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arXiv:2510.06725

Continuous measurement-based holonomic quantum computation

Anirudh Lanka, Juan Garcia-Nila, Todd A. Brun

Published Oct 8, 2025• 0 citations
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Abstract

We propose a scheme to generate holonomies using the Quantum Zeno effect, enabling logical unitary operations on quantum stabilizer codes purely through measurements. The quantum error-correcting code space is adiabatically rotated by measuring a succession of rotated stabilizer generators. When the rotation is sufficiently slow, the state remains confined to the instantaneous code space by the Zeno effect; otherwise, measurement-induced jumps can occur into a rotated orthogonal subspace. If the rotation completes a closed loop, the code state is transformed by a holonomy: a logical unitary transformation. We analytically derive the sequence of rotated stabilizer generators that produce a desired holonomy, and find the total time required to implement this procedure with a given success probability. If a measurement moves the state to the orthogonal subspace, we present a method to alter the path of the rotated observables to return the state either to the original code or the original error space with the desired holonomy; in the latter case, the holonomy is emulated. Finally, we establish conditions on the code and the measured observables that preserve the correctability of a given error set. When a code fails to meet the error-correcting conditions, our protocol can be applicable by augmenting the code with ancilla qubits.

DOI
10.48550/arxiv.2510.06725

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Simulation code for "Holonomic quantum gates via continuous measurement in bosonic codes: GKP and cat states"

Todd A. Brun, Anirudh Lanka

This repository contains the numerical simulation code supporting the paper “Holonomic quantum gates via continuous measurement in bosonic codes: GKP and cat states” by Juan Garcia-Nila, Anirudh Lanka, and Todd A. Brun. The code implements measurement-based holonomic logical gates for four-component cat codes and Gottesman–Kitaev–Preskill (GKP) codes. It includes simulations of the logical gate trajectories, Wigner-function evolution, target-state fidelities, holonomy infidelities, and leakage under finite-strength continuous measurement. The associated paper develops explicit continuously monitored holonomic protocols for non-Clifford logical gates in bosonic codes. Time-dependent stabilizer measurements Zeno-confine the state to an evolving codespace, producing logical operations through geometric holonomies while retaining access to error-syndrome information. Associated preprint: https://arxiv.org/abs/2608.11369

•Aug 11, 2026•0 citations
We propose a scheme to generate holonomies using the Quantum Zeno effect, enabling logical unitary operations on quantum stabilizer codes purely through measurements. The quantum error-correcting code space is adiabatically rotated by measuring a succession of rotated stabilizer generators. When the rotation is sufficiently slow, the state remains confined to the instantaneous code space by the Zeno effect; otherwise, measurement-induced jumps can occur into a rotated orthogonal subspace. If the rotation completes a closed loop, the code state is transformed by a holonomy: a logical unitary transformation. We analytically derive the sequence of rotated stabilizer generators that produce a desired holonomy, and find the total time required to implement this procedure with a given success probability. If a measurement moves the state to the orthogonal subspace, we present a method to alter the path of the rotated observables to return the state either to the original code or the original error space with the desired holonomy; in the latter case, the holonomy is emulated. Finally, we establish conditions on the code and the measured observables that preserve the correctability of a given error set. When a code fails to meet the error-correcting conditions, our protocol can be applicable by augmenting the code with ancilla qubits.
Continuous measurement-based holonomic quantum computation provides a route to universal logical computation in quantum error correcting codes. We introduce a fault-tolerant framework for implementing measurement-based holonomic gates that leverages continuous measurements with real-time feedback. We show that non-Markovian decoherence is intrinsically suppressed through the quantum Zeno effect, while Markovian errors are identified by the decoding of measurement records to reveal the rotated syndrome subspace populated during the evolution. This information enables steering holonomic paths mid-flight to ensure that the final evolution realizes the target logical gate. We further demonstrate that non-adiabatic effects give rise to measurement-induced errors, and we show that these can also be corrected by an analogous protocol. This approach relaxes the stringent adiabaticity requirement and enables faster implementation of holonomic gates.
We apply continuous measurement-based holonomic quantum computation (CMHQC) to bosonic quantum error-correcting codes and develop explicit protocols for both four-component cat codes and Gottesman-Kitaev-Preskill (GKP) codes. In this framework, a continuously monitored time-dependent codespace undergoes a closed trajectory on the Grassmannian manifold while Zeno confinement suppresses departures from the instantaneous code subspace. For cat codes, we construct a family of squeezed-cat trajectories whose projected Wilczek-Zee connection generates arbitrary logical Z rotations, including non-Clifford T-gates. For GKP codes, we introduce a translated-lattice trajectory that realizes the logical GKP T gate through a purely geometric holonomy. We derive the corresponding time-dependent projectors, analytically evaluate the projected connections, and show that the resulting holonomies reproduce the desired logical operations without Hamiltonian control. Furthermore, we analyze the error-correcting capabilities of the instantaneous codespaces by establishing dressed Knill-Laflamme conditions for the relevant bosonic error models and derive analytical estimates for leakage induced by finite-strength continuous measurements. Our results provide a concrete realization of measurement-induced holonomic control in experimentally relevant bosonic platforms and establish a full fault-tolerant logical gate implementation.
Abstract We apply continuous measurement-based holonomic quantum computation (CMHQC) \cite{Lanka2025,Lanka2026} to bosonic quantum error-correcting codes and develop explicit protocols for both four-component cat codes and Gottesman–Kitaev–Preskill (GKP) codes. In this framework, a continuously monitored time-dependent codespace undergoes a closed trajectory on the Grassmannian manifold while Zeno confinement suppresses departures from the instantaneous code subspace. For cat codes, we construct a family of squeezed-cat trajectories whose projected Wilczek–Zee connection generates arbitrary logical ZLZ_LZL​ rotations, including non-Clifford TLT_LTL​-gates. For GKP codes, we introduce a translated-lattice trajectory that realizes the logical TGKPT_\mathrm{GKP}TGKP​ gate through a purely geometric holonomy. We derive the corresponding time-dependent projectors, analytically evaluate the projected connections, and show that the resulting holonomies reproduce the desired logical operations without Hamiltonian control. Furthermore, we analyze the error-correcting capabilities of the instantaneous codespaces by establishing dressed Knill–Laflamme conditions for the relevant bosonic error models and derive analytical estimates for leakage induced by finite-strength continuous measurements. Our results provide a concrete realization of measurement-induced holonomic control in experimentally relevant bosonic platforms and establish a full fault-tolerant logical gate implementation.
Holonomic quantum computation exploits a quantum state's nontrivial, matrix-valued geometric phase (holonomy) to perform fault-tolerant computation. Holonomies arising from systems where the Hamiltonian traces a continuous path through parameter space have been well researched. Discrete holonomies, on the other hand, where the state jumps from point to point in state space, have had little prior investigation. Using a sequence of incomplete projective measurements of the spin operator, we build an explicit approach to universal quantum computation. We show that quantum error correction codes integrate naturally in our scheme, providing a model for measurement-based quantum computation that combines the passive error resilience of holonomic quantum computation and active error correction techniques. In the limit of dense measurements we recover known continuous-path holonomies.
Measurements and feedforward enhance the power of shallow quantum circuits, enabling the deterministic implementation of global unitary operations and the preparation of long-range entangled states. We establish a general correspondence between all such protocols and quantum error-correcting codes: the circuit preceding the measurements acts as an encoder, and unitary feedforward can eliminate post-selection if and only if the measurement projectors are detectable errors on the codespace. This correspondence provides a common framework for state preparation and implementation of global unitaries with measurements and feedforward, turning both into a code-design problem. For stabilizer codes with Pauli measurements, although the resulting operation can be non-Clifford, we show that its nonstabilizerness originates entirely from the encoder. We overcome this restriction using non-Pauli measurements or non-additive codes, constructing protocols that generate long-range nonstabilizerness while requiring only single-qubit unitary corrections.
This repository contains the numerical simulation code supporting the paper “Holonomic quantum gates via continuous measurement in bosonic codes: GKP and cat states” by Juan Garcia-Nila, Anirudh Lanka, and Todd A. Brun. The code implements measurement-based holonomic logical gates for four-component cat codes and Gottesman–Kitaev–Preskill (GKP) codes. It includes simulations of the logical gate trajectories, Wigner-function evolution, target-state fidelities, holonomy infidelities, and leakage under finite-strength continuous measurement. The associated paper develops explicit continuously monitored holonomic protocols for non-Clifford logical gates in bosonic codes. Time-dependent stabilizer measurements Zeno-confine the state to an evolving codespace, producing logical operations through geometric holonomies while retaining access to error-syndrome information. Associated preprint: https://arxiv.org/abs/2608.11369