samples. Relative to virtual distillation, our techniquecan compute powers of the density matrix without additional copies of quantumstates or quantum memory. We present numerical evidence using logical statesencoded with up to sixty physical qubits and show fast convergence toerror-free expectation values with only $10^5$ samples under 1% depolarizingnoise.

Logical shadow tomography: Efficient estimation of error-mitigated observables