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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. Single-Shot Decoding of Good Quantum LDPC Codes

   https://doi.org/10.1007/s00220-024-04951-6  

   Gu, Shouzhen ; Tang, Eugene ; Caha, Libor ; Choe, Shin Ho ; He, Zhiyang ; Kubica, Aleksander ( March 2024 , Communications in Mathematical Physics)

   Quantum Tanner codes constitute a family of quantum low-density parity-check codes with good parameters, i.e., constant encoding rate and relative distance. In this article, we prove that quantum Tanner codes also facilitate single-shot quantum error correction (QEC) of adversarial noise, where one measurement round (consisting of constant-weight parity checks) suffices to perform reliable QEC even in the presence of measurement errors. We establish this result for both the sequential and parallel decoding algorithms introduced by Leverrier and Zémor. Furthermore, we show that in order to suppress errors over multiple repeated rounds of QEC, it suffices to run the parallel decoding algorithm for constant time in each round. Combined with good code parameters, the resulting constant-time overhead of QEC and robustness to (possibly time-correlated) adversarial noise make quantum Tanner codes alluring from the perspective of quantum fault-tolerant protocols.

    

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

   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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