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Entanglement Purification with Quantum LDPC Codes and Iterative Decoding

Narayanan Rengaswamy, Nithin Raveendran, Ankur Raina, Bane Vasić

TL;DR

This work introduces entanglement purification protocols based on quantum LDPC codes to enable distributed fault-tolerant quantum computing, focusing on GHZ-state distillation. It develops a GHZ-map framework that induces joint codes across networked nodes and leverages a min-sum iterative decoder to achieve high thresholds at a fixed asymptotic yield $0.118$, with benchmark thresholds around $p_{th}oxed{≈0.107}$ and input fidelities near $0.7974$ for GHZ purification. Two protocols are proposed: Protocol I (general CSS-capable, with a two-round measurement burden) and Protocol II (CSS-focused, reduced overhead), both supporting scalable GHZ purification for multiple parties and networks. A purification-inspired algorithm for deriving logical Pauli operators, plus a detailed comparison to existing GHZ purification schemes, positions LDPC-based entanglement purification as a practical, high-threshold approach for quantum networks and distributed quantum computing. The results are complemented by a GHZ-state matrix identity, the extension to arbitrary $ ext{GHZ}^ ext{ } ext{ell}$, and open-source implementation resources.

Abstract

Recent constructions of quantum low-density parity-check (QLDPC) codes provide optimal scaling of the number of logical qubits and the minimum distance in terms of the code length, thereby opening the door to fault-tolerant quantum systems with minimal resource overhead. However, the hardware path from nearest-neighbor-connection-based topological codes to long-range-interaction-demanding QLDPC codes is a challenging one. Given the practical difficulty in building a monolithic architecture for quantum computers based on optimal QLDPC codes, it is worth considering a distributed implementation of such codes over a network of interconnected quantum processors. In such a setting, all syndrome measurements and logical operations must be performed using high-fidelity shared entangled states between the processing nodes. Since probabilistic many-to-1 distillation schemes for purifying entanglement are inefficient, we investigate quantum error correction based entanglement purification in this work. Specifically, we employ QLDPC codes to distill GHZ states, as the resulting high-fidelity logical GHZ states can interact directly with the code used to perform distributed quantum computing (DQC), e.g. for fault-tolerant Steane syndrome extraction. This protocol is applicable beyond DQC since entanglement purification is a quintessential task of any quantum network. We use the min-sum algorithm (MSA) based iterative decoder for distilling $3$-qubit GHZ states using a rate $0.118$ family of lifted product QLDPC codes and obtain an input threshold of $\approx 0.7974$ under i.i.d. single-qubit depolarizing noise. This represents the best threshold for a yield of $0.118$ for any GHZ purification protocol. Our results apply to larger size GHZ states as well, where we extend our technical result about a measurement property of $3$-qubit GHZ states to construct a scalable GHZ purification protocol.

Entanglement Purification with Quantum LDPC Codes and Iterative Decoding

TL;DR

This work introduces entanglement purification protocols based on quantum LDPC codes to enable distributed fault-tolerant quantum computing, focusing on GHZ-state distillation. It develops a GHZ-map framework that induces joint codes across networked nodes and leverages a min-sum iterative decoder to achieve high thresholds at a fixed asymptotic yield , with benchmark thresholds around and input fidelities near for GHZ purification. Two protocols are proposed: Protocol I (general CSS-capable, with a two-round measurement burden) and Protocol II (CSS-focused, reduced overhead), both supporting scalable GHZ purification for multiple parties and networks. A purification-inspired algorithm for deriving logical Pauli operators, plus a detailed comparison to existing GHZ purification schemes, positions LDPC-based entanglement purification as a practical, high-threshold approach for quantum networks and distributed quantum computing. The results are complemented by a GHZ-state matrix identity, the extension to arbitrary , and open-source implementation resources.

Abstract

Recent constructions of quantum low-density parity-check (QLDPC) codes provide optimal scaling of the number of logical qubits and the minimum distance in terms of the code length, thereby opening the door to fault-tolerant quantum systems with minimal resource overhead. However, the hardware path from nearest-neighbor-connection-based topological codes to long-range-interaction-demanding QLDPC codes is a challenging one. Given the practical difficulty in building a monolithic architecture for quantum computers based on optimal QLDPC codes, it is worth considering a distributed implementation of such codes over a network of interconnected quantum processors. In such a setting, all syndrome measurements and logical operations must be performed using high-fidelity shared entangled states between the processing nodes. Since probabilistic many-to-1 distillation schemes for purifying entanglement are inefficient, we investigate quantum error correction based entanglement purification in this work. Specifically, we employ QLDPC codes to distill GHZ states, as the resulting high-fidelity logical GHZ states can interact directly with the code used to perform distributed quantum computing (DQC), e.g. for fault-tolerant Steane syndrome extraction. This protocol is applicable beyond DQC since entanglement purification is a quintessential task of any quantum network. We use the min-sum algorithm (MSA) based iterative decoder for distilling -qubit GHZ states using a rate family of lifted product QLDPC codes and obtain an input threshold of under i.i.d. single-qubit depolarizing noise. This represents the best threshold for a yield of for any GHZ purification protocol. Our results apply to larger size GHZ states as well, where we extend our technical result about a measurement property of -qubit GHZ states to construct a scalable GHZ purification protocol.
Paper Structure (29 sections, 6 theorems, 44 equations, 5 figures, 4 algorithms)

This paper contains 29 sections, 6 theorems, 44 equations, 5 figures, 4 algorithms.

Table of Contents

  1. Introduction
  2. Main Results and Discussion
  3. Purifying Bell Pairs with QLDPC Codes
  4. New Protocols to Purify GHZ States with QLDPC Codes
  5. Discussion and Connections to Existing GHZ Purification Protocols
  6. Decoding QLDPC Codes under Realistic Noise Models
  7. Purification-Inspired Algorithm to Generate Logical Pauli Operators
  8. Notation and Background
  9. Revisiting the Bell Pair Distillation Protocol
  10. Bell Pair Distillation using the 5-Qubit Code
  11. Distillation of Greenberger-Horne-Zeilinger (GHZ) States
  12. GHZ State Matrix Identity
  13. Protocol I
  14. Distillation-Inspired Algorithm to Generate Logical Pauli Operators
  15. Output Fidelity of GHZ Distillation Protocol
  16. ...and 14 more sections

Key Result

Lemma 3

Let $M = \sum_{x,y \in \mathbb{F}_2^n} M_{xy} \IfNoValueTF{y} {\left\lvert x \right\rangle \left\langle x \right\vert} {\left\lvert x \right\rangle \left\langle y \right\rvert} \in \mathbb{C}^{2^n \times 2^n}$ be any matrix acting on Alice's qubits. Then,

Figures (5)

  • Figure 1: (top) The performance of a family of lifted product QLDPC codes with asymptotic rate $0.118$ using the sequential schedule of the min-sum algorithm (MSA) based decoder. Each data point is obtained by counting $100$ logical errors. (bottom) The threshold is about $10.6$-$10.7\%$. These results apply to Bell pair purification, up to a rescaling of the depolarizing probabilities.
  • Figure 2: Protocol II for GHZ purification using CSS codes. The protocol can be extended to general stabilizer codes through additional diagonal Clifford operations as in Protocol I. Alice generates $n$ copies of the ideal $3$-qubit GHZ state and marks one qubit of each triple as 'A', another as 'B', and the third as 'C'. She measures the stabilizers of the QLDPC code on qubits 'A' and classically communicates the results through a noiseless channel to Charlie. She also uses the results to appropriately measure stabilizers of (a potentially equivalent) QLDPC code on qubits 'B' and communicates these results to both Bob and Charlie, again through noiseless classical channels. She sends qubits 'B' to Bob and qubits 'C' to Charlie. Finally, both Bob and Charlie make stabilizer measurements, correct errors, and then all three parties invert the encoding to convert the logical GHZ states to physical GHZ states. Note that Bob and Charlie can perform their operations asynchronously.
  • Figure 3: (top) Protocol II performance of a family of lifted product QLDPC codes with asymptotic rate $0.118$ using the sequential schedule of the min-sum algorithm (MSA) based decoder. Each data point is obtained by counting almost $10^4$ logical errors except depolarizing probability $0.09$, which was obtained from $10^3$ logical errors. (bottom) The threshold is about $10.7\%$.
  • Figure 4: The QEC-based entanglement distillation protocol of Wilde et al. Wilde-isit10. Figure adapted from Wilde-isit10.
  • Figure 5: Performance of variations of the GHZ distillation protocol using the $[\![ 5,1,3 ]\!]$ perfect code, and comparison with the standard QEC performance of the same code on the depolarizing channel. The decoder employs a maximum likelihood decoding scheme that identifies a minimal weight error pattern matching the syndrome.

Theorems & Definitions (12)

  • Remark 1
  • Remark 2
  • Lemma 3
  • Lemma 4
  • Lemma 5
  • Theorem 6
  • Example 1
  • Theorem 7
  • Remark 8
  • Lemma 9
  • ...and 2 more