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Holonomic quantum gates via continuous measurement in bosonic codes: GKP and cat states

This paper proposes and analyzes a continuous measurement-based holonomic quantum computation framework that enables the implementation of arbitrary logical gates, including non-Clifford operations, for both cat and GKP bosonic codes through geometric trajectories and Zeno confinement, while simultaneously establishing their fault-tolerant error-correcting capabilities.

Original authors: Juan Garcia Nila, Anirudh Lanka, Todd Brun

Published 2026-09-24

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Juan Garcia Nila, Anirudh Lanka, Todd Brun

Original paper licensed under CC BY 4.0 (https://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Building a computer that can solve problems beyond the reach of today's machines requires a fundamental shift in how we handle information. In the quantum realm, data is stored in fragile states that can easily collapse if disturbed by the slightest noise. To protect this information, scientists use error-correcting codes, which spread a single piece of data across a larger system so that if one part fails, the whole does not. A major hurdle in this field has been performing the necessary calculations, or logic gates, without introducing new errors. Traditional methods often rely on complex, precisely timed pulses of energy to steer the system, but this approach demands extreme control and can be difficult to scale. An alternative idea involves using geometry, where the path a system takes matters more than the speed or force used to get there. This concept, known as holonomic computation, suggests that if a system is guided along a specific loop, it can return to its starting point with its information transformed in a useful way, much like how a traveler returning to their starting city after a long journey might find their compass pointing in a different direction.

Recently, researchers have explored a variation of this geometric idea that relies on continuous observation rather than direct pushing. In this framework, a system is constantly monitored, and the act of watching it keeps it confined to a specific safe zone, a phenomenon known as the quantum Zeno effect. By carefully changing what is being watched over time, the system is forced to evolve along a closed loop, accumulating the desired transformation purely through the geometry of its path. While this method has been proposed in theory, its application to the most promising physical systems for quantum computing has remained unclear. A team of researchers at the University of Southern California has now bridged this gap by demonstrating how this measurement-based approach can be applied to two leading types of quantum error-correcting codes: the cat code and the Gottesman-Kitaev-Preskill, or GKP, code. These codes store information in the vibrations of light or sound waves, known as bosonic modes, which offer a natural and efficient way to build quantum processors.

The team developed specific protocols to guide these bosonic systems through the necessary loops without using the traditional, complex Hamiltonian controls that usually drive quantum gates. For the cat code, which encodes information in superpositions of different wave patterns, the researchers designed a path that involves squeezing and rotating the wave pattern in a precise sequence. By continuously measuring the state of the system during this process, they ensured it stayed within the safe, error-corrected space. The result was a logical operation that performed a specific, non-standard rotation on the data. This is a significant achievement because standard quantum computers can easily perform a set of basic operations, but they struggle with a particular type of gate needed to make the computer truly powerful and universal. The team showed that their method could generate this difficult gate purely through the geometry of the measurement path, effectively creating a new tool for quantum processing without the need for intricate external driving forces.

For the GKP code, which uses a grid-like structure in the phase space of the oscillator to protect against small shifts in position or momentum, the researchers took a different approach. They constructed a path that effectively translated the entire grid of the code in a specific way. As the system moved along this trajectory under the watchful eye of continuous measurement, it accumulated a geometric phase that corresponded to a logical gate. This gate is essential for performing the complex calculations required for fault-tolerant quantum computing. The researchers simulated these processes and found that the system successfully performed the intended logical operations with high fidelity. In their simulations, the process reached a state of perfect accuracy at the end of the protocol, demonstrating that the geometric method works as intended. They also analyzed how the system would behave in a real-world scenario where the measurement is not infinitely strong, calculating the probability that the system might leak out of the safe zone. Their analysis showed that with sufficiently strong monitoring, the system remains confined, and the error-correcting properties of the code are preserved throughout the operation.

The study confirms that continuous measurement can serve as a robust mechanism for controlling quantum information in bosonic systems. By replacing direct, complex control with a strategy of constant observation and geometric evolution, the researchers have provided a concrete blueprint for implementing fault-tolerant gates. This work connects the abstract concepts of geometric phases and continuous monitoring with the practical engineering of quantum error correction. It suggests that future quantum processors might not need to rely solely on the precise timing of external pulses but could instead use the stabilizing power of measurement to guide their operations. The findings offer a new pathway toward building scalable quantum computers, showing that the geometry of a measurement path can be just as powerful as the forces used to drive a system, provided the system is kept under constant, careful watch.

Technical Summary: Holonomic Quantum Gates via Continuous Measurement in Bosonic Codes

Problem Statement
Fault-tolerant quantum computation requires a universal set of logical gates that remain compatible with quantum error correction (QEC). A central challenge is realizing non-Clifford gates, which are essential for universality but difficult to implement without significant experimental overhead or stringent control requirements. Conventional approaches often rely on Hamiltonian engineering, gate teleportation, or magic-state injection. While holonomic quantum computation (HQC) offers a robust alternative by utilizing geometric phases rather than dynamical phases, its application to experimentally relevant bosonic error-correcting codes has remained largely unexplored. Specifically, existing continuous measurement-based holonomic quantum computation (CMHQC) frameworks are naturally expressed in terms of stabilizer measurements for qubit codes, often requiring high-weight many-body operators that are difficult to implement. Bosonic codes, such as cat codes and Gottesman–Kitaev–Preskill (GKP) codes, utilize large Hilbert spaces and experimentally accessible observables (e.g., parity and phase-space displacements), presenting a natural setting to address these challenges, yet explicit protocols for CMHQC in these systems were lacking.

Methodology
The authors extend the CMHQC framework to bosonic quantum error-correcting codes. In this paradigm, logical gates emerge from the continuous monitoring of a time-dependent code subspace. A family of projectors traces a closed path on the Grassmannian manifold of encoded subspaces, while repeated measurements enforce Zeno confinement to the instantaneous code space. The resulting logical operation is determined by the associated Wilczek–Zee holonomy rather than explicit Hamiltonian driving.

The paper develops explicit protocols for two specific bosonic architectures:

Four-Component Cat Codes: The authors construct a family of "squeezed-cat" trajectories. They define a time-dependent unitary $V(t)$ involving a quarter-rotation operator and a two-photon squeezing operator $S(\xi(t))$ . The trajectory is designed such that the projected Wilczek–Zee connection generates arbitrary logical $Z_L$ rotations, including the non-Clifford $T_L$ gate.
GKP Codes: The authors introduce a "translated-lattice" trajectory. They define a unitary $V(t)$ involving a cubic phase function $f(Q/\sqrt{\pi})$ and a displacement operator. This trajectory realizes the logical $T_{GKP}$ gate through a purely geometric holonomy.

For both codes, the authors derive the corresponding time-dependent projectors and analytically evaluate the projected connections. They further analyze the error-correcting capabilities of the instantaneous codespaces by establishing "dressed" Knill–Laflamme conditions for relevant bosonic error models (e.g., photon loss for cat codes, small displacements for GKP codes). Finally, they derive analytical estimates for leakage induced by finite-strength continuous measurements using a Lindblad master equation approach in a rotating frame.

Key Contributions and Results

Explicit Protocols: The paper provides concrete constructions for holonomic paths in both cat and GKP codes. For cat codes, the protocol generates arbitrary $Z_L$ rotations and the $T_L$ gate. For GKP codes, it realizes the $T_{GKP}$ gate.
Measurement-Induced Control: Unlike previous bosonic holonomic proposals that rely on Hamiltonian control of the oscillator manifold, these operations are generated through continuous measurement and Zeno confinement.
Error Correction Analysis: The authors demonstrate that the instantaneous codespaces satisfy dressed Knill–Laflamme conditions for the relevant error models. This establishes that the codes retain their error-correcting properties throughout the holonomic evolution.
Leakage Bounds: Analytical upper bounds for leakage probability are derived as a function of measurement strength ( $\kappa$ ) and protocol duration ( $t_f$ ). The results show that in the strong-measurement regime (large $\kappa t_f$ ), leakage is suppressed, and the system remains confined to the codespace.
Numerical Validation:
- Cat Codes: Simulations show that for large coherent state amplitudes ( $|\alpha| \gtrsim 3$ ), the process infidelity for the $T_L$ gate falls below $10^{-3}$ . Fidelity reaches unity at the end of the protocol.
- GKP Codes: Numerical simulations using truncated finite-energy GKP states (via the Strawberry Fields library) show excellent agreement with analytical bounds. For finite-energy parameters $\epsilon \approx 0.05$ , the process infidelity is approximately $1.4 \times 10^{-7}$ . Leakage probability is shown to scale as $1/(\kappa t_f)$ , confirming the onset of the quantum Zeno regime.

Significance
The paper provides a concrete realization of measurement-induced holonomic control in experimentally relevant bosonic platforms. By establishing a direct connection between continuous-measurement holonomies and bosonic quantum error correction, the work connects geometric phases, continuous quantum measurement, and bosonic quantum error correction within a unified framework. The authors posit that bosonic codes serve as a natural platform for this type of control, offering a route toward robust, geometric quantum computation that leverages the inherent error-correcting properties of these codes while avoiding the experimental overhead of high-weight stabilizer measurements or complex Hamiltonian engineering. The work establishes the theoretical foundation for implementing universal gate sets in bosonic systems via continuous monitoring, with future research directions including the development of multi-qubit holonomic gates capable of supporting a complete fault-tolerant universal gate set.

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