Exploring large-scale entanglement in quantum simulation.

Entanglement is a distinguishing feature of quantum many-body systems, and uncovering the entanglement structure for large particle numbers in quantum simulation experiments is a fundamental challenge in quantum information science1. Here we perform experimental investigations of entanglement on the basis of the entanglement Hamiltonian (EH)2 as an effective description of the reduced density operator for large subsystems. We prepare ground and excited states of a one-dimensional XXZ Heisenberg chain on a 51-ion programmable quantum simulator3 and perform sample-efficient 'learning' of the EH for subsystems of up to 20 lattice sites4. Our experiments provide compelling evidence for a local structure of the EH. To our knowledge, this observation marks the first instance of confirming the fundamental predictions of quantum field theory by Bisognano and Wichmann5,6, adapted to lattice models that represent correlated quantum matter. The reduced state takes the form of a Gibbs ensemble, with a spatially varying temperature profile as a signature of entanglement2. Our results also show the transition from area- to volume-law scaling7 of von Neumann entanglement entropies from ground to excited states. As we venture towards achieving quantum advantage, we anticipate that our findings and methods have wide-ranging applicability to revealing and understanding entanglement in many-body problems with local interactions including higher spatial dimensions.

Practical quantum advantage in quantum simulation.

The development of quantum computing across several technologies and platforms has reached the point of having an advantage over classical computers for an artificial problem, a point known as 'quantum advantage'. As a next step along the development of this technology, it is now important to discuss 'practical quantum advantage', the point at which quantum devices will solve problems of practical interest that are not tractable for traditional supercomputers. Many of the most promising short-term applications of quantum computers fall under the umbrella of quantum simulation: modelling the quantum properties of microscopic particles that are directly relevant to modern materials science, high-energy physics and quantum chemistry. This would impact several important real-world applications, such as developing materials for batteries, industrial catalysis or nitrogen fixing. Much as aerodynamics can be studied either through simulations on a digital computer or in a wind tunnel, quantum simulation can be performed not only on future fault-tolerant digital quantum computers but also already today through special-purpose analogue quantum simulators. Here we overview the state of the art and future perspectives for quantum simulation, arguing that a first practical quantum advantage already exists in the case of specialized applications of analogue devices, and that fully digital devices open a full range of applications but require further development of fault-tolerant hardware. Hybrid digital-analogue devices that exist today already promise substantial flexibility in near-term applications.

Optimal metrology with programmable quantum sensors.

Quantum sensors are an established technology that has created new opportunities for precision sensing across the breadth of science. Using entanglement for quantum enhancement will allow us to construct the next generation of sensors that can approach the fundamental limits of precision allowed by quantum physics. However, determining how state-of-the-art sensing platforms may be used to converge to these ultimate limits is an outstanding challenge. Here we merge concepts from the field of quantum information processing with metrology, and successfully implement experimentally a programmable quantum sensor operating close to the fundamental limits imposed by the laws of quantum mechanics. We achieve this by using low-depth, parametrized quantum circuits implementing optimal input states and measurement operators for a sensing task on a trapped-ion experiment. With 26 ions, we approach the fundamental sensing limit up to a factor of 1.45 ± 0.01, outperforming conventional spin-squeezing with a factor of 1.87 ± 0.03. Our approach reduces the number of averages to reach a given Allan deviation by a factor of 1.59 ± 0.06 compared with traditional methods not using entanglement-enabled protocols. We further perform on-device quantum-classical feedback optimization to 'self-calibrate' the programmable quantum sensor with comparable performance. This ability illustrates that this next generation of quantum sensor can be used without previous knowledge of the device or its noise environment.

Analogue quantum chemistry simulation.

Computing the electronic structure of molecules with high precision is a central challenge in the field of quantum chemistry. Despite the success of approximate methods, tackling this problem exactly with conventional computers remains a formidable task. Several theoretical1,2 and experimental3-5 attempts have been made to use quantum computers to solve chemistry problems, with early proof-of-principle realizations done digitally. An appealing alternative to the digital approach is analogue quantum simulation, which does not require a scalable quantum computer and has already been successfully applied to solve condensed matter physics problems6-8. However, not all available or planned setups can be used for quantum chemistry problems, because it is not known how to engineer the required Coulomb interactions between them. Here we present an analogue approach to the simulation of quantum chemistry problems that relies on the careful combination of two technologies: ultracold atoms in optical lattices and cavity quantum electrodynamics. In the proposed simulator, fermionic atoms hopping in an optical potential play the role of electrons, additional optical potentials provide the nuclear attraction, and a single-spin excitation in a Mott insulator mediates the electronic Coulomb repulsion with the help of a cavity mode. We determine the operational conditions of the simulator and test it using a simple molecule. Our work opens up the possibility of efficiently computing the electronic structures of molecules with analogue quantum simulation.

Quantum Kibble-Zurek mechanism and critical dynamics on a programmable Rydberg simulator.

Quantum phase transitions (QPTs) involve transformations between different states of matter that are driven by quantum fluctuations1. These fluctuations play a dominant part in the quantum critical region surrounding the transition point, where the dynamics is governed by the universal properties associated with the QPT. Although time-dependent phenomena associated with classical, thermally driven phase transitions have been extensively studied in systems ranging from the early Universe to Bose-Einstein condensates2-5, understanding critical real-time dynamics in isolated, non-equilibrium quantum systems remains a challenge6. Here we use a Rydberg atom quantum simulator with programmable interactions to study the quantum critical dynamics associated with several distinct QPTs. By studying the growth of spatial correlations when crossing the QPT, we experimentally verify the quantum Kibble-Zurek mechanism (QKZM)7-9 for an Ising-type QPT, explore scaling universality and observe corrections beyond QKZM predictions. This approach is subsequently used to measure the critical exponents associated with chiral clock models10,11, providing new insights into exotic systems that were not previously understood and opening the door to precision studies of critical phenomena, simulations of lattice gauge theories12,13 and applications to quantum optimization14,15.

Essay: Quantum Sensing with Atomic, Molecular, and Optical Platforms for Fundamental Physics.

Atomic, molecular, and optical (AMO) physics has been at the forefront of the development of quantum science while laying the foundation for modern technology. With the growing capabilities of quantum control of many atoms for engineered many-body states and quantum entanglement, a key question emerges: what critical impact will the second quantum revolution with ubiquitous applications of entanglement bring to bear on fundamental physics? In this Essay, we argue that a compelling long-term vision for fundamental physics and novel applications is to harness the rapid development of quantum information science to define and advance the frontiers of measurement physics, with strong potential for fundamental discoveries. As quantum technologies, such as fault-tolerant quantum computing and entangled quantum sensor networks, become much more advanced than today's realization, we wonder what doors of basic science can these tools unlock. We anticipate that some of the most intriguing and challenging problems, such as quantum aspects of gravity, fundamental symmetries, or new physics beyond the minimal standard model, will be tackled at the emerging quantum measurement frontier. Part of a series of Essays which concisely present author visions for the future of their field.

Quantum and Classical Spin-Network Algorithms for q-Deformed Kogut-Susskind Gauge Theories.

Treating the infinite-dimensional Hilbert space of non-Abelian gauge theories is an outstanding challenge for classical and quantum simulations. Here, we employ q-deformed Kogut-Susskind lattice gauge theories, obtained by deforming the defining symmetry algebra to a quantum group. In contrast to other formulations, this approach simultaneously provides a controlled regularization of the infinite-dimensional local Hilbert space while preserving essential symmetry-related properties. This enables the development of both quantum as well as quantum-inspired classical spin-network algorithms for q-deformed gauge theories. To be explicit, we focus on SU(2)_{k} gauge theories with k∈N that are controlled by the deformation parameter q=e^{2πi/(k+2)}, a root of unity, and converge to the standard SU(2) Kogut-Susskind model as k→∞. In particular, we demonstrate that this formulation is well suited for efficient tensor network representations by variational ground-state simulations in 2D, providing first evidence that the continuum limit can be reached with k=O(10). Finally, we develop a scalable quantum algorithm for Trotterized real-time evolution by analytically diagonalizing the SU(2)_{k} plaquette interactions. Our work gives a new perspective for the application of tensor network methods to high-energy physics and paves the way for quantum simulations of non-Abelian gauge theories far from equilibrium where no other methods are currently available.

Fermionic Correlation Functions from Randomized Measurements in Programmable Atomic Quantum Devices.

We provide an efficient randomized measurement protocol to estimate two- and four-point fermionic correlations in ultracold atom experiments. Our approach is based on combining random atomic beam splitter operations, which can be realized with programmable optical landscapes, with high-resolution imaging systems such as quantum gas microscopes. We illustrate our results in the context of the variational quantum eigensolver algorithm for solving quantum chemistry problems.

Chiral quantum optics.

Advanced photonic nanostructures are currently revolutionizing the optics and photonics that underpin applications ranging from light technology to quantum-information processing. The strong light confinement in these structures can lock the local polarization of the light to its propagation direction, leading to propagation-direction-dependent emission, scattering and absorption of photons by quantum emitters. The possibility of such a propagation-direction-dependent, or chiral, light-matter interaction is not accounted for in standard quantum optics and its recent discovery brought about the research field of chiral quantum optics. The latter offers fundamentally new functionalities and applications: it enables the assembly of non-reciprocal single-photon devices that can be operated in a quantum superposition of two or more of their operational states and the realization of deterministic spin-photon interfaces. Moreover, engineered directional photonic reservoirs could lead to the development of complex quantum networks that, for example, could simulate novel classes of quantum many-body systems.

Hardware Efficient Quantum Simulation of Non-Abelian Gauge Theories with Qudits on Rydberg Platforms.

Non-Abelian gauge theories underlie our understanding of fundamental forces in nature, and developing tailored quantum hardware and algorithms to simulate them is an outstanding challenge in the rapidly evolving field of quantum simulation. Here we take an approach where gauge fields, discretized in spacetime, are represented by qudits and are time evolved in Trotter steps with multiqudit quantum gates. This maps naturally and hardware efficiently to an architecture based on Rydberg tweezer arrays, where long-lived internal atomic states represent qudits, and the required quantum gates are performed as holonomic operations supported by a Rydberg blockade mechanism. We illustrate our proposal for a minimal digitization of SU(2) gauge fields, demonstrating a significant reduction in circuit depth and gate errors in comparison to a traditional qubit-based approach, which puts simulations of non-Abelian gauge theories within reach of NISQ devices.

Real-time dynamics of lattice gauge theories with a few-qubit quantum computer.

Gauge theories are fundamental to our understanding of interactions between the elementary constituents of matter as mediated by gauge bosons. However, computing the real-time dynamics in gauge theories is a notorious challenge for classical computational methods. This has recently stimulated theoretical effort, using Feynman's idea of a quantum simulator, to devise schemes for simulating such theories on engineered quantum-mechanical devices, with the difficulty that gauge invariance and the associated local conservation laws (Gauss laws) need to be implemented. Here we report the experimental demonstration of a digital quantum simulation of a lattice gauge theory, by realizing (1 + 1)-dimensional quantum electrodynamics (the Schwinger model) on a few-qubit trapped-ion quantum computer. We are interested in the real-time evolution of the Schwinger mechanism, describing the instability of the bare vacuum due to quantum fluctuations, which manifests itself in the spontaneous creation of electron-positron pairs. To make efficient use of our quantum resources, we map the original problem to a spin model by eliminating the gauge fields in favour of exotic long-range interactions, which can be directly and efficiently implemented on an ion trap architecture. We explore the Schwinger mechanism of particle-antiparticle generation by monitoring the mass production and the vacuum persistence amplitude. Moreover, we track the real-time evolution of entanglement in the system, which illustrates how particle creation and entanglement generation are directly related. Our work represents a first step towards quantum simulation of high-energy theories using atomic physics experiments-the long-term intention is to extend this approach to real-time quantum simulations of non-Abelian lattice gauge theories.

Importance Sampling of Randomized Measurements for Probing Entanglement.

We show that combining randomized measurement protocols with importance sampling allows for characterizing entanglement in significantly larger quantum systems and in a more efficient way than in previous work. A drastic reduction of statistical errors is obtained using classical techniques of machine learning and tensor networks using partial information on the quantum state. In current experimental settings of engineered many-body quantum systems this significantly increases the (sub-)system sizes for which entanglement can be measured. In particular, we show an exponential reduction of the required number of measurements to estimate the purity of product states and GHZ states.

Quantum Variational Learning of the Entanglement Hamiltonian.

Learning the structure of the entanglement Hamiltonian (EH) is central to characterizing quantum many-body states in analog quantum simulation. We describe a protocol where spatial deformations of the many-body Hamiltonian, physically realized on the quantum device, serve as an efficient variational ansatz for a local EH. Optimal variational parameters are determined in a feedback loop, involving quench dynamics with the deformed Hamiltonian as a quantum processing step, and classical optimization. We simulate the protocol for the ground state of Fermi-Hubbard models in quasi-1D geometries, finding excellent agreement of the EH with Bisognano-Wichmann predictions. Subsequent on-device spectroscopy enables a direct measurement of the entanglement spectrum, which we illustrate for a Fermi Hubbard model in a topological phase.

Many-Body Chern Number from Statistical Correlations of Randomized Measurements.

One of the main topological invariants that characterizes several topologically ordered phases is the many-body Chern number (MBCN). Paradigmatic examples include several fractional quantum Hall phases, which are expected to be realized in different atomic and photonic quantum platforms in the near future. Experimental measurement and numerical computation of this invariant are conventionally based on the linear-response techniques that require having access to a family of states, as a function of an external parameter, which is not suitable for many quantum simulators. Here, we propose an ancilla-free experimental scheme for the measurement of this invariant, without requiring any knowledge of the Hamiltonian. Specifically, we use the statistical correlations of randomized measurements to infer the MBCN of a wave function. Remarkably, our results apply to disklike geometries that are more amenable to current quantum simulator architectures.

Quantum Information Scrambling in a Trapped-Ion Quantum Simulator with Tunable Range Interactions.

In ergodic many-body quantum systems, locally encoded quantum information becomes, in the course of time evolution, inaccessible to local measurements. This concept of "scrambling" is currently of intense research interest, entailing a deep understanding of many-body dynamics such as the processes of chaos and thermalization. Here, we present first experimental demonstrations of quantum information scrambling on a 10-qubit trapped-ion quantum simulator representing a tunable long-range interacting spin system, by estimating out-of-time ordered correlators (OTOCs) through randomized measurements. We also analyze the role of decoherence in our system by comparing our measurements to numerical simulations and by measuring Rényi entanglement entropies.

Mixed-State Entanglement from Local Randomized Measurements.

We propose a method for detecting bipartite entanglement in a many-body mixed state based on estimating moments of the partially transposed density matrix. The estimates are obtained by performing local random measurements on the state, followed by postprocessing using the classical shadows framework. Our method can be applied to any quantum system with single-qubit control. We provide a detailed analysis of the required number of experimental runs, and demonstrate the protocol using existing experimental data [Brydges et al., Science 364, 260 (2019)SCIEAS0036-807510.1126/science.aau4963].

Cross-Platform Verification of Intermediate Scale Quantum Devices.

We describe a protocol for cross-platform verification of quantum simulators and quantum computers. We show how to measure directly the overlap Tr[ρ_{1}ρ_{2}] and the purities Tr[ρ_{1,2}^{2}], and thus a fidelity of two, possibly mixed, quantum states ρ_{1} and ρ_{2} prepared in separate experimental platforms. We require only local measurements in randomized product bases, which are communicated classically. As a proof of principle, we present the measurement of experiment-theory fidelities for entangled 10-qubit quantum states in a trapped ion quantum simulator.

Universal photonic quantum computation via time-delayed feedback.

We propose and analyze a deterministic protocol to generate two-dimensional photonic cluster states using a single quantum emitter via time-delayed quantum feedback. As a physical implementation, we consider a single atom or atom-like system coupled to a 1D waveguide with a distant mirror, where guided photons represent the qubits, while the mirror allows the implementation of feedback. We identify the class of many-body quantum states that can be produced using this approach and characterize them in terms of 2D tensor network states.

Nondestructive Cooling of an Atomic Quantum Register via State-Insensitive Rydberg Interactions.

We propose a protocol for sympathetically cooling neutral atoms without destroying the quantum information stored in their internal states. This is achieved by designing state-insensitive Rydberg interactions between the data-carrying atoms and cold auxiliary atoms. The resulting interactions give rise to an effective phonon coupling, which leads to the transfer of heat from the data atoms to the auxiliary atoms, where the latter can be cooled by conventional methods. This can be used to extend the lifetime of quantum storage based on neutral atoms and can have applications for long quantum computations. The protocol can also be modified to realize state-insensitive interactions between the data and the auxiliary atoms but tunable and nontrivial interactions among the data atoms, allowing one to simultaneously cool and simulate a quantum spin model.

Variational Spin-Squeezing Algorithms on Programmable Quantum Sensors.

Arrays of atoms trapped in optical tweezers combine features of programmable analog quantum simulators with atomic quantum sensors. Here we propose variational quantum algorithms, tailored for tweezer arrays as programmable quantum sensors, capable of generating entangled states on demand for precision metrology. The scheme is designed to generate metrological enhancement by optimizing it in a feedback loop on the quantum device itself, thus preparing the best entangled states given the available quantum resources. We apply our ideas to the generation of spin-squeezed states on Sr atom tweezer arrays, where finite-range interactions are generated through Rydberg dressing. The complexity of experimental variational optimization of our quantum circuits is expected to scale favorably with system size. We numerically show our approach to be robust to noise, and surpassing known protocols.