Table of contents

Although quantum coherence is a basic trait of quantum mechanics, the presence of coherences in the quantum description of a certain phenomenon does not rule out the possibility to give an alternative description of the same phenomenon in purely classical terms. Here, we give definite criteria to determine when and to what extent quantum coherence is equivalent to non-classicality. We prove that a Markovian multi-time statistics obtained from repeated measurements of a non-degenerate observable cannot be traced back to a classical statistics if and only if the dynamics generates coherences and subsequently turns them into populations. Furthermore, we show with simple examples that such connection between quantum coherence and non-classicality is generally absent if the statistics is non-Markovian.

We present a collection of methods to simulate entangled dynamics of open quantum systems governed by the Lindblad master equation with tensor network methods. Tensor network methods using matrix product states (MPSs) have been proven very useful to simulate many-body quantum systems and have driven many innovations in research. Since the MPS design is tailored for closed one-dimensional systems governed by the Schrödinger equation, the next step for many-body quantum dynamics is the simulation of one-dimensional open quantum systems. We review the three dominant approaches to the simulation of open quantum systems via the Lindblad master equation: quantum trajectories, matrix product density operators (MPDOs), and locally purified tensor networks. Selected examples guide possible applications of the methods and serve moreover as a benchmark between the techniques. These examples include the finite temperature states of the transverse quantum Ising model, the dynamics of an exciton traveling under the influence of spontaneous emission and dephasing, and a double-well potential simulated with the Bose–Hubbard model including dephasing. We analyze which approach is favorable leading to the conclusion that a complete set of all three methods is most beneficial, pushing the limits of different scenarios. The convergence studies using analytical results for macroscopic variables and exact diagonalization methods as comparison, show, for example, that MPDOs are favorable for the exciton problem in our study. All three methods access the same library, i.e., the software package Open Source Matrix Product States, allowing us to have a meaningful comparison between the different approaches based on the selected examples. For example, tensor operations are accessed from the same subroutines and with the same optimization eliminating one possible bias in a comparison of such numerical methods.

Variational autoencoders (VAEs) are powerful generative models with the salient ability to perform inference. Here, we introduce a quantum variational autoencoder (QVAE): a VAE whose latent generative process is implemented as a quantum Boltzmann machine (QBM). We show that our model can be trained end-to-end by maximizing a well-defined loss-function: a 'quantum' lower-bound to a variational approximation of the log-likelihood. We use quantum Monte Carlo (QMC) simulations to train and evaluate the performance of QVAEs. To achieve the best performance, we first create a VAE platform with discrete latent space generated by a restricted Boltzmann machine. Our model achieves state-of-the-art performance on the MNIST dataset when compared against similar approaches that only involve discrete variables in the generative process. We consider QVAEs with a smaller number of latent units to be able to perform QMC simulations, which are computationally expensive. We show that QVAEs can be trained effectively in regimes where quantum effects are relevant despite training via the quantum bound. Our findings open the way to the use of quantum computers to train QVAEs to achieve competitive performance for generative models. Placing a QBM in the latent space of a VAE leverages the full potential of current and next-generation quantum computers as sampling devices.

We employ metastable ultracold ¹⁷³Yb atoms to study dynamics in the 1D dissipative Fermi–Hubbard model experimentally and theoretically, and observe a complete inhibition of two-body losses after initial fast transient dynamics. We attribute the suppression of particle loss to the dynamical generation of a highly entangled Dicke state. For several lattice depths and for two- and six-spin component mixtures we find very similar dynamics, showing that the creation of strongly correlated states is a robust and universal phenomenon. This offers interesting opportunities for precision measurements.

The investigation of macroscopic quantum phenomena is a current active area of research that offers significant promise to advance the forefronts of both fundamental and applied quantum science. Utilising the exquisite precision and control of quantum optics provides a powerful toolset for generating such quantum states where the types and 'size' of the states that can be generated are set by the experimental parameter regime available and the resourcefulness of the protocol applied. In this work we present a new multistep scheme to 'grow' macroscopic superposition states of motion of a mechanical oscillator via cavity quantum optomechanics. The scheme consists of a series of optical pulses interacting with a mechanical mode via radiation-pressure followed by photon-counting measurements. The multistep nature of our protocol allows macroscopic superposition states to be prepared with a relaxed requirement for the single-photon optomechanical coupling strength. To illustrate the experimental feasibility of our proposal, we quantify how initial mechanical thermal occupation and mechanical decoherence affects the non-classicality and macroscopicity of the states generated and show that our scheme is resilient to optical loss. The advantages of this protocol provide a promising path to grow non-classical mechanical quantum states to a macroscopic scale under realistic experimental conditions.

Trapped-ion systems are among the most promising hardware candidates for large scale quantum computing and quantum simulation. In order to scale up such devices, it is necessary to engineer extreme-high vacuum (XHV) environments to prevent background gas from disrupting the ion crystal. Here we present a new cryogenic ion trapping system designed for long time storage of large ion chains. Our apparatus is based on a segmented-blade ion trap enclosed in a 4 K cryostat, which enables us to routinely trap and hold over 100 ¹⁷¹Yb⁺ ions for hours in a linear configuration, due to low background gas pressure from differential cryo-pumping. We characterize the XHV cryogenic environment measuring pressures below ${10}^{-11}\,{\rm{Torr}}$ by recording both inelastic and elastic collisions between the ion chain and the molecular background gas. We also demonstrate coherent one and two-qubit operations and nearly equidistant ion spacing for chains of up to 44 ions using anharmonic axial potentials, in order to enable better detection and single ion addressing in large ion arrays. We anticipate that this reliable production and lifetime enhancement of large linear ion chains will enable quantum simulation of models that are intractable with classical computer modeling.

Photonic materials are an emerging platform to explore quantum matter (Carusotto and Ciuti 2013 Rev. Mod. Phys. 85 299; Sommer et al 2015 arXiv:1506.00341) and quantum dynamics (Peyronel et al 2012 Nature488 57). The development of Rydberg electromagnetically induced transparency (Weatherill et al 2008 J. Phys. B: At. Mol. Opt. Phys.41 201002; Petrosyan et al 2011 Phys. Rev. Lett.107 213601) provided a clear route to strong interactions between individual optical photons. In conjunction with carefully designed optical resonators, it is now possible to achieve extraordinary control of the properties of individual photons, introducing tunable gauge fields (Schine et al 2016 Nature534 671) whilst imbuing the photons with mass and embedding them on curved spatial manifolds (Sommer et al 2016 New J. Phys.18 035008). Building on work formalizing Rydberg mediated interactions between propagating photons (Gorshkov et al 2011 Phys. Rev. Lett.107 133602; Gullans et al 2016 Phys. Rev. Lett.117 113601), we develop a theory of interacting Rydberg polaritons in multimode optical resonators, where the strong interactions are married with tunable single particle properties to build and probe exotic matter. In the presence of strong coupling between the resonator field and a Rydberg-dressed atomic ensemble, a quasiparticle called the 'cavity Rydberg polariton' emerges. We investigate its properties, finding that it inherits both the fast dynamics of its photonic constituents and the strong interactions of its atomic constituents. We develop tools to properly renormalize the interactions when polaritons approach each other, and investigate the impact of atomic motion on the coherence of multi-mode polaritons, showing that most channels for atom-polariton cross-thermalization are strongly suppressed. Finally, we propose to harness the repeated diffraction and refocusing of the optical resonator to realize interactions which are local in momentum-space. This work points the way to efficient modeling of polaritonic quantum materials in properly renormalized strongly interacting effective theories, thereby enabling experimental studies of photonic fractional quantum Hall fluids and crystals (Sommer et al 2015 arXiv:1506.00341; Umucalılar et al 2014 Phys. Rev. A 89 023803; Grusdt and Fleischhauer 2013 Phys. Rev. A 87 043628), plus photonic quantum information processors and repeaters (Jaksch et al 2000 Phys. Rev. Lett.85 2208; Saffman et al 2010 Rev. Mod. Phys.82 2313; Han et al 2010 Phys. Rev. A 81, 052311).

The semi-classical discrete truncated Wigner approximation has recently been proposed as a simulation method for spin-1/2 systems. While it appears to provide a powerful approach which shows promising results in higher dimensions and for systems with long-range interactions, its performance is still not well understood in general. Here we perform a systematic benchmarking on the one-dimensional transverse-field Ising model and point to limitations of the approximation arising after sudden quenches into the quantum critical regime. Our procedure allows to identify the limitations of the semi-classical simulations and with that to determine the regimes and questions where quantum simulators can provide information which is inaccessible to semi-classics.

The quantum autoencoder is a recent paradigm in the field of quantum machine learning, which may enable an enhanced use of resources in quantum technologies. To this end, quantum neural networks with less nodes in the inner than in the outer layers were considered. Here, we propose a useful connection between quantum autoencoders and quantum adders, which approximately add two unknown quantum states supported in different quantum systems. Specifically, this link allows us to employ optimized approximate quantum adders, obtained with genetic algorithms, for the implementation of quantum autoencoders for a variety of initial states. Furthermore, we can also directly optimize the quantum autoencoders via genetic algorithms. Our approach opens a different path for the design of quantum autoencoders in controllable quantum platforms.

The variational quantum eigensolver (VQE) algorithm combines the ability of quantum computers to efficiently compute expectation values with a classical optimization routine in order to approximate ground state energies of quantum systems. In this paper, we study the application of VQE to the simulation of molecular energies using the unitary coupled cluster (UCC) ansatz. We introduce new strategies to reduce the circuit depth for the implementation of UCC and improve the optimization of the wavefunction based on efficient classical approximations of the cluster amplitudes. Additionally, we propose an analytical method to compute the energy gradient that reduces the sampling cost for gradient estimation by several orders of magnitude compared to numerical gradients. We illustrate our methodology with numerical simulations for a system of four hydrogen atoms that exhibit strong correlation and show that the circuit depth of VQE using a UCC ansatz can be reduced without introducing significant loss of accuracy in the final wavefunctions and energies.

Topological insulators are exotic material that possess conducting surface states protected by the topology of the system. They can be classified in terms of their properties under discrete symmetries and are characterized by topological invariants. The latter has been measured experimentally for several models in one, two and three dimensions in both condensed matter and quantum simulation platforms. The recent progress in quantum simulation opens the road to the simulation of higher dimensional Hamiltonians and in particular of the 4D quantum Hall effect. These systems are characterized by the second Chern number, a topological invariant that appears in the quantization of the transverse conductivity for the non-linear response to both external magnetic and electric fields. This quantity cannot always be computed analytically and there is therefore a need of an algorithm to compute it numerically. In this work, we propose an efficient algorithm to compute the second Chern number in 4D systems. We construct the algorithm with the help of lattice gauge theory and discuss the convergence to the continuous gauge theory. We benchmark the algorithm on several relevant models, including the 4D Dirac Hamiltonian and the 4D quantum Hall effect and verify numerically its rapid convergence.

Polar molecules offer a new platform for quantum simulation of systems with long-range interactions, based on the electrostatic interaction between their electric dipole moments. Here, we report the development of coherent quantum state control using microwave fields in ⁴⁰Ca¹⁹F and ⁸⁷Rb¹³³Cs molecules, a crucial ingredient for many quantum simulation applications. We perform Ramsey interferometry measurements with fringe spacings of ∼1 kHz and investigate the dephasing time of a superposition of N = 0 and N = 1 rotational states when the molecules are confined. For both molecules, we show that a judicious choice of molecular hyperfine states minimises the impact of spatially varying transition-frequency shifts across the trap. For magnetically trapped ⁴⁰Ca¹⁹F we use a magnetically insensitive transition and observe a coherence time of 0.61(3) ms. For optically trapped ⁸⁷Rb¹³³Cs we exploit an avoided crossing in the AC Stark shifts and observe a maximum coherence time of 0.75(6) ms.

Optomechanical devices operated at their quantum limit open novel perspectives for the ultrasensitive determination of mass and displacement, and also in the broader field of quantum technologies. The access to higher frequencies implies operation at higher temperatures and stronger immunity to environmental noise. We propose and demonstrate here a new concept of quantum well photoelastic comb for the efficient electrostrictive coupling of light to optomechanical resonances at hundreds of GHz in semiconductor hybrid resonators. A purposely designed ultra-high resolution Raman spectroscopy set-up is exploited to evidence the transfer of spectral weight from the mode at 60 GHz to modes at 190–230 GHz, corresponding to the 8th and 10th overtone of the fundamental breathing mode of the light-sound cavities. The coupling to mechanical frequencies two orders of magnitude larger than alternative approaches is attained without reduction of the optomechanical constant g₀. The wavelength dependence of the optomechanical coupling further proves the role of resonant photoelastic interaction, highlighting the potentiality to access strong-coupling regimes. The experimental results show that electrostrictive forces allow for the design of devices optimized to selectively couple to specific mechanical modes. Our proposal opens up exciting opportunities towards the implementation of novel approaches applicable in quantum and ultra-high frequency information technologies.

We introduce a simulator of a quantum internet with the specific goal to support software development. A quantum internet consists of local quantum processors, which are interconnected by quantum communication channels that enable the transmission of qubits between the different processors. While many simulators exist for local quantum processors, there is presently no simulator for a quantum internet tailored towards software development. Quantum internet protocols require both classical as well as quantum information to be exchanged between the network nodes, next to the execution of gates and measurements on a local quantum processor. This requires quantum internet software to integrate classical communication programming practises with novel quantum ones. SimulaQron is built to enable application development and explore software engineering practises for a quantum internet. SimulaQron can be run on one or more classical computers to simulate local quantum processors, which are transparently connected in the background to enable the transmission of qubits or the generation of entanglement between remote processors. Application software can access the simulated local quantum processors to execute local quantum instructions and measurements, but also to transmit qubits to remote nodes in the network. SimulaQron features a modular design that performs a distributed simulation based on any existing simulation of a quantum computer capable of integrating with Python. Programming libraries for Python and C are provided to facilitate application development.

We study the problem of $\mathrm{CNOT}$-optimal quantum circuit synthesis over gate sets consisting of $\mathrm{CNOT}$ and Z-basis rotations of arbitrary angles. We show that the circuit-polynomial correspondence relates such circuits to Fourier expansions of pseudo-Boolean functions, and that for certain classes of functions this expansion uniquely determines the minimum $\mathrm{CNOT}$ cost of an implementation. As a corollary we prove that $\mathrm{CNOT}$ minimization over $\mathrm{CNOT}$ and phase gates is at least as hard as synthesizing a $\mathrm{CNOT}$-optimal circuit computing a set of parities of its inputs. We then show that this problem is NP-complete for two restricted cases where all $\mathrm{CNOT}$ gates are required to have the same target, and where the circuit inputs are encoded in a larger state space. The latter case has applications to $\mathrm{CNOT}$ optimization over more general Clifford+T circuits. We further present an efficient heuristic algorithm for synthesizing circuits over $\mathrm{CNOT}$ and Z-basis rotations with small $\mathrm{CNOT}$ cost. Our experiments show a 23% reduction of $\mathrm{CNOT}$ gates on average across a suite of Clifford+T benchmark circuits, with a maximum reduction of 43%.

The fundamental process limiting the coherence of quantum-dot based single-photon sources is the interaction with phonons. We study theoretically the effect of phonon decoherence on the indistinguishability of single photons emitted from a quantum dot embedded in a suspended nanobeam waveguide. At low temperatures, the indistinguishability is limited by the coupling between the quantum dot and the fundamental vibrational modes of the waveguide and is sensitive to the quantum-dot position within the nanobeam cross-section. We show that this decoherence channel can be efficiently suppressed by clamping the waveguide with a low refractive index cladding material deposited on the waveguide. With only a few microns of cladding material, the coherence of the emitted single photons is drastically improved. We show that the degree of indistinguishability can reach near unity and become independent of the quantum-dot position. We finally show that the cladding material may serve dual purposes since it can also be applied as a means to efficiently outcouple single photons from the nanophotonic waveguide into an optical fiber. Our proposal paves the way for a highly efficient fiber-coupled source of indistinguishable single photons based on a planar nanophotonic platform.

Before executing a quantum algorithm, one must first decompose the algorithm into machine-level instructions compatible with the architecture of the quantum computer, a process known as quantum compiling. There are many different quantum circuit decompositions for the same algorithm but it is desirable to compile leaner circuits. A fundamentally important cost metric is the T count—the number of T gates in a circuit. For the single qubit case, optimal compiling is essentially a solved problem. However, multi-qubit compiling is a harder problem with optimal algorithms requiring classical runtime exponential in the number of qubits. Here, we present and compare several efficient quantum compilers for multi-qubit Clifford + T circuits. We implemented our compilers in C++ and benchmarked them on random circuits, from which we determine that our TODD compiler yields the lowest T counts on average. We also benchmarked TODD on a library of reversible logic circuits that appear in quantum algorithms and found that it reduced the T count for 97% of the circuits with an average T-count saving of 20% when compared against the best of all previous circuit decompositions.

Quantum error correction (QEC) and fault-tolerant (FT) mechanisms are essential for reliable quantum computing. However, QEC considerably increases the computation size up to four orders of magnitude. Moreover, FT implementation has specific requirements on qubit layouts, causing both resource and time overhead. Reducing spatial-temporal costs becomes critical since it is beneficial to decrease the failure rate of quantum computation. To this purpose, scalable qubit plane architectures and efficient mapping passes including placement and routing of qubits as well as scheduling of operations are needed. This paper proposes a full mapping process to execute lattice surgery-based quantum circuits on two surface code architectures, namely a checkerboard and a tile-based one. We show that the checkerboard architecture is 2x qubit-efficient but the tile-based one requires lower communication overhead in terms of both operation overhead (up to ∼86%) and latency overhead (up to ∼79%).

We analyze the quantum information processing capability of a superconducting transmon circuit used to mediate interactions between quantum information stored in a collection of phononic crystal cavity resonators. Having only a single processing element to be controlled externally makes this approach significantly less hardware-intensive than traditional architectures with individual control of each qubit. Moreover, when compared with the commonly considered alternative approach using coplanar waveguide or 3d cavity microwave resonators for storage, the nanomechanical resonators offer both very long lifetime and small size—two conflicting requirements for microwave resonators. A detailed gate error analysis leads to an optimal value for the qubit-resonator coupling rate as a function of the number of mechanical resonators in the system. For a given set of system parameters, a specific amount of coupling and number of resonators is found to optimize the quantum volume, an approximate measure for the computational capacity of a system. We see this volume is higher in the proposed hybrid nanomechanical architecture than in the competing on-chip electromagnetic approach.

A hybrid quantum register consisting of nuclear spins in a solid-state platform coupled to a central electron spin is expected to combine the advantages of its elements. However, the potential to exploit long nuclear spin coherence times is severely limited by magnetic noise from the central electron spin during external interrogation. We overcome this obstacle and present protocols for addressing a decoherence-free nuclear spin subspace which was not accessible by previously existing methods. We demonstrate the efficacy of our protocols using detailed numerical simulations of a nitrogen-vacancy centre with nearby ¹³C nuclei, and show that the resulting hybrid quantum register is immune to electron spin noise and external magnetic field drifts. Our work takes an important step toward realising robust quantum registers that can be easily manipulated, entangled, and, at the same time, well isolated from external noise, with applications from quantum information processing and communication to quantum sensing.

In bosonic quantum metrology, the estimate of a loss parameter is typically performed by means of pure states, such as coherent, squeezed or entangled states, while mixed thermal probes are discarded for their inferior performance. Here we show that thermal sources with suitable correlations can be engineered in such a way to approach, or even surpass, the error scaling of coherent states in the presence of general Gaussian decoherence. This decoherence is modeled by including bosonic loss, thermal noise, and even the possibility of environmental correlations, so as to simulate non-Markovian dynamics. Our findings pave the way for practical quantum metrology with thermal sources in optical instruments (e.g., photometers) or at different wavelengths (e.g., far infrared, microwave or x-ray) where the generation of quantum features, such as coherence, squeezing or entanglement, may be extremely challenging.

With experimental quantum computing technologies now in their infancy, the search for efficient means of testing the correctness of these quantum computations is becoming more pressing. An approach to the verification of quantum computation within the framework of interactive proofs has been fruitful for addressing this problem. Specifically, an untrusted agent (prover) alleging to perform quantum computations can have his claims verified by another agent (verifier) who only has access to classical computation and a small quantum device for preparing or measuring single qubits. However, when this quantum device is prone to errors, verification becomes challenging and often existing protocols address this by adding extra assumptions, such as requiring the noise in the device to be uncorrelated with the noise on the prover's devices. In this paper, we present a simple protocol for verifying quantum computations, in the presence of noisy devices, with no extra assumptions. This protocol is based on post hoc techniques for verification, which allow for the prover to know the desired quantum computation and its input. We also perform a simulation of the protocol, for a one-qubit computation, and find the error thresholds when using the qubit repetition code as well as the Steane code.

Coherent control of large entangled graph states enables a wide variety of quantum information processing tasks, including error-corrected quantum computation. The linear optical approach offers excellent control and coherence, but today most photon sources and entangling gates—required for the construction of large graph states—are probabilistic and rely on postselection. In this work, we provide proofs and heuristics to aid experimental design using postselection. We introduce a versatile design rule for postselectable experiments: drawn as a graph, with qubit as vertices and gates and photon-pair sources as edges, an experiment may only contain cycles with an odd number of sources. We analyse experiments that use photons from postselected photon-pair sources, and lower bound the number of accessible classes of graph state entanglement in the non-degenerate case—graph state entanglement classes that contain a tree are are always accessible. The proportion of graph states accessible by postselection shrinks rapidly, however. We list accessible classes for various resource states up to 9 qubits. Finally, we apply these methods to near-term multi-photon experiments.

We report results of a ground-state entanglement protocol for a pair of Cs atoms separated by 6 μm, combining the Rydberg blockade mechanism with a two-photon Raman transition to prepare the $| {{\rm{\Psi }}}^{+}\rangle =(| 10\rangle +| 01\rangle )/\sqrt{2}$ Bell state with a loss-corrected fidelity of 0.81(5), equal to the best demonstrated fidelity for atoms trapped in optical tweezers but without the requirement for dynamically adjustable interatomic spacing. Qubit state coherence is also critical for quantum information applications, and we characterise both ground-state and ground-Rydberg dephasing rates using Ramsey spectroscopy. We demonstrate transverse dephasing times ${T}_{2}^{* }=10(1)$ ms and ${T}_{2}^{\prime} =0.14(1)$ s for the qubit levels and achieve long ground-Rydberg coherence times of ${T}_{2}^{* }=17(2)\,\mu {\rm{s}}$ as required for implementing high-fidelity multi-qubit gate sequences where a control atom remains in the Rydberg state while applying local operations on neighbouring target qubits.

Satellite-based quantum communication (QC) is an invaluable resource for the realization of a quantum network at the global scale. In this regard, the use of satellites well beyond the low Earth orbit gives the advantage of long communication time with a ground station. However, high-orbit satellites pose a great technological challenge due to the high diffraction losses of the optical channel, and the experimental investigation of such quantum channels is still lacking. Here, we report on the first experimental exchange of single photons from a global navigation satellite system (GNSS) at a slant distance of 20 000 km, by exploiting the retroreflector array mounted on GLONASS satellites. We also observed the predicted temporal spread of the reflected pulses due to the geometrical shape of the array. Finally, we estimated the requirements needed for an active source on a GNSS satellite, aiming towards QC from GNSS with state-of-the-art technology.

The negative-AND (NAND) gate is universal for classical computation making it an important target for development. A seminal quantum computing algorithm by Farhi, Goldstone and Gutmann has demonstrated its realization by means of quantum scattering yielding a quantum algorithm that evaluates the output faster than any classical algorithm. Here, we derive the NAND outputs analytically from scattering theory using a tight-binding (TB) model and show the restrictions on the TB parameters in order to still maintain the NAND gate function. We map the quantum NAND tree onto a conjugated molecular system, and compare the NAND output with non-equilibrium Green's function transport calculations using density functional theory and TB Hamiltonians for the electronic structure. Further, we extend our molecular platform to show other classical gates that can be realized for quantum computing by scattering on graphs.

We report a proof-of-principle experimental demonstration of the quantum speed-up for learning agents utilizing a small-scale quantum information processor based on radiofrequency-driven trapped ions. The decision-making process of a quantum learning agent within the projective simulation paradigm for machine learning is implemented in a system of two qubits. The latter are realized using hyperfine states of two frequency-addressed atomic ions exposed to a static magnetic field gradient. We show that the deliberation time of this quantum learning agent is quadratically improved with respect to comparable classical learning agents. The performance of this quantum-enhanced learning agent highlights the potential of scalable quantum processors taking advantage of machine learning.