While it has been widely recognized that genuine quantum advantage for practical problems might only be achieved with fault-tolerant quantum computers, it is still not entirely clear whether the required quantum error correction will be physically feasible. In the present work, we carefully analyze the required energy resources to encode the logical qubit states for repetition, perfect, and Steane codes. We find that there is a universal trade-off between the target precision and the required energetic resources. Importantly, we find that the energetic resources intimately depend on the specific physical realization of a quantum error correction code, and that the required resources scale exponentially with the targeted precision of the encoding.
While it has been widely recognized that genuine quantum advantage for practical problems might only be achieved with fault-tolerant quantum computers, it is still not entirely clear whether the required quantum error correction will be physically feasible. In the present work, we carefully analyze the required energy resources to encode the logical qubit states for repetition, perfect, and Steane codes. We find that there is a universal trade-off between the target precision and the required energetic resources. Importantly, we find that the energetic resources intimately depend on the specific physical realization of a quantum error correction code, and that the required resources scale exponentially with the targeted precision of the encoding.
While it has been widely recognized that genuine quantum advantage for practical problems might only be achieved with fault-tolerant quantum computers, it is still not entirely clear whether the required quantum error correction will be physically feasible. In the present work, we carefully analyze the required energy resources to encode the logical qubit states for repetition, perfect, and Steane codes. We find that there is a universal trade-off between the target precision and the required energetic resources. Importantly, we find that the energetic resources intimately depend on the specific physical realization of a quantum error correction code, and that the required resources scale exponentially with the targeted precision of the encoding.
Quantum error correction allows one to make quantum computers fault-tolerant against unavoidable errors due to decoherence and imperfect physical gate operations. However, the fault-tolerant quantum computation requires impractically large computational resources for useful applications. This is a current major obstacle to the realization of a quantum computer. In particular, magic state distillation, which is a standard approach to universality, consumes the most resources in fault-tolerant quantum computation. For the resource problem, here we propose step-by-step magic state encoding for concatenated quantum codes, where magic states are encoded step by step from the physical level to the logical one. To manage errors during the encoding, we carefully use error detection. Since the sizes of intermediate codes are small, it is expected that the resource overheads will become lower than previous approaches based on the distillation at the logical level. Our simulation results suggest that the resource requirements for a logical magic state will become comparable to those for a single logical controlled-NOT gate. Thus, the present method opens a new possibility for efficient fault-tolerant quantum computation.
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
The seven-qubit quantum error-correcting code originally proposed by Steane is one of the best known quantum codes. The Steane code has a desirable property that most basic operations can be performed easily in a fault-tolerant manner. A major obstacle to fault-tolerant quantum computation with the Steane code is fault-tolerant preparation of encoded states, which requires large computational resources. Here we propose efficient state preparation methods for zero and magic states encoded with the Steane code, where the zero state is one of the computational basis states and the magic state allows us to achieve universality in fault-tolerant quantum computation. The methods minimize resource overheads for the fault-tolerant state preparation, and therefore reduce necessary resources for quantum computation with the Steane code. Thus, the present results will open a new possibility for efficient fault-tolerant quantum computation.
Modularization promises to break down the design and implementation complexity of large scale quantum processors into smaller manageable subtasks. In this approach, quantum channels, realized for instance through physical rerouting of qubits or quantum teleportation, connect multiple modules. Each of those modules hosts a subset of qubits, e.g. multiple logical qubits, and provides quantum operations on them. In this work, we implement for the first time all logical operations required for modular fault-tolerant universal quantum computing with a non-Calderbank-Shor-Steane (CSS) code, the perfect [[5,1,3]] code, on a trapped-ion quantum computer. This code is the smallest quantum error-correcting (QEC) code capable of correcting any single-qubit error, making it a compact alternative to larger CSS codes. We demonstrate logical state teleportation and a full suite of fault-tolerant operations required for universal logical control, including logical state preparation, QEC with real-time feedback, logical measurements, magic-state preparation, logical entangling operations, and magic-state injection. Moreover, we characterize the logical spectator error picked up by idling logical qubits during quantum operations on distinct qubit registers and demonstrate a logical Pauli quantum process tomography that minimizes required sampling resources for logical tomography.
This thesis deals with the problematics of the scalability of fault-tolerant quantum computing. This question is studied under the angle of estimating the resources needed to set up such computers. What we call a resource is, in principle, very general; it could be the power, the energy, the total bandwidth allocated to the different qubits... However, we mainly focus on the energetic cost of quantum computing. In particular, we develop an inter-disciplinary approach that allows to minimize the resources required to implement algorithms on quantum computers. By asking to find the minimum amount of resources required to perform a computation under the constraint that the algorithm provides a correct answer with a targeted accuracy, it is possible to optimize the whole computer in order to minimize the resources spent, while being sure to have a correct answer with a high probability. We apply this approach to a complete model fault-tolerant quantum computer based on superconducting qubits. Our results indicate that for algorithms implemented on thousands of logical qubits, our method makes it possible to reduce the energetic cost by orders of magnitudes in regimes where, without optimizing, the power consumption could be close to the gigawatt. This work illustrates that the energetic cost of quantum computing should be a criterion in itself, allowing to evaluate the scaling potential of a given quantum computer technology. It also illustrates that optimizing the architecture of a quantum computer, via inter-disciplinary methods, including algorithms, error correction, qubit physics, engineering aspects, such as the ones that we propose, can prove to be a powerful tool, clearly improving the scaling potential of quantum computers. Finally, we provide general hints about how to make fault-tolerant quantum computers energy efficient.