We present a general approach to error detection of bosonic quantum error-correction codes via an adaptive quantum phase estimation algorithm assisted by a single ancilla qubit. The approach is applicable to a broad class of bosonic codes whose error syndromes are described by symmetry or stabilizer operators, including the rotation-symmetric codes and Gottesman-Kitaev-Preskill (GKP) codes. The detection precision scales inversely with the total evolution time and thus reaches the Heisenberg limit. We numerically demonstrate the approach for several examples, such as detecting bosonic excitation loss errors in high-order cat or binomial codes and displacement errors in finite-energy GKP codes. We also extend the approach to efficiently generate arbitrary Fock states. Our schemes are feasible in present-day experiments.
We present a general approach to error detection of bosonic quantum error-correction codes via an adaptive quantum phase estimation algorithm assisted by a single ancilla qubit. The approach is applicable to a broad class of bosonic codes whose error syndromes are described by symmetry or stabilizer operators, including the rotation-symmetric codes and Gottesman-Kitaev-Preskill (GKP) codes. The detection precision scales inversely with the total evolution time and thus reaches the Heisenberg limit. We numerically demonstrate the approach for several examples, such as detecting bosonic excitation loss errors in high-order cat or binomial codes and displacement errors in finite-energy GKP codes. We also extend the approach to efficiently generate arbitrary Fock states. Our schemes are feasible in present-day experiments.
Bosonic rotation codes, introduced here, are a broad class of bosonic error-correcting codes based on phase-space rotation symmetry. We present a universal quantum computing scheme applicable to a subset of this class-number-phase codes-which includes the well-known cat and binomial codes, among many others. The entangling gate in our scheme is code agnostic and can be used to interface different rotation-symmetric encodings. In addition to a universal set of operations, we propose a teleportation-based error-correction scheme that allows recoveries to be tracked entirely in software. Focusing on cat and binomial codes as examples, we compute average gate fidelities for error correction under simultaneous loss and dephasing noise and show numerically that the error-correction scheme is close to optimal for error-free ancillae and ideal measurements. Finally, we present a scheme for fault-tolerant, universal quantum computing based on the concatenation of number-phase codes and Bacon-Shor subsystem codes.
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.
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.
Bosonic quantum error-correcting codes offer a viable direction towards reducing the hardware overhead required for fault-tolerant quantum information processing.A broad class of bosonic codes, namely rotation-symmetric codes, can be characterized by their phase-space rotation symmetry.However, their performance has been examined to date only within an idealistic noise model.Here, we further analyze the error-correction capabilities of rotation-symmetric codes using a teleportation-based error-correction circuit.To this end, we numerically compute the average gate fidelity, including measurement errors into the noise model of the data qubit.Focusing on physical measurement models, we assess the performance of heterodyne and adaptive homodyne detection in comparison to the previously studied canonical phase measurement.This setting allows us to shed light on the role of different currently available measurement schemes when decoding the encoded information.We find that with the currently achievable measurement efficiencies in microwave optics, bosonic rotation codes undergo a substantial decrease in their break-even potential.In addition, we perform a detailed analysis of Gottesman-Kitaev-Preskill (GKP) codes using a similar error-correction circuit that allows us to analyze the effect of realistic measurement models on different codes.In comparison to RSB codes, we find for GKP codes an even greater reduction in performance together with a vulnerability to photon-number dephasing.Our results show that highly efficient measurement protocols constitute a crucial building block towards error-corrected quantum information processing with bosonic continuous-variable systems.
The rotation symmetric bosonic code (RSBC) is a unified framework of practical bosonic codes that have rotation symmetries, such as cat codes and binomial codes. While cat and binomial codes achieve the breakeven point in which the coherence time of the encoded qubits exceeds that of unencoded qubits, the state preparation fidelity needs to be improved for practical quantum computing. Concerning this problem, we investigate the framework of symmetry expansion, a class of quantum error mitigation that virtually projects the state onto the noise-free symmetric subspace by exploiting the system's intrinsic symmetries and postprocessing of measurement outcomes. Although symmetry expansion has been limited to error mitigation of quantum states immediately before measurement, we successfully generalize symmetry expansion for state preparation. Then, we consider two types of stabilization: photon number and phase stabilization. Photon number stabilization can be performed by leveraging the rotation operators. On the other hand, the phase errors can be suppressed via the extension towards phase direction by leveraging the recently proposed projective squeezing method. To implement our method, we use an ancilla qubit and a randomly generated controlled gate between the bosonic code states and the ancilla qubit. Our novel error mitigation method will significantly enhance computation accuracy in the near-term bosonic quantum computing.
A promising route towards fault-tolerant quantum error correction is the concatenation of a Gottesman-Kitaev-Preskill (GKP) code with a qubit code. Development of such concatenated codes requires simulation tools which realistically model noise, while being able to simulate the dynamics of many modes. However, so far, large-scale simulation tools for concatenated GKP codes have been limited to idealized noise models and GKP code implementations. Here, we introduce the Bosonic Pauli+ model (BP+), which can be simulated efficiently for a large number of modes, while capturing the rich dynamics in the bosonic multi-mode Hilbert space. We demonstrate the method by simulating a hybrid surface code, where the data qubits are finite-energy GKP qubits stabilized using the small-Big-small (sBs) protocol, and the syndrome qubits are standard two-level systems. Using BP+, we present logical error rates of such an implementation. Confidence in the accuracy of the method is gained by comparing its predictions with full time evolution simulations for several relevant quantum circuits. While developed specifically for GKP qubits stabilized using the sBs protocol, the mathematical structure of BP+ is generic and may be applicable also to the simulation of concatenations using other bosonic codes.