Real-time quantum error correction beyond break-even.

The ambition of harnessing the quantum for computation is at odds with the fundamental phenomenon of decoherence. The purpose of quantum error correction (QEC) is to counteract the natural tendency of a complex system to decohere. This cooperative process, which requires participation of multiple quantum and classical components, creates a special type of dissipation that removes the entropy caused by the errors faster than the rate at which these errors corrupt the stored quantum information. Previous experimental attempts to engineer such a process1-7 faced the generation of an excessive number of errors that overwhelmed the error-correcting capability of the process itself. Whether it is practically possible to utilize QEC for extending quantum coherence thus remains an open question. Here we answer it by demonstrating a fully stabilized and error-corrected logical qubit whose quantum coherence is substantially longer than that of all the imperfect quantum components involved in the QEC process, beating the best of them with a coherence gain of G = 2.27 ± 0.07. We achieve this performance by combining innovations in several domains including the fabrication of superconducting quantum circuits and model-free reinforcement learning.

Quantum error correction of a qubit encoded in grid states of an oscillator.

The accuracy of logical operations on quantum bits (qubits) must be improved for quantum computers to outperform classical ones in useful tasks. One method to achieve this is quantum error correction (QEC), which prevents noise in the underlying system from causing logical errors. This approach derives from the reasonable assumption that noise is local, that is, it does not act in a coordinated way on different parts of the physical system. Therefore, if a logical qubit is encoded non-locally, we can-for a limited time-detect and correct noise-induced evolution before it corrupts the encoded information1. In 2001, Gottesman, Kitaev and Preskill (GKP) proposed a hardware-efficient instance of such a non-local qubit: a superposition of position eigenstates that forms grid states of a single oscillator2. However, the implementation of measurements that reveal this noise-induced evolution of the oscillator while preserving the encoded information3-7 has proved to be experimentally challenging, and the only realization reported so far relied on post-selection8,9, which is incompatible with QEC. Here we experimentally prepare square and hexagonal GKP code states through a feedback protocol that incorporates non-destructive measurements that are implemented with a superconducting microwave cavity having the role of the oscillator. We demonstrate QEC of an encoded qubit with suppression of all logical errors, in quantitative agreement with a theoretical estimate based on the measured imperfections of the experiment. Our protocol is applicable to other continuous-variable systems and, in contrast to previous implementations of QEC10-14, can mitigate all logical errors generated by a wide variety of noise processes and facilitate fault-tolerant quantum computation.

Deterministic teleportation of a quantum gate between two logical qubits.

A quantum computer has the potential to efficiently solve problems that are intractable for classical computers. However, constructing a large-scale quantum processor is challenging because of the errors and noise that are inherent in real-world quantum systems. One approach to addressing this challenge is to utilize modularity-a strategy used frequently in nature and engineering to build complex systems robustly. Such an approach manages complexity and uncertainty by assembling small, specialized components into a larger architecture. These considerations have motivated the development of a quantum modular architecture, in which separate quantum systems are connected into a quantum network via communication channels1,2. In this architecture, an essential tool for universal quantum computation is the teleportation of an entangling quantum gate3-5, but such teleportation has hitherto not been realized as a deterministic operation. Here we experimentally demonstrate the teleportation of a controlled-NOT (CNOT) gate, which we make deterministic by using real-time adaptive control. In addition, we take a crucial step towards implementing robust, error-correctable modules by enacting the gate between two logical qubits, encoding quantum information redundantly in the states of superconducting cavities6. By using such an error-correctable encoding, our teleported gate achieves a process fidelity of 79 per cent. Teleported gates have implications for fault-tolerant quantum computation3, and when realized within a network can have broad applications in quantum communication, metrology and simulations1,2,7. Our results illustrate a compelling approach for implementing multi-qubit operations on logical qubits and, if integrated with quantum error-correction protocols, indicate a promising path towards fault-tolerant quantum computation using a modular architecture.

Tracking photon jumps with repeated quantum non-demolition parity measurements.

Quantum error correction is required for a practical quantum computer because of the fragile nature of quantum information. In quantum error correction, information is redundantly stored in a large quantum state space and one or more observables must be monitored to reveal the occurrence of an error, without disturbing the information encoded in an unknown quantum state. Such observables, typically multi-quantum-bit parities, must correspond to a special symmetry property inherent in the encoding scheme. Measurements of these observables, or error syndromes, must also be performed in a quantum non-demolition way (projecting without further perturbing the state) and more quickly than errors occur. Previously, quantum non-demolition measurements of quantum jumps between states of well-defined energy have been performed in systems such as trapped ions, electrons, cavity quantum electrodynamics, nitrogen-vacancy centres and superconducting quantum bits. So far, however, no fast and repeated monitoring of an error syndrome has been achieved. Here we track the quantum jumps of a possible error syndrome, namely the photon number parity of a microwave cavity, by mapping this property onto an ancilla quantum bit, whose only role is to facilitate quantum state manipulation and measurement. This quantity is just the error syndrome required in a recently proposed scheme for a hardware-efficient protected quantum memory using Schrödinger cat states (quantum superpositions of different coherent states of light) in a harmonic oscillator. We demonstrate the projective nature of this measurement onto a region of state space with well-defined parity by observing the collapse of a coherent state onto even or odd cat states. The measurement is fast compared with the cavity lifetime, has a high single-shot fidelity and has a 99.8 per cent probability per single measurement of leaving the parity unchanged. In combination with the deterministic encoding of quantum information in cat states realized earlier, the quantum non-demolition parity tracking that we demonstrate represents an important step towards implementing an active system that extends the lifetime of a quantum bit.

Gated Conditional Displacement Readout of Superconducting Qubits.

We have realized a new interaction between superconducting qubits and a readout cavity that results in the displacement of a coherent state in the cavity, conditioned on the state of the qubit. This conditional state, when it reaches the cavity-following, phase-sensitive amplifier, matches its measured observable, namely, the in phase quadrature. In a setup where several qubits are coupled to the same readout resonator, we show it is possible to measure the state of a target qubit with minimal dephasing of the other qubits. Our results suggest novel directions for faster readout of superconducting qubits and implementations of bosonic quantum error-correcting codes.

Autonomously stabilized entanglement between two superconducting quantum bits.

Quantum error correction codes are designed to protect an arbitrary state of a multi-qubit register from decoherence-induced errors, but their implementation is an outstanding challenge in the development of large-scale quantum computers. The first step is to stabilize a non-equilibrium state of a simple quantum system, such as a quantum bit (qubit) or a cavity mode, in the presence of decoherence. This has recently been accomplished using measurement-based feedback schemes. The next step is to prepare and stabilize a state of a composite system. Here we demonstrate the stabilization of an entangled Bell state of a quantum register of two superconducting qubits for an arbitrary time. Our result is achieved using an autonomous feedback scheme that combines continuous drives along with a specifically engineered coupling between the two-qubit register and a dissipative reservoir. Similar autonomous feedback techniques have been used for qubit reset, single-qubit state stabilization, and the creation and stabilization of states of multipartite quantum systems. Unlike conventional, measurement-based schemes, the autonomous approach uses engineered dissipation to counteract decoherence, obviating the need for a complicated external feedback loop to correct errors. Instead, the feedback loop is built into the Hamiltonian such that the steady state of the system in the presence of drives and dissipation is a Bell state, an essential building block for quantum information processing. Such autonomous schemes, which are broadly applicable to a variety of physical systems, as demonstrated by the accompanying paper on trapped ion qubits, will be an essential tool for the implementation of quantum error correction.

Hot Nonequilibrium Quasiparticles in Transmon Qubits.

Nonequilibrium quasiparticle excitations degrade the performance of a variety of superconducting circuits. Understanding the energy distribution of these quasiparticles will yield insight into their generation mechanisms, the limitations they impose on superconducting devices, and how to efficiently mitigate quasiparticle-induced qubit decoherence. To probe this energy distribution, we systematically correlate qubit relaxation and excitation with charge-parity switches in an offset-charge-sensitive transmon qubit, and find that quasiparticle-induced excitation events are the dominant mechanism behind the residual excited-state population in our samples. By itself, the observed quasiparticle distribution would limit T_{1} to ≈200 µs, which indicates that quasiparticle loss in our devices is on equal footing with all other loss mechanisms. Furthermore, the measured rate of quasiparticle-induced excitation events is greater than that of relaxation events, which signifies that the quasiparticles are more energetic than would be predicted from a thermal distribution describing their apparent density.

Deterministic Remote Entanglement of Superconducting Circuits through Microwave Two-Photon Transitions.

Large-scale quantum information processing networks will most probably require the entanglement of distant systems that do not interact directly. This can be done by performing entangling gates between standing information carriers, used as memories or local computational resources, and flying ones, acting as quantum buses. We report the deterministic entanglement of two remote transmon qubits by Raman stimulated emission and absorption of a traveling photon wave packet. We achieve a Bell state fidelity of 73%, well explained by losses in the transmission line and decoherence of each qubit.

Realization of three-qubit quantum error correction with superconducting circuits.

Quantum computers could be used to solve certain problems exponentially faster than classical computers, but are challenging to build because of their increased susceptibility to errors. However, it is possible to detect and correct errors without destroying coherence, by using quantum error correcting codes. The simplest of these are three-quantum-bit (three-qubit) codes, which map a one-qubit state to an entangled three-qubit state; they can correct any single phase-flip or bit-flip error on one of the three qubits, depending on the code used. Here we demonstrate such phase- and bit-flip error correcting codes in a superconducting circuit. We encode a quantum state, induce errors on the qubits and decode the error syndrome--a quantum state indicating which error has occurred--by reversing the encoding process. This syndrome is then used as the input to a three-qubit gate that corrects the primary qubit if it was flipped. As the code can recover from a single error on any qubit, the fidelity of this process should decrease only quadratically with error probability. We implement the correcting three-qubit gate (known as a conditional-conditional NOT, or Toffoli, gate) in 63 nanoseconds, using an interaction with the third excited state of a single qubit. We find 85 ± 1 per cent fidelity to the expected classical action of this gate, and 78 ± 1 per cent fidelity to the ideal quantum process matrix. Using this gate, we perform a single pass of both quantum bit- and phase-flip error correction and demonstrate the predicted first-order insensitivity to errors. Concatenation of these two codes in a nine-qubit device would correct arbitrary single-qubit errors. In combination with recent advances in superconducting qubit coherence times, this could lead to scalable quantum technology.

Continuous Quantum Nondemolition Measurement of the Transverse Component of a Qubit.

Quantum jumps of a qubit are usually observed between its energy eigenstates, also known as its longitudinal pseudospin component. Is it possible, instead, to observe quantum jumps between the transverse superpositions of these eigenstates? We answer positively by presenting the first continuous quantum nondemolition measurement of the transverse component of an individual qubit. In a circuit QED system irradiated by two pump tones, we engineer an effective Hamiltonian whose eigenstates are the transverse qubit states, and a dispersive measurement of the corresponding operator. Such transverse component measurements are a useful tool in the driven-dissipative operation engineering toolbox, which is central to quantum simulation and quantum error correction.

Preparation and measurement of three-qubit entanglement in a superconducting circuit.

Traditionally, quantum entanglement has been central to foundational discussions of quantum mechanics. The measurement of correlations between entangled particles can have results at odds with classical behaviour. These discrepancies grow exponentially with the number of entangled particles. With the ample experimental confirmation of quantum mechanical predictions, entanglement has evolved from a philosophical conundrum into a key resource for technologies such as quantum communication and computation. Although entanglement in superconducting circuits has been limited so far to two qubits, the extension of entanglement to three, eight and ten qubits has been achieved among spins, ions and photons, respectively. A key question for solid-state quantum information processing is whether an engineered system could display the multi-qubit entanglement necessary for quantum error correction, which starts with tripartite entanglement. Here, using a circuit quantum electrodynamics architecture, we demonstrate deterministic production of three-qubit Greenberger-Horne-Zeilinger (GHZ) states with fidelity of 88 per cent, measured with quantum state tomography. Several entanglement witnesses detect genuine three-qubit entanglement by violating biseparable bounds by 830 ± 80 per cent. We demonstrate the first step of basic quantum error correction, namely the encoding of a logical qubit into a manifold of GHZ-like states using a repetition code. The integration of this encoding with decoding and error-correcting steps in a feedback loop will be the next step for quantum computing with integrated circuits.

Phase-preserving amplification near the quantum limit with a Josephson ring modulator.

Recent progress in solid-state quantum information processing has stimulated the search for amplifiers and frequency converters with quantum-limited performance in the microwave range. Depending on the gain applied to the quadratures of a single spatial and temporal mode of the electromagnetic field, linear amplifiers can be classified into two categories (phase sensitive and phase preserving) with fundamentally different noise properties. Phase-sensitive amplifiers use squeezing to reduce the quantum noise, but are useful only in cases in which a reference phase is attached to the signal, such as in homodyne detection. A phase-preserving amplifier would be preferable in many applications, but such devices have not been available until now. Here we experimentally realize a proposal for an intrinsically phase-preserving, superconducting parametric amplifier of non-degenerate type. It is based on a Josephson ring modulator, which consists of four Josephson junctions in a Wheatstone bridge configuration. The device symmetry greatly enhances the purity of the amplification process and simplifies both its operation and its analysis. The measured characteristics of the amplifier in terms of gain and bandwidth are in good agreement with analytical predictions. Using a newly developed noise source, we show that the upper bound on the total system noise of our device under real operating conditions is three times the quantum limit. We foresee applications in the area of quantum analog signal processing, such as quantum non-demolition single-shot readout of qubits, quantum feedback and the production of entangled microwave signal pairs.

Single-Photon-Resolved Cross-Kerr Interaction for Autonomous Stabilization of Photon-Number States.

Quantum states can be stabilized in the presence of intrinsic and environmental losses by either applying an active feedback condition on an ancillary system or through reservoir engineering. Reservoir engineering maintains a desired quantum state through a combination of drives and designed entropy evacuation. We propose and implement a quantum-reservoir engineering protocol that stabilizes Fock states in a microwave cavity. This protocol is realized with a circuit quantum electrodynamics platform where a Josephson junction provides direct, nonlinear coupling between two superconducting waveguide cavities. The nonlinear coupling results in a single-photon-resolved cross-Kerr effect between the two cavities enabling a photon-number-dependent coupling to a lossy environment. The quantum state of the microwave cavity is discussed in terms of a net polarization and is analyzed by a measurement of its steady state Wigner function.

Demonstration of two-qubit algorithms with a superconducting quantum processor.

Quantum computers, which harness the superposition and entanglement of physical states, could outperform their classical counterparts in solving problems with technological impact-such as factoring large numbers and searching databases. A quantum processor executes algorithms by applying a programmable sequence of gates to an initialized register of qubits, which coherently evolves into a final state containing the result of the computation. Building a quantum processor is challenging because of the need to meet simultaneously requirements that are in conflict: state preparation, long coherence times, universal gate operations and qubit readout. Processors based on a few qubits have been demonstrated using nuclear magnetic resonance, cold ion trap and optical systems, but a solid-state realization has remained an outstanding challenge. Here we demonstrate a two-qubit superconducting processor and the implementation of the Grover search and Deutsch-Jozsa quantum algorithms. We use a two-qubit interaction, tunable in strength by two orders of magnitude on nanosecond timescales, which is mediated by a cavity bus in a circuit quantum electrodynamics architecture. This interaction allows the generation of highly entangled states with concurrence up to 94 per cent. Although this processor constitutes an important step in quantum computing with integrated circuits, continuing efforts to increase qubit coherence times, gate performance and register size will be required to fulfil the promise of a scalable technology.

Non-Poissonian quantum jumps of a fluxonium qubit due to quasiparticle excitations.

As the energy relaxation time of superconducting qubits steadily improves, nonequilibrium quasiparticle excitations above the superconducting gap emerge as an increasingly relevant limit for qubit coherence. We measure fluctuations in the number of quasiparticle excitations by continuously monitoring the spontaneous quantum jumps between the states of a fluxonium qubit, in conditions where relaxation is dominated by quasiparticle loss. Resolution on the scale of a single quasiparticle is obtained by performing quantum nondemolition projective measurements within a time interval much shorter than T1, using a quantum-limited amplifier (Josephson parametric converter). The quantum jump statistics switches between the expected Poisson distribution and a non-Poissonian one, indicating large relative fluctuations in the quasiparticle population, on time scales varying from seconds to hours. This dynamics can be modified controllably by injecting quasiparticles or by seeding quasiparticle-trapping vortices by cooling down in a magnetic field.

Demonstrating a driven reset protocol for a superconducting qubit.

Qubit reset is crucial at the start of and during quantum information algorithms. We present the experimental demonstration of a practical method to force qubits into their ground state, based on driving appropriate qubit and cavity transitions. Our protocol, called the double drive reset of population, is tested on a superconducting transmon qubit in a three-dimensional cavity. Using a new method for measuring population, we show that we can prepare the ground state with a fidelity of at least 99.5% in less than 3 µs; faster times and higher fidelity are predicted upon parameter optimization.

Coupling superconducting qubits via a cavity bus.

Superconducting circuits are promising candidates for constructing quantum bits (qubits) in a quantum computer; single-qubit operations are now routine, and several examples of two-qubit interactions and gates have been demonstrated. These experiments show that two nearby qubits can be readily coupled with local interactions. Performing gate operations between an arbitrary pair of distant qubits is highly desirable for any quantum computer architecture, but has not yet been demonstrated. An efficient way to achieve this goal is to couple the qubits to a 'quantum bus', which distributes quantum information among the qubits. Here we show the implementation of such a quantum bus, using microwave photons confined in a transmission line cavity, to couple two superconducting qubits on opposite sides of a chip. The interaction is mediated by the exchange of virtual rather than real photons, avoiding cavity-induced loss. Using fast control of the qubits to switch the coupling effectively on and off, we demonstrate coherent transfer of quantum states between the qubits. The cavity is also used to perform multiplexed control and measurement of the qubit states. This approach can be expanded to more than two qubits, and is an attractive architecture for quantum information processing on a chip.

Resolving photon number states in a superconducting circuit.

Electromagnetic signals are always composed of photons, although in the circuit domain those signals are carried as voltages and currents on wires, and the discreteness of the photon's energy is usually not evident. However, by coupling a superconducting quantum bit (qubit) to signals on a microwave transmission line, it is possible to construct an integrated circuit in which the presence or absence of even a single photon can have a dramatic effect. Such a system can be described by circuit quantum electrodynamics (QED)-the circuit equivalent of cavity QED, where photons interact with atoms or quantum dots. Previously, circuit QED devices were shown to reach the resonant strong coupling regime, where a single qubit could absorb and re-emit a single photon many times. Here we report a circuit QED experiment in the strong dispersive limit, a new regime where a single photon has a large effect on the qubit without ever being absorbed. The hallmark of this strong dispersive regime is that the qubit transition energy can be resolved into a separate spectral line for each photon number state of the microwave field. The strength of each line is a measure of the probability of finding the corresponding photon number in the cavity. This effect is used to distinguish between coherent and thermal fields, and could be used to create a photon statistics analyser. As no photons are absorbed by this process, it should be possible to generate non-classical states of light by measurement and perform qubit-photon conditional logic, the basis of a logic bus for a quantum computer.

Generating single microwave photons in a circuit.

Microwaves have widespread use in classical communication technologies, from long-distance broadcasts to short-distance signals within a computer chip. Like all forms of light, microwaves, even those guided by the wires of an integrated circuit, consist of discrete photons. To enable quantum communication between distant parts of a quantum computer, the signals must also be quantum, consisting of single photons, for example. However, conventional sources can generate only classical light, not single photons. One way to realize a single-photon source is to collect the fluorescence of a single atom. Early experiments measured the quantum nature of continuous radiation, and further advances allowed triggered sources of photons on demand. To allow efficient photon collection, emitters are typically placed inside optical or microwave cavities, but these sources are difficult to employ for quantum communication on wires within an integrated circuit. Here we demonstrate an on-chip, on-demand single-photon source, where the microwave photons are injected into a wire with high efficiency and spectral purity. This is accomplished in a circuit quantum electrodynamics architecture, with a microwave transmission line cavity that enhances the spontaneous emission of a single superconducting qubit. When the qubit spontaneously emits, the generated photon acts as a flying qubit, transmitting the quantum information across a chip. We perform tomography of both the qubit and the emitted photons, clearly showing that both the quantum phase and amplitude are transferred during the emission. Both the average power and voltage of the photon source are characterized to verify performance of the system. This single-photon source is an important addition to a rapidly growing toolbox for quantum optics on a chip.

Two-mode correlation of microwave quantum noise generated by parametric down-conversion.

In this Letter, we report the observation of the correlation between two modes of microwave radiation resulting from the amplification of quantum noise by the Josephson parametric converter. This process, seen from the pump, can be viewed as parametric down-conversion. The correlation is measured by an interference experiment displaying a contrast better than 99% with a number of photons per mode greater than 250,000. Dispersive measurements of mesoscopic systems and quantum encryption can benefit from this development.