Decoding quantum errors with subspace expansions

Abstract

With rapid developments in quantum hardware comes a push towards the first practical applications. While fully fault-tolerant quantum computers are not yet realized, there may exist intermediate forms of error correction that enable practical applications. In this work, we consider the idea of post-processing error decoders using existing quantum codes, which mitigate errors on logical qubits using post-processing without explicit syndrome measurements or additional qubits beyond the encoding overhead. This greatly simplifies the experimental exploration of quantum codes on real, near-term devices, removing the need for locality of syndromes or fast feed-forward. We develop the theory of the method and demonstrate it on an example with the perfect [[5, 1, 3]] code, which exhibits a pseudo-threshold of p ≈ 0.50 under a single qubit depolarizing channel applied to all qubits. We also provide a demonstration of improved performance on an unencoded hydrogen molecule.

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.

Fig. 2. Cartoon schematic of error correction vs error projection in a stabilizer code.

We sketch the quantum space as divided into the blue code space, defined by  +1 eigenvalues of stabilizers and the red non-code space as defined by having  −1 eigenvalues for some of the stabilizers. In traditional error correction, the stabilizers are measured, the errors decoded, and recovery operations are applied to return one to the code space. In error projection, we use projectors based on stabilizers to remove sections of non-code space using only simple Pauli measurements and post-processing. We can also combine this technique with forms of recovery, but the effective difference is depicted by the discarding of large parts of errant Hilbert space.

Fig. 3. Pseudo-threshold crossover for recovery using the [[5, 1, 3]] code.

The model examines the impact of errors under an uncorrelated depolarizing channel showing a p ≈ 0.50 pseudo threshold for the full correction procedure. We plot the logical infidelity 1 −;F_L where F_L is the logical fidelity of a selected state in the code space of the [[5, 1, 3]] code as a function of the depolarizing probability for each qubit p. The label l denotes the number of products from the stabilizer generators used in the expansion operator set. The physical line depicts the same error if the logical state is encoded in a single qubit in the standard way. The “bare'' line indicates the logical error rate with no recovery procedure applied. The starred lines close to each level of the hierarchy show an approximation to that level of projection using the QSE projection with 2 less check operators to demonstrate the smooth performance of the subspace procedure.

Fig. 4. Error suppression using natural molecular symmetries.

Errors are plotted as a function of depolarizing probability and included projectors for an H₂ molecule at a bond length of 1.50 ${}^{{}^{\circ }}A$ with a physical encoding. F_L here is the logical fidelity, which is the same as physical for this encoding. At this bond length, the ground state wavefunction requires considerable entanglement to be qualitatively correct. The stabilizer elements used to perform the projection here are the α- and β- number parity as well as the total X operator, which need not be an exact symmetry, given in the Jordan-Wigner encoding as S = {Z₀Z₂, Z₁Z₃, X₀X₁X₂X₃}. The depolarizing channel is applied individually to each qubit on the exact ground state. As there is no error correction encoding overhead in this case, the expected improvement over the uncorrected solution is always positive. Hence it is always advantageous in these cases where simple symmetries are known to include these measurements and corrections.