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.
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.
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.
Fault-tolerant state preparation is essential for reliable quantum error correction, particularly in Steane-type error correction, which relies on robust ancilla states for syndrome readout. One method of fault-tolerant state preparation is to initialize multiple ancilla states and check them against each other to detect problematic errors. In the worst case, the number of states required for successful initialization grows polynomially with the code distance, but it has been shown that this can be reduced to constant ancilla overhead-in the best case, only four states are required. However, existing techniques for finding low-overhead initialization schemes are limited to codes with large symmetry groups, such as the Golay code. In this work, we propose a general, automated synthesis methodology for Steane-type fault-tolerant state preparation circuits that applies to arbitrary Calderbank-Shor-Steane (CSS) codes and does not rely on code symmetries. We apply the proposed methods to various CSS codes up to a distance of seven and simulate the successful fault-tolerant initialization of logical basis states under circuit-level depolarizing noise. The circuits synthesized using the proposed methodology provide an important step towards experimental realizations of high-fidelity ancilla states for near-term demonstration of fault-tolerant quantum computation.
We propose a method for universal fault-tolerant quantum computation using concatenated quantum error correcting codes. The concatenation scheme exploits the transversal properties of two different codes, combining them to provide a means to protect against low-weight arbitrary errors. We give the required properties of the error correcting codes to ensure universal fault tolerance and discuss a particular example using the 7-qubit Steane and 15-qubit Reed-Muller codes. Namely, other than computational basis state preparation as required by the DiVincenzo criteria, our scheme requires no special ancillary state preparation to achieve universality, as opposed to schemes such as magic state distillation. We believe that optimizing the codes used in such a scheme could provide a useful alternative to state distillation schemes that exhibit high overhead costs.
Fault-tolerant preparation of non-Clifford resource states is a major contributor to the overhead of quantum computation, motivating protocols that achieve high output fidelity with minimal qubit and circuit costs. We introduce a low-cost magic-state preparation protocol in which the choice of stabilizer generators is co-designed with the flag gadgets, allowing the syndrome-extraction circuit itself to filter correlated faults across a non-Clifford layer. The protocol prepares a logical plus state in the 15-qubit quantum Reed-Muller code, applies a transversal T gate, and gauge-fixes the same register into the seven-qubit Steane code. By reorganizing equivalent Z-type stabilizer generators into jointly flagged measurement groups, the protocol eliminates all accepted logical-error contributions arising from one or two circuit faults under destructive error detection. Under a uniform circuit-level depolarizing noise model, the postselected infidelity is 210.2p3+O(p4). At p=10−3, exact low-order enumeration combined with stratified sampling bounds the infidelity by 2.2×10−7 at 99.9% joint confidence, while retaining an acceptance probability of 86.9%. The complete circuit requires only 19 qubits and 82 CNOT gates. These results demonstrate that stabilizer-generator design can substantially reduce the cost of postselected magic-state preparation, although corrected operation and the fidelity of an unmeasured output block require separate analysis.
Steane code is one of the most widely studied quantum error-correction codes, which is a natural choice for fault-tolerant quantum computation (FTQC). However, the original Steane code is not fault-tolerant because the CNOT gates in an encoded block may cause error propagation. In this paper, we first propose a fault-tolerant encoding and decoding scheme, which analyzes all possible errors caused by each quantum gate in an error-correction period. In this scheme, we combine the results of measuring redundant qubits with those of syndrome measurements to identify specific errors for different types of errors. But due to the error propagation, there may be cases where different errors produce the same measurement results. Therefore, we introduce the "flag qubits" scheme (providing its usage conditions) to reduce error interference as much as possible, and we consider the errors caused by the introduced quantum gates, realizing the truly fault-tolerant Steane code. Afterwards, we provide the fault-tolerant scheme of the universal quantum gate set, including fault-tolerant preparation and verification of ancillary states. This is the first time that fault tolerance has been considered for every process of FTQC. Finally, We propose an algorithm for a more accurate estimation of thresholds and optimal error-correction period selection. Our simulation results based on this entire scheme demonstrate the effectiveness of this algorithm, satisfying the threshold theorem and the currently widely recognized threshold. We analyze the relationship among the maximum threshold, concatenated levels, and quantum logical depth, showing that quantum operations play a crucial role in increasing the threshold. Furthermore, we analyze the computational theoretical limits of quantum computers from the perspectives of attack and active defense based on our FTQC scheme, thereby assessing the security of a system.
Magic state distillation (MSD) is an essential element for universal fault-tolerant quantum computing, which distills a high-fidelity magic state from noisy magic states using ideal (error-corrected) Clifford operations. For ideal Clifford operations, it needs to be performed on the logical qubits and hence incurs a large spatiotemporal overhead, which is one of the major bottlenecks for the realization of fault-tolerant quantum computers (FTQCs). Here we propose zero-level distillation, which prepares a high-fidelity logical magic state at the physical level, namely , using physical qubits and nearest-neighbor two-qubit gates on a square lattice. We develop a zero-level distillation circuit and show that distillation can be made even more efficient than the conventional sophisticated approaches with logical level distillations. The key idea involves the Knill -type distillation using the Steane code and its careful mapping to the square-lattice architecture with error detection. The distilled magic state on the Steane-code state is then teleported or converted to surface codes. We numerically find that the error rate of the logical magic state scales as approximately 100 × p 2 in terms of the physical error rate p . For example, with a physical error rate of p = 10 − 4 ( 10 − 3 ), the logical error rate is reduced to p L = 10 − 6 ( 10 − 4 ), resulting in an improvement of 2 (1) orders of magnitude. This contributes to reducing both space and time overhead for early FTQC as well as full-fledged FTQC combined with conventional multilevel distillation protocols.