Performance Analysis of a Repetition Cat Code Architecture: Computing 256-bit Elliptic Curve Logarithm in 9 Hours with 126 133 Cat Qubits.

Cat qubits provide appealing building blocks for quantum computing. They exhibit a tunable noise bias yielding an exponential suppression of bit flips with the average photon number and a protection against the remaining phase errors can be ensured by a simple repetition code. We here quantify the cost of a repetition code and provide valuable guidance for the choice of a large scale architecture using cat qubits by realizing a performance analysis based on the computation of discrete logarithms on an elliptic curve with Shor's algorithm. By focusing on a 2D grid of cat qubits with neighboring connectivity, we propose to implement 2-qubit gates via lattice surgery and Toffoli gates with off-line fault-tolerant preparation of magic states through projective measurements and subsequent gate teleportations. All-to-all connectivity between logical qubits is ensured by routing qubits. Assuming a ratio between single- and two-photon losses of 10^{-5} and a cycle time of 500 ns, we show concretely that such an architecture can compute a 256-bit elliptic curve logarithm in 9 h with 126 133 cat qubits and on average 19 photons by cat state. We give the details of the realization of Shor's algorithm so that the proposed performance analysis can be easily reused to guide the choice of architecture for others platforms.

Loophole-free Bell inequality violation with superconducting circuits.

Superposition, entanglement and non-locality constitute fundamental features of quantum physics. The fact that quantum physics does not follow the principle of local causality1-3 can be experimentally demonstrated in Bell tests4 performed on pairs of spatially separated, entangled quantum systems. Although Bell tests, which are widely regarded as a litmus test of quantum physics, have been explored using a broad range of quantum systems over the past 50 years, only relatively recently have experiments free of so-called loopholes5 succeeded. Such experiments have been performed with spins in nitrogen-vacancy centres6, optical photons7-9 and neutral atoms10. Here we demonstrate a loophole-free violation of Bell's inequality with superconducting circuits, which are a prime contender for realizing quantum computing technology11. To evaluate a Clauser-Horne-Shimony-Holt-type Bell inequality4, we deterministically entangle a pair of qubits12 and perform fast and high-fidelity measurements13 along randomly chosen bases on the qubits connected through a cryogenic link14 spanning a distance of 30 metres. Evaluating more than 1 million experimental trials, we find an average S value of 2.0747 ± 0.0033, violating Bell's inequality with a P value smaller than 10-108. Our work demonstrates that non-locality is a viable new resource in quantum information technology realized with superconducting circuits with potential applications in quantum communication, quantum computing and fundamental physics15.

Factoring 2048-bit RSA Integers in 177 Days with 13 436 Qubits and a Multimode Memory.

We analyze the performance of a quantum computer architecture combining a small processor and a storage unit. By focusing on integer factorization, we show a reduction by several orders of magnitude of the number of processing qubits compared with a standard architecture using a planar grid of qubits with nearest-neighbor connectivity. This is achieved by taking advantage of a temporally and spatially multiplexed memory to store the qubit states between processing steps. Concretely, for a characteristic physical gate error rate of 10^{-3}, a processor cycle time of 1 microsecond, factoring a 2 048-bit RSA integer is shown to be possible in 177 days with 3D gauge color codes assuming a threshold of 0.75% with a processor made with 13 436 physical qubits and a memory that can store 28 million spatial modes and 45 temporal modes with 2 hours' storage time. By inserting additional error-correction steps, storage times of 1 second are shown to be sufficient at the cost of increasing the run-time by about 23%. Shorter run-times (and storage times) are achievable by increasing the number of qubits in the processing unit. We suggest realizing such an architecture using a microwave interface between a processor made with superconducting qubits and a multiplexed memory using the principle of photon echo in solids doped with rare-earth ions.

Bell correlations between light and vibration at ambient conditions.

Time-resolved Raman spectroscopy techniques offer various ways to study the dynamics of molecular vibrations in liquids or gases and optical phonons in crystals. While these techniques give access to the coherence time of the vibrational modes, they are not able to reveal the fragile quantum correlations that are spontaneously created between light and vibration during the Raman interaction. Here, we present a scheme leveraging universal properties of spontaneous Raman scattering to demonstrate Bell correlations between light and a collective molecular vibration. We measure the decay of these hybrid photon-phonon Bell correlations with sub-picosecond time resolution and find that they survive over several hundred oscillations at ambient conditions. Our method offers a universal approach to generate entanglement between light and molecular vibrations. Moreover, our results pave the way for the study of quantum correlations in more complex solid-state and molecular systems in their natural state.

Setting Up Experimental Bell Tests with Reinforcement Learning.

Finding optical setups producing measurement results with a targeted probability distribution is hard, as a priori the number of possible experimental implementations grows exponentially with the number of modes and the number of devices. To tackle this complexity, we introduce a method combining reinforcement learning and simulated annealing enabling the automated design of optical experiments producing results with the desired probability distributions. We illustrate the relevance of our method by applying it to a probability distribution favouring high violations of the Bell-Clauser-Horne-Shimony-Holt (CHSH) inequality. As a result, we propose new unintuitive experiments leading to higher Bell-CHSH inequality violations than the best currently known setups. Our method might positively impact the usefulness of photonic experiments for device-independent quantum information processing.

A gated quantum dot strongly coupled to an optical microcavity.

The strong-coupling regime of cavity quantum electrodynamics (QED) represents the light-matter interaction at the fully quantum level. Adding a single photon shifts the resonance frequencies-a profound nonlinearity. Cavity QED is a test bed for quantum optics1-3 and the basis of photon-photon and atom-atom entangling gates4,5. At microwave frequencies, cavity QED has had a transformative effect6, enabling qubit readout and qubit couplings in superconducting circuits. At optical frequencies, the gates are potentially much faster; the photons can propagate over long distances and can be easily detected. Following pioneering work on single atoms1-3,7, solid-state implementations using semiconductor quantum dots are emerging8-15. However, miniaturizing semiconductor cavities without introducing charge noise and scattering losses remains a challenge. Here we present a gated, ultralow-loss, frequency-tunable microcavity device. The gates allow both the quantum dot charge and its resonance frequency to be controlled electrically. Furthermore, cavity feeding10,11,13-17, the observation of the bare-cavity mode even at the quantum dot-cavity resonance, is eliminated. Even inside the microcavity, the quantum dot has a linewidth close to the radiative limit. In addition to a very pronounced avoided crossing in the spectral domain, we observe a clear coherent exchange of a single energy quantum between the 'atom' (the quantum dot) and the cavity in the time domain (vacuum Rabi oscillations), whereas decoherence arises mainly via the atom and photon loss channels. This coherence is exploited to probe the transitions between the singly and doubly excited photon-atom system using photon-statistics spectroscopy18. The work establishes a route to the development of semiconductor-based quantum photonics, such as single-photon sources and photon-photon gates.

Certifying the Building Blocks of Quantum Computers from Bell's Theorem.

Bell's theorem has been proposed to certify, in a device-independent and robust way, blocks either producing or measuring quantum states. In this Letter, we provide a method based on Bell's theorem to certify coherent operations for the storage, processing, and transfer of quantum information. This completes the set of tools needed to certify all building blocks of a quantum computer. Our method distinguishes itself by its robustness to experimental imperfections, and so could be used to certify that today's quantum devices are qualified for usage in future quantum computers.

Witnessing Optomechanical Entanglement with Photon Counting.

The ability to coherently control mechanical systems with optical fields has made great strides over the past decade, and now includes the use of photon counting techniques to detect the nonclassical nature of mechanical states. These techniques may soon be used to perform an optomechanical Bell test, hence highlighting the potential of cavity optomechanics for device-independent quantum information processing. Here, we propose a witness which reveals optomechanical entanglement without any constraint on the global detection efficiencies in a setup allowing one to test a Bell inequality. While our witness relies on a well-defined description and correct experimental calibration of the measurements, it does not need a detailed knowledge of the functioning of the optomechanical system. A feasibility study including dominant sources of noise and loss shows that it can readily be used to reveal optomechanical entanglement in present-day experiments with photonic crystal nanobeam resonators.

Two-Color Pump-Probe Measurement of Photonic Quantum Correlations Mediated by a Single Phonon.

We propose and demonstrate a versatile technique to measure the lifetime of the one-phonon Fock state using two-color pump-probe Raman scattering and spectrally resolved, time-correlated photon counting. Following pulsed laser excitation, the n=1 phonon Fock state is probabilistically prepared by projective measurement of a single Stokes photon. The detection of an anti-Stokes photon generated by a second, time-delayed laser pulse probes the phonon population with subpicosecond time resolution. We observe strongly nonclassical Stokes-anti-Stokes correlations, whose decay maps the single phonon dynamics. Our scheme can be applied to any Raman-active vibrational mode. It can be modified to measure the lifetime of n≥1 Fock states or the phonon quantum coherences through the preparation and detection of two-mode entangled vibrational states.

Noise-Resistant Device-Independent Certification of Bell State Measurements.

Device-independent certification refers to the characterization of an apparatus without reference to the internal description of other devices. It is a trustworthy certification method, free of assumption on the underlying Hilbert space dimension and on calibration methods. We show how it can be used to quantify the quality of a Bell-state measurement, whether deterministic, partial, or probabilistic. Our certification is noise resistant and opens the way towards the device-independent self-testing of Bell-state measurements in existing experiments.

Bell Correlations in a Many-Body System with Finite Statistics.

A recent experiment reported the first violation of a Bell correlation witness in a many-body system [Science 352, 441 (2016)]. Following discussions in this Letter, we address here the question of the statistics required to witness Bell correlated states, i.e., states violating a Bell inequality, in such experiments. We start by deriving multipartite Bell inequalities involving an arbitrary number of measurement settings, two outcomes per party and one- and two-body correlators only. Based on these inequalities, we then build up improved witnesses able to detect Bell correlated states in many-body systems using two collective measurements only. These witnesses can potentially detect Bell correlations in states with an arbitrarily low amount of spin squeezing. We then establish an upper bound on the statistics needed to convincingly conclude that a measured state is Bell correlated.

Generation of single photons with highly tunable wave shape from a cold atomic ensemble.

The generation of ultra-narrowband, pure and storable single photons with widely tunable wave shape is an enabling step toward hybrid quantum networks requiring interconnection of remote disparate quantum systems. It allows interaction of quantum light with several material systems, including photonic quantum memories, single trapped ions and opto-mechanical systems. Previous approaches have offered a limited tuning range of the photon duration of at most one order of magnitude. Here we report on a heralded single photon source with controllable emission time based on a cold atomic ensemble, which can generate photons with temporal durations varying over three orders of magnitude up to 10 µs without a significant change of the readout efficiency. We prove the nonclassicality of the emitted photons, show that they are emitted in a pure state, and demonstrate that ultra-long photons with nonstandard wave shape can be generated, which are ideally suited for several quantum information tasks.

Demonstration of Light-Matter Micro-Macro Quantum Correlations.

Quantum mechanics predicts microscopic phenomena with undeniable success. Nevertheless, current theoretical and experimental efforts still do not yield conclusive evidence that there is or is not a fundamental limitation on the possibility to observe quantum phenomena at the macroscopic scale. This question prompted several experimental efforts producing quantum superpositions of large quantum states in light or matter. We report on the observation of quantum correlations, revealed using an entanglement witness, between a single photon and an atomic ensemble of billions of ions frozen in a crystal. The matter part of the state involves the superposition of two macroscopically distinguishable solid-state components composed of several tens of atomic excitations. Assuming the insignificance of the time ordering our experiment indirectly shows light-matter micro-macro entanglement. Our approach leverages from quantum memory techniques and could be used in other systems to expand the size of quantum superpositions in matter.

Bell correlations in a Bose-Einstein condensate.

Characterizing many-body systems through the quantum correlations between their constituent particles is a major goal of quantum physics. Although entanglement is routinely observed in many systems, we report here the detection of stronger correlations--Bell correlations--between the spins of about 480 atoms in a Bose-Einstein condensate. We derive a Bell correlation witness from a many-particle Bell inequality involving only one- and two-body correlation functions. Our measurement on a spin-squeezed state exceeds the threshold for Bell correlations by 3.8 standard deviations. Our work shows that the strongest possible nonclassical correlations are experimentally accessible in many-body systems and that they can be revealed by collective measurements.

Heralded amplification of photonic qubits.

We demonstrate postselection free heralded qubit amplification for Time-Bin qubits and single photon states in an all-fibre, telecom-wavelength, scheme that highlights the simplicity, stability and potential for fully integrated photonic solutions. Exploiting high-efficiency superconducting detectors, the gain, fidelity and the performance of the amplifier are studied as a function of loss. We also demonstrate the first heralded single photon amplifier with independent sources. This provides a significant advance towards demonstrating device-independent quantum key distribution as well as fundamental tests of quantum mechanics over extended distances.

How difficult is it to prove the quantumness of macroscropic states?

General wisdom tells us that if two quantum states are "macroscopically distinguishable" then their superposition should be hard to observe. We make this intuition precise and general by quantifying the difficulty to observe the quantum nature of a superposition of two states that can be distinguished without microscopic accuracy. First, we quantify the distinguishability of any given pair of quantum states with measurement devices lacking microscopic accuracy, i.e., measurements suffering from limited resolution or limited sensitivity. Next, we quantify the required stability that has to be fulfilled by any measurement setup able to distinguish their superposition from a mere mixture. Finally, by establishing a relationship between the stability requirement and the "distinguishability with inaccurate measurements" of the two superposed states, we demonstrate that, indeed, the more distinguishable the states are, the more demanding the stability requirements.

Heralded single-phonon preparation, storage, and readout in cavity optomechanics.

We show how to use the radiation pressure optomechanical coupling between a mechanical oscillator and an optical cavity field to generate in a heralded way a single quantum of mechanical motion (a Fock state). Starting with the oscillator close to its ground state, a laser pumping the upper motional sideband produces correlated photon-phonon pairs via optomechanical parametric down-conversion. Subsequent detection of a single scattered Stokes photon projects the macroscopic oscillator into a single-phonon Fock state. The nonclassical nature of this mechanical state can be demonstrated by applying a readout laser on the lower sideband to map the phononic state to a photonic mode and performing an autocorrelation measurement. Our approach proves the relevance of cavity optomechanics as an enabling quantum technology.

Witnessing trustworthy single-photon entanglement with local homodyne measurements.

Single-photon entangled states, i.e., states describing two optical paths sharing a single photon, constitute the simplest form of entanglement. Yet they provide a valuable resource in quantum information science. Specifically, they lie at the heart of quantum networks, as they can be used for quantum teleportation, swapped, and purified with linear optics. The main drawback of such entanglement is the difficulty in measuring it. Here, we present and experimentally test an entanglement witness allowing one to say whether a given state is path entangled and also that entanglement lies in the subspace, where the optical paths are each filled with one photon at most, i.e., refers to single-photon entanglement. It uses local homodyning only and relies on no assumption about the Hilbert space dimension of the measured system. Our work provides a simple and trustworthy method for verifying the proper functioning of future quantum networks.

Faithful entanglement swapping based on sum-frequency generation.

We show that an entanglement swapping operation performed with spontaneous parametric down-conversion can be made faithful without postselection using sum-frequency generation. This invites us to revisit the sum-frequency process and from a proof-of-principle experiment, we demonstrate that it provides a realistic solution for nonlinear optics at the single-photon level. This opens the way to attractive alternatives to six-photon protocols based on linear optics used, e.g., for the heralded creation of maximally entangled pairs or for device-independent quantum key distribution.

Quantum storage of photonic entanglement in a crystal.

Entanglement is the fundamental characteristic of quantum physics-much experimental effort is devoted to harnessing it between various physical systems. In particular, entanglement between light and material systems is interesting owing to their anticipated respective roles as 'flying' and stationary qubits in quantum information technologies (such as quantum repeaters and quantum networks). Here we report the demonstration of entanglement between a photon at a telecommunication wavelength (1,338 nm) and a single collective atomic excitation stored in a crystal. One photon from an energy-time entangled pair is mapped onto the crystal and then released into a well-defined spatial mode after a predetermined storage time. The other (telecommunication wavelength) photon is sent directly through a 50-metre fibre link to an analyser. Successful storage of entanglement in the crystal is proved by a violation of the Clauser-Horne-Shimony-Holt inequality by almost three standard deviations (S = 2.64 ± 0.23). These results represent an important step towards quantum communication technologies based on solid-state devices. In particular, our resources pave the way for building multiplexed quantum repeaters for long-distance quantum networks.