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arXiv:1909.10075 · Physical Review A

Realizing modular quadrature measurements via a tunable photon-pressure coupling in circuit QED

Daniel J. Weigand, Barbara M. Terhal

Published May 18, 2020• 28 citations
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Abstract

One of the most direct preparations of a Gottesman-Kitaev-Preskill (GKP) qubit in an oscillator uses a tunable photon-pressure (also called optomechanical) coupling of the form $\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{q}{\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{b}}^{\ifmmode\dagger\else\textdagger\fi{}}\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{b}$, enabling us to imprint the modular value of the position $\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{q}$ of one oscillator onto the state of an ancilla oscillator. We analyze the practical feasibility of executing such modular quadrature measurements in a parametric circuit-QED realization of this coupling. We provide estimates for the expected GKP squeezing induced by the protocol and discuss the effect of photon loss and other errors on the resulting squeezing.

DOI
10.1103/physreva.101.053840

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One of the most direct preparations of a Gottesman-Kitaev-Preskill (GKP) qubit in an oscillator uses a tunable photon-pressure (also called optomechanical) coupling of the form q\ifmmode^\elseˆ\fib\ifmmode^\elseˆ\fi\ifmmode†\else\textdagger\fib\ifmmode^\elseˆ\fi\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{q}{\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{b}}^{\ifmmode\dagger\else\textdagger\fi{}}\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{b}q\ifmmode^\elseˆ\fi​b\ifmmode^\elseˆ\fi​\ifmmode†\else\textdagger\fib\ifmmode^\elseˆ\fi​, enabling us to imprint the modular value of the position q\ifmmode^\elseˆ\fi\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{q}q\ifmmode^\elseˆ\fi​ of one oscillator onto the state of an ancilla oscillator. We analyze the practical feasibility of executing such modular quadrature measurements in a parametric circuit-QED realization of this coupling. We provide estimates for the expected GKP squeezing induced by the protocol and discuss the effect of photon loss and other errors on the resulting squeezing.
Gottesman, Kitaev, and Preskill have formulated a way of encoding a qubit into an oscillator such that the qubit is protected against small shifts (translations) in phase space. The idea underlying this encoding is that error processes of low rate can be expanded into small shift errors. The qubit space is defined as an eigenspace of two mutually commuting displacement operators Sp{S}_{p}Sp​ and Sq{S}_{q}Sq​ which act as large shifts or translations in phase space. We propose and analyze the approximate creation of these qubit states by coupling the oscillator to a sequence of ancilla qubits. This preparation of the states uses the idea of phase estimation where the phase of the displacement operator, say Sp{S}_{p}Sp​, is approximately determined. We consider several possible forms of phase estimation. We analyze the performance of repeated and adaptive phase estimation as the simplest and experimentally most viable schemes given a realistic upper limit on the number of photons in the oscillator. We propose a detailed physical implementation of this protocol using the dispersive coupling between a transmon ancilla qubit and a cavity mode in circuit QED. We provide an estimate that in a current experimental setup one can prepare a good code state from a squeezed vacuum state using eight rounds of adaptive phase estimation, lasting in total about $4\phantom{\rule{0.28em}{0ex}}\ensuremath{\mu}\text{s}$, with 9494%94 (heralded) chance of success.
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.
Encoding a qubit in the continuous degrees of freedom of an oscillator is a significant pursuit of quantum computation. One advantageous way to achieve this is through the 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 the paper we investigate the impact of imperfect GKP states on computational circuits independently of the physical architecture. To this end we analyze the behaviour 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 the imperfections in the GKP states, we are able to gauge the utility of these states for NISQ (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 the others. We produce detailed numerical results for the preparation of GKP states along with estimating the resource requirements in practical settings and probing the quality of the resulting states with the tools we develop.
We extend the controlled displacement interaction between a qubit and a harmonic oscillator to the multi-qubit (qudit) case. We define discrete quadratures of the qudit and show how the qudit state can be displaced in these quadratures controlled by an oscillator quadrature. Using this interaction, a periodic repetition of the state encoded in the qudit, can be deterministically mapped onto the oscillator, which is initialized in a squeezed state. Based on this controlled displacement interaction, we present a full circuit that encodes quantum information in a superposition of qudit quadrature states, and successively prepares the oscillator in the corresponding superposition of approximate Gottesman-Kitaev-Preskill (GKP) states. This preparation scheme is found to be similar to phase estimation, with the addition of a disentanglement gate. Our protocol for GKP state preparation is efficient in the sense, that the set of qubits need only interact with the oscillator through two time-independent interactions, and in the sense that the squeeze factor (in dB) of the produced GKP state grows linearly in the number of qubits used.
Gottesman-Kitaev-Preskill (GKP) states are a central resource for fault-tolerant optical continuous-variable quantum computing. However, their realization in the optical domain remains to be demonstrated. Here we propose a method for preparing GKP states using a cavity QED system that can be realized in several platforms, such as trapped atoms, quantum dots, or diamond color centers. We then further combine the protocol with the previously proposed breeding protocol by Vasconcelos et al. to relax the demands on the quality of the QED system, finding that GKP states with more than 10 dB squeezing could be achieved in near-future experiments.
Gottesman-Kitaev-Preskill (GKP) states are a central resource for fault-tolerant optical continuous-variable quantum computing and communication. However, their realization in the optical domain remains to be demonstrated. Here we propose a method for preparing GKP states using a cavity QED system as the only non-Gaussian element. This system can be realized in several platforms such as trapped atoms, quantum dots or diamond color centers. We simulate the protocol with finite-cooperativity systems to evaluate the achievable quality of the produced GKP states in realistic noisy systems. We further combine the protocol with the previously proposed breeding protocol by Vasconcelos et al. to relax the demands on the quality of the QED system. We find that GKP states with more than 10 dB squeezing could be achieved in near-future experiments. Additionally, the approach can be made fully deterministic, thus providing a more scalable and hardware-efficient approach compared to probabilistic approaches based on photon counting.