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The promise of quantum computers is that certain computational tasks might be executed exponentially faster on a quantum processor than on a classical processor 1 . A fundamental challenge is to build a high-fidelity processor capable of running quantum algorithms in an exponentially large computational space. Here we report the use of a processor with programmable superconducting qubits 2 – 7 to create quantum states on 53 qubits, corresponding to a computational state-space of dimension 2 53 (about 10 16 ). Measurements from repeated experiments sample the resulting probability distribution, which we verify using classical simulations. Our Sycamore processor takes about 200 seconds to sample one instance of a quantum circuit a million times—our benchmarks currently indicate that the equivalent task for a state-of-the-art classical supercomputer would take approximately 10,000 years. This dramatic increase in speed compared to all known classical algorithms is an experimental realization of quantum supremacy 8 – 14 for this specific computational task, heralding a much-anticipated computing paradigm. Quantum supremacy is demonstrated using a programmable superconducting processor known as Sycamore, taking approximately 200 seconds to sample one instance of a quantum circuit a million times, which would take a state-of-the-art supercomputer around ten thousand years to compute.

Realizing the potential of quantum computing requires sufficiently low logical error rates 1 . Many applications call for error rates as low as 10 −15 (refs. 2 – 9 ), but state-of-the-art quantum platforms typically have physical error rates near 10 −3 (refs. 10 – 14 ). Quantum error correction 15 – 17 promises to bridge this divide by distributing quantum logical information across many physical qubits in such a way that errors can be detected and corrected. Errors on the encoded logical qubit state can be exponentially suppressed as the number of physical qubits grows, provided that the physical error rates are below a certain threshold and stable over the course of a computation. Here we implement one-dimensional repetition codes embedded in a two-dimensional grid of superconducting qubits that demonstrate exponential suppression of bit-flip or phase-flip errors, reducing logical error per round more than 100-fold when increasing the number of qubits from 5 to 21. Crucially, this error suppression is stable over 50 rounds of error correction. We also introduce a method for analysing error correlations with high precision, allowing us to characterize error locality while performing quantum error correction. Finally, we perform error detection with a small logical qubit using the 2D surface code on the same device 18 , 19 and show that the results from both one- and two-dimensional codes agree with numerical simulations that use a simple depolarizing error model. These experimental demonstrations provide a foundation for building a scalable fault-tolerant quantum computer with superconducting qubits. Repetition codes running many cycles of quantum error correction achieve exponential suppression of errors with increasing numbers of qubits.

The leakage of quantum information out of the two computational states of a qubit into other energy states represents a major challenge for quantum error correction. During the operation of an error-corrected algorithm, leakage builds over time and spreads through multi-qubit interactions. This leads to correlated errors that degrade the exponential suppression of the logical error with scale, thus challenging the feasibility of quantum error correction as a path towards fault-tolerant quantum computation. Here, we demonstrate a distance-3 surface code and distance-21 bit-flip code on a quantum processor for which leakage is removed from all qubits in each cycle. This shortens the lifetime of leakage and curtails its ability to spread and induce correlated errors. We report a tenfold reduction in the steady-state leakage population of the data qubits encoding the logical state and an average leakage population of less than 1 × 10−3 throughout the entire device. Our leakage removal process efficiently returns the system back to the computational basis. Adding it to a code circuit would prevent leakage from inducing correlated error across cycles. With this demonstration that leakage can be contained, we have resolved a key challenge for practical quantum error correction at scale.Physical realizations of qubits are often vulnerable to leakage errors, where the system ends up outside the basis used to store quantum information. A leakage removal protocol can suppress the impact of leakage on quantum error-correcting codes.

Measurement has a special role in quantum theory: by collapsing the wavefunction it can enable phenomena such as teleportation and thereby alter the \"arrow of time\" that constrains unitary evolution. When integrated in many-body dynamics, measurements can lead to emergent patterns of quantum information in space-time that go beyond established paradigms for characterizing phases, either in or out of equilibrium. On present-day NISQ processors, the experimental realization of this physics is challenging due to noise, hardware limitations, and the stochastic nature of quantum measurement. Here we address each of these experimental challenges and investigate measurement-induced quantum information phases on up to 70 superconducting qubits. By leveraging the interchangeability of space and time, we use a duality mapping, to avoid mid-circuit measurement and access different manifestations of the underlying phases -- from entanglement scaling to measurement-induced teleportation -- in a unified way. We obtain finite-size signatures of a phase transition with a decoding protocol that correlates the experimental measurement record with classical simulation data. The phases display sharply different sensitivity to noise, which we exploit to turn an inherent hardware limitation into a useful diagnostic. Our work demonstrates an approach to realize measurement-induced physics at scales that are at the limits of current NISQ processors.

We demonstrate a high dynamic range Josephson parametric amplifier (JPA) in which the active nonlinear element is implemented using an array of rf-SQUIDs. The device is matched to the 50 \\(\\) environment with a Klopfenstein-taper impedance transformer and achieves a bandwidth of 250-300 MHz, with input saturation powers up to -95 dBm at 20 dB gain. A 54-qubit Sycamore processor was used to benchmark these devices, providing a calibration for readout power, an estimate of amplifier added noise, and a platform for comparison against standard impedance matched parametric amplifiers with a single dc-SQUID. We find that the high power rf-SQUID array design has no adverse effect on system noise, readout fidelity, or qubit dephasing, and we estimate an upper bound on amplifier added noise at 1.6 times the quantum limit. Lastly, amplifiers with this design show no degradation in readout fidelity due to gain compression, which can occur in multi-tone multiplexed readout with traditional JPAs.

Systems of correlated particles appear in many fields of science and represent some of the most intractable puzzles in nature. The computational challenge in these systems arises when interactions become comparable to other energy scales, which makes the state of each particle depend on all other particles. The lack of general solutions for the 3-body problem and acceptable theory for strongly correlated electrons shows that our understanding of correlated systems fades when the particle number or the interaction strength increases. One of the hallmarks of interacting systems is the formation of multi-particle bound states. In a ring of 24 superconducting qubits, we develop a high fidelity parameterizable fSim gate that we use to implement the periodic quantum circuit of the spin-1/2 XXZ model, an archetypal model of interaction. By placing microwave photons in adjacent qubit sites, we study the propagation of these excitations and observe their bound nature for up to 5 photons. We devise a phase sensitive method for constructing the few-body spectrum of the bound states and extract their pseudo-charge by introducing a synthetic flux. By introducing interactions between the ring and additional qubits, we observe an unexpected resilience of the bound states to integrability breaking. This finding goes against the common wisdom that bound states in non-integrable systems are unstable when their energies overlap with the continuum spectrum. Our work provides experimental evidence for bound states of interacting photons and discovers their stability beyond the integrability limit.

Inherent symmetry of a quantum system may protect its otherwise fragile states. Leveraging such protection requires testing its robustness against uncontrolled environmental interactions. Using 47 superconducting qubits, we implement the one-dimensional kicked Ising model which exhibits non-local Majorana edge modes (MEMs) with \\(Z_2\\) parity symmetry. Remarkably, we find that any multi-qubit Pauli operator overlapping with the MEMs exhibits a uniform late-time decay rate comparable to single-qubit relaxation rates, irrespective of its size or composition. This characteristic allows us to accurately reconstruct the exponentially localized spatial profiles of the MEMs. Furthermore, the MEMs are found to be resilient against certain symmetry-breaking noise owing to a prethermalization mechanism. Our work elucidates the complex interplay between noise and symmetry-protected edge modes in a solid-state environment.

Leakage of quantum information out of computational states into higher energy states represents a major challenge in the pursuit of quantum error correction (QEC). In a QEC circuit, leakage builds over time and spreads through multi-qubit interactions. This leads to correlated errors that degrade the exponential suppression of logical error with scale, challenging the feasibility of QEC as a path towards fault-tolerant quantum computation. Here, we demonstrate the execution of a distance-3 surface code and distance-21 bit-flip code on a Sycamore quantum processor where leakage is removed from all qubits in each cycle. This shortens the lifetime of leakage and curtails its ability to spread and induce correlated errors. We report a ten-fold reduction in steady-state leakage population on the data qubits encoding the logical state and an average leakage population of less than \\(1 10^-3\\) throughout the entire device. The leakage removal process itself efficiently returns leakage population back to the computational basis, and adding it to a code circuit prevents leakage from inducing correlated error across cycles, restoring a fundamental assumption of QEC. With this demonstration that leakage can be contained, we resolve a key challenge for practical QEC at scale.

Practical quantum computing will require error rates that are well below what is achievable with physical qubits. Quantum error correction offers a path to algorithmically-relevant error rates by encoding logical qubits within many physical qubits, where increasing the number of physical qubits enhances protection against physical errors. However, introducing more qubits also increases the number of error sources, so the density of errors must be sufficiently low in order for logical performance to improve with increasing code size. Here, we report the measurement of logical qubit performance scaling across multiple code sizes, and demonstrate that our system of superconducting qubits has sufficient performance to overcome the additional errors from increasing qubit number. We find our distance-5 surface code logical qubit modestly outperforms an ensemble of distance-3 logical qubits on average, both in terms of logical error probability over 25 cycles and logical error per cycle (\\(2.914\\% 0.016\\%\\) compared to \\(3.028\\% 0.023\\%\\)). To investigate damaging, low-probability error sources, we run a distance-25 repetition code and observe a \\(1.710^-6\\) logical error per round floor set by a single high-energy event (\\(1.610^-7\\) when excluding this event). We are able to accurately model our experiment, and from this model we can extract error budgets that highlight the biggest challenges for future systems. These results mark the first experimental demonstration where quantum error correction begins to improve performance with increasing qubit number, illuminating the path to reaching the logical error rates required for computation.

Quantum many-body systems display rich phase structure in their low-temperature equilibrium states. However, much of nature is not in thermal equilibrium. Remarkably, it was recently predicted that out-of-equilibrium systems can exhibit novel dynamical phases that may otherwise be forbidden by equilibrium thermodynamics, a paradigmatic example being the discrete time crystal (DTC). Concretely, dynamical phases can be defined in periodically driven many-body localized systems via the concept of eigenstate order. In eigenstate-ordered phases, the entire many-body spectrum exhibits quantum correlations and long-range order, with characteristic signatures in late-time dynamics from all initial states. It is, however, challenging to experimentally distinguish such stable phases from transient phenomena, wherein few select states can mask typical behavior. Here we implement a continuous family of tunable CPHASE gates on an array of superconducting qubits to experimentally observe an eigenstate-ordered DTC. We demonstrate the characteristic spatiotemporal response of a DTC for generic initial states. Our work employs a time-reversal protocol that discriminates external decoherence from intrinsic thermalization, and leverages quantum typicality to circumvent the exponential cost of densely sampling the eigenspectrum. In addition, we locate the phase transition out of the DTC with an experimental finite-size analysis. These results establish a scalable approach to study non-equilibrium phases of matter on current quantum processors.