Abstract This study proposes a logical qubit behavior model (LQBM) that calculates the state of the logical qubit based on a surface code after a logical qubit operation. LQBM can simulate the surface code whose distance exceeds 5, which is impossible with conventional quantum simulators. A complex error syndrome measurement must not be executed in this model. LQBM works at the mixed level of a logical and physical qubit. Probability amplitudes and the number of state vectors are designed at the physical qubit level. The state vectors of physical qubits constituting a logical qubit are abstracted to the value of the logical qubit level. LQBM converts all logical qubit operations, which include an initialization, a state injection, universal gates, and lattice surgery operations (e.g., a merge and a split) to simple calculations. LQBM provides a better computational complexity of O(log(distance)) than O(distance $$^2$$ 2 ). Through simulation experiments, it was found that LQBM performs logical qubit operations up to 24 million times faster than existing quantum simulators in the case of distance = 5.
Abstract This study proposes a logical qubit behavior model (LQBM) that calculates the state of the logical qubit based on a surface code after a logical qubit operation. LQBM can simulate the surface code whose distance exceeds 5, which is impossible with conventional quantum simulators. A complex error syndrome measurement must not be executed in this model. LQBM works at the mixed level of a logical and physical qubit. Probability amplitudes and the number of state vectors are designed at the physical qubit level. The state vectors of physical qubits constituting a logical qubit are abstracted to the value of the logical qubit level. LQBM converts all logical qubit operations, which include an initialization, a state injection, universal gates, and lattice surgery operations (e.g., a merge and a split) to simple calculations. LQBM provides a better computational complexity of O(log(distance)) than O(distance 2 2 ). Through simulation experiments, it was found that LQBM performs logical qubit operations up to 24 million times faster than existing quantum simulators in the case of distance = 5.
Resource consumption of the conventional surface code is expensive, in part due to the need to separate the defects that create the logical qubit far apart on the physical qubit lattice. We propose that instantiating the deformation-based surface code using superstabilizers will make it possible to detect short error chains connecting the superstabilizers, allowing us to place logical qubits close together. Additionally, we demonstrate the process of conversion from the defect-based surface code, which works as arbitrary state injection, and a lattice-surgery-like controlled not (cnot) gate implementation that requires fewer physical qubits than the braiding cnot gate. Finally, we propose a placement design for the deformation-based surface code and analyze its resource consumption; large-scale quantum computation requires 425d2+170d+289 physical qubits per logical qubit, where d is the code distance of the standard surface code, whereas the planar code requires $16{d}^{2}\ensuremath{-}16d+4$ physical qubits per logical qubit, for a reduction of about 50%.
Abstract Quantum error correction is likely to be key in obtaining near term quantum advantage. We propose a novel method for providing multiple logical qubits in the correction of quantum errors using classical computers. The core idea of our work is built upon two main pillars: dividing the Hilbert space into reduced Hilbert spaces with individual logical qubits and synthesizing the reduced Hilbert spaces through a mathematical collaborating between classical bits and logical quantum states. We demonstrate that our method supports at least 20 logical qubits in a surface code with a code distance of 3. Furthermore, we generate entangled states of multiple logical qubits from lattice surgery-based surface codes using only physical qubit operations. This approach enables classical computers to support a larger number of logical qubits using less memory and perform faster simulations.
The demonstration of quantum error correction (QEC) is one of the most important milestones in the realization of fully-fledged quantum computers. Toward this, QEC experiments using the surface codes have recently been actively conducted. However, it has not yet been realized to protect logical quantum information beyond the physical coherence time. In this work, we performed a full simulation of QEC for the rotated surface codes with a code distance 5, which employs 49 qubits and is within reach of the current state-of-the-art quantum computers. In particular, we evaluate the logical error probability in a realistic noise model that incorporates not only stochastic Pauli errors but also coherent errors due to a systematic control error or unintended interactions. While a straightforward simulation of 49 qubits is not tractable within a reasonable computational time, we reduced the number of qubits required to 26 qubits by delaying the syndrome measurement in simulation. This and a fast quantum computer simulator, , implemented on GPU allows us to simulate full QEC with an arbitrary local noise within reasonable simulation time. Based on the numerical results, we also construct and verify an effective model to incorporate the effect of the coherent error into a stochastic noise model. This allows us to understand what the effect coherent error has on the logical error probability on a large scale without full simulation based on the detailed full simulation of a small scale. The present simulation framework and effective model, which can handle arbitrary local noise, will play a vital role in clarifying the physical parameters that future experimental QEC should target. Published by the American Physical Society 2024
Fault-tolerant quantum computation requires logical operations that manipulate encoded information while preserving quantum error-correction protection. In planar surface-code architectures, code deformation and lattice surgery provide a local, measurement-based route to such operations. Here we experimentally realize key elements of patch-based surface-code logical processing on a 107-qubit superconducting quantum processor. We first implement a reusable primitive layer comprising merge and split, patch expansion and shrinkage, and deformations mediated by domain walls and twist defects. We then compose these primitives to realize logical state routing, the logical controlled-NOT gate, and the single-qubit Hadamard and phase gates, which together form a Clifford-generating set. All operations are implemented on distance-three rotated surface-code patches with multi-round syndrome extraction and neural-network decoding, without post-selection. Our results advance superconducting surface-code experiments from protected logical memory to active, patch-based fault-tolerant logical operations.
Abstract. In recent years, surface codes have become the preferred method for quantum error correction in large scale computational and communications architectures. Their comparatively high fault-tolerant thresholds and their natural 2-dimensional nearest neighbour (2DNN) structure make them an obvious choice for large scale designs in experimentally realistic systems. While fundamentally based on the toric code of Kitaev, there are many variants, two of which are the planar-and defect- based codes. Planar codes require fewer qubits to implement (for the same strength of error correction), but are restricted to encoding a single qubit of information. Interactions between encoded qubits are achieved via transversal operations, thus destroying the inherent 2DNN nature of the code. In this paper we introduce a new technique enabling the coupling of two planar codes without transversal operations, maintaining the 2DNN of the encoded computer. Our lattice surgery technique comprises splitting and merging planar code surfaces, and enables us to perform universal quantum computation (including magic state injection) while removing the need for braided logic in a strictly 2DNN design, and hence reduces the overall qubit resources for logic operations. We show how lattice surgery allows us to distribute encoded GHZ states in a more direct (and overhead friendly) manner, and how a demonstration of an encoded CNOT between two distance 3 logical states is possible with 53 physical qubits, half of that required in any other known construction in 2D. ar
Surface-code quantum error correction (QEC) is pivotal for fault-tolerant quantum computing, but its energy efficiency remains underexplored. This work derives a practical lower bound on the energy cost per logical-qubit operation, accounting for gate-fidelity decay's impact on code distance and physical-qubit overhead. By modeling the energy consumption of gate operations and error correction, we express the total cost as $ E = (E_g N_g + E_c N_c)/N_l $, where $ E_g $, $ E_c $, $ N_g $, $ N_c $, and $ N_l $ represent gate and physical-qubit energy costs, gate counts, physical-qubit counts, and logical qubit counts, respectively. Gate-fidelity decay increases the required code distance $ d $, scaling $ N_c $ and thus $ E $. We propose normalizing computational cost in joules per logical-qubit operation (JPCUB), isolating logical work from error-correction overhead. Quantitative predictions for $ E_g $, $ E_c $, and $ N_c $ remain [to verify], but the functional dependence on $ d $ and gate fidelities is established. This framework informs energy-efficient QEC design and benchmarks. ## 1. Introduction Surface codes enable fault-tolerant quantum computation by encoding logical qubits in a 2D lattice of physical qubits, correcting errors through syndrome measurements. However, the energy cost per logical operation—critical for scalability—remains poorly quantified. While theoretical foundations exist (Raussendorf & Terhal, 2006; Kitaev, 2003), prior work focuses on error-correction thresholds rather than energy efficiency. Here, we address this gap by deriving a lower bound on energy cost, considering how gate-fidelity decay necessitates larger code distances $ d $, increasing physical-qubit overhead $ N_c $. This analysis connects to the joules-per-compute (JPCUB) benchmark, advocating for energy normalization per logical qubit to disentangle computational work from error-correction costs. ## 2. Background Surface codes encode logical qubits using a 2D grid of