Abstract Gottesman–Kitaev–Preskill (GKP) states appear to be amongst the leading candidates for correcting errors when encoding qubits into oscillators. However the preparation of GKP states remains a significant theoretical and experimental challenge. Until now, no clear definitions for fault-tolerantly preparing GKP states have been provided. Without careful consideration, a small number of faults can lead to large uncorrectable shift errors. After proposing a metric to compare approximate GKP states, we provide rigorous definitions of fault-tolerance and introduce a fault-tolerant phase estimation protocol for preparing such states. The fault-tolerant protocol uses one flag qubit and accepts only a subset of states in order to prevent measurement readout errors from causing large shift errors. We then show how the protocol can be implemented using circuit QED. In doing so, we derive analytic expressions which describe the leading order effects of the nonlinear dispersive shift and Kerr nonlinearity. Using these expressions, we show that to mitigate the nonlinear dispersive shift and Kerr terms would require the protocol to be implemented on time scales four orders of magnitude longer than the time scales relevant to the protocol for physically motivated parameters. Despite these restrictions, we numerically show that a subset of the accepted states of the fault-tolerant phase estimation protocol maintain good error correcting capabilities even in the presence of noise.
Abstract Gottesman–Kitaev–Preskill (GKP) states appear to be amongst the leading candidates for correcting errors when encoding qubits into oscillators. However the preparation of GKP states remains a significant theoretical and experimental challenge. Until now, no clear definitions for fault-tolerantly preparing GKP states have been provided. Without careful consideration, a small number of faults can lead to large uncorrectable shift errors. After proposing a metric to compare approximate GKP states, we provide rigorous definitions of fault-tolerance and introduce a fault-tolerant phase estimation protocol for preparing such states. The fault-tolerant protocol uses one flag qubit and accepts only a subset of states in order to prevent measurement readout errors from causing large shift errors. We then show how the protocol can be implemented using circuit QED. In doing so, we derive analytic expressions which describe the leading order effects of the nonlinear dispersive shift and Kerr nonlinearity. Using these expressions, we show that to mitigate the nonlinear dispersive shift and Kerr terms would require the protocol to be implemented on time scales four orders of magnitude longer than the time scales relevant to the protocol for physically motivated parameters. Despite these restrictions, we numerically show that a subset of the accepted states of the fault-tolerant phase estimation protocol maintain good error correcting capabilities even in the presence of noise.
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.
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.
Fault-tolerant quantum error correction is essential for implementing quantum algorithms of significant practical importance.In this work, we propose a highly effective use of the surface Gottesman-Kitaev-Preskill (GKP) code, i.e., the surface code consisting of bosonic GKP qubits instead of bare two-level qubits.In our proposal, we use error-corrected two-qubit gates between GKP qubits and introduce a maximum-likelihood decoding strategy for correcting shift errors in the two-GKP-qubit gates.Our proposed decoding reduces the total CNOT failure rate of the GKP qubits, e.g., from 0.87% to 0.36% at a GKP squeezing of 12 dB, compared to the case where the simple closest-integer decoding is used.Then, by concatenating the GKP code with the surface code, we find that the threshold GKP squeezing is given by 9.9 dB under the the assumption that finite squeezing of the GKP states is the dominant noise source.More importantly, we show that a low logical failure rate p L < 10 -7 can be achieved with moderate hardware requirements, e.g., 291 modes and 97 qubits at a GKP squeezing of 12 dB as opposed to 1457 bare qubits for the standard rotated surface code at an equivalent noise level (i.e., p = 0.36%).Such a low failure rate of our surface-GKP code is possible through the use of space-time correlated edges in the matching graphs of the surface-code decoder.Further, all edge weights in the matching graphs are computed dynamically based on analog information from the GKP error correction using the full history of all syndrome measurement rounds.We also show that a highly squeezed GKP state of GKP squeezing 12 dB can be experimentally realized by using a dissipative stabilization method, namely, the big-small-big method, with fairly conservative experimental parameters.Lastly, we introduce a three-level ancilla scheme to mitigate ancilla decay errors during a GKP state preparation.
The Gottesman-Kitaev-Preskill (gkp) quantum error-correcting code attracts much attention in continuous variable (CV) quantum computation and CV quantum communication due to the simplicity of error-correcting routines and the high tolerance against Gaussian errors. Since the gkp code state should be regarded as a limit of physically meaningful approximate ones, various approximations have been developed until today, but explicit relations among them are still unclear. In this paper, we rigorously prove the equivalence of these approximate gkp codes with an explicit correspondence of the parameters. We also propose a standard form of the approximate code states in the position representation, which enables us to derive closed-form expressions for the Wigner function, inner products, and the average photon number in terms of the theta functions. Our results serve as fundamental tools for further analyses of fault-tolerant quantum computation and channel coding using approximate gkp codes.
The Gottesman-Kitaev-Preskill (GKP) code is an exciting route to fault-tolerant quantum computing since Gaussian resources and GKP Pauli-eigenstate preparation are sufficient to achieve universal quantum computing. However, there is a disconnect between the noise model that GKP qubits are in theory designed to correct—uniform random displacement errors—and the conditions that affect GKP qubits in superconducting devices in practice: realistic noise channels, logical gates, and inefficient measurements. In this work we bridge this gap in three ways. First, we approximate the effect loss and dephasing on approximate GKP codestates using a random displacement channel, and show that this approximation matches well with numerics. Second, we analyze the error-spreading properties of GKP Clifford gates and describe how a modification in the decoder following the implementation of each gate can reduce the gate infidelity by multiple orders of magnitude. Finally, we consider the effect of homodyne measurement inefficiencies on logical state readout and analyze a scheme to improve the measurement efficiency using the theory of quantum trajectories.