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The intriguing many-body phases of quantum matter arise from the interplay of particle interactions, spatial symmetries, and external fields. Generating these phases in an engineered system could provide deeper insight into their nature. Using superconducting qubits, we simultaneously realize synthetic magnetic fields and strong particle interactions, which are among the essential elements for studying quantum magnetism and fractional quantum Hall phenomena. The artificial magnetic fields are synthesized by sinusoidally modulating the qubit couplings. In a closed loop formed by the three qubits, we observe the directional circulation of photons, a signature of broken time-reversal symmetry. We demonstrate strong interactions through the creation of photon vacancies, or ‘holes’, which circulate in the opposite direction. The combination of these key elements results in chiral ground-state currents. Our work introduces an experimental platform for engineering quantum phases of strongly interacting photons. Superconducting circuits, coupled to form a ring in which a photonic excitation can circulate between sites, are established as a versatile platform for studying the interplay of strong particle interactions and external fields.

The realization of a quantum kicked top provides evidence for ergodic dynamics and thermalization in a small quantum system consisting of three superconducting qubits. Statistical mechanics is founded on the assumption that all accessible configurations of a system are equally likely. This requires dynamics that explore all states over time, known as ergodic dynamics. In isolated quantum systems, however, the occurrence of ergodic behaviour has remained an outstanding question 1 , 2 , 3 , 4 . Here, we demonstrate ergodic dynamics in a small quantum system consisting of only three superconducting qubits. The qubits undergo a sequence of rotations and interactions and we measure the evolution of the density matrix. Maps of the entanglement entropy show that the full system can act like a reservoir for individual qubits, increasing their entropy through entanglement. Surprisingly, these maps bear a strong resemblance to the phase space dynamics in the classical limit; classically, chaotic motion coincides with higher entanglement entropy. We further show that in regions of high entropy the full multi-qubit system undergoes ergodic dynamics. Our work illustrates how controllable quantum systems can investigate fundamental questions in non-equilibrium thermodynamics.

Superconducting quantum circuits are used to directly observe and characterize topological phase transitions; this approach promises to be a powerful and general platform for characterizing topological phenomena in quantum systems. Lab demonstrations of the topological Haldane model The quantum Hall effect leads to topologically protected edge states, and for a long time was thought to exclusively emerge in the presence of an external magnetic field. But in 1988, Duncan Haldane proposed a model in which this exotic electronics structure arises without this requirement. He proposed that, in a honeycomb lattice with a staggered flux, the necessary ingredients for a quantum Hall effect would be inherent in the material itself. The principles behind this concept were later recruited to design topological insulators, but in its original expression, the Haldane model has not been observed in the laboratory. In this issue of Nature , two groups report on progress connected to the Haldane model. Gregor Jotzu et al . report the first realization of the Haldane model and Pedram Roushan et al . show how it can be precisely measured. Jotzu et al . use ultracold fermions to realize the breaking of time-reversal and inversion symmetry — the two main requirements of the model — by implementing a circular modulation of the lattice position and an energy offset between neighbouring sites. Roushan et al . use superconducting quantum circuits — a Josephson junction sandwiched between superconducting electrodes — to realize a non-interacting form of the Haldane model with a single qubit and an interacting two-qubit model through a new experimental setup called 'gmon' coupling architecture. Their setup allows them to characterize both cases by measuring the Berry curvature, a feature that all topological structures have in common. Topology, with its abstract mathematical constructs, often manifests itself in physics and has a pivotal role in our understanding of natural phenomena. Notably, the discovery of topological phases in condensed-matter systems has changed the modern conception of phases of matter 1 , 2 , 3 , 4 , 5 . The global nature of topological ordering, however, makes direct experimental probing an outstanding challenge. Present experimental tools are mainly indirect and, as a result, are inadequate for studying the topology of physical systems at a fundamental level. Here we employ the exquisite control afforded by state-of-the-art superconducting quantum circuits to investigate topological properties of various quantum systems. The essence of our approach is to infer geometric curvature by measuring the deflection of quantum trajectories in the curved space of the Hamiltonian 6 . Topological properties are then revealed by integrating the curvature over closed surfaces, a quantum analogue of the Gauss–Bonnet theorem. We benchmark our technique by investigating basic topological concepts of the historically important Haldane model 7 after mapping the momentum space of this condensed-matter model to the parameter space of a single-qubit Hamiltonian. In addition to constructing the topological phase diagram, we are able to visualize the microscopic spin texture of the associated states and their evolution across a topological phase transition. Going beyond non-interacting systems, we demonstrate the power of our method by studying topology in an interacting quantum system. This required a new qubit architecture 8 , 9 that allows for simultaneous control over every term in a two-qubit Hamiltonian. By exploring the parameter space of this Hamiltonian, we discover the emergence of an interaction-induced topological phase. Our work establishes a powerful, generalizable experimental platform to study topological phenomena in quantum systems.

Quantized eigenenergies and their associated wave functions provide extensive information for predicting the physics of quantum many-body systems. Using a chain of nine superconducting qubits, we implement a technique for resolving the energy levels of interacting photons. We benchmark this method by capturing the main features of the intricate energy spectrum predicted for two-dimensional electrons in a magnetic field—the Hofstadter butterfly. We introduce disorder to study the statistics of the energy levels of the system as it undergoes the transition from a thermalized to a localized phase. Our work introduces a many-body spectroscopy technique to study quantum phases of matter.

Quantum information scientists are getting closer to building a quantum computer that can perform calculations that a classical computer cannot. It has been estimated that such a computer would need around 50 qubits, but scaling up existing architectures to this number is tricky. Neill et al. explore how increasing the number of qubits from five to nine affects the quality of the output of their superconducting qubit device. If, as the number of qubits grows further, the error continues to increase at the same rate, a quantum computer with about 60 qubits and reasonable fidelity might be achievable with current technologies. Science , this issue p. 195 Scaling of errors and output with the number of qubits is explored in a five- to nine-qubit device. A key step toward demonstrating a quantum system that can address difficult problems in physics and chemistry will be performing a computation beyond the capabilities of any classical computer, thus achieving so-called quantum supremacy. In this study, we used nine superconducting qubits to demonstrate a promising path toward quantum supremacy. By individually tuning the qubit parameters, we were able to generate thousands of distinct Hamiltonian evolutions and probe the output probabilities. The measured probabilities obey a universal distribution, consistent with uniformly sampling the full Hilbert space. As the number of qubits increases, the system continues to explore the exponentially growing number of states. Extending these results to a system of 50 qubits has the potential to address scientific questions that are beyond the capabilities of any classical computer.

A universal set of logic gates in a superconducting quantum circuit is shown to have gate fidelities at the threshold for fault-tolerant quantum computing by the surface code approach, in which the quantum bits are distributed in an array of planar topology and have only nearest-neighbour couplings. Error-free quantum computing in prospect Quantum computers can only work in practice if, like conventional computers, they are fault-tolerant. This means that a system has to be in place to detect any errors and correct them. For quantum error correction such a system involves entangling several quantum bits (qubits) with each other. In the so-called surface code error-correction architecture, qubits are placed in a lattice and are entangled with four nearest neighbours. Rami Barends et al . report the construction of such a surface code system with five qubits in a row made from superconducting devices. This system performs with fidelity that is at the threshold for quantum error correction, suggesting that error-free quantum computing should be possible. The platform lends itself to scaling up to larger numbers of qubits and two-dimensional architecture. A quantum computer can solve hard problems, such as prime factoring 1 , 2 , database searching 3 , 4 and quantum simulation 5 , at the cost of needing to protect fragile quantum states from error. Quantum error correction 6 provides this protection by distributing a logical state among many physical quantum bits (qubits) by means of quantum entanglement. Superconductivity is a useful phenomenon in this regard, because it allows the construction of large quantum circuits and is compatible with microfabrication. For superconducting qubits, the surface code approach to quantum computing 7 is a natural choice for error correction, because it uses only nearest-neighbour coupling and rapidly cycled entangling gates. The gate fidelity requirements are modest: the per-step fidelity threshold is only about 99 per cent. Here we demonstrate a universal set of logic gates in a superconducting multi-qubit processor, achieving an average single-qubit gate fidelity of 99.92 per cent and a two-qubit gate fidelity of up to 99.4 per cent. This places Josephson quantum computing at the fault-tolerance threshold for surface code error correction. Our quantum processor is a first step towards the surface code, using five qubits arranged in a linear array with nearest-neighbour coupling. As a further demonstration, we construct a five-qubit Greenberger–Horne–Zeilinger state 8 , 9 using the complete circuit and full set of gates. The results demonstrate that Josephson quantum computing is a high-fidelity technology, with a clear path to scaling up to large-scale, fault-tolerant quantum circuits.

A digitized approach to adiabatic quantum computing, combining the generality of the adiabatic algorithm with the universality of the digital method, is implemented using a superconducting circuit to find the ground states of arbitrary Hamiltonians. A demonstration of quantum computing Adiabatic quantum computers are analogue machines that, with the help of quantum tunnelling, slowly evolve from a simple input to the desired, more complicated output. Although adiabiatic quantum computers can be very fast at specific tasks, they are limited by noise and errors that cannot be corrected during the computation. In contrast, universal quantum computers are digital devices that use logic gates and allow for error correction. Here, Rami Barends et al . combine the advantages of adiabiatic and universal quantum computers by digitizing an adiabiatic quantum computation. This approach allows for encoding non-stoquastic Hamiltonians, which are crucial for simulating physical and chemical systems with interacting fermions. Quantum mechanics can help to solve complex problems in physics 1 and chemistry 2 , provided they can be programmed in a physical device. In adiabatic quantum computing 3 , 4 , 5 , a system is slowly evolved from the ground state of a simple initial Hamiltonian to a final Hamiltonian that encodes a computational problem. The appeal of this approach lies in the combination of simplicity and generality; in principle, any problem can be encoded. In practice, applications are restricted by limited connectivity, available interactions and noise. A complementary approach is digital quantum computing 6 , which enables the construction of arbitrary interactions and is compatible with error correction 7 , 8 , but uses quantum circuit algorithms that are problem-specific. Here we combine the advantages of both approaches by implementing digitized adiabatic quantum computing in a superconducting system. We tomographically probe the system during the digitized evolution and explore the scaling of errors with system size. We then let the full system find the solution to random instances of the one-dimensional Ising problem as well as problem Hamiltonians that involve more complex interactions. This digital quantum simulation 9 , 10 , 11 , 12 of the adiabatic algorithm consists of up to nine qubits and up to 1,000 quantum logic gates. The demonstration of digitized adiabatic quantum computing in the solid state opens a path to synthesizing long-range correlations and solving complex computational problems. When combined with fault-tolerance, our approach becomes a general-purpose algorithm that is scalable.

A quantum error correction scheme is demonstrated in a system of superconducting qubits, and repeated quantum non-demolition measurements are used to track errors and reduce the failure rate; increasing the system size from five to nine qubits improves the failure rate further. A milestone in quantum error correction Quantum states are fragile and easily destroyed, which presents a major obstacle in quantum computing. Quantum error correction can mitigate this problem by identifying and correcting environmentally induced quantum errors. Here, the authors demonstrate aspects of a quantum error correction scheme in a system of superconducting qubits. Using repeated quantum non-demolition measurements, they track bit-flip errors and manage to reduce the failure rate. Increasing the system size from five to nine qubits improves the failure rate further, and the authors demonstrate that an increase in code complexity enhances the fidelity. Although many more developments are still necessary for quantum error correction schemes to be applicable in a sizeable quantum computer, this work is an important step in this direction. Quantum computing becomes viable when a quantum state can be protected from environment-induced error. If quantum bits (qubits) are sufficiently reliable, errors are sparse and quantum error correction (QEC) 1 , 2 , 3 , 4 , 5 , 6 is capable of identifying and correcting them. Adding more qubits improves the preservation of states by guaranteeing that increasingly larger clusters of errors will not cause logical failure—a key requirement for large-scale systems. Using QEC to extend the qubit lifetime remains one of the outstanding experimental challenges in quantum computing. Here we report the protection of classical states from environmental bit-flip errors and demonstrate the suppression of these errors with increasing system size. We use a linear array of nine qubits, which is a natural step towards the two-dimensional surface code QEC scheme 7 , and track errors as they occur by repeatedly performing projective quantum non-demolition parity measurements. Relative to a single physical qubit, we reduce the failure rate in retrieving an input state by a factor of 2.7 when using five of our nine qubits and by a factor of 8.5 when using all nine qubits after eight cycles. Additionally, we tomographically verify preservation of the non-classical Greenberger–Horne–Zeilinger state. The successful suppression of environment-induced errors will motivate further research into the many challenges associated with building a large-scale superconducting quantum computer.

We report the first electronic structure calculation performed on a quantum computer without exponentially costly precompilation. We use a programmable array of superconducting qubits to compute the energy surface of molecular hydrogen using two distinct quantum algorithms. First, we experimentally execute the unitary coupled cluster method using the variational quantum eigensolver. Our efficient implementation predicts the correct dissociation energy to within chemical accuracy of the numerically exact result. Second, we experimentally demonstrate the canonical quantum algorithm for chemistry, which consists of Trotterization and quantum phase estimation. We compare the experimental performance of these approaches to show clear evidence that the variational quantum eigensolver is robust to certain errors. This error tolerance inspires hope that variational quantum simulations of classically intractable molecules may be viable in the near future.

One of the key applications of quantum information is simulating nature. Fermions are ubiquitous in nature, appearing in condensed matter systems, chemistry and high energy physics. However, universally simulating their interactions is arguably one of the largest challenges, because of the difficulties arising from anticommutativity. Here we use digital methods to construct the required arbitrary interactions, and perform quantum simulation of up to four fermionic modes with a superconducting quantum circuit. We employ in excess of 300 quantum logic gates, and reach fidelities that are consistent with a simple model of uncorrelated errors. The presented approach is in principle scalable to a larger number of modes, and arbitrary spatial dimensions. Quantum simulation offers an unparalleled computational resource, but realizing it for fermionic systems is challenging due to their particle statistics. Here the authors report on the time evolutions of fermionic interactions implemented with digital techniques on a nine-qubit superconducting circuit.