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Electronic structure calculations of molecular systems are among the most promising applications for fault-tolerant quantum computing (FTQC) in quantum chemistry and drug design. However, while recent algorithmic advancements such as qubitization and Tensor Hypercontraction (THC) have significantly reduced the complexity of such calculations, they do not yet achieve computational runtimes short enough to be practical for industrially relevant use cases. In this work, we introduce several advances to electronic structure calculation for molecular systems, resulting in a two-orders-of-magnitude speedup of estimated runtimes over prior-art algorithms run on comparable quantum devices. One of these advances is a novel framework for block-invariant symmetry-shifted Tensor Hypercontraction (BLISS-THC), with which we achieve the tightest Hamiltonian factorizations reported to date. We compile our algorithm for an Active Volume (AV) architecture, a technical layout that has recently been proposed for fusion-based photonic quantum hardware. AV compilation contributes towards a lower runtime of our computation by eliminating overheads stemming from connectivity issues in the underlying surface code. We present a detailed benchmark of our approach, focusing primarily on the computationally challenging benchmark molecule P450. Leveraging a number of hardware tradeoffs in interleaving-based photonic FTQC, we estimate runtimes for the electronic structure calculation of P450 as a function of the device footprint.

Classical shadow tomography has become a powerful tool in learning about quantum states prepared on a quantum computer. Recent works have used classical shadows to variationally enforce N-representability conditions on the 2-particle reduced density matrix. In this paper, we build upon previous research by choice of an improved estimator within classical shadow tomography and rephrasing the optimisation constraints, resulting in an overall enhancement in performance under comparable measurement shot budgets. We further explore the specific regimes where these methods outperform the unbiased estimator of the standalone classical shadow protocol and quantify the potential savings in numerical studies.

The quantum phase estimation algorithm stands as the primary method for determining the ground state energy of a molecular electronic Hamiltonian on a quantum computer. In this context, the ability to initialize a classically tractable state that has a strong overlap with the desired ground state is critical as it directly affects the runtime of the algorithm. However, several numerical studies have shown that this overlap decays exponentially with system size. In this work, we demonstrate that this decay can be alleviated by optimizing the molecular orbital basis, for an initial state constructed from a single Slater determinant. We propose a practical method to achieve this optimization without knowledge of the true molecular ground state and test this method numerically. By comparing the resulting optimized orbitals to the natural orbitals, we find improved overlap. Specifically, for four iron-sulfur molecules, which are known to suffer from the mentioned decay, we show that our method yields one to two orders of magnitude improvement compared to localized molecular orbitals.

The computational cost of quantum algorithms for physics and chemistry is closely linked to the spectrum of the Hamiltonian, a property that manifests in the necessary rescaling of its eigenvalues. The typical approach of using the 1-norm as an upper bound to the spectral norm to rescale the Hamiltonian suits the most general case of bounded Hermitian operators but neglects the influence of symmetries commonly found in chemical systems. In this work, we introduce a hierarchy of symmetry-aware spectral bounds that provide a unified understanding of the performance of quantum phase estimation algorithms using block-encoded electronic structure Hamiltonians. We present a variational and numerically tractable method for computing these bounds, based on orbital optimization, to demonstrate that the computed bounds are smaller than conventional spectral bounds for a variety of molecular benchmark systems. We also highlight the unique analytical and numerical scaling behavior of these bounds in the thermodynamic and complete basis set limits. Our work shows that there is room for improvement in reducing the 1-norm, not yet achieved through methods like double factorization and tensor hypercontraction, but highlights potential challenges in improving the performance of current quantum algorithms beyond small constant factors through 1-norm reduction techniques alone., Comment: 11 pages, 3 figures, 1 table

Quantum phase estimation based on qubitization is the state-of-the-art fault-tolerant quantum algorithm for computing ground-state energies in chemical applications. In this context, the 1-norm of the Hamiltonian plays a fundamental role in determining the total number of required iterations and also the overall computational cost. In this work, we introduce the symmetry-compressed double factorization (SCDF) approach, which combines a compressed double factorization of the Hamiltonian with the symmetry shift technique, significantly reducing the 1-norm value. The effectiveness of this approach is demonstrated numerically by considering various benchmark systems, including the FeMoco molecule, cytochrome P450, and hydrogen chains of different sizes. To compare the efficiency of SCDF to other methods in absolute terms, we estimate Toffoli gate requirements, which dominate the execution time on fault-tolerant quantum computers. For the systems considered here, SCDF leads to a sizeable reduction of the Toffoli gate count in comparison to other variants of double factorization or even tensor hypercontraction, which is usually regarded as the most efficient approach for qubitization.

We provide a method for estimating the expectation value of an operator that can utilize prior knowledge to accelerate the learning process on a quantum computer. Specifically, suppose we have an operator that can be expressed as a concise sum of projectors whose expectation values we know a priori to be $O(\epsilon)$. In that case, we can estimate the expectation value of the entire operator within error $\epsilon$ using a number of quantum operations that scales as $O(1/\sqrt{\epsilon})$. We then show how this can be used to reduce the cost of learning a potential energy surface in quantum chemistry applications by exploiting information gained from the energy at nearby points. Furthermore, we show, using Newton-Cotes methods, how these ideas can be exploited to learn the energy via integration of derivatives that we can estimate using a priori knowledge. This allows us to reduce the cost of energy estimation if the block-encodings of directional derivative operators have a smaller normalization constant than the Hamiltonian of the system., Comment: 23 pages, v2: additional explanations to clarify the assumptions and results

We present the first hardware implementation of electrostatic interaction energies using a trapped-ion quantum computer. As test system for our computation, we focus on the reduction of $\mathrm{NO}$ to $\mathrm{N}_2\mathrm{O}$ catalyzed by a nitric oxide reductase (NOR). The quantum computer is used to generate an approximate ground state within the NOR active space. To efficiently measure the necessary one-particle density matrices, we incorporate fermionic basis rotations into the quantum circuit without extending the circuit length, laying the groundwork for further efficient measurement routines using factorizations. Measurements in the computational basis are then used as inputs for computing the electrostatic interaction energies on a classical computer. Our experimental results strongly agree with classical noise-less simulations of the same circuits, finding electrostatic interaction energies within chemical accuracy despite hardware noise. This work shows that algorithms tailored to specific observables of interest, such as interaction energies, may require significantly fewer quantum resources than individual ground state energies would in the straightforward supermolecular approach.

We present a super-polynomial improvement in the precision scaling of quantum simulations for coupled classical-quantum systems in this paper. Such systems are found, for example, in molecular dynamics simulations within the Born-Oppenheimer approximation. By employing a framework based on the Koopman-von Neumann formalism, we express the Liouville equation of motion as unitary dynamics and utilize phase kickback from a dynamical quantum simulation to calculate the quantum forces acting on classical particles. This approach allows us to simulate the dynamics of these particles without the overheads associated with measuring gradients and solving the equations of motion on a classical computer, resulting in a super-polynomial advantage at the price of increased space complexity. We demonstrate that these simulations can be performed in both microcanonical and canonical ensembles, enabling the estimation of thermodynamic properties from the prepared probability density., Comment: 19 + 51 pages

The efficient computation of observables beyond the total energy is a key challenge and opportunity for fault-tolerant quantum computing approaches in quantum chemistry. Here we consider the symmetry-adapted perturbation theory (SAPT) components of the interaction energy as a prototypical example of such an observable. We provide a guide for calculating this observable on a fault-tolerant quantum computer while optimizing the required computational resources. Specifically, we present a quantum algorithm that estimates interaction energies at the first-order SAPT level with a Heisenberg-limited scaling. To this end, we exploit a high-order tensor factorization and block encoding technique that efficiently represents each SAPT observable. To quantify the computational cost of our methodology, we provide resource estimates in terms of the required number of logical qubits and Toffoli gates to execute our algorithm for a range of benchmark molecules, also taking into account the cost of the eigenstate preparation and the cost of block encoding the SAPT observables. Finally, we perform the resource estimation for a heme and artemisinin complex as a representative large-scale system encountered in drug design, highlighting our algorithm's performance in this new benchmark study and discussing possible bottlenecks that may be improved in future work., Comment: 55 pages, 18 figures, 4 tables

Over the past three decades significant reductions have been made to the cost of estimating ground-state energies of molecular Hamiltonians with quantum computers. However, comparatively little attention has been paid to estimating the expectation values of other observables with respect to said ground states, which is important for many industrial applications. In this work we present a novel expectation value estimation (EVE) quantum algorithm which can be applied to estimate the expectation values of arbitrary observables with respect to any of the system's eigenstates. In particular, we consider two variants of EVE: std-EVE, based on standard quantum phase estimation, and QSP-EVE, which utilizes quantum signal processing (QSP) techniques. We provide rigorous error analysis for both both variants and minimize the number of individual phase factors for QSPEVE. These error analyses enable us to produce constant-factor quantum resource estimates for both std-EVE and QSP-EVE across a variety of molecular systems and observables. For the systems considered, we show that QSP-EVE reduces (Toffoli) gate counts by up to three orders of magnitude and reduces qubit width by up to 25% compared to std-EVE. While estimated resource counts remain far too high for the first generations of fault-tolerant quantum computers, our estimates mark a first of their kind for both the application of expectation value estimation and modern QSP-based techniques., Comment: 36 pages, 9 figures

Quantum computers promise to impact industrial applications, for which quantum chemical calculations are required, by virtue of their high accuracy. This perspective explores the challenges and opportunities of applying quantum computers to drug design, discusses where they could transform industrial research and elaborates on what is needed to reach this goal.

The future development of quantum technologies relies on creating and manipulating quantum systems of increasing complexity, with key applications in computation, simulation and sensing. This poses severe challenges in the efficient control, calibration and validation of quantum states and their dynamics. Although the full simulation of large-scale quantum systems may only be possible on a quantum computer, classical characterization and optimization methods still play an important role. Here, we review different approaches that use classical post-processing techniques, possibly combined with adaptive optimization, to learn quantum systems, their correlation properties, dynamics and interaction with the environment. We discuss theoretical proposals and successful implementations across different multiple-qubit architectures such as spin qubits, trapped ions, photonic and atomic systems, and superconducting circuits. This Review provides a brief background of key concepts recurring across many of these approaches with special emphasis on the Bayesian formalism and neural networks., Comment: Review. 20 pages, 4 figures and 2 boxes (reformatted)

The computation of interaction energies on noisy intermediate-scale quantum (NISQ) computers appears to be challenging with straightforward application of existing quantum algorithms. For example, use of the standard supermolecular method with the variational quantum eigensolver (VQE) would require extremely precise resolution of the total energies of the fragments to provide for accurate subtraction to the interaction energy. Here we present a symmetry-adapted perturbation theory (SAPT) method that may provide interaction energies with high quantum resource efficiency. Of particular note, we present a quantum extended random-phase approximation (ERPA) treatment of the SAPT second-order induction and dispersion terms, including exchange counterparts. Together with previous work on first-order terms, this provides a recipe for complete SAPT(VQE) interaction energies up to second order. The SAPT interaction energy terms are computed as first-level observables with no subtraction of monomer energies invoked, and the only quantum observations needed are the the VQE one- and two-particle density matrices. We find empirically that SAPT(VQE) can provide accurate interaction energies even with coarsely optimized, low circuit depth wavefunctions from the quantum computer, simulated through ideal statevectors. The errors on the total interaction energy are orders of magnitude lower than the corresponding VQE total energy errors of the monomer wavefunctions., Comment: 38 pages, 6 primary figures

Accurate models of real quantum systems are important for investigating their behaviour, yet are difficult to distill empirically. Here, we report an algorithm -- the Quantum Model Learning Agent (QMLA) -- to reverse engineer Hamiltonian descriptions of a target system. We test the performance of QMLA on a number of simulated experiments, demonstrating several mechanisms for the design of candidate Hamiltonian models and simultaneously entertaining numerous hypotheses about the nature of the physical interactions governing the system under study. QMLA is shown to identify the true model in the majority of instances, when provided with limited a priori information, and control of the experimental setup. Our protocol can explore Ising, Heisenberg and Hubbard families of models in parallel, reliably identifying the family which best describes the system dynamics. We demonstrate QMLA operating on large model spaces by incorporating a genetic algorithm to formulate new hypothetical models. The selection of models whose features propagate to the next generation is based upon an objective function inspired by the Elo rating scheme, typically used to rate competitors in games such as chess and football. In all instances, our protocol finds models that exhibit $F_1$-score $\geq 0.88$ when compared with the true model, and it precisely identifies the true model in 72% of cases, whilst exploring a space of over $250,000$ potential models. By testing which interactions actually occur in the target system, QMLA is a viable tool for both the exploration of fundamental physics and the characterisation and calibration of quantum devices., Comment: 29 pages, 7 figures

While most work on the quantum simulation of chemistry has focused on computing energy surfaces, a similarly important application requiring subtly different algorithms is the computation of energy derivatives. Almost all molecular properties can be expressed an energy derivative, including molecular forces, which are essential for applications such as molecular dynamics simulations. Here, we introduce new quantum algorithms for computing molecular energy derivatives with significantly lower complexity than prior methods. Under cost models appropriate for noisy-intermediate scale quantum devices we demonstrate how low rank factorizations and other tomography schemes can be optimized for energy derivative calculations. We perform numerics revealing that our techniques reduce the number of circuit repetitions required by many orders of magnitude for even modest systems. In the context of fault-tolerant algorithms, we develop new methods of estimating energy derivatives with Heisenberg limited scaling incorporating state-of-the-art techniques for block encoding fermionic operators. Our results suggest that the calculation of forces on a single nucleus may be of similar cost to estimating energies of chemical systems, but that further developments are needed for quantum computers to meaningfully assist with molecular dynamics simulations., Comment: 48 pages, 14 page appendix, 10 figures. v2 contains updated lambdas (rescaling factors) for sparse FT encodings and some NISQ methods, obtained by localizing orbitals

We explore the use of symmetry-adapted perturbation theory (SAPT) as a simple and efficient means to compute interaction energies between large molecular systems with a hybrid method combing NISQ-era quantum and classical computers. From the one- and two-particle reduced density matrices of the monomer wavefunctions obtained by the variational quantum eigensolver (VQE), we compute SAPT contributions to the interaction energy [SAPT(VQE)]. At first order, this energy yields the electrostatic and exchange contributions for non-covalently bound systems. We empirically find from ideal statevector simulations that the SAPT(VQE) interaction energy components display orders of magnitude lower absolute errors than the corresponding VQE total energies. Therefore, even with coarsely optimized low-depth VQE wavefunctions, we still obtain sub kcal/mol accuracy in the SAPT interaction energies. In SAPT(VQE), the quantum requirements, such as qubit count and circuit depth, are lowered by performing computations on the separate molecular systems. Furthermore, active spaces allow for large systems containing thousands of orbitals to be reduced to a small enough orbital set to perform the quantum portions of the computations. We benchmark SAPT(VQE) (with the VQE component simulated by ideal state-vector simulators) against a handful of small multi-reference dimer systems and the iron center containing human cancer-relevant protein lysine-specific demethylase 5 (KDM5A)., Comment: 16 pages, 4 figures, plus supplemental data

Photons are natural carriers of high-dimensional quantum information, and, in principle, can benefit from higher quantum information capacity and noise-resilience. However, schemes to generate the resources required for high-dimensional quantum computing have so far been lacking in linear optics. Here, we show how to generate GHZ states in arbitrary dimensions and numbers of photons using linear optical circuits described by Fourier transform matrices. Combining our results with recent schemes for qudit Bell measurements, we show that universal linear optical quantum computing can be performed in arbitrary dimensions.

An isolated system of interacting quantum particles is described by a Hamiltonian operator. Hamiltonian models underpin the study and analysis of physical and chemical processes throughout science and industry, so it is crucial they are faithful to the system they represent. However, formulating and testing Hamiltonian models of quantum systems from experimental data is difficult because it is impossible to directly observe which interactions the quantum system is subject to. Here, we propose and demonstrate an approach to retrieving a Hamiltonian model from experiments, using unsupervised machine learning. We test our methods experimentally on an electron spin in a nitrogen-vacancy interacting with its spin bath environment, and numerically, finding success rates up to 86%. By building agents capable of learning science, which recover meaningful representations, we can gain further insight on the physics of quantum systems., Comment: 27 pages, 8 figures

Exploiting semiconductor fabrication techniques, natural carriers of quantum information such as atoms, electrons, and photons can be embedded in scalable integrated devices. Integrated optics provides a versatile platform for large-scale quantum information processing and transceiving with photons. Scaling up the integrated devices for quantum applications requires highperformance single-photon generation and photonic qubit-qubit entangling operations. However, previous demonstrations report major challenges in producing multiple bright, pure and identical single-photons, and entangling multiple photonic qubits with high fidelity. Another notable challenge is to noiselessly interface multiphoton sources and multiqubit operators in a single device. Here we demonstrate on-chip genuine multipartite entanglement and quantum teleportation in silicon, by coherently controlling an integrated network of microresonator nonlinear single-photon sources and linear-optic multiqubit entangling circuits. The microresonators are engineered to locally enhance the nonlinearity, producing multiple frequencyuncorrelated and indistinguishable single-photons, without requiring any spectral filtering. The multiqubit states are processed in a programmable linear circuit facilitating Bell-projection and fusion operation in a measurement-based manner. We benchmark key functionalities, such as intra-/inter-chip teleportation of quantum states, and generation of four-photon Greenberger-HorneZeilinger entangled states. The production, control, and transceiving of states are all achieved in micrometer-scale silicon chips, fabricated by complementary metal-oxide-semiconductor processes. Our work lays the groundwork for scalable on-chip multiphoton technologies for quantum computing and communication.

In quantum communications, quantum states are employed for the transmission of information between remote parties. This usually requires sharing knowledge of the measurement bases through a classical public channel in the sifting phase of the protocol. Here, we demonstrate a quantum communication scheme where the information on the bases is shared "non-classically", by encoding this information in the same photons used for carrying the data. This enhanced capability is achieved by exploiting the localization of the photonic wave function, observed when the photons are prepared and measured in the same quantum basis. We experimentally implement our scheme by using a multi-mode optical fiber coupled to an adaptive optics setup. We observe a decrease in the error rate for higher dimensionality, indicating an improved resilience against noise., Comment: 9 pages, 5 figures

Implementing large instances of quantum algorithms requires the processing of many quantum information carriers in a hardware platform that supports the integration of different components. While established semiconductor fabrication processes can integrate many photonic components, the generation and algorithmic processing of many photons has been a bottleneck in integrated photonics. Here we report the on-chip generation and processing of quantum states of light with up to eight photons in quantum sampling algorithms. Switching between different optical pumping regimes, we implement the Scattershot, Gaussian and standard boson sampling protocols in the same silicon chip, which integrates linear and nonlinear photonic circuitry. We use these results to benchmark a quantum algorithm for calculating molecular vibronic spectra. Our techniques can be readily scaled for the on-chip implementation of specialised quantum algorithms with tens of photons, pointing the way to efficiency advantages over conventional computers.

Future quantum computers require a scalable architecture on a scalable technology---one that supports millions of high-performance components. Measurement-based protocols, based on graph states, represent the state of the art in architectures for optical quantum computing. Silicon photonics offers enormous scale and proven quantum optical functionality. Here we report the first demonstration of photonic graph states on a mass-manufactured chip using four on-chip generated photons. We generate both star- and line-type graph states, implementing a basic measurement-based protocol, and measure heralded interference of the chip's four photons. We develop a model of the device and bound the dominant sources of error using Bayesian inference. The two-photon barrier, which has constrained chip-scale quantum optics, is now broken; future increases in on-chip photon number now depend solely on reducing loss, and increasing rates. This experiment, combining silicon technology with a graph-based architecture, illuminates one path to a large-scale quantum future.

Integrated optics is an engineering solution proposed for exquisite control of photonic quantum information. Here we use silicon photonics and the linear combination of quantum operators scheme to realise a fully programmable two-qubit quantum processor. The device is fabricated with readily available CMOS based processing and comprises four nonlinear photon-sources, four filters, eighty-two beam splitters and fifty-eight individually addressable phase shifters. To demonstrate performance, we programmed the device to implement ninety-eight various two-qubit unitary operations (with average quantum process fidelity of 93.2$\pm$4.5%), a two-qubit quantum approximate optimization algorithm and efficient simulation of Szegedy directed quantum walks. This fosters further use of the linear combination architecture with silicon photonics for future photonic quantum processors., Comment: 23 pages, 13 figures

Nitrogen-vacancy (NV) centres in diamond are appealing nano-scale quantum sensors for temperature, strain, electric fields and, most notably, for magnetic fields. However, the cryogenic temperatures required for low-noise single-shot readout that have enabled the most sensitive NV-magnetometry reported to date, are impractical for key applications, e.g. biological sensing. Overcoming the noisy readout at room-temperature has until now demanded repeated collection of fluorescent photons, which increases the time-cost of the procedure thus reducing its sensitivity. Here we show how machine learning can process the noisy readout of a single NV centre at room-temperature, requiring on average only one photon per algorithm step, to sense magnetic field strength with a precision comparable to those reported for cryogenic experiments. Analysing large data sets from NV centres in bulk diamond, we report absolute sensitivities of $60$ nT s$^{1/2}$ including initialisation, readout, and computational overheads. We show that dephasing times can be simultaneously estimated, and that time-dependent fields can be dynamically tracked at room temperature. Our results dramatically increase the practicality of early-term single spin sensors., Comment: including supplementary informations

The ability to control multidimensional quantum systems is key for the investigation of fundamental science and for the development of advanced quantum technologies. Here we demonstrate a multidimensional integrated quantum photonic platform able to robustly generate, control and analyze high-dimensional entanglement. We realize a programmable bipartite entangled system with dimension up to $15 \times 15$ on a large-scale silicon-photonics quantum circuit. The device integrates more than 550 photonic components on a single chip, including 16 identical photon-pair sources. We verify the high precision, generality and controllability of our multidimensional technology, and further exploit these abilities to demonstrate key quantum applications experimentally unexplored before, such as quantum randomness expansion and self-testing on multidimensional states. Our work provides a prominent experimental platform for the development of multidimensional quantum technologies., Comment: Science, (2018)

Quantum phase estimation is a fundamental subroutine in many quantum algorithms, including Shor's factorization algorithm and quantum simulation. However, so far results have cast doubt on its practicability for near-term, non-fault tolerant, quantum devices. Here we report experimental results demonstrating that this intuition need not be true. We implement a recently proposed adaptive Bayesian approach to quantum phase estimation and use it to simulate molecular energies on a Silicon quantum photonic device. The approach is verified to be well suited for pre-threshold quantum processors by investigating its superior robustness to noise and decoherence compared to the iterative phase estimation algorithm. This shows a promising route to unlock the power of quantum phase estimation much sooner than previously believed.

Efficiently characterising quantum systems, verifying operations of quantum devices and validating underpinning physical models, are central challenges for the development of quantum technologies and for our continued understanding of foundational physics. Machine-learning enhanced by quantum simulators has been proposed as a route to improve the computational cost of performing these studies. Here we interface two different quantum systems through a classical channel - a silicon-photonics quantum simulator and an electron spin in a diamond nitrogen-vacancy centre - and use the former to learn the latter's Hamiltonian via Bayesian inference. We learn the salient Hamiltonian parameter with an uncertainty of approximately $10^{-5}$. Furthermore, an observed saturation in the learning algorithm suggests deficiencies in the underlying Hamiltonian model, which we exploit to further improve the model itself. We go on to implement an interactive version of the protocol and experimentally show its ability to characterise the operation of the quantum photonic device. This work demonstrates powerful new quantum-enhanced techniques for investigating foundational physical models and characterising quantum technologies.

Imperfections in integrated photonics manufacturing have a detrimental effect on the maximal achievable visibility in interferometric architectures. These limits have profound implications for further photonics technological developments and in particular for quantum photonics technologies. Active optimisation approaches, together with reconfigurable photonics, have been proposed as a solution to overcome this. In this paper, we demonstrate an ultra-high (>60 dB) extinction ratio in a silicon photonic device consisting of cascaded Mach-Zehnder interferometers, in which additional interferometers function as variable beamsplitters. The imperfections of fabricated beamsplitters are compensated using an automated progressive optimization algorithm with no requirement for pre-calibration. This work shows the possibility of integrating and accurately controlling linear-optical components for large-scale quantum information processing and other applications., Comment: 4 pages, 3 figures

Integrated photonics has enabled much progress towards quantum technologies. Many applications, including quantum communication, sensing, and distributed and cloud quantum computing, will require coherent photonic interconnection between separate chip-based sub-systems. Large-scale quantum computing systems and architectures may ultimately require quantum interconnects to enable scaling beyond the limits of a single wafer and towards "multi-chip" systems. However, coherently interconnecting separate chips is challenging due to the fragility of these quantum states and the demanding challenges of transmitting photons in at least two media within a single coherent system. Distribution and manipulation of qubit entanglement between multiple devices is one of the most stringent requirements of the interconnected system. Here, we report a quantum photonic interconnect demonstrating high-fidelity entanglement distribution and manipulation between two separate chips, implemented using state-of-the-art silicon photonics. Path-entangled states are generated and manipulated on-chip, and distributed between the chips by interconverting between path-encoding and polarisation-encoding. We use integrated state analysers to confirm a Bell-type violation of $S$=2.638+-0.039 between two chips. With improvements in loss, this quantum interconnect will provide new levels of flexible systems and architectures for quantum technologies., Comment: 12 pages, 10 figures

Photonic qubits constitute a leading platform to disruptive quantum technologies due to their unique low-noise properties. The cost of the photonic approach is the non-deterministic nature of many of the processes, including single-photon generation, which arises from parametric sources and negligible interaction between photons. Active temporal multiplexing - repeating a generation process in time and rerouting to single modes using an optical switching network - is a promising approach to overcome this challenge and will likely be essential for large-scale applications with greatly reduced resource complexity and system sizes. Requirements include the precise synchronization of a system of low-loss switches, delay lines, fast photon detectors, and feed-forward. Here we demonstrate temporal multiplexing of 8 'bins' from a double-passed heralded photon source and observe an increase in the heralding and heralded photon rates. This system points the way to harnessing temporal multiplexing in quantum technologies, from single-photon sources to large-scale computation., Comment: Minor revisions

Entanglement--one of the most delicate phenomena in nature--is an essential resource for quantum information applications. Large entangled cluster states have been predicted to enable universal quantum computation, with the required single- qubit measurements readily implemented with photons. Useful large-scale systems must generate and control qubit entanglement on-chip, where quantum information is naturally encoded in photon path. Here we report a silicon photonic chip which integrates resonant-enhanced sources, filters, and reconfigurable optics to generate a path-entangled two-qubit state--the smallest non-trivial cluster state--and analyse its entanglement. We show that ring-resonator-based spontaneous four-wave mixing sources can be made highly indistinguishable, despite their nonlinear dynamics, and the first evidence that their frequency correlations are small, as predicted. We use quantum state tomography, and the strict Bell-CHSH inequality to quantify entanglement in the device, confirming its high performance. This work integrates essential components for building devices and systems to harness quantum entanglement on the large scale.

From bird flocks to fish schools, animal groups often seem to react to environmental perturbations as if of one mind. Most studies in collective animal behaviour have aimed to understand how a globally ordered state may emerge from simple behavioural rules. Less effort has been devoted to understanding the origin of collective response, namely the way the group as a whole reacts to its environment. Yet collective response is the adaptive key to survivor, especially when strong predatory pressure is present. Here we argue that collective response in animal groups is achieved through scale-free behavioural correlations. By reconstructing the three-dimensional position and velocity of individual birds in large flocks of starlings, we measured to what extent the velocity fluctuations of different birds are correlated to each other. We found that the range of such spatial correlation does not have a constant value, but it scales with the linear size of the flock. This result indicates that behavioural correlations are scale-free: the change in the behavioural state of one animal affects and is affected by that of all other animals in the group, no matter how large the group is. Scale-free correlations extend maximally the effective perception range of the individuals, thus compensating for the short-range nature of the direct inter-individual interaction and enhancing global response to perturbations. Our results suggest that flocks behave as critical systems, poised to respond maximally to environmental perturbations., Comment: Submitted to PNAS