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arXiv:1911.11141 · Physical Review Research

Cost-reduced all-Gaussian universality with the Gottesman-Kitaev-Preskill code: Resource-theoretic approach to cost analysis

Hayata Yamasaki, Takaya Matsuura, Masato Koashi

Published Jun 2, 2020• 50 citations
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Abstract

The authors present a new mechanism to prepare magic states of the Gottesman-Kitaev-Preskill (GKP) continuous-variable quantum error-correcting code from two perspectives. The paper proposes two methods: a cost-reduced scheme of universal quantum computation by experimentally tractable Gaussian operations using the GKP magic states as a single resource, and the formulation of a resource-theoretical method for quantitatively analyzing the usefulness of magic states of the GKP code.

DOI
10.1103/physrevresearch.2.023270

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The authors present a new mechanism to prepare magic states of the Gottesman-Kitaev-Preskill (GKP) continuous-variable quantum error-correcting code from two perspectives. The paper proposes two methods: a cost-reduced scheme of universal quantum computation by experimentally tractable Gaussian operations using the GKP magic states as a single resource, and the formulation of a resource-theoretical method for quantitatively analyzing the usefulness of magic states of the GKP code.
Although the similarity between nonstabilizer states—also known as magic states—in discrete-variable systems and non-Gaussian states in continuous-variable systems has widely been recognized, the precise connections between these two notions have still been unclear. We establish a fundamental link between these two quantum resources via the Gottesman-Kitaev-Preskill (GKP) encoding. We show that the negativity of the continuous-variable Wigner function for an encoded GKP state coincides with a magic measure we introduce, which matches the negativity of the discrete Wigner function for odd dimensions. We also provide a continuous-variable representation of the stabilizer Rényi entropy—a recent proposal for a magic measure for multiqubit states. We further provide the magic measure with an operational interpretation as a runtime of a classical simulation algorithm. In addition, we employ our results to prove that implementing a multiqubit logical non-Clifford operation in the GKP code subspace requires a non-Gaussian operation even at the limit of perfect encoding, despite the fact that the ideal GKP states already come with a large amount of non-Gaussianity.
The Gottesman-Kitaev-Preskill (GKP) encoding of a qubit within an oscillator is particularly appealing for fault-tolerant quantum computing with bosons because Gaussian operations on encoded Pauli eigenstates enable Clifford quantum computing with error correction. We show that applying GKP error correction to Gaussian input states, such as vacuum, produces distillable magic states, achieving universality without additional non-Gaussian elements. Fault tolerance is possible with sufficient squeezing and low enough external noise. Thus, Gaussian operations are sufficient for fault-tolerant, universal quantum computing given a supply of GKP-encoded Pauli eigenstates.
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.
Gate teleportation, together with magic state distillation, is a promising route towards fault-tolerant, universal computation. In the context of bosonic quantum computation, Baragiola et al. PRL 123(20).200502 (2019) showed that within the framework of Gottesman--Kitaev--Preskill codes, encoded magic states suitable for distillation can be produced by error correcting Gaussian states, such as the vacuum. Here, we show that applying the same framework to simple non-Gaussian input states can significantly improve the quality of the magic states obtained, reducing the overall resources for the complete distillation procedure. We focus on superpositions of coherent states or Fock states, showing that many can lead to improvements in the generation of high-quality encoded magic states, which in some cases reduces the resources required for magic state distillation by about a factor 333. We also investigate the primary source of these improvements and find that, unlike what was previously conjectured, the suitability of input states is not fully explained by symmetry arguments. Instead, the best states seem to avoid projection near stabilizer states as a result of the error correction procedure.
The Gottesman-Kitaev-Preskill (GKP) encoding of a qubit into a bosonic mode is a promising bosonic code for quantum computation due to its tolerance for noise and all-Gaussian gate set. We present a toolkit for phase-space description and manipulation of GKP encodings that includes Wigner functions for ideal and approximate GKP states, for various types of mixed GKP states, and for GKP-encoded operators. One advantage of a phase-space approach is that Gaussian unitaries, required for computation with GKP codes, correspond to simple transformations on the arguments of Wigner functions. We use this fact and our toolkit to describe GKP error correction, including magic-state preparation, entirely in phase space using operations on Wigner functions. While our focus here is on the square-lattice GKP code, we provide a general framework for GKP codes defined on any lattice.
In the field of fault-tolerant quantum computing, continuous-variable systems can be utilized to protect quantum information from noise through the use of bosonic codes. These codes map qubit-type quantum information onto the larger bosonic Hilbert space, and can be divided into two main categories: translational-symmetric codes, such as Gottesman-Kitaev-Preskill (GKP) codes, and rotational-symmetric codes, including cat and binomial codes. The relationship between these families of codes has not yet been fully understood. We present an iterative protocol for converting between two instances of these codes---generalized GKP (so-called qunaught) states and fourfold-symmetric binomial states corresponding to a zero-logical encoded qubit---using only Gaussian operations. This conversion demonstrates the potential for universality of binomial states for all-Gaussian quantum computation and provides a method for the heralded preparation of GKP states. Through numerical simulation, we obtain GKP qunaught states with a fidelity of over 9898%98 and a probability of approximately 3.143.14%3.14, after only two steps of our iterative protocol, though higher fidelities can be achieved with additional iterations at the cost of lower success probabilities.