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

TL;DR

This work introduces the composable logical gate error as a unified metric for assessing how well approximate quantum error-correcting codes implement logical gates, accounting for leakage and logical inaccuracy. It provides computable, matrix-element–based bounds and ties the error to the Crawford inner numerical radius, enabling circuit-wide error budgeting without energy-bounded norms in CV settings. Focusing on approximate GKP codes with symmetric squeezing, the authors derive concrete upper bounds showing Pauli gates can achieve errors scaling linearly with the squeezing parameter, while certain Clifford implementations fail under the same physical constraints, even in the infinite-squeezing limit. A no-go theorem demonstrates that standard linear-optics gate implementations cannot universally realize Clifford gates accurately in symmetric approximate GKP codes, underscoring a gap between ideal GKP intuition and physically realizable codes and motivating hybrid or alternative approaches for fault-tolerant CV quantum computation.

Abstract

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.

Figures (5)

• Figure 1: Illustration of the construction of the circuit $U^{(3)} U^{(2)} U^{(1)}$ from the graph $G$ as described in \ref{['it:circuitgraphconstruction']}. Input vertices are marked in red, interior vertices in green and output vertices in orange. To each interior vertex $v_t$ for $t \in \{1,2,3\}$ we associate a unitary $U^{(t)}$. The (input and output) spaces $\mathcal{K}_{e_j}$ associated with each edge $e_j$ for $j \in \{1,\dots,7\}$ satisfy Eq. \ref{['eq:inputoutputspace']}, e.g., $\mathcal{K}_{e_1} \otimes \mathcal{K}_{e_2} \simeq \mathcal{K}_{e_4}$.
• Figure 2: Diagrammatic illustration of the definition of $B$.
• Figure 3: The approximate GKP-state $\ket{\mathsf{GKP}_{\kappa, \Delta}^\varepsilon}$ in position-space. The red line represents the envelope $\eta_\kappa(x)\propto e^{-\kappa^2x^2/2}$ of the state, a Gaussian with variance $\kappa^{-2}$. According to our convention, this function has $\varepsilon$-truncated Gaussian peaks of variance $\Delta^2$ centered at all integers.
• Figure 4: Illustration of the relationship between the different types of integer-spaced GKP states. The state $\ket{\mathsf{GKP}_{\kappa,\Delta}}$ is the "standard" approximate GKP state, a superposition of Gaussians centered on integers with a (peak-wise) Gaussian envelope. The state $\ket{\mathsf{GKP}^\varepsilon_{\kappa,\Delta}}$ is obtained from it by replacing the Gaussians by truncated Gaussians, and has support only near each integer. Similarly, the state $\ket{\mathsf{gkp}_{\kappa,\Delta}}$ is a superposition of Gaussians but with a Gaussian envelope which is applied point-wise. The truncated state $\ket{\mathsf{gkp}^\varepsilon_{\kappa,\Delta}}$ is obtained from it by replacing the Gaussians by truncated Gaussians, and has support only near each integer. Finally, peakwise and point-wise GKP states $\ket{\mathsf{GKP}_{\kappa,\Delta}}$ and $\ket{\mathsf{gkp}_{\kappa,\Delta}}$ are related by the Fourier transform, which essentially swaps the role of $\kappa$ and $\Delta$, see Lemma \ref{['thm:fouriertransformapproximate']} for a detailed statement.
• Figure 5: The four states $\ket{\mathsf{GKP}_{\kappa,\Delta}}$, $\ket{\mathsf{GKP}^\varepsilon_{\kappa,\Delta}}$, $\ket{\mathsf{gkp}_{\kappa,\Delta}}$ and $\ket{\mathsf{gkp}^\varepsilon_{\kappa,\Delta}}$ are close for suitable choices of parameters $(\kappa,\Delta,\varepsilon)$. Quantitative bounds on the overlap of pairs of states are established in the Lemmas indicated.

Theorems & Definitions (116)

• Lemma 2.1: Subadditivity of the logical gate error
• proof
• Lemma 2.2
• proof
• Lemma 2.3
• proof
• Lemma 3.1
• proof
• Definition 3.2
• Lemma 3.3: Logical gate error of unitary implementations