Quantum Error Correction for Surface Codes using FPGA

FPGA-Based Surface Code Decoding

Scalable Quantum Error Correction with FPGA

• Scalable Quantum Error Correction for Surface Codes using FPGA [1]: This paper discusses the necessity for a fault-tolerant quantum computer to decode and correct errors faster than they occur. It implies the use of FPGAs to achieve the speed required for real-time error correction in quantum computing systems.

Learning to Decode Surface Codes

• Learning to Decode the Surface Code with a Recurrent, Transformer [2]: The study highlights the use of neural networks with a nearly constant number of parameters across different code distances, suggesting a potential application for FPGAs in implementing such neural network decoders for surface codes.

Multi-Round Subsystem Quantum Error Correction

• Demonstrating multi-round subsystem quantum error correction using [3]: This research extends to decoding strategies that could be implemented on FPGAs. It describes a decoder as an algorithm that inputs syndrome measurements from an error-correcting code and outputs a correction, which is a task suitable for FPGA implementation.

LILLIPUT: Low-Latency FPGA Decoder

• LILLIPUT: A Lightweight Low-Latency Lookup-Table Decoder for Near [4]: The paper proposes LILLIPUT, a low-latency lookup table decoder for small surface codes that requires less than 7% of logic elements on FPGAs. It offers high accuracy and can be seamlessly integrated with existing quantum devices, making it a promising solution for near-term quantum error correction.

FPGA-Oriented LDPC Decoder

• FPGA-Oriented LDPC Decoder for Cyber-Physical Systems [6]: Although this paper focuses on the hardware implementation of a decoder for Quasi-Cyclic Low-Density Parity-Check (QC-LDPC) codes, the principles and techniques discussed could be relevant for the implementation of surface code decoders on FPGAs.

Scalable and Fast ANN Syndrome Decoder

• A scalable and fast artificial neural network syndrome decoder for [7]: This study reports on a scalable and fast decoder implementation for surface code syndrome processing, which could potentially be implemented on FPGAs given their capability to handle complex computations efficiently.

Real-Time Decoding for Fault-Tolerant Quantum Computing

• Real-time decoding for fault-tolerant quantum computing [16]: This paper reviews the state of real-time decoding for fault-tolerant quantum computing, including the use of FPGAs. It discusses the challenges and potential solutions for implementing decoders on FPGAs, which are expected to be an integral part of the control systems for quantum computers.

Techniques for Combining Fast Local Decoders with Global Decoders

• Techniques for combining fast local decoders with global decoders [18]: The paper provides a review of the rotated surface code and discusses the need for fast decoders, which could be implemented on FPGAs. It also estimates the resource cost for implementing neural network decoders on FPGAs.

Challenges

Real-Time Decoding Requirements

• Speed and Latency: Surface code decoding must be performed at speeds that match or exceed the rate at which errors occur in a quantum computer. FPGAs must be able to process error syndromes and apply corrections in real-time to prevent the accumulation of errors[1][7][10][11][17][19].

• Backlog Problem: If the decoder cannot keep up with the error syndromes generated by the qubits, a backlog of unprocessed data can build up, leading to an exponential slowdown of the quantum computer[11][17].

Resource Constraints and Optimization

• Resource Utilization: FPGAs have limited resources in terms of logic elements, memory, and processing power. Efficient use of these resources is crucial, especially when dealing with larger code distances and more complex decoding algorithms[4][13][14].

• Memory Overhead: Some decoding algorithms, such as lookup table (LUT) based decoders, can incur significant memory overhead. Compressed LUTs and other optimization techniques may be necessary to reduce memory requirements[4][13].

Algorithmic Complexity

• Complex Decoding Algorithms: Surface code decoding algorithms, such as the Union-Find (UF) decoder, can be complex and require careful implementation to ensure they operate efficiently on FPGAs[1][8].

• Parallelism and Scalability: Exploiting the parallel computing capabilities of FPGAs is essential for achieving the necessary speedup for decoding. This requires a scalable architecture that can handle increasing code distances[1][8].

Integration with Quantum Systems

• Integration with Qubit Control Systems: The FPGA-based decoders need to be seamlessly integrated with the control systems of quantum computers to ensure timely application of corrections[5][7][17].

• Handling Continuous Error Correction: Some quantum error correction schemes, like continuous quantum error correction, require the FPGA to actively correct errors as they are detected, which poses additional challenges for real-time operation[2].

Noise and Error Rates

• Error Syndrome Density: Decoding strategies must be able to handle high error syndrome densities, which can be challenging for both local and global decoders[18][20].

• Threshold Values: Decoders must maintain error rates below certain threshold values to ensure fault tolerance. This requires careful consideration of noise models and decoding strategies[18][20].

Technological Advancements

• Advancements in FPGA Technology: The state of FPGA technology may limit the practical implementation of certain quantum error correction systems. Real implementation results are necessary to determine if current or near-future FPGA technology can meet the timing constraints set by quantum processors[14][15].

Cryogenic Operation

• Cryogenic Operation: For solid-state qubit platforms, the electronics, including the decoder, may eventually need to operate at cryogenic temperatures, which presents additional engineering challenges[7].

Future Developments

• ASIC vs. FPGA: While FPGAs offer reconfigurability, there is a question of whether Application-Specific Integrated Circuits (ASICs) might eventually be necessary for scalability and speed comparable to ASIC solutions, especially when co-integrated with qubit control and readout electronics[7].

fpgas compare to other hardware platforms

FPGAs vs. ASICs

• Design Flexibility: FPGAs offer high design flexibility, allowing for modifications post-manufacturing, which is particularly useful for rapid prototyping and adaptive designs. ASICs, on the other hand, have limited flexibility once fabricated, as changes require a new design and fabrication process[1][7].

• Performance: ASICs generally provide higher performance due to their custom design, optimized for specific applications. FPGAs may have somewhat lower performance due to the reconfigurable architecture, which introduces a performance overhead[1][4][7].

• Power Consumption: ASICs typically have lower power consumption due to their optimized design, while FPGAs generally consume more power, especially in large designs[1][7].

• Time-to-Market: FPGAs have a shorter time-to-market since they do not require the lengthy fabrication process that ASICs do. Design changes can be quickly implemented by reprogramming the FPGA[1].

• Cost: ASICs have a lower per-unit cost for high-volume production but higher Non-Recurring Engineering (NRE) costs. FPGAs have higher per-unit costs for low-volume production but lower NRE costs[1].

FPGAs vs. CPUs

• Parallel Processing: FPGAs excel in parallel processing, which is advantageous for tasks that require a lot of data processing quickly, such as real-time surface code decoding. CPUs are more suited to general computing tasks and may not offer the same level of parallelism[3].

• Adaptability: FPGAs are adaptable and can be reconfigured for various functions throughout their lifecycle, whereas CPUs have a fixed architecture[3].

• Real-Time Performance: FPGAs are capable of fast, high-quality calculations in real-time, making them suitable for edge computing applications like surface code decoding[3].

FPGAs vs. GPUs

• Parallelism: Both FPGAs and GPUs are capable of parallel processing, but GPUs are designed with many smaller cores that work in parallel, making them ideal for tasks like AI training that involve a ton of simultaneous calculations. FPGAs consist of programmable logic blocks and interconnects that can be configured for a wide range of digital circuits[2][5].

• Flexibility: FPGAs are noted for their flexibility and can serve various functions, while GPUs are more suitable for high processing power workloads[5].

• Energy Efficiency: GPUs have higher energy consumption compared to FPGAs, which can be a consideration for surface code decoding where power efficiency is important[5].

General Considerations for Surface Code Decoding

• Real-Time Decoding: Surface code decoding requires real-time operation to meet the tight delay constraints of quantum error correction. FPGAs are well-suited for this task due to their ability to perform high-speed processing and programmability[4][5][6].

• Integration with Quantum Systems: FPGAs are already integrated within the architecture of some quantum computers for control tasks, making them a realistic evaluation platform for surface code decoding[6][17].

• Cryogenic Operation: FPGAs have been studied in cryogenic temperatures, which is relevant for superconducting qubit platforms. They do not experience significant changes in latency at these temperatures, which is critical for real-time decoding[17].

plus: FPGAs vs. SFQ

FPGA-Based Decoding

• Real-Time Decoding: FPGAs are well-suited for real-time surface code decoding due to their ability to perform high-speed processing and programmability[1][4].

• Parallel Processing: FPGAs excel in parallel processing, which is advantageous for tasks that require a lot of data processing quickly, such as surface code decoding[5].

• Adaptability: FPGAs are adaptable and can be reconfigured for various functions throughout their lifecycle, making them suitable for edge computing applications[5].

• Integration with Quantum Systems: FPGAs are already integrated within the architecture of some quantum computers for control tasks, making them a realistic evaluation platform for surface code decoding[3].

• Cryogenic Operation: FPGAs have been studied in cryogenic temperatures, which is relevant for superconducting qubit platforms. They do not experience significant changes in latency at these temperatures, which is critical for real-time decoding[3].

SFQ-Based Decoding

• Ultra-Fast and Low-Power Performance: SFQ logic is a digital circuit composed of superconductor devices, offering ultra-fast and low-power performance compared to CMOS[3].

• Cryogenic Operation: SFQ-based decoders can operate in a cryogenic environment, making them suitable for integration with superconducting qubit platforms[3].

• Memory-Efficient Decoding Algorithms: Memory-intensive decoding algorithms, such as MWPM or UF, are unsuitable for SFQ circuits because a large amount of RAM is expensive in SFQ implementations. Thus, memory-efficient decoding algorithms are required for SFQ circuit execution[3].

• Custom Design: SFQ-based decoders need to be designed and developed by hand, which can be a downside for complex algorithms. SPICE simulations are then needed to prove that the output is as expected for given input conditions[3].

• Limited EDA Tool Support: SFQ logic needs to be fully supported by commercial EDA tools (Electronic Design Automation) that do not exist yet. Scaling this development methodology to complex algorithms requires the support of these tools[3].

Implementing FPGA for Surface Code Decoding

Scalable Quantum Error Correction for Surface Codes using FPGA

• Distributed Union-Find (UF) Decoder: The study proposes a distributed version of the Union-Find (UF) decoder that exploits parallel computing resources for further speedup. Using an FPGA-based implementation, the authors empirically show that this distributed UF decoder has a sublinear average time complexity with regard to the code distance, given parallel computing resources[1].

• Helios Architecture: The implementation employs a scalable architecture called Helios that organizes parallel computing resources into a hybrid tree-grid structure. The Helios architecture can implement surface codes up to a distance of 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[1].

Hardware System Design of Weighted Iterative Greedy Decoder for Surface Code

• Look-Up Table for Shortest Paths: The key idea of this design is constructing a look-up table for calculating the shortest paths between nodes in a weighted graph. This approach enables parallel processing and efficient decoding of surface codes[2].

Decoders for Quantum Error Correction (APS March Meeting 2024)

• FPGA and ASIC Implementations: The authors implement their decoder on both an FPGA and ASIC, the latter ultimately being necessary for any cost-effective scalable solution. They simulate a logical memory experiment using their decoder and present the results at the APS March Meeting 2024[3].

The Path to Useful Quantum Computing Needs a QEC Roadmap

• Collision Clustering Algorithm: The authors at Riverlane developed the Collision Clustering algorithm, which works by growing clusters of errors and quickly evaluating whether they collide or not. This code not only provides a speed advantage but also balances the speed, accuracy, cost, hardware, and power requirements to provide a practical route to error-corrected quantum computing. To implement Collision Clustering, they coded in Verilog, which enables it to load onto an FPGA[4].

LILLIPUT: A Lightweight Low-Latency Lookup-Table Decoder for Near-Term Quantum Error Correction

• Compressed Look-Up Tables (CLUTs): The authors propose LILLIPUT, a lightweight low latency lookup table decoder for small surface codes. To address the challenge of high memory overhead to store the LUTs, they propose Compressed Look-Up Tables (CLUTs). CLUTs exploit the fact that not all entries of a LUT are accessed with equal probability, reducing the memory requirement of LILLIPUT by up to 107x (from 148 MB to 1.3 MB)[6].

practices

1. Utilize Parallel Computing Resources

• Distributed Union-Find Decoder: Implementing a distributed version of the Union-Find (UF) decoder that exploits parallel computing resources can significantly speed up the decoding process. This approach has been shown to have a sublinear average time complexity with regard to the code distance, given sufficient parallel computing resources[2].

2. Employ Scalable Architectures

• Helios Architecture: Using a scalable architecture like Helios, which organizes parallel computing resources into a hybrid tree-grid structure, can enable the implementation of surface codes up to a certain distance with efficient decoding time per measurement round. This architecture has demonstrated the capability to decode surface codes of arbitrarily large distances without a growing backlog[2].

3. Optimize Memory Usage

• Compressed Look-Up Tables (CLUTs): To address the challenge of high memory overhead for storing Look-Up Tables (LUTs), employing Compressed Look-Up Tables (CLUTs) can significantly reduce memory requirements. CLUTs leverage the insight that not all entries of a LUT are accessed with equal probability, thus only storing entries corresponding to addresses of low Hamming weights that are most likely to occur[3].

4. Design for Low Latency

• LILLIPUT Decoder: Implementing a lightweight low latency Look-Up Table decoder like LILLIPUT for small surface codes can achieve high accuracy while maintaining low decoding latency. LILLIPUT directly interfaces with the qubit readout circuits, detects, and corrects errors in real-time, and computes the logical error efficiently[3].

5. Leverage Heuristic-Based Decoders

• Feedforward Neural Networks: For small code distances, employing heuristic-based decoders, such as those using feedforward neural networks, can decrease decoding time. These decoders are easily programmable to hardware, offering a practical solution for near-term implementations[1].

6. Ensure Real-Time Error Correction

• Integration with Quantum Systems: The FPGA-based decoders need to be seamlessly integrated with the control systems of quantum computers to ensure timely application of corrections. This integration is crucial for maintaining the fidelity of quantum operations[1][2][3].

7. Conduct Design Space Exploration

• Reconfigurability Advantage: The reconfigurability of FPGAs allows for design space exploration across various decoder configurations. This exploration can help identify the optimal balance between accuracy, latency, and resource utilization for specific surface code parameters[3].

8. Focus on Error Detection and Correction Efficiency

• Error Detection Events Translation: Implementing systems that efficiently translate syndromes every Quantum Error Correction (QEC) cycle into error detection events, assign errors to each data qubit from a Look-Up Table, and maintain an error log in real-time is essential for effective error correction[3].