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State-adaptive quantum error correction and fault-tolerant quantum computing

D. -S. Wang

TL;DR

The paper introduces state-adaptive quantum error correction (SAQEC), where prior knowledge of the input quantum state enables error correction without requiring decoupling from the environment. It establishes the state-adaptive quantum capacity as $Q_{SA}(\Phi)=\frac{1}{2}\max_{\rho} I(\rho,\Phi)$, via simulation of entanglement-assisted coding and a direct Petz-map based argument, and shows SA can function as an analogue to classical channel capacity in the quantum setting. It then proposes a practical fault-tolerant universal quantum computing scheme using stabilizer codes, gate teleportation, and code switching to realize universal Clifford+$T$ computation with reduced overhead, potentially benefiting current photonic and solid-state platforms. The work connects quantum channel capacities with state-aware strategies, opening avenues for channel-adaptive and state-adaptive coding and providing a unified view of SA and EA roles in quantum information processing.

Abstract

We present a theoretical framework for state-adaptive quantum error correction that bridges the gap between quantum computing and error correction paradigms. By incorporating knowledge of quantum states into the error correction process, we establish a new capacity regime governed by quantum mutual information rather than coherent information. This approach reveals a fundamental connection to entanglement-assisted protocols. We demonstrate practical applications in fault-tolerant quantum computation, showing how state-adaptivity enables enhanced error correction without additional measurement overhead. The framework provides insights into quantum channel capacities while offering implementation advantages for current quantum computing platforms.

State-adaptive quantum error correction and fault-tolerant quantum computing

TL;DR

The paper introduces state-adaptive quantum error correction (SAQEC), where prior knowledge of the input quantum state enables error correction without requiring decoupling from the environment. It establishes the state-adaptive quantum capacity as , via simulation of entanglement-assisted coding and a direct Petz-map based argument, and shows SA can function as an analogue to classical channel capacity in the quantum setting. It then proposes a practical fault-tolerant universal quantum computing scheme using stabilizer codes, gate teleportation, and code switching to realize universal Clifford+ computation with reduced overhead, potentially benefiting current photonic and solid-state platforms. The work connects quantum channel capacities with state-aware strategies, opening avenues for channel-adaptive and state-adaptive coding and providing a unified view of SA and EA roles in quantum information processing.

Abstract

We present a theoretical framework for state-adaptive quantum error correction that bridges the gap between quantum computing and error correction paradigms. By incorporating knowledge of quantum states into the error correction process, we establish a new capacity regime governed by quantum mutual information rather than coherent information. This approach reveals a fundamental connection to entanglement-assisted protocols. We demonstrate practical applications in fault-tolerant quantum computation, showing how state-adaptivity enables enhanced error correction without additional measurement overhead. The framework provides insights into quantum channel capacities while offering implementation advantages for current quantum computing platforms.

Paper Structure

This paper contains 12 sections, 1 theorem, 11 equations, 2 figures, 1 table.

Table of Contents

  1. Introduction
  2. Channel capacity
  3. State-adaptive quantum error correction
  4. Equivalence with EA setting
  5. Direct proof of state-adaptive quantum capacity
  6. Relation among capacities
  7. Transversal fault-tolerant universal quantum computing via coding switching
  8. Conclusion
  9. Acknowledgement
  10. Channel capacity
  11. Examples of noise channels
  12. Physical implementation

Key Result

Theorem 1

The state-adaptive quantum capacity $Q_{SA}(\Phi)$ of a quantum channel $\Phi$ is half of the maximal quantum mutual information

Figures (2)

  • Figure 1: The diagram to show the relation among the coding models and their capacities, with $C_H \geq P \geq Q$, $C_{EA} \geq C_H$, $Q_{EA} \geq Q$, and $C_{EA} = 2Q_{EA}$. Here $C_H$, $P$, $Q$ are the Holevo, private, and quantum capacity of a quantum channel, respectively.
  • Figure 2: Schematics for the fault-tolerant quantum computing scheme based on gate teleportation and code switching. A transversal logical Clifford gate $G$ is stored in its Choi state $|G\rangle$, protected by SAQEC and executed by gate teleportation. A transversal logical gate $T$ is performed by a back and forth code switching (the triangles) and transversal $T$ on the new code, protected by usual QEC and executed by gate teleportation. The arrows show the time flow.

Theorems & Definitions (3)

  • Theorem 1
  • proof
  • proof