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arXiv:2412.02442

The fabulous world of GKP codes

Jonathan Conrad

Published Jan 1, 2024• 0 citations

Abstract

Quantum error correction is an essential ingredient in the development of quantum technologies. Its subject is to investigate ways to embed quantum Hilbert spaces into a physical system such that this subspace is robust against small imperfections in the physical systems. This task is exceedingly complex: for one, this is due to the vast diversity of possible physical systems with different structure to use. For another, every physical setting also comes with its own imperfections that need to be protected against. Bred by the complexity of a technological ambition, research on quantum error correction has developed into a large field of research that ranges from engineering of small systems with a single photon to the creation of macroscopic topological phases of matter and models of complex emergent physics. A quintessential tool in quantum error correction is the stabilizer formalism, which tames quantum systems by enforcing symmetries. A Gottesman-Kitaev-Preskill (GKP) code is a stabilizer code that creates a logical subspace within an infinite dimensional Hilbert space by endowing it with translational symmetries. While in practice the infinitude of the Hilbert space, as well as the infinitude of the translational symmetry group are considered as obstacles for implementation, in theory these are precisely the features that make the theory of GKP codes particularly rich, well behaved and well-connected to fascinating topics in mathematics. The purpose of this thesis is to explore these connections: to understand the coding theoretic and practical properties of GKP codes, utilizing its rich mathematical foundation, and to provide a foundation for future research. Along this journey we discover -- through the looking glass of GKP codes -- how quantum error correction fits into a fabulous mathematical world and formulate a series of dreams about possible directions of research.

DOI: 10.17169/refubium-45505

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Quantum error correction is an essential ingredient in the development of quantum technologies. Its subject is to investigate ways to embed quantum Hilbert spaces into a physical system such that this subspace is robust against small imperfections in the physical systems. This task is exceedingly complex: for one, this is due to the vast diversity of possible physical systems with different structure to use. For another, every physical setting also comes with its own imperfections that need to be protected against. Bred by the complexity of a technological ambition, research on quantum error correction has developed into a large field of research that ranges from engineering of small systems with a single photon to the creation of macroscopic topological phases of matter and models of complex emergent physics. A quintessential tool in quantum error correction is the stabilizer formalism, which tames quantum systems by enforcing symmetries. A Gottesman-Kitaev-Preskill (GKP) code is a stabilizer code that creates a logical subspace within an infinite dimensional Hilbert space by endowing it with translational symmetries. While in practice the infinitude of the Hilbert space, as well as the infinitude of the translational symmetry group are considered as obstacles for implementation, in theory these are precisely the features that make the theory of GKP codes particularly rich, well behaved and well-connected to fascinating topics in mathematics. The purpose of this thesis is to explore these connections: to understand the coding theoretic and practical properties of GKP codes, utilizing its rich mathematical foundation, and to provide a foundation for future research. Along this journey we discover -- through the looking glass of GKP codes -- how quantum error correction fits into a fabulous mathematical world and formulate a series of dreams about possible directions of research.

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

We examine general Gottesman-Kitaev-Preskill (GKP) codes for continuous-variable quantum error correction, including concatenated GKP codes, through the lens of lattice theory, in order to better understand the structure of this class of stabilizer codes. We derive formal bounds on code parameters, show how different decoding strategies are precisely related, propose new ways to obtain GKP codes by means of glued lattices and the tensor product of lattices and point to natural resource savings that have remained hidden in recent approaches. We present general results that we illustrate through examples taken from different classes of codes, including scaled self-dual GKP codes and the concatenated surface-GKP code.

In order to solve practical problems on a quantum computer, it is necessary to use fault-tolerant quantum error correction schemes to overcome noise: the errors arising from imperfections in physical components. The Gottesman-Kitaev-Preskill (GKP) code aims at achieving this in a hardware efficient manner by encoding finite-dimensional logical subspaces in the Hilbert space of one or more continuous variable modes. In near term implementations, however, it is not feasible to eliminate errors entirely, so it is natural to also employ alternative error mitigation techniques together with error correction. In this work, we study a quantum error mitigation method known as probabilistic error cancellation in the context of the GKP code. We compare Steane-type and teleportation-based GKP error correction, and calculate the sampling overheads associated with the technique for square and hexagonal GKP codes. We employ the stabilizer subsystem decomposition for GKP codes to obtain an effective logical channel for noisy operations that is used to express the ideal one of a target logical unitary. We consider noise from finite squeezing on the data and the two ancilla modes, needed for error correction, and examine the relationship between the sampling overhead and the noise for different decoding methods. Our results are calculated for single- and two-qubit GKP Clifford gates and one round of error correction, and show the nontrivial combination of error correction and mitigation in continuous variable codes.

We examine general Gottesman-Kitaev-Preskill (GKP) codes for continuous-variable quantum error correction, including concatenated GKP codes, through the lens of lattice theory, in order to better understand the structure of this class of stabilizer codes. We derive formal bounds on code parameters, show how different decoding strategies are precisely related, propose new ways to obtain GKP codes by means of glued lattices and the tensor product of lattices and point to natural resource savings that have remained hidden in recent approaches. We present general results that we illustrate through examples taken from different classes of codes, including scaled self-dual GKP codes and the concatenated surface-GKP code.

We study a class of quantum error-correcting codes through the geometry of complex abelian varieties. These codes, introduced by Gottesman--Kitaev--Preskill, are built from symplectically integral lattices and therefore naturally define polarized complex abelian varieties. We give a precise mathematical formulation of this relationship and extend it to a dictionary between the main structures of GKP code theory and classical objects in the theory of abelian varieties. For instance, under this dictionary, the finite-dimensional code space becomes the space of theta functions

H^0(X, L)

, logical Pauli gates arise from the theta group, passive logical Clifford gates correspond to automorphisms of the polarized abelian variety, and concatenation with stabilizer codes corresponds to isogeny. We also prove several key results that give precise mathematical formulations of statements about these codes that often appear in heuristic form in the physics literature. In particular, we prove that the encoding is asymptotically isometric, that every logical Clifford gate is realized by a Gaussian unitary, and that, for noise of small variance, the failure probability is governed to first order by the shortest nontrivial displacement in the kernel of the polarization isogeny, a systolic invariant of the underlying polarization. This leads naturally to optimization problems on the moduli space of polarized abelian varieties.

The Gottesman-Kitaev-Preskill (GKP) code is an important type of bosonic quantum error-correcting code. Since the GKP code only protects against small shift errors in $\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{p}$ and $\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{q}$ quadratures, it is necessary to concatenate the GKP code with a stabilizer code for the larger error correction. In this paper, we consider the concatenation of the single-mode GKP code with the two-dimensional (2D) color code (color-GKP code) on the square-octagon lattice. We use the Steane-type scheme with a maximum-likelihood estimation for GKP error correction and show its advantage for the concatenation. In our main paper, the minimum-weight perfect matching algorithm is applied to decode the color-GKP code. Complemented with the continuous-variable information from the GKP code, the threshold of the 2D color code is improved. If only data GKP qubits are noisy, the threshold reaches $\ensuremath{\sigma}\ensuremath{\approx}0.59(\overline{p}\ensuremath{\approx}13.3%)$ compared with

\overline{p}=10.2%

of the normal 2D color code. If measurements are also noisy, we introduce the generalized restriction decoder on the three-dimensional space-time graph for decoding. The threshold reaches $\ensuremath{\sigma}\ensuremath{\approx}0.46$ when measurements in the GKP error correction are noiseless, and $\ensuremath{\sigma}\ensuremath{\approx}0.24$ when all measurements are noisy. Lastly, the good performance of the generalized restriction decoder is also shown on the normal 2D color code giving the threshold at

3.1%

under the phenomenological error model.