Computing logical error thresholds with the Pauli Frame Sparse Representation

Abstract

We introduce a sparse classical representation, a truncation strategy and a shot-efficient sampling method to push the classical prediction of quantum error correction thresholds beyond Clifford operations and Pauli errors. As two illustrations of the potential of our method, we first show that coherent noise error thresholds, when computed at the circuit level (i.e taking into account full syndrome circuits) for distances up to d=9, are systematically overestimated (by a factor of about 4) by a Pauli-twirling approximation of the noise. We then apply our method to the recently introduced magic-state cultivation protocol. We show, through shot-efficient importance sampling, that, at distance d=5, the multiplicative factor between the T-gate and the S-gate injection error rate is not the one conjectured from low-d computations: it can be as large as 7.

Figures (13)

• Figure 1: Overview of the Pauli Frame Sparse Representation, and how different operations affect the different components. a) an instance of PFSR is composed of a stabilizer frame, and a sparse vector of populated basis kets, with amplitude, labels, and Pauli histories. b) When applying a Clifford operator, one may simply conjugate both the stabilizer frame and the Pauli histories, leaving the size of the sparse vector unchanged. c) When applying a single Pauli operator, one may also choose to leave the stabilizer frame unchanged by composing the Pauli histories and relabelling accordingly. d) Any general unitary can be decomposed as a linear combination of Paulis that can be applied to the basis kets individually, before adding and merging the basis kets. In the worst case scenario, this will multiply the size of the vector by the number of Paulis in the linear combination. e) When performing a projective measurement, we distinguish two cases: if the measured Pauli commutes with all stabilizers of the stabilizer frame, the latter does not need to be updated, and we can simply perform the projection by deleting the basis kets that are outside of the eigenspace we project to. This will on average divide the size of the sparse vector by two. If the measured Pauli anticommutes with at least one stabilizer of the stabilizer frame, one must update the stabilizer frame and the Pauli histories according to the method described in \ref{['subsubsec:anticommuting_proj']}. The size of the sparse vector will usually stay the same, up to merging of some Pauli histories.
• Figure 2: Patch of rotated surface code with distance $d=5$. Black dots represent physical data qubits, and orange (blue) faces represent $X$-type ($Z$-type) stabilizers. The dotted red (blue) contours represent logical operators $X_\mathrm{L}$ ($Z_\mathrm{L})$. Each plaquette and each edge hosts an additional ancilla qubit (not shown) used to measure the corresponding stabilizer.
• Figure 3: Layered noise application applied to simulation of phenomenological noise on rotated surface code.
• Figure 4: Maximal number of populated basis kets reached through the simulation of d+1 rounds of quantum error correction using the Pauli Frame Sparse Representation. Computed for a patch of rotated surface code submitted to phenomenological-level amplitude damping noise with $\gamma = 0.15$, under the layered noise application scheme.
• Figure 5: Logical error rate for the rotated surface code against amplitude damping noise at the phenomenological level. a) shows exact simulation of amplitude damping using the Pauli Frame Sparse Representation, while b) corresponds to the Pauli-twirled approximation simulated with Stim. Each point corresponds to $10^5$ trajectores in a) and $10^6$ trajectories in b). Both families of simulation show similar thresholds (indicated by the black dashed lines) $\gamma_\mathrm{exact} \approx \gamma_\mathrm{PTA} \approx 0.072$.