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Societal challenges are inherently complex and multi-tiered, arising from the interplay of diverse stakeholders with a spectrum of purposes and different perceptions and expectations, interdependent systems, and dynamic contextual factors that transcend single domains or disciplines. This paper presents a novel approach to developing a Reference Model of Governance tailored to a specific complex, multi-organisational enterprise facing socially complex challenges. Drawing on Angyal’s systems framework, the model introduces a three-dimensional structure with vertical, progression, and transverse dimensions, integrated within a holistic contextual whole. By mapping selected systems methodologies, including Soft Systems Methodology (SSM), Viable System Model (VSM), System Dynamics (SD), and dependency modelling, to these dimensions, the model offers a pragmatic, structured way to explore and regulate complex organisational behaviour. It enables collaborative inquiry, supports adaptive governance, and enhances the enterprise’s ability to address dynamic societal problems such as health, education, and public service delivery. The result is a governance reference model that captures both the operational and contextual realities of the enterprise, providing actionable insight for strategic design or diagnostic intervention. The novel approach is grounded in systemic and critical systems thinking and emphasises the use of methods for understanding to develop a common and shared understanding of the enterprise context and to surface multiple stakeholder perspectives.

Practical quantum computing will require error rates well below those achievable with physical qubits. Quantum error correction 1 , 2 offers a path to algorithmically relevant error rates by encoding logical qubits within many physical qubits, for which 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 for logical performance to improve with increasing code size. Here we report the measurement of logical qubit performance scaling across several code sizes, and demonstrate that our system of superconducting qubits has sufficient performance to overcome the additional errors from increasing qubit number. We find that our distance-5 surface code logical qubit modestly outperforms an ensemble of distance-3 logical qubits on average, in terms of both 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 1.7 × 10 −6 logical error per cycle floor set by a single high-energy event (1.6 × 10 −7 excluding this event). We accurately model our experiment, extracting error budgets that highlight the biggest challenges for future systems. These results mark an experimental demonstration in which quantum error correction begins to improve performance with increasing qubit number, illuminating the path to reaching the logical error rates required for computation. A study demonstrating increasing error suppression with larger surface code logical qubits, implemented on a superconducting quantum processor.

Quantum many-body systems display rich phase structure in their low-temperature equilibrium states 1 . However, much of nature is not in thermal equilibrium. Remarkably, it was recently predicted that out-of-equilibrium systems can exhibit novel dynamical phases 2 – 8 that may otherwise be forbidden by equilibrium thermodynamics, a paradigmatic example being the discrete time crystal (DTC) 7 , 9 – 15 . Concretely, dynamical phases can be defined in periodically driven many-body-localized (MBL) systems via the concept of eigenstate order 7 , 16 , 17 . In eigenstate-ordered MBL 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, or from regimes in which the dynamics of a few select states can mask typical behaviour. Here we implement tunable controlled-phase (CPHASE) gates on an array of superconducting qubits to experimentally observe an MBL-DTC and demonstrate its characteristic spatiotemporal response for generic initial states 7 , 9 , 10 . Our work employs a time-reversal protocol to quantify the impact of external decoherence, and leverages quantum typicality to circumvent the exponential cost of densely sampling the eigenspectrum. Furthermore, we locate the phase transition out of the DTC with an experimental finite-size analysis. These results establish a scalable approach to studying non-equilibrium phases of matter on quantum processors. A study establishes a scalable approach to engineer and characterize a many-body-localized discrete time crystal phase on a superconducting quantum processor.

The work of Berezinskii, Kosterlitz and Thouless in the 1970s 1 , 2 revealed exotic phases of matter governed by the topological properties of low-dimensional materials such as thin films of superfluids and superconductors. A hallmark of this phenomenon is the appearance and interaction of vortices and antivortices in an angular degree of freedom—typified by the classical XY model—owing to thermal fluctuations. In the two-dimensional Ising model this angular degree of freedom is absent in the classical case, but with the addition of a transverse field it can emerge from the interplay between frustration and quantum fluctuations. Consequently, a Kosterlitz–Thouless phase transition has been predicted in the quantum system—the two-dimensional transverse-field Ising model—by theory and simulation 3 – 5 . Here we demonstrate a large-scale quantum simulation of this phenomenon in a network of 1,800 in situ programmable superconducting niobium flux qubits whose pairwise couplings are arranged in a fully frustrated square-octagonal lattice. Essential to the critical behaviour, we observe the emergence of a complex order parameter with continuous rotational symmetry, and the onset of quasi-long-range order as the system approaches a critical temperature. We describe and use a simple approach to statistical estimation with an annealing-based quantum processor that performs Monte Carlo sampling in a chain of reverse quantum annealing protocols. Observations are consistent with classical simulations across a range of Hamiltonian parameters. We anticipate that our approach of using a quantum processor as a programmable magnetic lattice will find widespread use in the simulation and development of exotic materials. A large-scale programmable quantum simulation is described, using a D-Wave quantum processor to simulate a two-dimensional magnetic lattice in the vicinity of a topological phase transition.

IntroductionUrban green and blue space (UGBS) interventions, such as the development of an urban greenway, have the potential to provide public health benefits and multiple co-benefits in the realms of the environment, economy and society. This paper presents the protocol for a 5-year follow-up evaluation of the public health benefits and co-benefits of an urban greenway in Belfast, UK.Methods and analysisThe natural experiment evaluation uses a range of systems-oriented and mixed-method approaches. First, using group model building methods, we codeveloped a causal loop diagram with stakeholders to inform the evaluation framework. We will use other systems methods including viable systems modelling and soft systems methodology to understand the context of the system (ie, the intervention) and the stakeholders involved in the development, implementation and maintenance phases. The effectiveness evaluation includes a repeat cross-sectional household survey with a random sample of 1200 local residents (adults aged ≥16 years old) who live within 1 mile of the greenway. The survey is complemented with administrative data from the National Health Service. For the household survey, outcomes include physical activity, mental well-being, quality of life, social capital, perceptions of environment and biodiversity. From the administrative data, outcomes include prescription medications for a range of non-communicable diseases such as cardiovascular disease, type II diabetes mellitus, chronic respiratory and mental health conditions. We also investigate changes in infectious disease rates, including COVID-19, and maternal and child health outcomes such as birth weight and gestational diabetes. A range of economic evaluation methods, including a cost-effectiveness analysis and social return on investment (SROI), will be employed. Findings from the household survey and administrative data analysis will be further explored in focus groups with a subsample of those who complete the household survey and the local community to explore possible mechanistic pathways and other impacts beyond those measured. Process evaluation methods include intercept surveys and direct observation of the number and type of greenway visitors using the Systems for Observing Play and Recreation in Communities tool. Finally, we will use methods such as weight of evidence, simulation and group model building, each embedding participatory engagement with stakeholders to help us interpret, triangulate and synthesise the findings.Ethics and disseminationTo our knowledge, this is one of the first natural experiments with a 5-year follow-up evaluation of an UGBS intervention. The findings will help inform future policy and practice on UGBS interventions intended to bring a range of public health benefits and co-benefits. Ethics approval was obtained from the Medicine, Health and Life Sciences Research Ethics Committee prior to the commencement of the study. All participants in the household survey and focus group workshops will provide written informed consent before taking part in the study. Findings will be reported to (1) participants and stakeholders; (2) funding bodies supporting the research; (3) local, regional and national governments to inform policy; (4) presented at local, national and international conferences and (5) disseminated by peer-review publications.

Quantum error correction 1 , 2 , 3 – 4 provides a path to reach practical quantum computing by combining multiple physical qubits into a logical qubit, in which the logical error rate is suppressed exponentially as more qubits are added. However, this exponential suppression only occurs if the physical error rate is below a critical threshold. Here we present two below-threshold surface code memories on our newest generation of superconducting processors, Willow: a distance-7 code and a distance-5 code integrated with a real-time decoder. The logical error rate of our larger quantum memory is suppressed by a factor of Λ  = 2.14 ± 0.02 when increasing the code distance by 2, culminating in a 101-qubit distance-7 code with 0.143% ± 0.003 per cent error per cycle of error correction. This logical memory is also beyond breakeven, exceeding the lifetime of its best physical qubit by a factor of 2.4 ± 0.3. Our system maintains below-threshold performance when decoding in real time, achieving an average decoder latency of 63 microseconds at distance 5 up to a million cycles, with a cycle time of 1.1 microseconds. We also run repetition codes up to distance 29 and find that logical performance is limited by rare correlated error events, occurring approximately once every hour or 3 × 10 9 cycles. Our results indicate device performance that, if scaled, could realize the operational requirements of large-scale fault-tolerant quantum algorithms. Two below-threshold surface code memories on superconducting processors markedly reduce logical error rates, achieving high efficiency and real-time decoding, indicating potential for practical large-scale fault-tolerant quantum algorithms.

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.

Lattice gauge theories (LGTs) can be employed to understand a wide range of phenomena, from elementary particle scattering in high-energy physics to effective descriptions of many-body interactions in materials. Studying dynamical properties of emergent phases can be challenging as it requires solving many-body problems that are generally beyond perturbative limits. 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 which 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 via 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, while for strong confinement transverse fluctuations are effectively frozen. In addition, we demonstrate a resonance condition at which dynamical string breaking is facilitated. Our LGT implementation on a quantum processor presents a novel set of techniques for investigating emergent excitations and string dynamics.

Quantum error correction is essential for bridging the gap between the error rates of physical devices and the extremely low logical error rates required for quantum algorithms. Recent error-correction demonstrations on superconducting processors have focused primarily on the surface code, which offers a high error threshold but poses limitations for logical operations. In contrast, the color code enables much more efficient logic, although it requires more complex stabilizer measurements and decoding techniques. Measuring these stabilizers in planar architectures such as superconducting qubits is challenging, and so far, realizations of color codes have not addressed performance scaling with code size on any platform. Here, we present a comprehensive demonstration of the color code on a superconducting processor, achieving logical error suppression and performing logical operations. Scaling the code distance from three to five suppresses logical errors by a factor of \\(_3/5\\) = 1.56(4). Simulations indicate this performance is below the threshold of the color code, and furthermore that the color code may be more efficient than the surface code with modest device improvements. Using logical randomized benchmarking, we find that transversal Clifford gates add an error of only 0.0027(3), which is substantially less than the error of an idling error correction cycle. We inject magic states, a key resource for universal computation, achieving fidelities exceeding 99% with post-selection (retaining about 75% of the data). Finally, we successfully teleport logical states between distance-three color codes using lattice surgery, with teleported state fidelities between 86.5(1)% and 90.7(1)%. This work establishes the color code as a compelling research direction to realize fault-tolerant quantum computation on superconducting processors in the near future.