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arXiv:2102.05276 · Physical Review A

Estimation of Gaussian random displacement using non-Gaussian states

Fumiya Hanamura, Warit Asavanant, Kosuke Fukui, Shunya Konno, Akira Furusawa

Published Dec 1, 2021• 17 citations
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Abstract

In continuous-variable quantum information processing, quantum error correction of Gaussian errors requires simultaneous estimation of both quadrature components of displacements in phase space. However, quadrature operators $x$ and $p$ are noncommutative conjugate observables, whose simultaneous measurement is prohibited by the uncertainty principle. Gottesman-Kitaev-Preskill (GKP) error correction deals with this problem using complex non-Gaussian states called GKP states. On the other hand, simultaneous estimation of displacement using experimentally feasible non-Gaussian states has not been well studied. In this paper, we consider a multiparameter estimation problem of displacements assuming an isotropic Gaussian prior distribution and allowing postselection of measurement outcomes. We derive a lower bound for the estimation error when only Gaussian operations are used and show that even simple non-Gaussian states such as single-photon states can beat this bound. Based on Ghosh's bound, we also obtain a lower bound for the estimation error when the maximum photon number of the input state is given. Our results reveal the role of non-Gaussianity in the estimation of displacements and pave the way toward the error correction of Gaussian errors using experimentally feasible non-Gaussian states.

DOI
10.1103/physreva.104.062601

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In continuous-variable quantum information processing, quantum error correction of Gaussian errors requires simultaneous estimation of both quadrature components of displacements in phase space. However, quadrature operators xxx and ppp are noncommutative conjugate observables, whose simultaneous measurement is prohibited by the uncertainty principle. Gottesman-Kitaev-Preskill (GKP) error correction deals with this problem using complex non-Gaussian states called GKP states. On the other hand, simultaneous estimation of displacement using experimentally feasible non-Gaussian states has not been well studied. In this paper, we consider a multiparameter estimation problem of displacements assuming an isotropic Gaussian prior distribution and allowing postselection of measurement outcomes. We derive a lower bound for the estimation error when only Gaussian operations are used and show that even simple non-Gaussian states such as single-photon states can beat this bound. Based on Ghosh's bound, we also obtain a lower bound for the estimation error when the maximum photon number of the input state is given. Our results reveal the role of non-Gaussianity in the estimation of displacements and pave the way toward the error correction of Gaussian errors using experimentally feasible non-Gaussian states.
Uncertainty principle prohibits the precise measurement of both components of displacement parameters in phase space. We have theoretically shown that this limit can be beaten using single-photon states, in a single-shot and single-mode setting [F. Hanamura et al., Estimation of gaussian random displacement using non-gaussian states, Phys. Rev. A 104, 062601 (2021).PLRAAN2469-992610.1103/PhysRevA.104.062601]. In this Letter, we validate this by experimentally beating the classical limit. In optics, this is the first experiment to estimate both parameters of displacement using non-Gaussian states. This result is related to many important applications, such as quantum error correction.
Determining the quantum capacity of a noisy quantum channel is an important problem in the field of quantum communication theory. In this work, we consider the Gaussian random-displacement channel ${\mathcal{N}}_{\ensuremath{\sigma}}$, a type of bosonic Gaussian channels relevant in various bosonic quantum information processing systems. In particular, we attempt to make progress on the problem of determining the quantum capacity of a Gaussian random-displacement channel by analyzing the error-correction performance of several families of multimode Gottesman-Kitaev-Preskill (GKP) codes. In doing so we analyze the surface-square GKP codes using an efficient and exact maximum-likelihood decoder (MLD) up to a large code distance of d=39d=39d=39. We find that the error threshold of the surface-square GKP code is remarkably close to $\ensuremath{\sigma}=1/\sqrt{e}\ensuremath{\simeq}0.6065$ at which the best-known lower bound of the quantum capacity of ${\mathcal{N}}_{\ensuremath{\sigma}}$ vanishes. We also analyze the performance of color-hexagonal GKP codes up to a code distance of d=13d=13d=13 using a tensor-network decoder serving as an approximate MLD. By focusing on multimode GKP codes that encode just one logical qubit over multiple bosonic modes, we show that GKP codes can achieve nonzero quantum state transmission rates for a Gaussian random-displacement channel ${\mathcal{N}}_{\ensuremath{\sigma}}$ at larger values of $\ensuremath{\sigma}$ than previously demonstrated. Thus, our work reduces the gap between the quantum communication theoretic bounds and the performance of explicit bosonic quantum error-correcting codes in regards to the quantum capacity of a Gaussian random-displacement channel.
We examine the performance of the single-mode Gottesman-Kitaev-Preskill (GKP) code and its concatenation with the toric code for a noise model of Gaussian shifts, or displacement errors. We show how one can optimize the tracking of errors in repeated noisy error correction for the GKP code. We do this by examining the maximum-likelihood problem for this setting and its mapping onto a 1D Euclidean path-integral modeling a particle in a random cosine potential. We demonstrate the efficiency of a minimum-energy decoding strategy as a proxy for the path integral evaluation. In the second part of this paper, we analyze and numerically assess the concatenation of the GKP code with the toric code. When toric code measurements and GKP error correction measurements are perfect, we find that by using GKP error information the toric code threshold improves from 1010%10 to 1414%14. When only the GKP error correction measurements are perfect we observe a threshold at 66%6. In the more realistic setting when all error information is noisy, we show how to represent the maximum likelihood decoding problem for the toric-GKP code as a 3D compact QED model in the presence of a quenched random gauge field, an extension of the random-plaquette gauge model for the toric code. We present a decoder for this problem which shows the existence of a noise threshold at shift-error standard deviation ${\ensuremath{\sigma}}_{0}\ensuremath{\approx}0.243$ for toric code measurements, data errors and GKP ancilla errors. If the errors only come from having imperfect GKP states, then this corresponds to states with just four photons or more. Our last result is a no-go result for linear oscillator codes, encoding oscillators into oscillators. For the Gaussian displacement error model, we prove that encoding corresponds to squeezing the shift errors. This shows that linear oscillator codes are useless for quantum information protection against Gaussian shift errors.
Abstract Continuous-variable quantum-computing is the most scalable implementation of QC to date but requires non-Gaussian resources to allow exponential speedup and quantum correction, using error encoding such as Gottesman–Kitaev–Preskill (GKP) states. However, GKP state generation is still an experimental challenge. We show theoretically that photon catalysis, the interference of coherent states with single-photon states followed by photon-number-resolved detection, is a powerful enabler for non-Gaussian quantum state engineering such as exactly displaced single-photon states and M -symmetric superpositions of squeezed vacuum (SSV), including squeezed cat states ( M = 2). By including photon-counting based state breeding, we demonstrate the potential to enlarge SSV states and produce GKP states.
Abstract The central issue in quantum parameter estimation is to find out the optimal measurement setup that leads to the ultimate lower bound of an estimation error. We address here a question of whether a Gaussian measurement scheme can achieve the ultimate bound for phase estimation in single-mode Gaussian metrology that exploits single-mode Gaussian probe states in a Gaussian environment. We identify three types of optimal Gaussian measurement setups yielding the maximal Fisher information depending on displacement, squeezing, and thermalization of the probe state. We show that the homodyne measurement attains the ultimate bound for both displaced thermal probe states and squeezed vacuum probe states, whereas for the other single-mode Gaussian probe states, the optimized Gaussian measurement cannot be the optimal setup, although they are sometimes nearly optimal. We then demonstrate that the measurement on the basis of the product quadrature operators X^P^+P^X^\hat X\hat P + \hat P\hat XX^P^+P^X^ X ̂ P ̂ + P ̂ X ̂ , i.e., a non-Gaussian measurement, is required to be fully optimal.