A hardware-native time-frequency GKP logical qubit toward fault-tolerant photonic operation

TL;DR

This work establishes a hardware-native Gottesman--Kitaev--Preskill (GKP) qubit encoded in the time--frequency (TF) phase space of propagating single photons, with a metrological optical-frequency-comb reference enforcing a square TF lattice via stabilizers $\\hat{S}_{\\tau}$ and $\\hat{S}_{\\Omega}$. Finite-energy TF grid states, modeled as a comb of Gaussians with widths $(\\sigma_{\\tau},\\sigma_{\\Omega})$, yield intrinsic protection against Gaussian TF displacements, while logical operations are implemented deterministically through phase and delay controls corresponding to TF displacements $\\bar{Z}=\\hat{D}(\\sqrt{\\pi},0)$ and $\\bar{X}=\\hat{D}(0,\\sqrt{\\pi})$. The platform supports scalable multiplexing across frequency-comb modes, enabling parallel TF--GKP qubits with a shared metrological reference, and outlines concrete steps toward active syndrome extraction via an ancillary TF grid state, TF beam-splitter coupling, and time--frequency-resolved detection. While not yet demonstrating repeated error correction or universal gates, the work provides the hardware-native stabilization, noise model, and displacement-based control required for integration into erasure-aware and fusion-based fault-tolerant photonic architectures, linking precision metrology with bosonic quantum information processing.

Abstract

We realize a hardware-native time--frequency Gottesman--Kitaev--Preskill (GKP) logical qubit encoded in the continuous phase space of single photons, establishing a propagating photonic implementation of bosonic grid encoding. Finite-energy grid states are generated deterministically using coherently driven entangled nonlinear biphoton sources that produce single-photon frequency-comb supermodes. An optical-frequency-comb reference anchors the time--frequency phase space and enforces commuting displacement stabilizers directly at the hardware level, continuously defining the logical subspace. Timing jitter, spectral drift, and phase noise map naturally onto Gaussian displacement errors within this lattice, yielding intrinsic correctability inside a stabilizer cell. Logical operations correspond to experimentally accessible phase and delay controls, enabling deterministic state preparation and manipulation. Building on the modal time--frequency GKP framework, we identify a concrete pathway toward active syndrome extraction and deterministic displacement recovery using ancillary grid states and interferometric time--frequency measurements. These primitives establish a hardware-compatible route for integrating the time--frequency GKP logical layer into erasure-aware and fusion-based fault-tolerant photonic architectures.

Figures (4)

• Figure 1: Hardware-native implementation of a time--frequency (TF) GKP logical qubit using coherently seeded entangled nonlinear biphoton sources. A stabilized optical frequency comb (OFC) with locked repetition rate $f_r$ and carrier-envelope offset frequency $f_{\mathrm{CEO}}$ pumps two coherently seeded entangled nonlinear biphoton sources (ENBS1/ENBS2) implemented in periodically poled lithium niobate (PPLN) crystals, thereby defining a metrologically anchored TF lattice with spacing $(T_r,\omega_r)$. A phase-locked idler seed laser selects pump-matched driven signal supermodes $\ket{\hat{A}_{\mathrm{SM},1}}$ and $\ket{\hat{A}_{\mathrm{SM},2}}$ (red), which constitute the physical bosonic modes encoding the logical states $\ket{0_L}=\ket{0_{\mathrm{TF}}}$ and $\ket{1_L}=\ket{1_{\mathrm{TF}}}$. Control parameters map directly to TF displacement operators: the pump--seed phase offset $\Delta\phi_{sd}$ implements $\bar{Z}$-type temporal translations, while an acousto-optic modulator (AOM) applies calibrated spectral displacements $\Delta\omega=\omega_r/2$ (equivalently $\sqrt{\pi}/T_r$ under the GKP mapping) corresponding to $\bar{X}$-type logical translations and stabilizer or syndrome increments. A TF interferometer with adjustable signal-phase shift $\Delta\phi_s$ enables stabilizer-sensitive interference measurements of $\langle S_\tau\rangle$ and $\langle S_\Omega\rangle$. The port-resolved signal outputs $N_{\mathrm{sig},1}(t)$ and $N_{\mathrm{sig},2}(t)$ are detected using single-photon detectors (SPD) at the signal wavelength (807 nm).
• Figure 2: From canonical GKP codes to TF--GKP encoding and finite-energy grid-state models. (a) Canonical square-lattice GKP code in the $(q,p)$ phase space, shown in units of $\sqrt{\pi}$ with lattice spacing $2\sqrt{\pi}$ and the logical unit cell highlighted. (b) Corresponding TF--GKP lattice in the dimensionless canonical variables $(\tau,\Omega)$ defined in Sec. \ref{['sec:TFphase']}, illustrating the isomorphic operator mapping $(\hat{q},\hat{p})\leftrightarrow(\hat{\tau},\hat{\Omega})$. (c) Representative Wigner-function illustration of a finite-energy TF grid state, showing localized lattice peaks and interference structure determined by finite effective squeezing. (d) Comb-of-Gaussians approximation with broadened lattice peaks characterized by widths $(\sigma_\tau,\sigma_\Omega)$, used to quantify intrinsic displacement tolerance and logical failure probability in Sec. \ref{['sec:errors']} and Fig. \ref{['fig:fig4']}(a).
• Figure 3: Operational mapping between laboratory controls and TF--GKP displacements and syndrome inference. (a) Control architecture: a stabilized pump optical frequency comb (OFC) and phase-locked seed laser drive two ENBS units, while an electro-optic modulator (EOM) and an acousto-optic modulator (AOM) implement programmable time--frequency (TF) displacement operations. (b) Displacement picture in the TF phase space $(\tau,\Omega)$: calibrated controls generate small logical or noise-induced displacements $(\delta\tau,\delta\Omega)$ within the shaded logical unit cell defined by the stabilizer lattice. (c) Summary of TF--GKP code elements (displacement operators, lattice spacing $2\sqrt{\pi}$, stabilizers, logical unit cell, correctable region) together with calibrated control-to-displacement mappings (AOM RF frequency $\rightarrow \delta\Omega$, EOM voltage $\rightarrow \delta\tau$, PZT/pump-phase offsets $\rightarrow$ phase translations). (d) Conceptual syndrome interpretation: a measured TF outcome is associated with the nearest lattice site within a stabilizer cell, and time- and frequency-resolved detection provide projections onto $\tau$ and $\Omega$ that supply the geometric information required for displacement decoding.
• Figure 4: Quantitative robustness and control feasibility for TF--GKP encoding. (a) Logical failure probability $P_{\mathrm{fail}}$ under nearest-lattice decoding of a finite-energy TF grid state subject to Gaussian displacement noise, plotted versus normalized noise widths $\sigma_{\tau}/\sqrt{\pi}$ and $\sigma_{\Omega}/\sqrt{\pi}$. The map exhibits a broad low-error region (blue), with contours of constant $P_{\mathrm{fail}}$ indicating the intrinsic correctability landscape for simultaneous timing and frequency fluctuations. The dashed line $\sigma_{\tau}=2\sigma_{\Omega}$ marks a representative anisotropic-noise (or anisotropic-squeezing) condition naturally compatible with the ENBS/SPFC platform, and the highlighted point indicates a realistic operating regime within the high-fidelity region. (b) Representative transfer functions $T(f)$ of experimentally available actuation channels (piezoelectric transducer (PZT), acousto-optic modulator (AOM), electro-optic modulator (EOM)), showing that the control bandwidth required to implement TF displacements for logical operations and stabilizer/syndrome steps is compatible with available phase and frequency modulation technology.