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arXiv:2409.19699 · Quantum Science and Technology

Efficient verification of stabilizer code subspaces with local measurements

Congcong Zheng, Xutao Yu, Zaichen Zhang, Ping Xu, Kun Wang

Published May 22, 2026• 2 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1088/2058-9565/ae723b

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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.
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.
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.
Protecting information in systems that have more than two basis states (qudits) not only offers a promising route for reducing the number of individual quantum locations that must be protected, while more accurately reflecting the structure of realistic quantum hardware, but also has some possibly enticing foundational strengths. While work in the past has largely focused on protecting information in quantum devices with locations that are some consistent local structure, this work considers coding-theoretic constraints on devices constructed from locations which may vary in their local structures -- these are mixed-register quantum devices. In this work we provide some general results for mixed-register Pauli operators, then identify some stabilizer encoded information forms that are forbidden. Building on these insights, we construct coding-theoretically optimal mixed-register stabilizer codes from sets of codes defined on coprime local-dimensions. The construction of such codes results in codes with logical subspaces that do not directly correspond to any of the constituent local-dimensions.
Abstract We develop explicit device-independent self-testing constructions for logical subspaces of quantum error-correcting codes. By constructing multipartite Bell inequalities tailored to stabilizer code structures, we show that maximal quantum violations uniquely certify that the underlying shared state resides within the logical subspace of a given code, up to local isometries. These constructions extend earlier stabilizer-subspace self-testing results beyond the five-qubit code to several representative stabilizer codes, including the six-qubit code, the Steane code, and Shor’s nine-qubit code. The tilted sum-of-squares variants further impose algebraic constraints on the relative weights of the logical sectors. Together, these results provide explicit examples of device-independent self-testing for representative stabilizer-code logical subspaces and suggest possible directions for verification and benchmarking of encoded quantum information.
Statistical verification of a quantum state aims to certify whether a given unknown state is close to the target state with confidence. So far, sample-optimal verification protocols based on local measurements have been found only for disparate groups of states: bipartite pure states, GHZ states, and antisymmetric basis states. In this work, we investigate systematically optimal verification of entangled stabilizer states using Pauli measurements. First, we provide a lower bound on the sample complexity of any verification protocol based on separable measurements, which is independent of the number of qubits and the specific stabilizer state. Then we propose a simple algorithm for constructing optimal protocols based on Pauli measurements. Our calculations suggest that optimal protocols based on Pauli measurements can saturate the above bound for all entangled stabilizer states, and this claim is verified explicitly for states up to seven qubits. Similar results are derived when each party can choose only two measurement settings, say X and Z. Furthermore, by virtue of the chromatic number, we provide an upper bound for the minimum number of settings required to verify any graph state, which is expected to be tight. For experimentalists, optimal protocols and protocols with the minimum number of settings are explicitly provided for all equivalent classes of stabilizer states up to seven qubits. For theorists, general results on stabilizer states (including graph states in particular) and related structures derived here may be of independent interest beyond quantum state verification.
Efficiently characterizing quantum error correcting codes is a key challenge on the path to fault-tolerant quantum computation. Stabilizer codes, a central class of such codes, are defined by a set of stabilizer generators. Here, we present an algorithm that uses random single-qubit measurements to learn the stabilizer generators of any stabilizer code from NNN copies of stabilizer states in its codespace, requiring no prior knowledge of the code's structure. This also enables verification that a device implements its intended code. We derive a lower bound on NNN needed to recover the stabilizer generators with high probability, together with a bound on the algorithm's overall probability of success. When applied to quantum low-density parity-check (qLDPC) codes, a leading candidate for practical fault-tolerant architectures, our approach requires a number of states that scales polylogarithmically with nnn, the number of qubits.