A unified optical platform for non-Gaussian and fault-tolerant Gottesman-Kitaev-Preskill states

TL;DR

This work presents a unified optical platform that uses Gaussian inputs, an optical parametric amplifier, and heralded photon detection to generate non-Gaussian resources without relying on high-photon-number Fock states. It demonstrates photon-added squeezed states with near-unit fidelity, approximate cubic-phase states with fidelities above $0.985$, and squeezed-cat states with fidelity above $0.99$, which can be iteratively bred into Gottesman-Kitaev-Preskill (GKP) grid states surpassing the fault-tolerance threshold of $9.75$ dB using input squeezing below $3$ dB. The approach yields higher heralding success probabilities than Fock-based schemes, provides a practical route to universal CV quantum computing, and offers a scalable path across quantum communication, metrology, and computation. By unifying cat-state generation, photon addition, cubic-phase resources, and GKP encoding within a single platform, the work paves the way for compact, hardware-friendly non-Gaussian state generation with broad applicability.

Abstract

Quantum technologies, encompassing communication, computation, and metrology, rely on the generation and control of non-Gaussian states of light. These states enable secure quantum communication, fault-tolerant quantum computation, and precision sensing beyond classical limits, yet their practical realisation remains a major challenge due to reliance on high-photon-number Fock states or strong non-linearities. Here we introduce a unified optical framework that removes this constraint, using only Gaussian inputs, optical parametric amplification, and heralded photon detection. Within a single architecture, we demonstrate the generation of photon-added squeezed states with near unit fidelity, cubic-phase-like states with strong non-linearities and fidelities above 98.5%, and squeezed-cat states exceeding 99% fidelity that can be iteratively bred into GKP grid states surpassing the 9.75 dB fault-tolerance threshold. Operating entirely below 3 dB of input squeezing, the approach provides a scalable, experimentally accessible platform that unites the state resources required for quantum communication, metrology, and computation within one coherent optical framework.

Figures (6)

• Figure 1: Schematic of the proposed non-Gaussian source.(a) Generic scheme for generating non-Gaussian states from a Gaussian input $\ket{\psi_{\mathrm{in}_1}}$. The input state is interfered with a Fock state $\ket{n}$ on a beamsplitter with transmissivity, $\tau$, and the desired non-Gaussian output, $\ket{\psi_{\mathrm{out}}}$, is heralded by a PNRD. (b) In the proposed non-Gaussian source, a Gaussian seed state, $\ket{\psi_{\mathrm{in}_2}}$, is injected into an optical parametric amplifier (OPA), generating two correlated output modes. PNRD on one mode heralds the generation of a non-Gaussian state in the other.
• Figure 2: Comparison of photon-addition schemes to squeezed states.(a) Wigner functions of photon-added squeezed states generated using the OPA (top row), discrete Fock-state addition (middle row), and the ideal photon-added states obtained by applying the creation operator, $\hat{a}^\dagger$ to the squeezed vacuum (bottom row), for photon numbers $n = 0$--$4$. A squeezed vacuum with $r = 0.3$ (2.61 dB) is passed through an OPA with gain $\kappa = 0.6190$, and conditioning on $n$ detected photons produces the photon-added state. The comparable Fock-addition scheme requires $r = 0.592$ (5.14 dB). For reference, the ideal photon-added squeezed state is shown for $r = 0.272$ (2.36 dB). (b) Success probability as a function of photon number for both schemes. The OPA scheme uses a squeezed input state with $r = 0.30$ and gain parameter $\kappa = 0.6190$, shown in solid blue line. For the Fock-state scheme, the red solid and dashed lines show success probabilities for Fock states generated with $\kappa = 0.6190$ and mixed with the squeezed state at $\tau = 0.5$ and $\tau = 0.1$, respectively. The orange lines show the same configuration using Fock states generated with $\kappa = 0.3023$, mixed at $\tau = 0.5$ (solid) and $\tau = 0.1$ (dashed). (c) Fidelity of the generated states with respect to the ideal photon-added squeezed states, shown for ideal (solid) and non-ideal (dashed) detection conditions. The non-ideal case uses a detector efficiency of $\eta = 0.95$ and a dark-count rate of 20 cps, corresponding to a probability of $4 \times 10^{-7}$ (see Methods). For the OPA scheme, the fidelity is optimised against the ideal photon-added squeezed state with squeezing $r = 0.272$. The discrete Fock-addition scheme is likewise optimised, which occurs for an input squeezing of $r = 0.592$ when compared to the same ideal target state with $r = 0.272$. Non-ideal fidelities are evaluated with respect to this same ideal target. Note that lines are shown as continuous lines for visual clarity; photon number is discrete and data are evaluated only at integer values.
• Figure 3: Comparison of Wigner functions of ideal and approximate cubic-phase states. Panels (a) and (b) depict the Wigner functions of ideal cubic-phase states with non-linearity strengths of (a)$\gamma=0.1$ and (b)$\gamma=0.2$, respectively. Panels (c) and (d) present the corresponding approximate cubic-phase states generated by seeding a coherent state into the OPA and heralding on $n=3$ photons, with OPA gain set to $\kappa=0.3023$. (c) The fidelity between the ideal cubic-phase state and the OPA output was optimised by varying the magnitude of the input coherent state to achieve a non-linearity of $\gamma = 0.1$. The optimal match is obtained for a coherent-state amplitude of $\alpha = -1.35i$, giving a fidelity of $F = 0.994$. (d) For a stronger non-linearity of $\gamma = 0.2$, the optimal fidelity, $F=0.984$, is obtained for a coherent-state amplitude of $\alpha = -1.1i$. Insets give top view of Wigner functions.
• Figure 4: Comparison of Wigner functions of ideal and approximate squeezed-cat states.(a) OPA scheme for generating squeezed-cat states. In the first round, the OPA is seeded with a squeezed vacuum, and one of the output modes is heralded using a PNRD. Upon a successful detection event, the output passes through an optical switch (OS) that feeds the signal back into the OPA input for the next round. If the detection fails, the OS redirects and discards the signal. After $k$ successful iterations, the OS routes the final output for further use, resulting in the desired squeezed-cat amplitude and squeezing. Panels (b)--(d) show the Wigner functions of the ideal squeezed-cat states with complex amplitudes $\alpha = 4.9$, $5.7$, and $6.33$, and squeezing parameters $r = 0$, $0$, and $0.24\approx2.08$ dB, respectively. Each ideal state in (b)--(d) corresponds to the approximate squeezed-cat states shown in (e)--(g), respectively. Panels (e)--(g) depict the Wigner functions of the approximate squeezed-cat states generated using the OPA scheme. In (e), the OPA is seeded with a squeezed state of $r = -1\approx-8.69$ dB and repeated for $k = 6$ rounds, with heralding on $n = 2$ photons, giving a fidelity of $F = 0.992$ with the corresponding ideal state in (b). In (f), the initial squeezing is $r = -1\approx-8.69$ dB with $k = 5$ rounds and heralding on $n = 3$ photons, giving $F = 0.992$ with the ideal state in (c). In (g), the initial squeezing is $r = -0.5\approx4.34$ dB with $k = 5$ rounds and heralding on $n = 5$ photons, achieving $F = 0.991$ with the ideal state in (d). Panels (h)--(j) show the $p$-quadrature probability distributions of the ideal states in (b)--(d) (orange lines) and their corresponding approximate states in (e)--(g) (blue lines), respectively. Panel (h) corresponds to an even cat state, while panels (i) and (j) correspond to odd cat states. The OPA gain for panels (e)--(g) is set to $\kappa=0.5322$.
• Figure 5: Wigner functions of the approximate GKP states.(a) Schematic of the protocol for generating approximate GKP states from squeezed cat states, following the approach of Refs. vasconcelos2010allweigand2018generating. Two identical squeezed-cat states interfere on a balanced (50:50) beam splitter. One output mode is subjected to homodyne detection, and conditioned on the measurement outcome, the remaining mode is kept. A final squeezing operation is applied to adjust the lattice spacing of the resulting state to match that of an ideal GKP grid. Panels (b) and (c) show the approximate GKP states obtained from the squeezed-cat states generated using the protocol in Fig. \ref{['fig:figure4']}(a). In both cases, a single-photon detection event is heralded by the PNRD, and the procedure is repeated $k=6$ and $k=7$ times, respectively, to reach the required level of squeezing in the cat state. In panel (b), a logical $\ket{0_L}$ GKP state is generated from an initial squeezed state with $r = 0.268 \approx 2.33~\text{dB}$, yielding a final grid state with $10~\text{dB}$ of squeezing. Panel (c) shows the corresponding logical $\ket{1_L}$ state, produced from an input squeezing of $r = 0.306 \approx 2.66~\text{dB}$, resulting in a GKP state with $10.54~\text{dB}$ of squeezing. (d) shows the effective symmetric GKP squeezing when the optical switch in Fig. \ref{['fig:figure4']}(a) is assumed to introduce loss. The dashed grey line indicates the $9.75~\text{dB}$ squeezing threshold required for fault-tolerant optical quantum computing. We compare the performance of the protocol for different numbers of PNRDs and varying numbers of OPA iterations, $k$. Note that the $n = 6, k = 1$ (in red) and $n = 7, k = 1$ (in maroon) cases require no optical switch, as no iterations are performed. These single-round schemes are therefore unaffected by switching loss and still achieve GKP squeezing of $10$ dB and $10.54$ dB from input squeezing of $r = 1.23$ dB and $r = 1.24$ dB, respectively. (e) summarises the protocol parameters for different combinations of $n$ and $k$, together with the corresponding squeezed-cat generation rates, assuming an optical switch operating at $50~\text{kHz}$. For the single-shot cases $k = 6$ and $k = 7$, where no optical switch is required, we instead assume detector-limited operation at $10~\text{MHz}$. The “maximum loss’’ indicates the highest optical loss that each resulting GKP state can tolerate while still maintaining the $9.75~\text{dB}$ squeezing threshold required for fault-tolerant optical quantum computing. The OPA gain used for the generation of all GKP states is fixed at $\kappa = 0.7082$.