Quantum simulation promises to advance materials discovery by accurately simulating complex states of matter, their microscopic excitations, and macroscopic response functions. The central challenge in resolving the underlying interacting dynamics is to combine high-fidelity evolution with the sophisticated control necessary to manipulate individual quasi-particles in quantum many-body states. Here, we report on high-precision simulation of both linear and non-linear response functions in a 2D XY spin-1/2 magnet using an analog-digital superconducting processor of up to 97 qubits. By interleaving digital gates with analog evolution precisely characterized via Hamiltonian learning, we selectively excite magnons at tunable energy densities. Measuring first the linear magnon response – a central probe in neutron-scattering experiments – we extract temperature-dependent spectra and lifetimes. Our results reveal stark variations in magnon decay rates across the Brillouin zone, with enhancement near van Hove singularities and suppression for edge-localized modes. Next, we perform a suite of nonlinear measurements, including the study of self-scattering mechanisms, as well as pump-probe spectroscopy to directly characterize the magnon interactions. While matrix-product state simulations capture the dynamics well in either small systems or at low temperatures, their predictions become inaccurate away from these limits. This work demonstrates precise simulation of the interacting dynamics in quantum magnets, and provides key insights into quasi-particles and their microscopic scattering mechanisms.

This perspective outlines promising pathways and critical obstacles on the road to developing useful quantum computing applications, drawing on insights from the Google Quantum AI team. We propose a five-stage framework for this process, spanning from theoretical explorations of quantum advantage to the practicalities of compilation and resource estimation. For each stage, we discuss key trends, milestones, and inherent scientific and sociological impediments. We argue that two central stages—identifying concrete problem instances expected to exhibit quantum advantage, and connecting such problems to real-world use cases—represent essential and currently under-resourced challenges. Throughout, we touch upon related topics, including the promise of generative artificial intelligence for aspects of this research, criteria for compelling demonstrations of quantum advantage, and the future of compilation as we enter the era of early fault-tolerant quantum computing.

Broadly applicable quantum advantage, particularly in classical data processing and machine learning, has been a fundamental open problem. In this work, we prove that a small quantum computer of polylogarithmic size can perform large-scale classification and dimension reduction on massive classical data by processing samples on the fly, whereas any classical machine achieving the same prediction performance requires exponentially larger size. Furthermore, classical machines that are exponentially larger yet below the required size need superpolynomially more samples and time. We validate these quantum advantages in real-world applications, including single-cell RNA sequencing and movie review sentiment analysis, demonstrating four to six orders of magnitude reduction in size with fewer than 60 logical qubits. These quantum advantages are enabled by quantum oracle sketching, an algorithm for accessing the classical world in quantum superposition using only random classical data samples. Combined with classical shadows, our algorithm circumvents the data loading and readout bottleneck to construct succinct classical models from massive classical data, a task provably impossible for any classical machine that is not exponentially larger than the quantum machine. These quantum advantages persist even when classical machines are granted unlimited time or if BPP=BQP, and rely only on the correctness of quantum mechanics. Together, our results establish machine learning on classical data as a broad and natural domain of quantum advantage and a fundamental test of quantum mechanics at the complexity frontier.

Disorder-induced phenomena in quantum many-body systems pose significant challenges for analytical methods and numerical simulations at relevant time and system scales. To reduce the cost of disorder-sampling, we investigate quantum circuits initialized in states tunable to superpositions over all disorder configurations. In a translationally-invariant lattice gauge theory (LGT), these states can be interpreted as a superposition over gauge sectors. We observe localization in this LGT in the absence of disorder in one and two dimensions: perturbations fail to diffuse despite fully disorder-free evolution and initial states. However, Rényi entropy measurements reveal that superposition-prepared states fundamentally differ from those obtained by direct disorder sampling. Leveraging superposition, we propose an algorithm with a polynomial speedup in sampling disorder configurations, a longstanding challenge in many-body localization studies.

Artificial intelligence and quantum computing have both seen tremendous progress in recent years, opening up new avenues for accelerating scientific discovery. Classical machine learning has already proven to be useful for addressing challenges in quantum computation; and hardware progress is well underway towards the early fault-tolerant regime. On the other hand, the emerging field of quantum machine learning is aimed at utilizing the strengths of both computational paradigms, via the development of learning algorithms or models that benefit from running on quantum devices or from training on quantum data. We therefore argue that both of these paradigms stand to benefit substantially from each other. Here we review the most salient opportunities for machine learning in quantum computing, hoping to inspire a virtuous cycle through which both fields can mutually benefit.

Variational quantum computing offers a flexible computational approach with a broad range of applications. However, a key obstacle to realizing their potential is the barren plateau (BP) phenomenon. When a model exhibits a BP, its parameter optimization landscape becomes exponentially flat and featureless as the problem size increases. Importantly, all the moving pieces of an algorithm — choices of ansatz, initial state, observable, loss function and hardware noise — can lead to BPs if they are ill-suited. As BPs strongly impact on trainability, researchers have dedicated considerable effort to develop theoretical and heuristic methods to understand and mitigate their effects. As a result, the study of BPs has become a thriving area of research, influencing and exchanging ideas with other fields such as quantum optimal control, tensor networks and learning theory. This article provides a review of the current understanding of the BP phenomenon. Barren plateaus are widely considered as one of the main limitations for variational quantum algorithms. This Review summarizes the latest understandings of barren plateaus, indicating its causes, architecture that will suffer from this phenomenon, and discusses strategies that can — and cannot — avoid it.

Understanding how interacting particles approach thermal equilibrium is a major challenge of quantum simulators1,2. Unlocking the full potential of such systems towards this goal requires flexible initial state preparation, precise time evolution and extensive probes for final state characterization. Here we present a quantum simulator comprising 69 superconducting qubits that supports both universal quantum gates and high-fidelity analogue evolution, with performance beyond the reach of classical simulation in cross-entropy benchmarking experiments. This hybrid platform features more versatile measurement capabilities compared with analogue-only simulators, which we leverage here to reveal a coarsening-induced breakdown of Kibble-Zurek scaling predictions3 in the XY model, as well as signatures of the classical Kosterlitz-Thouless phase transition4. Moreover, the digital gates enable precise energy control, allowing us to study the effects of the eigenstate thermalization hypothesis5-7 in targeted parts of the eigenspectrum. We also demonstrate digital preparation of pairwise-entangled dimer states, and image the transport of energy and vorticity during subsequent thermalization in analogue evolution. These results establish the efficacy of superconducting analogue-digital quantum processors for preparing states across many-body spectra and unveiling their thermalization dynamics.

Without large quantum computers to empirically evaluate performance, theoretical frameworks such as the quantum statistical query (QSQ) are a primary tool to study quantum algorithms for learning classical functions and search for quantum advantage in machine learning tasks. However, we only understand quantum advantage in this model at two extremes: either exponential advantages for uniform input distributions or no advantage for arbitrary distributions. Our work helps close the gap between these two regimes by designing an efficient quantum algorithm for learning periodic neurons in the QSQ model over a variety of non-uniform distributions and the first explicit treatment of real-valued functions. We prove that this problem is hard not only for classical gradient-based algorithms, which are the workhorses of machine learning, but also for a more general class of SQ algorithms, establishing an exponential quantum advantage.

Quantum observables in the form of few-point correlators are the key to characterizing the dynamics of quantum many-body systems. In dynamics with fast entanglement generation, quantum observables generally become insensitive to the details of the underlying dynamics at long times due to the effects of scrambling. In experimental systems, repeated time-reversal protocols have been successfully implemented to restore sensitivities of quantum observables. Using a 103-qubit superconducting quantum processor, we characterize ergodic dynamics using the second-order out-of-time-order correlators, OTOC^(2). In contrast to dynamics without time reversal, OTOC^(2) are observed to remain sensitive to the underlying dynamics at long time scales. Furthermore, by inserting Pauli operators during quantum evolution and randomizing the phases of Pauli strings in the Heisenberg picture, we observe substantial changes in OTOC^(2) values. This indicates that OTOC^(2) is dominated by constructive interference between Pauli strings that form large loops in configuration space. The observed interference mechanism endows OTOC^(2) with a high degree of classical simulation complexity, which culminates in a set of large-scale OTOC^(2) measurements exceeding the simulation capacity of known classical algorithms. Further supported by an example of Hamiltonian learning through OTOC^(2), our results indicate a viable path to practical quantum advantage.

A remarkable characteristic of quantum computing is the potential for reliable computation despite faulty qubits. This can be achieved through quantum error correction, which is typically implemented by repeatedly applying static syndrome checks, permitting correction of logical information. Recently, the development of time-dynamic approaches to error correction has uncovered new codes and new code implementations. In this work, we experimentally demonstrate three time-dynamic implementations of the surface code, each offering a unique solution to hardware design challenges and introducing flexibility in surface code realization. First, we embed the surface code on a hexagonal lattice, reducing the necessary couplings per qubit from four to three. Second, we walk a surface code, swapping the role of data and measure qubits each round, achieving error correction with built-in removal of accumulated non-computational errors. Finally, we realize the surface code using iSWAP gates instead of the traditional CNOT, extending the set of viable gates for error correction without additional overhead. We measure the error suppression factor when scaling from distance-3 to distance-5 codes of Λ_35,hex = 2.15(2), Λ_35,walk = 1.69(6), and Λ_35,iSWAP = 1.56(2), achieving state-of-the-art error suppression for each. With detailed error budgeting, we explore their performance trade-offs and implications for hardware design. This work demonstrates that dynamic circuit approaches satisfy the demands for fault-tolerance and opens new alternative avenues for scalable hardware design.

Classical data can be copied and re-used for computation, with adverse consequences economically and in terms of data privacy. Motivated by this, we formulate problems in one-way communication complexity where Alice holds some data x and Bob holds m inputs y(1),..., y(m). They want to compute m instances of a bipartite relation R( center dot, center dot) on every pair ( x, y(1)),..., ( x, y(m)). We call this the asymmetric direct sum question for one-way communication. We give examples where the quantum communication complexity of such problems scales polynomially with m, while the classical communication complexity depends at most logarithmically on m. Thus, for such problems, data behaves like a consumable resource that is effectively destroyed upon use when the owner stores and transmits it as quantum states, but not when transmitted classically. We show an application to a strategic data-selling game, and discuss other potential economic implications.

A remarkable characteristic of quantum computing is the potential for reliable computation despite faulty qubits. This can be achieved through quantum error correction, which is typically implemented by repeatedly applying static syndrome checks, permitting correction of logical information. Recently, the development of time-dynamic approaches to error correction has enabled different codes and implementations that do not rely on static syndrome measurements. Here we experimentally demonstrate three time-dynamic implementations of the surface code, each offering a distinct solution to hardware design challenges faced by surface code realizations. First, we embed the surface code on a hexagonal lattice, reducing the necessary couplings per qubit from four to three. Second, we walk a surface code, swapping the role of data and measure qubits each round, achieving error correction with built-in removal of accumulated non-computational errors. Finally, we realize the surface code using iSWAP gates instead of the traditional CNOT, extending the set of viable gates for error correction without additional overhead. We measure the error suppression factor when scaling from distance-3 to distance-5 codes of Lambda 35,hex = 2.15(2), Lambda 35,walk = 1.69(6) and Lambda 35,iSWAP = 1.56(2), achieving state-of-the-art error suppression for each. Our work demonstrates that dynamic circuit approaches meet the demands for fault tolerance and enable alternative strategies for scalable hardware design.

Recent breakthroughs in generative machine learning, powered by massive computational resources, have demonstrated unprecedented human-like capabilities. While beyond-classical quantum experiments can generate samples from classically intractable distributions, their complexity has thwarted all efforts toward efficient learning. This challenge has hindered demonstrations of generative quantum advantage: the ability of quantum computers to learn and generate desired outputs substantially better than classical computers. We resolve this challenge by introducing families of generative quantum models that are hard to simulate classically, are efficiently trainable, exhibit no barren plateaus or proliferating local minima, and can learn to generate distributions beyond the reach of classical computers. Using a 68-qubit superconducting quantum processor, we demonstrate these capabilities in two scenarios: learning classically intractable probability distributions and learning quantum circuits for accelerated physical simulation. Our results establish that both learning and sampling can be performed efficiently in the beyond-classical regime, opening new possibilities for quantum-enhanced generative models with provable advantage.

Lattice gauge theories (LGTs)1-4 can be used to understand a wide range of phenomena, from elementary particle scattering in high-energy physics to effective descriptions of many-body interactions in materials5-7. Studying dynamical properties of emergent phases can be challenging, as it requires solving many-body problems that are generally beyond perturbative limits8-10. Here we investigate the dynamics of local excitations in a Z 2 LGT using a two-dimensional lattice of superconducting qubits. We first construct a simple variational circuit that prepares low-energy states that have a large overlap with the ground state; then we create charge excitations with local gates and simulate their quantum dynamics by means of a discretized time evolution. As the electric field coupling constant is increased, our measurements show signatures of transitioning from deconfined to confined dynamics. For confined excitations, the electric field induces a tension in the string connecting them. Our method allows us to experimentally image string dynamics in a (2+1)D LGT, from which we uncover two distinct regimes inside the confining phase: for weak confinement, the string fluctuates strongly in the transverse direction, whereas for strong confinement, transverse fluctuations are effectively frozen11,12. We also demonstrate a resonance condition at which dynamical string breaking is facilitated. Our LGT implementation on a quantum processor presents a new set of techniques for investigating emergent excitations and string dynamics.

Quantum error correction1-4 is essential for bridging the gap between the error rates of physical devices and the extremely low error rates required for quantum algorithms. Recent error-correction demonstrations on superconducting processors5-8 have focused primarily on the surface code9, which offers a high error threshold but poses limitations for logical operations. The colour code10 enables more efficient logic, but it requires more complex stabilizer measurements and decoding. Measuring these stabilizers in planar architectures such as superconducting qubits is challenging, and realizations of colour codes11-19 have not addressed performance scaling with code size on any platform. Here we present a comprehensive demonstration of the colour code on a superconducting processor8. Scaling the code distance from three to five suppresses logical errors by a factor of Λ3/5 = 1.56(4). Simulations indicate this performance is below the threshold of the colour code, and the colour code may become more efficient than the surface code following modest device improvements. We test transversal Clifford gates with logical randomized benchmarking20 and inject magic states21, a key resource for universal computation, achieving fidelities exceeding 99% with post-selection. Finally, we teleport logical states between colour codes using lattice surgery22. This work establishes the colour code as a compelling research direction to realize fault-tolerant quantum computation on superconducting processors in the near future.

The quest to identify quantum advantages lies at the heart of quantum technology. While quantum devices promise extraordinary capabilities, from exponential computational speedups to unprecedented measurement precision, distinguishing genuine advantages from mere illusions remains a formidable challenge. In this endeavor, quantum theorists are like prophets attempting to foretell the future, yet the boundary between visionary insight and unfounded fantasy is perilously thin. In this perspective, we examine our mathematical tools for navigating the vast world of quantum advantages across computation, learning, sensing, and communication. We explore five keystone properties: predictability, typicality, robustness, verifiability, and usefulness that define an ideal quantum advantage, and envision what new quantum advantages could arise in a future with ubiquitous quantum technology. We prove that some quantum advantages are inherently unpredictable using classical resources alone, suggesting a landscape far richer than what we can currently foresee. While mathematical rigor remains our indispensable guide, the ultimate power of quantum technologies may emerge from advantages we cannot yet conceive.

Despite fundamental interests in learning quantum circuits, the existence of a computationally efficient algorithm for learning shallow quantum circuits remains an open question. Because shallow quantum circuits can generate distributions that are classically hard to sample from, existing learning algorithms do not apply. In this work, we present a polynomial-time classical algorithm for learning the description of any unknown n-qubit shallow quantum circuit U (with arbitrary unknown architecture) within a small diamond distance using single-qubit measurement data on the output states of U. We also provide a polynomial-time classical algorithm for learning the description of any unknown n-qubit state | ψ ⟩ = U | 0n ⟩ prepared by a shallow quantum circuit U (on a 2D lattice) within a small trace distance using single-qubit measurements on copies of | ψ ⟩. Our approach uses a quantum circuit representation based on local inversions and a technique to combine these inversions. This circuit representation yields an optimization landscape that can be efficiently navigated and enables efficient learning of quantum circuits that are classically hard to simulate.

Real-world applications of computing can be extremely time-sensitive. It would be valuable if we could accelerate such tasks by performing some of the work ahead of time. Motivated by this, we propose a cost model for quantum algorithms that allows quantum precomputation, i.e., for a polynomial amount of "free" computation before the input to an algorithm is fully specified, and methods for taking advantage of it. We analyze two families of unitaries that are asymptotically more efficient to implement in this cost model than in the standard one. The first example of quantum precomputation, based on density matrix exponentiation, could offer an exponential advantage under certain conditions. The second example uses a variant of gate teleportation to achieve a quadratic advantage when compared with implementing the unitaries directly. These examples hint that quantum precomputation may offer a new arena in which to seek quantum advantage.

Engineered dissipative reservoirs have the potential to steer many-body quantum systems toward correlated steady states useful for quantum simulation of high-temperature superconductivity or quantum magnetism. Using up to 49 superconducting qubits, we prepared low-energy states of the transverse-field Ising model through coupling to dissipative auxiliary qubits. In one dimension, we observed long-range quantum correlations and a ground-state fidelity of 0.86 for 18 qubits at the critical point. In two dimensions, we found mutual information that extends beyond nearest neighbors. Lastly, by coupling the system to auxiliaries emulating reservoirs with different chemical potentials, we explored transport in the quantum Heisenberg model. Our results establish engineered dissipation as a scalable alternative to unitary evolution for preparing entangled many-body states on noisy quantum processors.

Understanding universal aspects of quantum dynamics is an unresolved problem in statistical mechanics. In particular, the spin dynamics of the one-dimensional Heisenberg model were conjectured as to belong to the Kardar-Parisi-Zhang (KPZ) universality class based on the scaling of the infinite-temperature spin-spin correlation function. In a chain of 46 superconducting qubits, we studied the probability distribution of the magnetization transferred across the chain’s center, P M . The first two moments of P M show superdiffusive behavior, a hallmark of KPZ universality. However, the third and fourth moments ruled out the KPZ conjecture and allow for evaluating other theories. Our results highlight the importance of studying higher moments in determining dynamic universality classes and provide insights into universal behavior in quantum systems.