Entanglement Purification with Quantum LDPC Codes and Iterative Decoding

Narayanan Rengaswamy1, Nithin Raveendran1, Ankur Raina2, and Bane Vasić1

1Department of Electrical and Computer Engineering, University of Arizona, Tucson, Arizona 85721, USA
2Department of Electrical Engineering and Computer Sciences, Indian Institute of Science Education and Research, Bhopal, Madhya Pradesh 462066, India

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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 likely a challenging one. Given the practical difficulty in building a monolithic architecture for quantum systems, such as computers, based on optimal QLDPC codes, it is worth considering a distributed implementation of such codes over a network of interconnected medium-sized quantum processors. In such a setting, all syndrome measurements and logical operations must be performed through the use of 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 the application of DQC since entanglement distribution and purification is a quintessential task of any quantum network. We use the min-sum algorithm (MSA) based iterative decoder with a sequential schedule for distilling $3$-qubit GHZ states using a rate $0.118$ family of lifted product QLDPC codes and obtain an input fidelity 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.

Our software is available github and zenodo.

Quantum error correction is essential to build reliable and scalable quantum computers. The optimal quantum error correcting codes require a high amount of long-range connectivity between qubits in the hardware, which is difficult to implement. Given this practical challenge, a distributed implementation of these codes becomes a viable approach, where long-range connectivity can be realized via shared high-fidelity entangled states such as Greenberger-Horne-Zeilinger (GHZ) states. However, in this case, one needs an efficient mechanism to purify the noisy GHZ states generated in hardware and match the fidelity requirements of the distributed implementation of the optimal codes. In this work, we develop a new technical insight on GHZ states and use that to design a new protocol to efficiently distill high-fidelity GHZ states using the same optimal codes that would be used to build the distributed quantum computer. The minimum required input fidelity for our protocol is far better than any other protocol in the literature for GHZ states. Besides, the distilled GHZ states can seamlessly interact with the states of the distributed computer because they belong to the same optimal quantum error correcting code.

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