The generation of a logical qubit called the Gottesman-Kitaev-Preskill (GKP) qubit in an optical traveling wave is a major challenge for realizing large-scale universal fault-tolerant optical quantum computers. Recently, probabilistic generation of elementary GKP qubits has been demonstrated using photon number measurements and homodyne measurements. However, the generation rate is only a few Hz, and it will be difficult to generate fault-tolerant GKP qubits at a practical rate unless success probability is significantly improved. Here, we propose a method to efficiently synthesize GKP qubits from several quantum states by adaptive Gaussian operations. In the initial-state preparation that utilizes photon number measurements, an adaptive operation allows any measurement outcome above a certain threshold to be considered as a success. This threshold is lowered by utilizing the generalized photon subtraction method. The initial states are synthesized into a GKP qubit by homodyne measurements and a subsequent adaptive operation. As a result, the single-shot success probability of generating fault-tolerant GKP qubits in a realistic scale system exceeds $10%$, which is $1\ifmmode\times\else\texttimes\fi{}{10}^{6}$ times better than previous methods. This proposal will become a powerful tool for advancing optical quantum computers from the proof-of-principle stage to practical application.
The generation of a logical qubit called the Gottesman-Kitaev-Preskill (GKP) qubit in an optical traveling wave is a major challenge for realizing large-scale universal fault-tolerant optical quantum computers. Recently, probabilistic generation of elementary GKP qubits has been demonstrated using photon number measurements and homodyne measurements. However, the generation rate is only a few Hz, and it will be difficult to generate fault-tolerant GKP qubits at a practical rate unless success probability is significantly improved. Here, we propose a method to efficiently synthesize GKP qubits from several quantum states by adaptive Gaussian operations. In the initial-state preparation that utilizes photon number measurements, an adaptive operation allows any measurement outcome above a certain threshold to be considered as a success. This threshold is lowered by utilizing the generalized photon subtraction method. The initial states are synthesized into a GKP qubit by homodyne measurements and a subsequent adaptive operation. As a result, the single-shot success probability of generating fault-tolerant GKP qubits in a realistic scale system exceeds 10, which is 1\ifmmode×\else\texttimes\fi106 times better than previous methods. This proposal will become a powerful tool for advancing optical quantum computers from the proof-of-principle stage to practical application.
Gottesman-Kitaev-Preskill (GKP) qubits, known for their exceptional error-correction capabilities, are highly coveted in quantum computing. However, generating optical GKP qubits has been a significant challenge. Measurement-based methods, where a portion of entangled squeezed vacuum modes are measured with photon number resolving detectors heralding a desired state in the undetected modes, have emerged as leading candidates for optical GKP qubit generation due their minimal resource requirements. While the current measurement-based methods can produce high-quality GKP qubits, they suffer from low success probabilities limiting experimental realization. The heart of the problem lies in the duality of photon number resolving measurements, being both the source of nonlinearity needed to generate quality GKP qubits and the component driving down their probability of successful production. Our method, breeding approximate squeezed coherent state superpositions created by generalized photon subtraction, overcomes this problem by supplementing two photon number resolving measurements with a single high success probability homodyne measurement. This scheme achieves success probabilities $\ensuremath{\ge}{10}^{\ensuremath{-}5}$, two orders of magnitude higher than strictly photon number resolving measurement based methods, while still producing states with high fidelity, possessing error-correction capabilities equivalent to up to a 10 dB squeezed GKP qubit. This breakthrough significantly advances the practical use of the optical GKP qubit encoding.
To harness the potential of a quantum computer, quantum information must be protected against error by encoding it into a logical state that is suitable for quantum error correction. The Gottesman-Kitaev-Preskill (GKP) qubit is a promising candidate because the required multiqubit operations are readily available at optical frequency. To date, however, GKP qubits have been demonstrated only at mechanical and microwave frequencies. We realized a GKP state in propagating light at telecommunication wavelength and verified it through homodyne measurements without loss corrections. The generation is based on interference of cat states, followed by homodyne measurements. Our final states exhibit nonclassicality and non-Gaussianity, including the trident shape of faint instances of GKP states. Improvements toward brighter, multipeaked GKP qubits will be the basis for quantum computation with light.
Building a useful photonic quantum computer requires robust techniques to synthesize optical states that can encode qubits. Gottesman–Kitaev–Preskill (GKP) states1 offer one of the most attractive classes of such qubit encodings, as they enable the implementation of universal gate sets with straightforward, deterministic and room temperature-compatible Gaussian operations2. Existing pioneering demonstrations generating optical GKP states3 and other complex non-Gaussian states4–11 have relied on free-space optical components, hindering the scaling eventually required for a utility-scale system. Here we use an ultra-low-loss integrated photonic chip fabricated on a customized multilayer silicon nitride 300-mm wafer platform, coupled over fibre with high-efficiency photon number resolving detectors, to generate GKP qubit states. These states show critical mode-level features necessary for fault tolerance, including at least four resolvable peaks in both p and q quadratures, and a clear lattice structure of negative Wigner function regions, in this case a 3 × 3 grid. We also show that our GKP states show sufficient structure to indicate that the devices used to make them could, after further reduction in optical losses, yield states for the fault-tolerant regime. This experiment validates a key pillar of bosonic architectures for photonic quantum computing2,12, paving the way for arrays of GKP sources that will supply future fault-tolerant machines. An ultra-low-loss integrated photonic chip fabricated on a customized multilayer silicon nitride 300-mm wafer platform, coupled over fibre with high-efficiency photon number resolving detectors, is used to generate Gottesman–Kitaev–Preskill qubit states.
Gottesman-Kitaev-Preskill (GKP) states are essential non-Gaussian resources for fault-tolerant photonic quantum computing, enabling logical qubit encoding with intrinsic robustness against errors. Several approaches to GKP state preparation have been explored, including measurement-based protocols in circuit QED and trapped-ion systems, cat-state breeding, and photon-subtraction schemes. However, these methods are either restricted to specific platforms or require deep non-Gaussian resource chains with exponentially low success probabilities. Gaussian Boson Sampling (GBS) offers a compelling all-photonic alternative by generating non-Gaussian states through measurement-induced nonlinearity, without the need for matter-based ancilla or active feedforward. Nevertheless, its practical implementation is limited by the exponential computational cost of evaluating matrix hafnians-#P-complete functions that govern photon-number probabilities. To address this challenge, we introduce a two-stage Histogram Gradient Boosting surrogate pipeline that predicts, without any hafnian computation, the optimal heralding pattern, circuit fidelity, and post-selection probability for candidate GBS circuits, while reserving exact quantum simulation exclusively for surrogate-selected candidates. Trained on circuit configurations across 3-5 optical modes, the surrogate achieves 90.0% GKP-detection accuracy on a held-out set, representing a 23.7 percentage-point improvement over the baseline, with a fidelity mean absolute error of 0.032 and a log-scale post-selection probability R2=0.837, reducing the total simulation burden by approximately 90%.
Practical quantum computing requires robust encoding of logical qubits in physical systems to protect fragile quantum information. Currently, the lack of scalability limits the logical encoding in most physical systems, and thus the high scalability of propagating light can be a game changer for realizing a practical quantum computer. However, propagating light also has a drawback: the difficulty of logical encoding due to weak nonlinearity. Here, we propose Gaussian breeding that encodes arbitrary Gottesman-Kitaev-Preskill (GKP) qubits in propagating light. The key idea is the efficient and iterable generation of quantum superpositions by photon detectors, which is the most widely used nonlinear element in quantum propagating light. This formulation makes it possible to systematically create the desired qubits with minimal resources. Our simulations show that GKP qubits above a fault-tolerant threshold, including ``magic states'', can be generated with a high success probability and with a high fidelity exceeding 0.99. This result fills an important missing piece toward practical quantum computing.
Abstract Practical quantum computing requires robust encoding of logical qubits in physical systems to protect fragile quantum information. Currently, the lack of scalability limits the logical encoding in most physical systems, and thus the high scalability of propagating light can be a game changer. However, propagating light also has difficulty in logical encoding due to weak nonlinearity. Here, we propose a synthesizer that encodes Gottesman-Kitaev-Preskill (GKP) qubits in propagating light by exploiting the nonlinearity of photon detectors. This synthesizer is based on an approach what we call Gaussian breeding, leading to the following four advantages: (i) systematic and rigorous synthesis of arbitrary GKP qubits, (ii) use of minimal resources, (iii) high fidelity and high success probability, and (iv) robustness against loss. There has been no protocol that incorporates all these advantages, and thus the proposed synthesizer excels in both performance and feasibility. By employing our method, one can generate GKP qubits using a few to several squeezed light sources, beam splitters and photon detectors.