Gottesman-Kitaev-Preskill Code

• GKP Code is a continuous-variable quantum error-correcting code that embeds qubits into the infinite-dimensional space of harmonic oscillators using a lattice of phase-space symmetries.
• It corrects small displacement errors in both position and momentum by leveraging continuous syndrome measurements and maximum-likelihood decoding methodologies.
• GKP codes form the cornerstone of fault-tolerant quantum architectures, enabling higher noise thresholds and practical error correction with relaxed squeezing requirements.

The Gottesman–Kitaev–Preskill (GKP) code is a continuous-variable quantum error-correcting code that encodes discrete logical information (typically a qubit) into the infinite-dimensional Hilbert space of quantum harmonic oscillators. By imposing a lattice of translational symmetries in phase space, the GKP code achieves resilience against small shift (displacement) errors in both position and momentum quadratures. Original theoretical proposals have matured into experimentally viable protocols, and the GKP code is now central to leading architectures targeting high-threshold, hardware-efficient quantum computation and communication.

1. Encoding Mechanism and Stabilizer Structure

The GKP code encodes a finite-dimensional logical system (usually a qubit) into an oscillator by constraining codewords to be stabilized by highly nontrivial displacement operators. In the prototypical single-mode case, the ideal logical basis states in the position ($q$) representation are combs of Dirac delta functions positioned at

$$|0̄\rangle \propto \sum_{n\in\mathbb{Z}} \delta(q - 2\sqrt{\pi} n), \qquad |1̄\rangle \propto \sum_{n\in\mathbb{Z}} \delta(q - (2n+1)\sqrt{\pi})$$

These states are simultaneous eigenstates of a pair of commuting stabilizer operators: $$S_q = \exp(-i 2\sqrt{\pi} \hat{p}), \qquad S_p = \exp(i 2\sqrt{\pi} \hat{q})$$ which effect translations by $2\sqrt{\pi}$ in the conjugate quadratures. The GKP codespace is the common +1 eigenspace under both $S_q$ and $S_p$. In practice, ideal delta functions are replaced by narrow Gaussians (finite squeezing), resulting in "approximate" GKP states with a width parameter $\Delta$: $$|\tilde{0}\rangle \propto \sum_{s} \exp\left[-\frac{(q - 2s\sqrt{\pi})^2}{2\Delta^2}\right]$$ The code generalizes to multimode systems by constructing the stabilizer group from the translations defined by a full-rank lattice $\mathcal{L}$ in phase space. The requirement that all stabilizers commute imposes a symplectic integrality condition on $\mathcal{L}$: for basis vectors $\xi_i, \xi_j \in \mathcal{L}$, $\xi_i^T J \xi_j \in \mathbb{Z}$, with $J$ the standard symplectic matrix.

2. Resistance to Shift Errors and Error Correction

The GKP code is fundamentally designed to correct small displacement (shift) errors in both quadratures. Any physical error $E$ acting as a small displacement $e^{iu\hat{p}}e^{iv\hat{q}}$ will shift the code state by $(u, v)$ in phase space. Because the codewords are invariant under stabilizer displacements, such an error is correctable provided $|u|,|v| < \sqrt{\pi}/2$—half the lattice spacing. Syndrome measurement yields the displacement modulo the lattice vector, so correction consists of shifting back to the nearest lattice point.

The recovery process leverages the code's continuous-variable nature. Homodyne measurements (e.g., in the $q$ basis) yield real-valued outcomes $q_{out}$ that encode the actual shift modulo $\sqrt{\pi}$ (the lattice spacing). Rather than reducing this information to a discrete error (as in qubit codes), the full continuous outcome is used to update the error probability distribution via Bayes' theorem: $$P(u_1 \mid u_1 + u_2 = q_{out} + n\sqrt{\pi}) = \frac{P(u_2 = q_{out} + n\sqrt{\pi} - u_1) P(u_1)}{P(u_1 + u_2 = q_{out} + n\sqrt{\pi})}$$ This enables the assignment of a conditional logical error rate $P(\bar{X} \mid q_{out})$ to each qubit. In multiqubit settings or concatenated codes, the conditional probabilities are mapped to log-likelihood weights for use in decoders such as minimum-weight matching.

3. Decoding Strategies: Maximum-Likelihood Utilizing Continuous Syndromes

Traditionally, decoding with GKP codes has employed minimum-weight matching (or similar graph-based algorithms) using an average logical error rate per qubit. This work introduces and formalizes maximum-likelihood decoding (MLD) using all available analog syndrome data. For a set of measured outcomes $\{q_{out,j}\}$ and their associated error probabilities, MLD computes the total likelihood of each logical error equivalence class $G$: $P(G) = \sum_{g \in G} P(g)$ where $P(g)$ is the probability of a particular error configuration based on all individual $P(\bar{X} \mid q_{out,j})$. Decoding proceeds by selecting the equivalence class that maximizes this sum, minimizing the logical error probability. Specifically, in concatenated settings (e.g., GKP + repetition code or GKP + toric code), the weights assigned to physical qubits become

$$w_j = -\log \left[ \frac{1 - P(\bar{X} | q_{out, j})}{P(\bar{X} | q_{out, j})} \right]$$

This provides a refined, instance-specific decoding metric, in stark contrast with a uniform (average) error assignment. Simulations confirm that including analog syndrome information increases the noise threshold for logical error protection.

4. Simulation and Analytical Results for Concatenated Architectures

Detailed analytical and numerical studies of concatenated GKP codes (e.g., GKP + three-qubit repetition, GKP + toric code) demonstrate:

• Conditional logical error rates (from syndrome measurement) directly determine decoder weights.
• In the toric code, a fixed average error rate leads to a critical threshold at effective $p_{\mathrm{err}} \approx 10.3\%$, while using continuous GKP outcomes allows the threshold to be met even with $p_{\mathrm{err}} \approx 14\%$ (i.e., noisier GKP states with variance $\sigma \approx 0.6$).
• For the repetition code, maximum-likelihood decoding resolves syndrome ambiguities that cannot be addressed with uniform error probabilities.
• Theoretical models and simulations verify that maximum-likelihood decoding using the analog data relaxes the required state squeezing for practical error correction, making GKP codes more feasible in the presence of experimental imperfections.

Post-selection (discarding qubits with syndrome outcomes close to decision boundaries) further reduces logical error, although it is less compatible with full fault-tolerance.

5. Application to Fault-Tolerant Quantum Computation

Efficient use of the full continuous syndrome information strengthens the intrinsic fault tolerance of the GKP code. The update of conditional error probabilities following each correction round allows concatenated codes to "track" error likelihoods at every level, propagating analog information up through concatenated layers such as the surface code.

In these architectures, GKP codes serve as the inner (bosonic) code, primarily correcting shift errors, while outer stabilizer codes (e.g., surface or toric) handle residual logical qubit errors. The decoder for the outer code is modified so that the individual weight or likelihood for each "qubit" (actually a GKP-protected mode) is determined by its observed syndrome, thus optimally exploiting the analog information. This closes the gap between continuous-variable and discrete-variable error correction, enabling higher thresholds and reduced resource overhead per logical qubit.

6. Significance, Limitations, and Implementation Considerations

The described approach establishes that GKP codes, by leveraging the continuous information naturally produced by bosonic modes, achieve enhanced error thresholds compared to codes that discard that information. Maximum-likelihood decoding with analog data not only improves performance in simulation but also supports practical experimental designs with relaxed squeezing requirements.

Practical considerations include:

• Measurement precision: High-fidelity homodyne detection is critical for accurate syndrome readout and reliable conditional error probability estimation.
• Decoder complexity: Maximum-likelihood processing of analog syndromes increases classical decoder complexity relative to binary-weight decoders but remains computationally feasible for moderate system sizes.
• Ancilla quality: The presence of noise and finite squeezing in ancilla modes must be accounted for in the updated error model; the use of conditional probability updates ensures that the effect of ancilla-induced errors is properly propagated.
• Fault-tolerance: While post-selection can lower logical error rates in some scenarios, in full fault-tolerant schemes all outcomes must be accepted to preserve computational determinism.

7. Impact and Future Directions

The methodology introduced—assigning conditional error rates based on analog syndrome outcomes and adopting maximum-likelihood decoding for GKP-encoded qubits—has a foundational impact on the design and decoding of concatenated bosonic quantum codes. These concepts have been incorporated in subsequent high-threshold schemes and are central to practical proposals for fault-tolerant, hardware-efficient quantum computing. Future research directions include optimized decoding algorithms for large-scale systems, further integration with hybrid discrete/continuous-variable codes, and experimental implementations that make full use of continuous measurement data to reduce quantum hardware overhead and enhance noise tolerance.