Fig. 1: Algorithmic schematics of the stochastic and deterministic subspace expansions.

The goal is to use an expansion in a subspace around a prepared quantum state, ρ, to improve the expected value of the logical observable 〈Γ〉, without requiring ancilla based syndrome measurements or feedfoward. An observable in the logical space Γ is expressed as a sum of Pauli operators, Γ_k, while symmetries, M_k, either naturally dictated by a system or from the stabilizer group S are selected. In the stochastic case (a), the state ρ is re-prepared many times, and these measurements are used to assemble the corrected expectation value 〈Γ〉 by expanding the averaged result in the resulting subspace. In the deterministic case (b), we may expand the set of M_k to include non-symmetries, and the corresponding averages over ρ are evaluated with many repetitions to form the representations of the operators in the subspace around ρ. These matrices define an offline generalized eigenvalue problem whose solution, C, defines both an optimal projector in the basis of operators M_i, \({\overline{P}}_{c}\) and corrected expectation values 〈Γ〉 for desired observables. We note that a scheme for including recovery operations can be found in the methods section.