Composable logical gate error in approximate quantum error correction: reexamining gate implementations in Gottesman-Kitaev-Preskill codes

Quantifying the accuracy of logical gates is paramount in approximate error correction, where perfect implementations are often unachievable with the available set of physical operations. To this end, we introduce a single scalar quantity we call the (composable) logical gate error. It captures both the deviation of the logical action from the desired target gate as well as leakage out of the code space. It is subadditive under successive application of gates, providing a simple means for analyzing circuits. We show how to bound the composable logical gate error in terms of matrix elements of physical unitaries between (approximate) logical computational basis states. In the continuous-variable context, this sidesteps the need for computing energy-bounded norms. As an example, we study the composable logical gate error for linear optics implementations of Paulis and Cliffords in approximate Gottesman-Kitaev-Preskill (GKP) codes. We find that the logical gate error for implementations of Paulis depends linearly on the squeezing parameter. This implies that their accuracy improves monotonically with the amount of squeezing. For some Cliffords, however, linear optics implementations which are exact for ideal GKP codes fail in the approximate case: they have a constant logical gate error even in the limit of infinite squeezing. This is consistent with previous results about the limitations of certain gate implementations for approximate GKP codes. It shows that findings applicable to ideal GKP codes do not always translate to the realm of physically realizable approximate GKP codes.

💡 Research Summary

The paper tackles a central problem in fault‑tolerant quantum computation with approximate error‑correcting codes: how to quantify the accuracy of logical gates when perfect implementations are impossible with the available physical operations. The authors introduce a single scalar metric, the “composable logical gate error,” which simultaneously captures deviation from the intended logical action and leakage out of the code space. Defined as the diamond‑norm distance between the ideal logical channel (restricted to the code) and the actual physical channel followed by projection back onto the code, this quantity is zero only for a perfect implementation, and it is sub‑additive under sequential composition and tensor products. Consequently, the total error of a circuit can be bounded by the sum of the individual gate errors, greatly simplifying error analysis for complex circuits.

To make the metric computable, the authors express the error in terms of matrix elements of the physical unitary with respect to an orthonormal logical basis. They construct an auxiliary operator (B) that effectively encodes the overlap between the physical unitary and the target logical unitary after projection onto the code. The error then takes the simple closed form (\mathrm{err}=2\sqrt{1-c(B)^2}), where (c(B)) is the Crawford number (inner numerical radius) of (B). This formulation bypasses the need for energy‑bounded norms that appear in earlier continuous‑variable analyses, and it yields explicit polynomial‑time bounds that depend only on a few matrix elements, even for high‑dimensional logical spaces.

The framework is applied to the Gottesman‑Kitaev‑Preskill (GKP) code, a continuous‑variable stabilizer code whose ideal logical Pauli and Clifford operations can be realized exactly by Gaussian (linear‑optics) unitaries. Realistic GKP states are finitely squeezed, described by two parameters ((\kappa,\Delta)) that control the width of the overall envelope and the width of each peak. The authors focus on the symmetric squeezing regime (\Delta=\kappa/(2\pi d)), denoted ( \mathrm{GKP}^{\star}_\kappa