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Encoding a Qubit into an Oscillator with Near-Term Experimental Devices

Daniel J. Weigand

Published Nov 23, 2020• 1 citation
Publisher

Abstract

A universal, large-scale quantum computer would be a powerful tool with applications of high value to mankind. For example, such a computer could significantly speed up the search for new medications or materials. However, the error rates of current qubit designs are simply too large to enable interesting computations. Therefore, both error correction and improved designs of qubits are needed. In 2001, Gottesman, Kitaev and Preskill proposed an encoding (GKP code) where a qubit is stored in a harmonic oscillator — a system that can be controlled and manufactured with high precision, and therefore have comparatively high coherence times. Moreover, the code offers good protection against losses, a simple gate set, and error correction circuits that are comparatively easy to implement. The drawback is that encoding a qubit into a GKP code state is a challenging task. In this thesis, we develop efficient schemes to encode a GKP qubit. Bosonic codes, where a qubit is stored in an oscillator, and in particular the GKP code are still relatively unknown. Therefore, we will start the thesis with an overview of the field, and provide the reader with the tools to analyze a GKP code, as these are quite different from standard error correcting codes. A tool which is important to understand, and that describes a protocol that encodes a GKP qubit is the so-called phase estimation algorithm. This algorithm allows to measure the eigenvalue of any unitary operation, and is one of the cornerstones of quantum information. We will show how phase estimation can be applied to encode a GKP qubit, and what the requirements for an experiment attempting to do so are. A major advantage of the GKP code over other encodings is that it can tolerate significant photon loss before the encoded information is lost. In addition, states that are closely related to the GKP qubit can be used to violate Bell’s inequalities (i. e. prove the presence of entanglement), even in the presence of large noise. Both these applications make the code particularly interesting in the optical regime, where error correction usually cannot be done while the signal is travelling. In this thesis, we will analyze an encoding protocol originally proposed by H.M. Vasconcelos, L. Sanz, and S. Glancy, Optics Letters 35, 3261 (2010) that relied on post-selection, and show that any output state can be used as a GKP code state with a simple change of frame, providing an exponential speedup. In 2019, two separate experiments generated a GKP code state for the first time: C.Flühmann et al., Nature 566, 513 (2019) realized a GKP qubit in the motional mode of a trapped ion, while P. Campagne-Ibarcq et al., Nature 584, 368 (2020) realized it with a transmon qubit coupled to a microwave cavity. However, both these experiments employ phase estimation, which is slow because it requires many measurements in sequence. We propose a circuit that allows a single-shot measurement of the GKP stabilizers, and analyze the performance of such a measurement as well as the impact of noise.

DOI
10.4233/uuid:72abf99f-dd2d-42a1-8c59-7a83870c9d3c

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A universal, large-scale quantum computer would be a powerful tool with applications of high value to mankind. For example, such a computer could significantly speed up the search for new medications or materials. However, the error rates of current qubit designs are simply too large to enable interesting computations. Therefore, both error correction and improved designs of qubits are needed. In 2001, Gottesman, Kitaev and Preskill proposed an encoding (GKP code) where a qubit is stored in a harmonic oscillator — a system that can be controlled and manufactured with high precision, and therefore have comparatively high coherence times. Moreover, the code offers good protection against losses, a simple gate set, and error correction circuits that are comparatively easy to implement. The drawback is that encoding a qubit into a GKP code state is a challenging task. In this thesis, we develop efficient schemes to encode a GKP qubit. Bosonic codes, where a qubit is stored in an oscillator, and in particular the GKP code are still relatively unknown. Therefore, we will start the thesis with an overview of the field, and provide the reader with the tools to analyze a GKP code, as these are quite different from standard error correcting codes. A tool which is important to understand, and that describes a protocol that encodes a GKP qubit is the so-called phase estimation algorithm. This algorithm allows to measure the eigenvalue of any unitary operation, and is one of the cornerstones of quantum information. We will show how phase estimation can be applied to encode a GKP qubit, and what the requirements for an experiment attempting to do so are. A major advantage of the GKP code over other encodings is that it can tolerate significant photon loss before the encoded information is lost. In addition, states that are closely related to the GKP qubit can be used to violate Bell’s inequalities (i. e. prove the presence of entanglement), even in the presence of large noise. Both these applications make the code particularly interesting in the optical regime, where error correction usually cannot be done while the signal is travelling. In this thesis, we will analyze an encoding protocol originally proposed by H.M. Vasconcelos, L. Sanz, and S. Glancy, Optics Letters 35, 3261 (2010) that relied on post-selection, and show that any output state can be used as a GKP code state with a simple change of frame, providing an exponential speedup. In 2019, two separate experiments generated a GKP code state for the first time: C.Flühmann et al., Nature 566, 513 (2019) realized a GKP qubit in the motional mode of a trapped ion, while P. Campagne-Ibarcq et al., Nature 584, 368 (2020) realized it with a transmon qubit coupled to a microwave cavity. However, both these experiments employ phase estimation, which is slow because it requires many measurements in sequence. We propose a circuit that allows a single-shot measurement of the GKP stabilizers, and analyze the performance of such a measurement as well as the impact of noise.
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.
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.
A quantum computer with low-error, high-speed quantum operations and capability for interconnections is required for useful quantum computations. A logical qubit called Gottesman-Kitaev-Preskill (GKP) qubit in a single Bosonic harmonic oscillator is efficient for mitigating errors in a quantum computer. The particularly intriguing prospect of GKP qubits is that entangling gates as well as syndrome measurements for quantum error correction only require efficient, noise-robust linear operations. To date, however, GKP qubits have been only demonstrated at mechanical and microwave frequency in a highly nonlinear physical system. The physical platform that naturally provides the scalable linear toolbox is optics, including near-ideal loss-free beam splitters and near-unit efficiency homodyne detectors that allow to obtain the complete analog syndrome for optimized quantum error correction. Additional optical linear amplifiers and specifically designed GKP qubit states are then all that is needed for universal quantum computing. In this work, we realize a GKP state in propagating light at the telecommunication wavelength and demonstrate homodyne meausurements on the GKP states for the first time without any loss corrections. Our GKP states do not only show non-classicality and non-Gaussianity at room temperature and atmospheric pressure, but unlike the existing schemes with stationary qubits, they are realizable in a propagating wave system. This property permits large-scale quantum computation and interconnections, with strong compatibility to optical fibers and 5G telecommunication technology.
Bosonic quantum error correction codes encode logical qubits in the Hilbert space of one or multiple harmonic oscillators. A prominent class of bosonic codes is that of Gottesman-Kitaev-Preskill (GKP) codes of which implementations have been demonstrated with trapped ions and microwave cavities. In this paper, we investigate theoretically the preparation and error correction of a GKP qubit in a vibrational mode of a neutral atom stored in an optical dipole trap. This platform has recently shown remarkable progress in simultaneously controlling the motional and electronic degrees of freedom of trapped atoms. The protocols we develop make use of motional states and, additionally, internal electronic states of the trapped atom to serve as an ancilla qubit. We compare optical tweezer arrays and optical lattices and find that the latter provide more flexible control over the confinement in the out-of-plane direction, which can be utilized to optimize the conditions for the implementation of GKP codes. Concretely, the different frequency scales that the harmonic oscillators in the axial and radial lattice directions exhibit and a small oscillator anharmonicity prove to be beneficial for robust encodings of GKP states. Finally, we underpin the experimental feasibility of the proposed protocols by numerically simulating the preparation of GKP qubits in an optical lattice with realistic parameters.
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.
Encoding a qubit in the continuous degrees of freedom of an oscillator is a promising path to error-corrected quantum computation. One advantageous way to achieve this is through Gottesman-Kitaev-Preskill (GKP) grid states, whose symmetries allow for the correction of any small continuous error on the oscillator. Unfortunately, ideal grid states have infinite energy, so it is important to find finite-energy approximations that are realistic, practical, and useful for applications. In the first half of this work we investigate the impact of imperfect GKP states on computational circuits independently of the physical architecture. To this end, we analyze the behavior of the physical and logical content of normalizable GKP states through several figures of merit, employing a recently developed modular subsystem decomposition. By tracking the errors that enter into the computational circuit due to imperfections in the GKP states, we are able to gauge the utility of these states for noisy intermediate-scale quantum devices. In the second half, we focus on a state preparation approach in the photonic domain wherein photon-number-resolving measurements on some modes of Gaussian states produce non-Gaussian states in others. We produce detailed numerical results for the preparation of GKP states alongside estimating the resource requirements in practical settings and probing the quality of the resulting states with the tools we develop. Our numerical experiments indicate that we can generate any state in the GKP Bloch sphere with nearly equal resources, which has implications for magic state preparation overheads.