Improvements in quality of life and technological advancement are often accompanied by increasingly complex societal challenges. When formulated as computational problems, these challenges can exceed the processing capacity of classical computing. Quantum algorithms can solve these problems more efficiently by leveraging the principles of superposition and entanglement. However, in practice, quantum computation is highly susceptible to qubit errors, which can compromise the results. Various quantum error correction methods were proposed to eliminate these errors. In this paper, the application of one such method-the surface code-to the heavy-hexagonal quantum processor layout is studied and the future research directions are highlighted.
Improvements in quality of life and technological advancement are often accompanied by increasingly complex societal challenges. When formulated as computational problems, these challenges can exceed the processing capacity of classical computing. Quantum algorithms can solve these problems more efficiently by leveraging the principles of superposition and entanglement. However, in practice, quantum computation is highly susceptible to qubit errors, which can compromise the results. Various quantum error correction methods were proposed to eliminate these errors. In this paper, the application of one such method-the surface code-to the heavy-hexagonal quantum processor layout is studied and the future research directions are highlighted.
Topological quantum error correction is a milestone in the scaling roadmap of quantum computers, which targets circuits with trillions of gates that would allow running quantum algorithms for real-world problems. The square-lattice surface code has become the workhorse to address this challenge, as it poses milder requirements on current devices both in terms of required error rates and small local connectivities. In some platforms, however, the connectivities are kept even lower in order to minimise gate errors at the hardware level, which limits the error correcting codes that can be directly implemented on them. In this work, we make a comparative study of possible strategies to overcome this limitation for the heavy-hexagonal lattice, the architecture of current IBM superconducting quantum computers. We explore two complementary strategies: the search for an efficient embedding of the surface code into the heavy-hexagonal lattice, as well as the use of codes whose connectivity requirements are naturally tailored to this architecture, such as subsystem-type and Floquet codes. Using noise models of increased complexity, we assess the performance of these strategies for IBM devices in terms of their error thresholds and qubit footprints. An optimized SWAP-based embedding of the surface code is found to be the most promising strategy towards a near-term demonstration of quantum error correction advantage.
Implementing fault-tolerant quantum computing necessitates the realization of logical non-Clifford gates, which requires the preparation of specific quantum states known as magic states. However, IBM’s heavy-hexagon structure, which has limited qubit connectivity, presents challenges in adapting quantum error correction codes such as the surface code. Several methods have been proposed to address these challenges by adapting the surface code to the heavy-hexagon architecture. In this study, we implement the magic state injection process within two distinct implementations of surface codes (standard and rotated methods) suitable for the heavy-hexagon structure and compare their logical error rates. Furthermore, we propose initialization methods to enhance the performance of magic state injection in the heavy-hexagon structure, thereby efficiently achieving logical non-Clifford gates with reduced error rates.
Designs for quantum error correction depend strongly on the connectivity of the qubits. For solid-state qubits, the most straightforward approach is to have connectivity constrained to a planar graph. Practical considerations may also further restrict the connectivity, resulting in a relatively sparse graph such as the heavy-hexagonal (“heavy-hex”) architecture of current IBM Quantum devices. In such cases, it is hard to use all qubits to their full potential. Instead, in order to emulate the denser connectivity required to implement well-known quantum error-correcting codes, many qubits remain effectively unused. In this work, we show how this bug can be turned into a feature. By using the unused qubits of one code to execute another, two codes can be implemented on top of each other, allowing easy application of fault-tolerant entangling gates and measurements. We demonstrate this by realizing a surface code and a Bacon-Shor code on a 133-qubit IBM Quantum device. Using transversal controlled- X () gates and lattice surgery, we demonstrate entanglement between these logical qubits with code distance up to d = 4 and five rounds of stabilizer-measurement cycles. The nonplanar coupling between the qubits allows us to simultaneously measure the logical X X , Y Y , and Z Z observables. With this, we verify the violation of Bell’s inequality for both the d = 2 case with postselection featuring a fidelity of 94% and the d = 3 instance using only quantum error correction. Published by the American Physical Society 2024
Abstract 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.
Quantum computation exploits superposition, entanglement, interference, and measurement to process information.Research has moved from theoretical algorithms to functional devices where the performance is not only limited by qubit physics, but also by the control, compilation, error mitigation, and error correction.This review considers quantum computing on various levels of abstraction, including physical qubits, quantum algorithms, system architecture, fault-tolerance, and emerging applications.Study-specific evidence regarding scale, operational quality, connectivity, control and suppression of errors is compared across the different types of superconducting circuits, trapped ions, semiconductor spin qubits, neutral atoms and photonic processors.The review of the fundamental algorithms, noisy intermediate-scale quantum computing, variational and hybrid quantum-classical approaches, quantum simulation, quantum machine learning, and security implications is also discussed.Measurable progress has been demonstrated in published experiments for logical-qubit operation, below-threshold surface-code memory, programmable photonic sampling and platformspecific scaling.However, the reported metrics are not directly interchangeable, and practical advantage remains task-and baseline-dependent.Further progress will need to be accompanied by coordinated developments in hardware reliability, error correction, classical control, estimates of algorithmic resources, and clear classical method comparisons.
A comprehensive mathematical model formulation is presented, encompassing gate fidelity optimization, coherence time extension, stabilizer code evolution, and surface code implementation. The research demonstrates significant advancements in qubit stability, with a 7% increase in gate fidelity and a remarkable 50% extension in coherence time achieved through optimized gate operations and material improvements. Quantum error correction techniques, guided by the Lindblad master equation and the surface code, result in a 25% reduction in error rates, contributing to the overall stability of the quantum processor. The outcomes not only bring practical quantum computing closer to realization but also provide a foundation for future innovations. The research identifies avenues for continued optimization, including advanced gate designs, exploration of emerging qubit technologies, and the development of sophisticated error correction codes. Further interdisciplinary collaborations and investigations into scalable quantum architectures, materials science, and cryogenic engineering are essential for overcoming remaining challenges. The insights gained contribute to the advancement of fault-tolerant quantum computing systems, offering transformative capabilities for computation and technology.