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1. Scalable Quantum Error Correction for Surface Codes using FPGA

   https://doi.org/10.1109/FCCM57271.2023.00045  

   Liyanage, Namitha; Wu, Wu; Deters, Alexander; Zhong, Lin (September 2023, IEEE)

   A fault-tolerant quantum computer must decode and correct errors faster than they appear. The faster errors can be corrected, the more time the computer can do useful work. The Union-Find (UF) decoder is promising with an average time complexity slightly higher than O(d3). We report a distributed version of the UF decoder that exploits parallel computing resources for further speedup. Using an FPGA-based implementation, we empirically show that this distributed UF decoder has a sublinear average time complexity with regard to d, given O(d3) parallel computing resources. The decoding time per measurement round decreases as d increases, a first time for a quantum error decoder. The implementation employs a scalable architecture called Helios that organizes parallel computing resources into a hybrid tree-grid structure. We are able to implement d up to 21 with a Xilinx VCU129 FPGA, for which an average decoding time is 11.5 ns per measurement round under phenomenological noise of 0.1%, significantly faster than any existing decoder implementation. Since the decoding time per measurement round of Helios decreases with d, Helios can decode a surface code of arbitrarily large d without a growing backlog. 

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2. FPGA-Based Distributed Union-Find Decoder for Surface Codes

   https://doi.org/10.1109/TQE.2024.3467271  

   Liyanage, Namitha; Wu, Yue; Tagare, Siona; Zhong, Lin (January 2024, IEEE Transactions on Quantum Engineering)

   A fault-tolerant quantum computer must decode and correct errors faster than they appear to prevent exponential slowdown due to error correction. The Union-Find (UF) decoder is promising with an average time complexity slightly higher than $O(d^3)$. We report a distributed version of the UF decoder that exploits parallel computing resources for further speedup. Using an FPGA-based implementation, we empirically show that this distributed UF decoder has a sublinear average time complexity with regard to $$d$$, given $O(d^3)$ parallel computing resources. The decoding time per measurement round decreases as $$d$$ increases, the first time for a quantum error decoder. The implementation employs a scalable architecture called Helios that organizes parallel computing resources into a hybrid tree-grid structure. Using a Xilinx VCU129 FPGA, we successfully implement $$d$$ up to 21 with an average decoding time of 11.5 ns per measurement round under 0.1\% phenomenological noise, and 23.7 ns for $$d=17$$ under equivalent circuit-level noise. This performance is significantly faster than any existing decoder implementation. Furthermore, we show that \name can optimize for resource efficiency by decoding $$d=51$ on a Xilinx VCU129 FPGA with an average latency of 544 ns per measurement round. 

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3. Belief-Propagation with Quantum Messages for Polar Codes on Classical-Quantum Channels

   Mandal, Avijit; Brandsen; S. Pfister, Henry D. (January 2023, 2023 IEEE International Symposium on Information Theory (ISIT))

   This paper considers the design and decoding of polar codes for general classical-quantum (CQ) channels. It focuses on decoding via belief-propagation with quantum messages (BPQM) and, in particular, the idea of paired-measurement BPQM (PM-BPQM) decoding. Since the PM-BPQM decoder admits a classical density evolution (DE) analysis, one can use DE to design a polar code for any CQ channel and then efficiently compute the trade-off between code rate and error probability. We have also implemented and tested a classical simulation of our PM-BPQM decoder for polar codes. While the decoder can be implemented efficiently on a quantum computer, simulating the decoder on a classical computer actually has exponential complexity. Thus, simulation results for the decoder are somewhat limited and are included primarily to validate our theoretical results. 

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4. Belief-Propagation with Quantum Messages for Polar Codes on Classical-Quantum Channels, IEEE International Symposium on Information Theory, Taipei, Taiwan, June 25-30, 2023

   https://doi.org/10.1109/ISIT54713.2023.10206723  

   Mandal, Avijit; Brandsen, S; Pfister, Henry D. (June 2023, Abstracts of papers IEEE International Symposium on Information Theory)

   This paper considers the design and decoding of polar codes for general classical-quantum (CQ) channels. It focuses on decoding via belief-propagation with quantum messages (BPQM) and, in particular, the idea of paired-measurement BPQM (PM-BPQM) decoding. Since the PM-BPQM decoder admits a classical density evolution (DE) analysis, one can use DE to design a polar code for any CQ channel and then efficiently compute the trade-off between code rate and error probability. We have also implemented and tested a classical simulation of our PM-BPQM decoder for polar codes. While the decoder can be implemented efficiently on a quantum computer, simulating the decoder on a classical computer actually has exponential complexity. Thus, simulation results for the decoder are somewhat limited and are included primarily to validate our theoretical results. 

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5. Trapping Sets of Quantum LDPC Codes

   https://doi.org/10.22331/q-2021-10-14-562  

   Raveendran, Nithin; Vasić, Bane (October 2021, Quantum)

   Iterative decoders for finite length quantum low-density parity-check (QLDPC) codes are attractive because their hardware complexity scales only linearly with the number of physical qubits. However, they are impacted by short cycles, detrimental graphical configurations known as trapping sets (TSs) present in a code graph as well as symmetric degeneracy of errors. These factors significantly degrade the decoder decoding probability performance and cause so-called error floor. In this paper, we establish a systematic methodology by which one can identify and classify quantum trapping sets (QTSs) according to their topological structure and decoder used. The conventional definition of a TS from classical error correction is generalized to address the syndrome decoding scenario for QLDPC codes. We show that the knowledge of QTSs can be used to design better QLDPC codes and decoders. Frame error rate improvements of two orders of magnitude in the error floor regime are demonstrated for some practical finite-length QLDPC codes without requiring any post-processing. 

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