Quantum subspace verification for error correction codes

Abstract

Benchmarking the performance of quantum error correction codes in physical systems is crucial for achieving fault-tolerant quantum computing. Current methodologies, such as (shadow) tomography or direct fidelity estimation, fall short in efficiency due to the neglect of possible prior knowledge about quantum states. To address the challenge, we introduce a framework of quantum subspace verification, employing the knowledge of quantum error correction code subspaces to reduce the potential measurement budgets. Specifically, we give the sample complexity to estimate the fidelity to the target subspace under some confidence level. Building on the framework, verification operators are developed, which can be implemented with experiment-friendly local measurements for stabilizer codes and quantum low-density parity-check (QLDPC) codes. Our constructions require $O(n-k)$ local measurement settings for both, and the sample complexity of $O(n-k)$ for stabilizer codes and of $O((n-k{)}^2)$ for generic QLDPC codes, where $n$ and $k$ are the numbers of physical and logical qubits, respectively. Notably, for certain codes like the notable Calderbank-Shor-Steane codes and QLDPC stabilizer codes, the setting number and sample complexity can be significantly reduced and are even independent of $n$. In addition, by combining the proposed subspace verification and direct fidelity estimation, we construct a protocol to verify the fidelity of general magic logical states with exponentially smaller sample complexity than previous methods. Our finding facilitates efficient and feasible verification of quantum error correction codes and also magical states, advancing the realization in practical quantum platforms.

15 pages, 1 figure

References in corpus (42)

• Supplementary information for "Quantum supremacy using a programmable superconducting processor"
• Quantum cryptography: Public key distribution and coin tossing
• Encoding a qubit in an oscillator
• Universal Quantum Computation with ideal Clifford gates and noisy ancillas
• Predicting Many Properties of a Quantum System from Very Few Measurements
• Strong quantum computational advantage using a superconducting quantum processor
• Continuous-variable optical quantum state tomography
• Quantum state tomography via compressed sensing
• Quantum error correction below the surface code threshold
• Efficient quantum state tomography
• Surface code quantum computing by lattice surgery
• Direct Fidelity Estimation from Few Pauli Measurements
• Magic state distillation with low overhead
• Detecting Genuine Multipartite Entanglement with Two Local Measurements
• Realization of an Error-Correcting Surface Code with Superconducting Qubits
• Quantum LDPC codes with positive rate and minimum distance proportional to n^{1/2}
• Entanglement Detection in the Stabilizer Formalism
• Learning Quantum Systems
• Robust shadow estimation
• Toolbox for entanglement detection and fidelity estimation
• Theory of quantum system certification: a tutorial
• Optimal verification of entangled states with local measurements
• Efficient Verification of Pure Quantum States in the Adversarial Scenario
• A study of LOCC-detection of a maximally entangled state using hypothesis testing
• General framework for verifying pure quantum states in the adversarial scenario
• Efficient Verification of Hypergraph States
• Optimal verification of general bipartite pure states
• Efficient verification of Dicke states
• Verifiable fault-tolerance in measurement-based quantum computation
• Detecting multipartite entanglement structure with minimal resources
• Optimal verification of stabilizer states
• Partially Fault-tolerant Quantum Computing Architecture with Error-corrected Clifford Gates and Space-time Efficient Analog Rotations
• Efficient verification of quantum processes
• Efficient verification of quantum gates with local operations
• Approximate Quantum Error Correction Revisited: Introducing the Alpha-bit
• Quantum gate verification and its application in property testing
• Universally Optimal Verification of Entangled States with Nondemolition Measurements
• Learning logical Pauli noise in quantum error correction
• Efficient verification of Affleck-Kennedy-Lieb-Tasaki states
• Efficient Verification of Ground States of Frustration-Free Hamiltonians
• Efficient verification of entangled continuous-variable quantum states with local measurements
• Versatile fidelity estimation with confidence

Cited by in corpus (1)

• Optimal randomized measurements for a family of non-linear quantum properties