Abstract Quantum error correction (QEC) is the core mechanism enabling scalable and fault-tolerant quantum computation. Among various QEC schemes, Calderbank–Shor–Steane (CSS) codes play a central role. Efficiently verifying whether a quantum state remains within the target CSS code subspace is therefore essential both for understanding the theoretical performance of these codes and for assessing the error-correcting capability of practical quantum platforms. In this work, we develop an efficient framework for verifying whether a quantum system remains within the encoded subspace of a CSS code. We first show that the stabilizer structure of any CSS code can be mapped to a two-colorable graph, enabling the design of a hybrid verification strategy based solely on local Pauli measurements. We further analytically optimize the measurement probabilities and derive the resulting sample complexities for representative CSS codes. Specifically, for the Toric code, the efficiency of our hybrid strategy increases with lattice size and asymptotically approaches near-optimal performance. These results establish a unified and scalable approach to CSS code subspace verification, providing a foundation for certifiable and fault-tolerant quantum information processing.
Abstract Quantum error correction (QEC) is the core mechanism enabling scalable and fault-tolerant quantum computation. Among various QEC schemes, Calderbank–Shor–Steane (CSS) codes play a central role. Efficiently verifying whether a quantum state remains within the target CSS code subspace is therefore essential both for understanding the theoretical performance of these codes and for assessing the error-correcting capability of practical quantum platforms. In this work, we develop an efficient framework for verifying whether a quantum system remains within the encoded subspace of a CSS code. We first show that the stabilizer structure of any CSS code can be mapped to a two-colorable graph, enabling the design of a hybrid verification strategy based solely on local Pauli measurements. We further analytically optimize the measurement probabilities and derive the resulting sample complexities for representative CSS codes. Specifically, for the Toric code, the efficiency of our hybrid strategy increases with lattice size and asymptotically approaches near-optimal performance. These results establish a unified and scalable approach to CSS code subspace verification, providing a foundation for certifiable and fault-tolerant quantum information processing.
Abstract We address the task of verifying whether a quantum computer, designed to be protected by a specific stabilizer code, correctly encodes the corresponding logical qubits. To achieve this, we develop a general framework for subspace verification and explore several practically significant stabilizer code subspaces. First, we present two efficient verification strategies for general stabilizer code subspaces, using measurements of their stabilizer generators and stabilizer groups, respectively. Then, building on the observation that certain tests can be conducted in parallel due to specific structural properties of the subspace, we propose a coloring strategy tailored to graph code subspaces and an XZ strategy for Calderbank–Shor–Steane code subspaces. Compared to stabilizer-based strategies, these new strategies require significantly fewer measurement settings and consume fewer state copies, achieving near-global optimality. Notably, all the strategies employ a limited number of Pauli measurements, are non-adaptive, and work for mixed states, enabling efficient experimental certification of whether quantum states remain within the target code subspace in noisy quantum computers. This work presents the first systematic study of efficient verification of stabilizer code subspaces with local measurements.
Stabilizer codes lie at the heart of modern quantum-error-correcting codes (QECCs). Of particular importance is a class called Calderbank-Shor-Steane (CSS) codes, which includes many important examples such as toric codes, color codes, and fractons. Recent studies have revealed that the decoding transition for these QECCs could be intrinsically captured by calculating information-theoretic quantities from the mixed state. Here we perform a simple analytic calculation of the coherent information for general CSS codes under local incoherent Pauli errors via diagonalization of the density matrices and mapping to classical statistical mechanical (SM) models. Our result establishes a rigorous connection between the decoding transition of the quantum code and the phase transition in the random classical SM model. It is also directly confirmed for CSS codes that exact error correction is possible if and only if the maximum-likelihood decoder always succeeds in the thermodynamic limit. Thus, the fundamental threshold is saturated by the optimal decoder.
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.
Stabilizer states are extensively studied in quantum information theory for their structures based on the Pauli group. Calderbank-Shor-Steane (CSS) stabilizer states are of particular importance in their application to fault-tolerant quantum computation (FTQC). However, how to fault-tolerantly prepare arbitrary CSS stabilizer states for general CSS stabilizer codes is still unknown, and their preparation can be highly costly in computational resources. In this paper, we show how to prepare a large class of CSS stabilizer states useful for FTQC. We propose distillation protocols using syndrome encoding by classical codes or quantum CSS codes. Along the same lines, we show that classical coding techniques can reduce the ancilla consumption in Steane syndrome extraction by using additional transversal controlled-not gates and classical computing power. In the scenario of a fixed ancilla consumption rate, we can increase the frequency of quantum error correction and effectively lower the error rate.
Quantum error correction is pivotal in advancing toward large-scale quantum computation, and efficient verification is crucial for ensuring the high fidelity of code states. Traditional methodologies, such as state tomography, direct fidelity estimation, and state verification, either fall short in measurement efficiency, especially for large-scale systems, or are restricted to some specific states. In this work, we introduce a general framework for quantum subspace verification, enabling efficient and measurement-noise-robust fidelity estimation between a given state and the target subspace with a specified confidence level. By integrating the proposed subspace verification with direct fidelity estimation, we develop a composite protocol that significantly improves the efficiency of verifying the fidelity of general magic logical states, as demonstrated by intuitive numerical results. This improvement stems from the use of subspace verification, which leverages the knowledge of code subspaces to significantly reduce measurement costs. Additionally, we detail the construction of verification operators for typical error correction codes, including general stabilizer codes and quantum low-density parity-check codes, enabling their efficient implementation using practical local measurements. Notably, for certain codes, such as the Calderbank-Shor-Steane codes and quantum low-density parity-check stabilizer codes, we reduce the number of required measurement settings and sample complexity to a constant level using graphical methods. Our approach facilitates efficient and feasible verification of error correction codes and generic magic logical states, advancing their practical implementations on quantum platforms.
Quantum error correction (QEC) is an effective way to overcome quantum noise and de-coherence, meanwhile the fault tolerance of the encoding circuit, syndrome measurement circuit, and logical gate realization circuit must be ensured so as to achieve reliable quantum computing. Steane code is one of the most famous codes, proposed in 1996, however, the classical encoding circuit based on stabilizer implementation is not fault-tolerant. In this paper, we propose a method to design a fault-tolerant encoding circuit for Calderbank-Shor-Steane (CSS) code based on stabilizer implementation and “flag” bits. We use the Steane code as an example to depict in detail the fault-tolerant encoding circuit design process including the logical operation implementation, the stabilizer implementation, and the “flag” qubits design. The simulation results show that assuming only one quantum gate will be wrong with a certain probability p, the classical encoding circuit will have logic errors proportional to p; our proposed circuit is fault-tolerant as with the help of the “flag” bits, all types of errors in the encoding process can be accurately and uniquely determined, the errors can be fixed. If all the gates will be wrong with a certain probability p, which is the actual situation, the proposed encoding circuit will also be wrong with a certain probability, but its error rate has been reduced greatly from p to p2 compared with the original circuit. This encoding circuit design process can be extended to other CSS codes to improve the correctness of the encoding circuit.