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
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
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 ZL rotations, including non-Clifford TL-gates. For GKP codes, we introduce a translated-lattice trajectory that realizes the logical 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.
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.
Replication package — code, Jupyter notebooks, and precomputed data — for "Bosonic codes from compact phase spaces" (D. Roberts, A. Slipper, A. Parhizkar, V. V. Albert, and M. Hafezi). The paper develops the correspondence between Gottesman–Kitaev–Preskill (GKP) codes and elliptic curves — established concurrently by Conrad, Burchards & Flammia [1] and the present authors [2] — and extends it, via geometric quantization, to bosonic quantum codes associated with algebraic curves of arbitrary genus. Following geometric quantization, a quantized phase space is a family of coherent states holomorphically parametrized by a complex manifold; this work realizes the phase spaces parametrized by a compact Riemann surface. Code states are then holomorphic sections of a line bundle over that surface (automorphic forms), and the code dimension is fixed by Riemann–Roch — recovering GKP codes when the surface is an elliptic curve (genus 1) and yielding a new class of hyperbolic CV quantum codes at genus greater than or equal to 2. The package also numerically illustrates the central no-go result: an amenability dichotomy under which arbitrarily accurate approximate code words exist at genus one but not at any higher genus. The genus-two Bolza surface — the most symmetric genus-two curve — serves as the explicit worked example throughout, and is what the numerics compute. For that example the package: constructs the k=2 code space, verifies the Veronese factorization Y^2 = XZ that generates the full tower of code spaces, and renders the Husimi Q-functions of the code basis (main-text Fig. 2); builds the Poincaré theta series spanning the code space and studies their convergence; verifies the weight-2 canonical gate representation — V_R, V_U in the main text, ρ(R), ρ(U) in the notebooks — for the automorphism group Aut(C) = GL(2, F_3), symbolically and numerically; derives the circuit-QED conditional two-mode-squeezing Hamiltonian via a lab-frame dispersive Schrieffer–Wolff reduction; simulates the conditional-kick stabilization dynamics (main-text Fig. 3) and diagonalizes the stabilizer Hamiltonian to exhibit a spectral gap, a genus-two instance of the amenability dichotomy. Every computational figure in the main text and Supplemental Material is regenerated; the two non-computational schematics (main-text Fig. 1 and the SM circuit diagram) are illustrations, not code output. See README.md for the full file-to-figure map. [1] https://arxiv.org/abs/2407.03270 (2024)[2] https://github.com/droberts01/higher-genus-gkp (2024)
Bosonic codes offer hardware-efficient approaches to quantum error correction, with the best encodings offering effective protection of idle quantum information against loss and dephasing - particularly rotation-symmetric codes, which include the cat and binomial code families. However, rotation-symmetric codes are only naturally endowed with a single logical Pauli gate, while other logical gates require the use of non-linear operations, obstructing the utility of these codes for realizing quantum algorithms. Here, we balance error protection with controllability by introducing bosonic cyclic codes: a generalization of rotation-symmetric codes that enable the measured tradeoff of error protection properties for fault-tolerant logical phase gates. Through our general construction, we find that sacrificing the detectability of a single photon loss relative to a rotation-symmetric code can yield a number of logical phase gates commensurate with the original rotation symmetry order of the code, all achievable via passive Gaussian rotations. Giving the corresponding generalizations of cat and binomial codes - which we dub cyclic cat and Vandermonde codes, respectively - we further find that many of the desirable properties of these codes transfer to the bosonic cyclic code setting. We go on to discuss the larger SU(2) symmetry and rotation gates of the codes, which yield additional stabilizers and logical Pauli gates, as well as new non-Clifford gates for the smallest `kitten' binomial code, and provide a new error detection protocol. Finally, we introduce a general paradigm for converting higher-order stabilizers to logical gates, as in our generalization of rotation-symmetric codes, and apply it to several multimode bosonic codes.