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arXiv:2512.14811 · Physical Review A

Stabilizers may be poor bounds for fidelities

Aaron Z. Goldberg

quant-phSubmitted Dec 16, 2025• Announced Dec 18, 2025• 0 citations• Revised Sep 8, 2026 (v2)
v1 · Dec 16, 2025v2 · Sep 8, 2026
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Abstract

The defining feature of ideal Gottesman-Kitaev-Preskill (GKP) states is that they are unchanged by stabilizers, which allow them to detect and correct for common errors without destroying the quantum information encoded in the states. Given this property, can one use the amount to which a state is unchanged by the stabilizers as a proxy for the quality of a GKP state? This is shown to hold in the opposite manner to which it is routinely assumed, because in fact the fidelity a state has to an ideal GKP state is only \textit{upper} bounded by the stabilizer expectation values. This means that, for qubits encoded in harmonic oscillators via the GKP code, a good stabilizer expectation value does not guarantee proximity to an ideal GKP state in terms of any distance based on fidelity.

Journal reference
Phys. Rev. A (2026) 114, 032413
DOI
10.1103/d36t-6w76
Comments
10 pages, 6 figures; close to published version

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The defining feature of ideal Gottesman-Kitaev-Preskill (GKP) states is that they are unchanged by stabilizers, which allow them to detect and correct for common errors without destroying the quantum information encoded in the states. Given this property, can one use the amount to which a state is unchanged by the stabilizers as a proxy for the quality of a GKP state? This is shown to hold in the opposite manner to which it is routinely assumed, because in fact the fidelity a state has to an ideal GKP state is only \textit{upper} bounded by the stabilizer expectation values. This means that, for qubits encoded in harmonic oscillators via the GKP code, a good stabilizer expectation value does not guarantee proximity to an ideal GKP state in terms of any distance based on fidelity.
Gottesman, Kitaev and Preskill have proposed a scheme to encode a qubit in a harmonic oscillator, which is called the GKP code. It is designed to be resistant to small shift errors contained in momentum and position quadratures. Thus there's some intrinsic fault tolerance of the GKP code. In this thesis,we propose a method to utilize all the information contained in the continuous shifts, not just simply map a GKP-encoded qubit to a normal qubit. This method enables us to do maximum-likelihood decisions and thus increase fault-tolerance of the GKP code. This thesis shows that the continuous nature of the GKP code is quite useful for quantum error correction.
The Gottesman-Kitaev-Preskill (GKP) error-correcting code encodes a finite-dimensional logical space in one or more bosonic modes, and has recently been demonstrated in trapped ions and superconducting microwave cavities. In this work we introduce a new subsystem decomposition for GKP codes that we call the stabilizer subsystem decomposition, analogous to the usual approach to quantum stabilizer codes. The decomposition has the defining property that a partial trace over the nonlogical stabilizer subsystem is equivalent to an ideal decoding of the logical state, distinguishing it from previous GKP subsystem decompositions. We describe how to decompose arbitrary states across the subsystem decomposition using a set of transformations that move between the decompositions of different GKP codes. Besides providing a convenient theoretical view on GKP codes, such a decomposition is also of practical use. We use the stabilizer subsystem decomposition to efficiently simulate noise acting on single-mode GKP codes, and in contrast to more conventional Fock basis simulations, we are able to consider essentially arbitrarily large photon numbers for realistic noise channels, such as loss and dephasing. Published by the American Physical Society 2024
Straightforward logical operations contrasting with complex state preparation are the hallmarks of the bosonic encoding proposed by Gottesman, Kitaev and Preskill (GKP). The recently reported generation and error-correction of GKP qubits in trapped ions and superconducting circuits thus holds great promise for the future of quantum computing architectures based on such encoded qubits. However, these experiments rely on error-syndrome detection via an auxiliary physical qubit, whose noise may propagate and corrupt the encoded GKP qubit. We propose a simple module composed of two oscillators and a physical qubit, operated with two experimentally accessible quantum gates and elementary feedback controls to implement an error-corrected GKP qubit protected from such propagating errors. In the idealized setting of periodic GKP states, we develop efficient numerical methods to optimize our protocol parameters and show that errors of the encoded qubit stemming from flips of the physical qubit and diffusion of the oscillators state in phase-space may be exponentially suppressed as the noise strength over individual operations is decreased. Our approach circumvents the main roadblock towards fault-tolerant quantum computation with GKP qubits.
Straightforward logical operations contrasting with complex state preparation are the hallmarks of the bosonic encoding proposed by Gottesman, Kitaev, and Preskill (GKP). The recently reported generation and error correction of GKP qubits in trapped ions and superconducting circuits thus holds great promise for the future of quantum computing architectures based on such encoded qubits. However, these experiments rely on error-syndrome detection via an auxiliary physical qubit, whose noise may propagate and corrupt the encoded GKP qubit. We propose a simple module composed of two oscillators and a physical qubit, operated with two experimentally accessible quantum gates and elementary feedback controls to implement an error-corrected GKP qubit protected from such propagating errors. In the idealized setting of periodic GKP states, we develop efficient numerical methods to optimize our protocol parameters and show that errors of the encoded qubit stemming from flips of the physical qubit and diffusion of the oscillators state in phase space may be exponentially suppressed as the noise strength over individual operations is decreased. Our approach circumvents the main roadblock towards fault-tolerant quantum computation with GKP qubits.
Physical Gottesman-Kitaev-Preskill (GKP) states are inherently noisy as ideal ones would require infinite energy. While this is typically considered as a deficiency to be actively corrected, this work demonstrates that imperfect GKP stabilizer states can be leveraged in order to apply non-Clifford gates using only linear optical elements. In particular, Gaussian operations on normalizable GKP states, combined with homodyne measurements, permit two key primitives: clean projection onto Pauli eigenstates in the normalizable GKP codespace, thereby implementing Clifford gates with high fidelity; and probabilistic projection of unmeasured modes onto non-Pauli eigenstates. These results demonstrate that normalizable GKP stabilizer states combined with Gaussian operations provide a practical framework for computational universality within the measurement-based model of quantum computation in a realistic continuous-variable setting.