Approximate Gottesman-Kitaev-Preskill Codes

• The paper demonstrates how applying a Gaussian envelope to ideal grid states yields finite-energy approximate GKP codes, quantifying trade-offs in error correction and energy cost.
• Modular subsystem decomposition is used to separate logical qubits from gauge noise, enabling precise assessment of logical fidelity and error impacts.
• Photonic state preparation using displacement, squeezing, and PNR detection offers a practical route to generate non-Gaussian grid states for NISQ and fault-tolerant quantum devices.

Approximate Gottesman-Kitaev-Preskill (GKP) codes are a class of continuous-variable quantum error-correcting codes in which qubits are encoded into the infinite-dimensional Hilbert space of a bosonic mode, such as an optical or microwave oscillator. Ideal GKP codewords are nonphysical grid (or "lattice comb") states in phase space characterized by delta-function spikes with perfect translational symmetry, capable of correcting arbitrary small shift errors. Due to their infinite energy, physically realizable quantum experiments must employ finite-energy, "^^^^1^^^^" GKP codes, in which the ideal comb is replaced by a periodic array of Gaussian peaks enveloped by a broad, normalizing function. Theoretical characterization, practical state preparation, correction protocols, and performance metrics for these approximate codes are key for their deployment in near-term quantum architectures.

1. Structure and Characterization of Approximate GKP States

Approximate GKP codewords are constructed from ideal grid states through the action of an "envelope" Gaussian damping operator: $E(\epsilon) = \exp(–\epsilon\, n)$ where $n$ is the photon number operator, and $\epsilon$ controls the degree of damping. The finite-energy codewords are given by

$$|\psi_{(\epsilon)}\rangle = E(\epsilon) |\psi_{(I)}\rangle$$

where $|\psi_{(I)}\rangle$ is an ideal codeword. This operation broadens each delta spike into a finite-width Gaussian of width $\Delta$ and imposes a slowly varying envelope over the entire comb structure. As a result, the exact phase-space symmetries enjoyed by the ideal code are broken, and logical actions such as repeated displacements that are identity in the ideal space now degrade both the physical and logical fidelity.

To quantify the non-idealities, multiple figures of merit are employed:

• Physical fidelity $F^{(P)} = |\langle\psi|\phi\rangle|^2$: overlap of full physical states.
• Logical fidelity $F^{(L)}$: overlap after tracing out non-logical (gauge) degrees of freedom via modular subsystem decomposition.
• Distribution distance: $L^1$-distance between binned measurement distributions on the ideal and approximate state.
• Photon number statistics: mean and variance of photon number as energy cost proxies.

Increasing $\epsilon$ (or alternatively, increasing the spike width $\Delta$) reflects the trade-off between physical realizability and code performance, with narrower peaks and weaker envelopes yielding better error correction but higher energy requirements.

2. Modular Subsystem Decomposition and Logical Information

Approximate GKP states can be analyzed via a modular subsystem decomposition, wherein the total Hilbert space decomposes into a product of a logical (qubit, “L”) subsystem and a “gauge” (G) subsystem: $$|\psi_{(I)}\rangle = |\overline{\psi}\rangle_L \otimes |+_I\rangle_G$$ In the finite-energy case, this separation becomes imperfect, and logical information "leaks" into the gauge sector. The logical fidelity is then computed by tracing out the gauge subsystem, assessing how well the logical qubit survives noisy operations, especially in the presence of envelope-induced entanglement between sectors. The distortion of the logical Bloch sphere provides a diagnostic of the code’s resilience.

This decomposition allows the practitioner to distinctly assess errors impacting the encoded information separately from those affecting ancillary degrees of freedom—a tool crucial for designing error-correction circuits for NISQ devices.

3. Photonic Preparation Schemes for Approximate GKP States

A leading strategy for preparing approximate GKP states in the laboratory is via measurement-induced non-Gaussianity in multimode optical circuits, resembling a generalized boson-sampling architecture. The state preparation protocol is as follows:

1. Initial state: Displaced and squeezed vacuum states are prepared across multiple optical modes.
2. Interference: These modes are combined in a linear interferometer (network of beam splitters and phase shifters).
3. Photon-number resolving (PNR) detection: All but one output mode are measured using PNR detectors; successful heralding projects the remaining mode into a non-Gaussian state.
4. Core-state representation: The resulting single-mode output is written as

$$|\psi_{(A)}\rangle \approx S(\zeta) D(\beta) \sum_{n=0}^{n_\text{max}} c_n |n\rangle$$

where $S(\zeta)$ and $D(\beta)$ are the squeezing and displacement operators. The coefficients $c_n$ (and truncation order $n_\text{max}$) are optimized so that, after further Gaussian operations, the prepared state approximates the desired GKP target. Circuit parameters and postselection patterns are numerically optimized (e.g., via global optimization algorithms such as "basinhopping") to maximize fidelity and manage success probability.

This approach can generate GKP states anywhere on the Bloch sphere with nearly uniform resource requirements—a property with major implications for "magic state" distillation overheads.

4. Numerical Performance, Resource Estimation, and Trade-offs

Detailed numerical studies in this framework yield the following insights:

• Success probability: High-quality approximate GKP states are produced with low (typically sub-percent) probability due to the stochastic nature of PNR detection and postselection.
• Photon number scaling: Achieving higher squeezing (narrower peaks, greater correction ability) requires an increased average photon number (energy cost per qubit).
• Circuit robustness: Output fidelity is robust with respect to reasonable errors in squeezing parameters but is more sensitive to deviations in interferometer (beam splitter) angles.
• Loss and decoherence: Simulated optical loss degrades both the fidelity and non-Gaussianity (e.g., quantified by negative values in the Wigner function) of the desired states. Circuits can be numerically reoptimized to mitigate these effects.
• Resource balance: Any practical implementation must trade off the number of modes, achievable squeezing (e.g., up to 12 dB), and acceptable heralding probabilities to yield suitable rates for initializing logical qubits.

Empirically, a wider Fock “core” (larger $n_\text{max}$) and more input modes enhance the fidelity to the GKP target, especially as the target squeezing increases.

5. Relevance for NISQ and Fault-Tolerant Quantum Devices

Approximate GKP codes are well-matched for near-term devices due to their operational compatibility with NISQ-era photonic hardware:

• The modular subsystem decomposition enables logical information extraction, even from imperfect or highly mixed states.
• Figures of merit for logical fidelity and distribution distance provide tools for device benchmarking and error analysis.
• Preparation protocols exploit resources—displacement, squeezing, linear optics, and PNR detection—that are already routine in many laboratories.
• The produced approximate GKP states can serve as seeds for concatenated error-correcting codes, augmenting their tolerance to noise in realistic settings.

Although full, near-perfect error correction is precluded by practical limits on achievable squeezing and photon number, well-optimized states secure sufficient logical fidelity and error recovery for use in proof-of-principle and small-scale fault-tolerance demonstrations. The process of tuning envelope width and core truncation is essential to this robustness.

6. Representative Equations and Metrics

Key results and analytic expressions from this research include:

• Envelope application:

$$E(\epsilon) = \exp(-\epsilon n),\quad |\psi_{(\epsilon)}\rangle = E(\epsilon)|\psi_{(I)}\rangle$$

• Core-state expansion:

$$|\psi_{(A)}\rangle \approx S(\zeta) D(\beta) \sum_{n=0}^{n_\text{max}} \frac{c_n}{\mathcal{N}} |n\rangle$$

• Logical observable mapping:

$$\bar{X} = X(\sqrt{\pi}),\;\; \bar{Z} = Z(\sqrt{\pi})$$

with $X(\alpha) = \exp(-i\alpha p)$, $Z(\beta) = \exp(i\beta q)$.

• Projection operator and QEC matrix:

$$P_G = |0\rangle \langle 0| + |1\rangle \langle 1|;\;\; \gamma_{kk'} \equiv P_G E_k^\dagger E_{k'} P_G$$

• Wigner logarithmic negativity as a measure of non-Gaussianity:

$$W_N(\rho) = \log\left(\int dq\,dp\, |W_{\rho}(q, p)| - 1 + 1\right)$$

These quantities enable comparative benchmarking and the theoretical design of photonic circuits for state preparation.

7. Summary and Outlook

Approximate GKP codes are essential for practical quantum error correction in continuous-variable systems, bridging the physically unattainable ideal GKP codewords with the realities of finite energy, loss, and experimental limitations. The interplay between finite resource scaling, code performance metrics, and photonic state synthesis protocols underscores the practical feasibility and theoretical richness of this approach. The photonic preparation schemes using Gaussian resources and PNR detection provide a viable and reproducible path for initializing non-Gaussian grid states crucial to current and upcoming quantum computing architectures. By carefully optimizing envelope width, Fock basis truncation, and numerically tuning optical circuits, these approximate codes can maintain robust logical information and are poised for experimental deployment in NISQ photonic devices and beyond.