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arXiv:2002.01402 · Physical Review Letters

Universal Gate Set for Continuous-Variable Quantum Computation with Microwave Circuits

Timo Hillmann, Fernando Quijandría, Göran Johansson, Alessandro Ferraro, Simone Gasparinetti, Giulia Ferrini

Published Oct 12, 2020• 77 citations
arXivHTMLPDF

Abstract

We provide an explicit construction of a universal gate set for continuous-variable quantum computation with microwave circuits. Such a universal set has been first proposed in quantum-optical setups, but its experimental implementation has remained elusive in that domain due to the difficulties in engineering strong nonlinearities. Here, we show that a realistic three-wave mixing microwave architecture based on the superconducting nonlinear asymmetric inductive element [Frattini et al., Appl. Phys. Lett. 110, 222603 (2017)APPLAB0003-695110.1063/1.4984142] allows us to overcome this difficulty. As an application, we show that this architecture allows for the generation of a cubic phase state with an experimentally feasible procedure. This work highlights a practical advantage of microwave circuits with respect to optical systems for the purpose of engineering non-Gaussian states and opens the quest for continuous-variable algorithms based on few repetitions of elementary gates from the continuous-variable universal set.

DOI
10.1103/physrevlett.125.160501

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We provide an explicit construction of a universal gate set for continuous-variable quantum computation with microwave circuits. Such a universal set has been first proposed in quantum-optical setups, but its experimental implementation has remained elusive in that domain due to the difficulties in engineering strong nonlinearities. Here, we show that a realistic three-wave mixing microwave architecture based on the superconducting nonlinear asymmetric inductive element [Frattini et al., Appl. Phys. Lett. 110, 222603 (2017)APPLAB0003-695110.1063/1.4984142] allows us to overcome this difficulty. As an application, we show that this architecture allows for the generation of a cubic phase state with an experimentally feasible procedure. This work highlights a practical advantage of microwave circuits with respect to optical systems for the purpose of engineering non-Gaussian states and opens the quest for continuous-variable algorithms based on few repetitions of elementary gates from the continuous-variable universal set.
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Abstract Continuous variable (CV) quantum computation offers an alternative to qubit-based computing by exploiting the infinite-dimensional Hilbert space of bosonic modes. Despite recent progress, superconducting platforms have yet to demonstrate a scalable architecture capable of universal computation. Here, we design and numerically simulate a two-layer superconducting architecture that implements all five interactions of the universal CV gate set (rotation, displacement, squeezing, Kerr, and beam splitter) within experimentally accessible regimes. To this end, we employ a DC-SQUID as the bosonic mode, a fluxonium qubit to mediate nonlinear interactions, and two ancillary qubits that enable Gaussian and multi-mode operations. By tuning fluxes and frequencies, we achieve high fidelities ( ⩾ 98 % ) across all gates within state-of-the-art parameter ranges. The modular nature of the design allows straightforward scaling, establishing a feasible pathway toward high-fidelity, universal CV quantum computation based on superconducting circuits.
Quantum non-Gaussian gate is a missing piece to the realization of continuous-variable universal quantum operations in an optical system. In a measurement-based implementation of the cubic phase gate, a lowest-order non-Gaussian gate, non-Gaussian ancillary states that have a property we call nonlinear squeezing are required. This property, however, has never been experimentally verified. In this paper, we generate a superposition between a vacuum state and a single-photon state whose nonlinear squeezing is maximized by the optimization of the superposition coefficients. The nonlinear squeezing is observed via real-time quadrature measurements, meaning that the generated states are compatible with real-time feedforward and are suitable as ancillary states for the cubic phase gate in the time domain. Moreover, by increasing the number of photons, it is expected that nonlinear squeezing can be further improved. The idea presented here can be readily extended to higher-order phase gates [P. Marek et al., Phys. Rev. A 97, 022329 (2018)]. As such, this work presents an important step to extend continuous-variable quantum information processing from Gaussian regime to non-Gaussian regime.
Multimode superconducting parametric cavity is a flexible platform that has been used to study a variety of topics in microwave quantum optics ranging from parametric amplification, entanglement generation to higher order spontaneous parametric downconversion (SPDC). \nLeveraging the extensive toolbox of interactions available in this system, we can look to explore exciting applications in quantum computation and simulation. \nIn this thesis, we study the use of the parametric cavity to realize continuous variable (CV) quantum computation. \nWe propose and examine in detail the scheme to compute with the microwave photons in the orthogonal frequency modes of the cavity via successive application of parametric pump pulses or cavity drives. \nThe family of all Gaussian transformations can be accomplished easily with interactions already demonstrated in this system. \nFrom recent results and proposals involving higher order SPDC, there are also clear pathways towards realizing the non-Gaussian resources necessary for universal computation. \nCommon measurements on the system are accomplished with standard measurement techniques on the output state of the cavity and additional useful measurements may be implemented using available parametric interactions or new device designs involving a qubit as a nonlinear probe. \nUsing the parametric cavity, we experimentally implemented a hybrid quantum-classical machine learning algorithm called the Quantum Kitchen Sinks (QKS) as the first step towards developing this platform for quantum computation. \nThe algorithm is studied over two sets of experiments starting from partial experimental implementation of the quantum variational circuits to fully experimental implementation using multiple simultaneous continuous wave (CW) pumps. \nIn both cases, we find that the quantum part of the algorithm implemented in the parametric cavity improved the classification accuracy on a difficult synthetic data set up to 90.1% and 99.5% respectively when compared to a classical linear machine learning algorithm.
We propose an experimental design for universal continuous-variable quantum computation that incorporates recent innovations in linear-optics-based continuous-variable cluster state generation and cubic-phase gate teleportation. The first ingredient is a protocol for generating the bilayer-square-lattice cluster state (a universal resource state) with temporal modes of light. With this state, measurement-based implementation of Gaussian unitary gates requires only homodyne detection. Second, we describe a measurement device that implements an adaptive cubic-phase gate, up to a random phase-space displacement. It requires a two-step sequence of homodyne measurements and consumes a (non-Gaussian) cubic-phase state.
The motional modes of trapped ions are promising candidates for continuous-variable (CV) quantum computation and simulation. However, to achieve a universal gate set for CV quantum computation, anharmonic nonlinear unitaries are required. Here, we experimentally demonstrate a protocol that allows us to approximate the cubic phase gate. By numerically optimizing the phase angles and amplitudes of a set of state-dependent force pulses, we demonstrate the cubic phase gate for both vacuum and low-ααα coherent state inputs and verify its performance. Our protocol can also be extended to approximate other nonlinear unitaries.