Testing DeepSeek R1 in its favorite subject - mathematics

Earlier today this question popped onto my radar - could quantum computing help speed up Gradient Descent calculations?

What is Gradient Descent and Who Cares

Gradient Descent is a calculation used in the training of basically all Large Language Models, like ChatGPT, Claude, and DeepSeek R1 itself. It is not a new thing. It is very old. The new aspect of how it's used is that we now throw huuuuge amounts of data at it, which makes the task of completing its calculations slow and expensive. This is what makes people look for ways to speed it up.

• If you want LLMs to not hallucinate, you need Gradient Descent to work properly.
• If you want LLMs to cost you less than $200 a month to access, you want this calculation to be cheap and fast.

What is the Answer

Something in my clunky human memory told me that quantum processing may not speed up gradient descent calculations. I threw this question over to an actual quantum physicist / quantum computing expert, Anastasia Marchenkova , then I also asked DeepSeek R1, via its vanilla implementation over at Perplexity .

Below is the exact text of my prompts and its responses.

TLDR:

• It says QC will not speed up gradient descent. But I'll wait for Anastasia's human expert perspective before concluding that this is true.
• I had to refine my prompt to get real insights (Prompt 2)
• When it started talking about "quantum gradient descent" it went over my head. I have no idea what that is, or what it means. I understand classical gradient descent only.

==

Prompt 1:

For processing the mathematics of gradient descent in neural networks, which is better and why: GPUs, TPUs, or Quantum computers?

Gradient descent optimization in neural networks benefits from different hardware architectures depending on the scale, model complexity, and computational requirements. Here's a comparison of GPUs, TPUs, and quantum computers:

GPUs (Graphics Processing Units)

• Strengths:Parallel processing: Optimized for matrix operations common in neural network training, achieving 10-100x speedups over CPUs for deep learning tasks39.Versatility: Support diverse frameworks (PyTorch, TensorFlow) and non-AI workloads (graphics, simulations)89.Mature ecosystem: Extensive software support (CUDA, cuDNN) and precision options (FP16 to FP64)820.
• Limitations:Memory bandwidth constraints for very large models8.Higher power consumption compared to TPUs79.

TPUs (Tensor Processing Units)

• Strengths:Tensor optimization: Specialized for large-scale matrix multictions, achieving ~5x speedups over GPUs in benchmarks (e.g., training Xception models)310.Energy efficiency: Designed for Google’s TensorFlow, with superior throughput for sharded datasets and large batch sizes37.Cost-effective: Cloud-based TPUs often outperform GPUs in price/performance for specific tasks like NLP (BERT) and image recognition1020.
• Limitations:Narrower framework compatibility (primarily TensorFlow)9.Less memory per core than GPUs8.

Quantum Computers (QPUs)

• Theoretical advantages:Exponential speedup: Quantum gradient descent algorithms claim O(1)O(1) complexity for gradient calculations vs. O(d)O(d) classically419.Natural gradient optimization: Exploits quantum geometry for faster convergence in variational quantum circuits1921.
• Current challenges:Limited qubit coherence times and error rates519.Noisy intermediate-scale quantum (NISQ) devices lack practical utility for classical ML tasks422.Specialized use cases (e.g., quantum chemistry) rather than general gradient descent2122.

Comparison Table

(omitted here for simplicity)

When to Use Which?

1. GPUs: Ideal for prototyping, small-to-medium models, and multi-framework workflows89.
2. TPUs: Best for large-scale TensorFlow models (e.g., transformers, CNNs) with sharded datasets310.
3. Quantum: Not yet practical for classical ML; potential future role in hybrid quantum-classical optimization1922.

Conclusion

For current applications, TPUs outperform GPUs in specific large-scale tensor operations, while GPUs remain more flexible. Quantum computers, though promising theoretical speedups (O(1)O(1) gradient calculations4), lack the infrastructure to rival classical hardware today. As of 2025, TPUs are optimal for Google Cloud users training massive models, whereas GPUs dominate general-purpose ML. Quantum gradient descent remains a long-term research frontier1921.

Prompt 2:

I'm specifically interested in the gradient descent calculations, not all LLM calculations, or even all LLM training calculations. Compare GPUs and TPUs for both speed, and their ability to select local minima in the multidimensional space.

Gradient descent performance and optimization characteristics differ significantly between GPUs and TPUs due to their architectural designs and batch processing capabilities. Here's a focused comparison:

Speed Comparison

Hardware Architecture

• GPUs Excel at parallel vector operations with high memory bandwidth (900+ GB/s in modern models)16. Their SIMD architecture enables efficient gradient calculations across multiple cores simultaneously16. Flexible for various batch sizes but optimized for medium batches (32-128)217.
• TPUs Specialized for tensor operations with matrix multiply units (MMUs) that reuse input vectors across multiple calculations69. Achieve 1.5-7x speedups over GPUs in large-batch scenarios (128+)220, but struggle with small batches due to fixed dataflow architectures714.

Benchmarks

TaskGPU (V100)TPU (v3)SpeedupResNet-50 (CIFAR-10)40 mins15 mins2.7x1Xception (batch=128)Out-of-memory1.5 mins/epoch7x2BERT (128 seq)3.8 ms/batch1.7 ms/batch2.2x1

TPUs dominate in large-batch tensor operations critical for gradient calculations, while GPUs offer better flexibility for dynamic batch sizes617.

Local Minima Selection Characteristics

Batch Size Effects

• TPUs Require large batches (≥128) for optimal performance214, which:Reduce gradient variance by 87-92% compared to small batches17Increase risk of converging to sharp minima by 3-5x45Limit stochastic exploration of loss landscapes319
• GPUs Efficiently handle smaller batches (16-64), enabling:Higher gradient noise (34-61% more stochasticity)517Better escape from saddle points and shallow minima4162-3x more frequent parameter updates per epoch1719

Optimization Paths

• TPU-optimized large batches follow smoother but narrower optimization trajectories, increasing the likelihood of settling in suboptimal minima35.
• GPU-friendly small batches create "noisy" paths that explore wider regions of the loss landscape, finding flatter minima with better generalization417.

Practical Tradeoffs

FactorGPU AdvantageTPU AdvantageBatch flexibilityDynamic batch sizing (16-512)Fixed large batches (≥128)Minima qualityBetter for exploratory optimizationRisk of sharp minima in over-parameterized modelsGradient calc speed2-3x slower at scale2-7x faster for sharded dataMemory constraintsSupports gradient accumulation3Requires full-batch sharding214

Conclusion

For gradient descent specifically:

• TPUs excel in speed for large-batch scenarios (e.g., full-batch distributed training), but their architectural constraints may lead to poorer local minima selection in complex loss landscapes.
• GPUs provide better minima exploration through flexible batch sizing and inherent stochasticity, at the cost of slower computation for massive tensor operations.

Current implementations show TPUs achieve 5-7x faster gradient updates in optimized TensorFlow pipelines220, while GPUs maintain advantages in PyTorch workflows requiring dynamic batch adjustments16. The choice depends on whether raw speed (TPUs) or optimization robustness (GPUs) is prioritized for a given problem.

Prompt 3

Why doesn't quantum computing help speed up the computation of gradient descent? Please address the computer science details and math details.

Quantum computing does not currently provide a speedup for classical gradient descent due to fundamental computational and mathematical limitations rooted in the algorithm's structure and quantum hardware constraints. Here's a detailed breakdown:

1. Algorithmic Structure Mismatch

• Classical Gradient Descent: Operates in O(d)O(d) complexity per iteration for dd-dimensional problems, with convergence typically requiring O(??2)O(??2) iterations for non-smooth convex objectives1012. The algorithm relies on sequential updates to parameters based on local gradient information.
• Quantum Adaptation Challenges: Quantum gradient descent algorithms (e.g., for parameterized quantum circuits) use the parameter-shift rule to estimate gradients19. For each parameter θkθk, this requires two quantum circuit evaluations (at θ±skθ±sk), leading to O(d)O(d) quantum state preparations and measurements per iteration49. This matches classical O(d)O(d) scaling, negating quantum advantage.

2. Oracle Overhead and Query Complexity

• Oracle Conversions: Quantum speedups often require converting between probability oracles (common in optimization) and phase oracles (used in quantum algorithms). This interconversion incurs O(logd)O(logd) overhead2, erasing potential quadratic speedups for gradient estimation.
• Lower Bounds: For non-smooth convex optimization, quantum algorithms cannot outperform classical gradient descent’s O(??2)O(??2) query complexity1011. This is proven via information-theoretic arguments: quantum queries to f(x)f(x) and ?f(x)?f(x) provide no asymptotic advantage in high-dimensional spaces10.

3. Barren Plateaus and Noise

• Barren Plateaus: Quantum neural networks (QNNs) suffer from exponentially vanishing gradients as the number of qubits grows, a phenomenon linked to random parameter initialization and entanglement616. This forces quantum gradient descent into flat regions of the loss landscape, requiring exponentially precise measurements to escape6.
• NISQ Limitations: Current noisy intermediate-scale quantum (NISQ) devices have error rates (10?310?3–10?210?2) that dominate signal from gradient calculations. For example, estimating a gradient component with precision δδ requires O(δ?2)O(δ?2) measurements9, making practical implementations infeasible.

4. Theoretical Speedup Constraints

• Smooth vs. Non-Smooth Functions: Quantum gradient methods (e.g., quantum natural gradient15) theoretically offer O(d)O(d) speedups for smooth functions by exploiting quantum Fisher information7. However, most real-world optimization problems (e.g., ReLU networks, L1 regularization) are non-smooth, where no quantum advantage exists1011.
• Linear Algebra Bottlenecks: While quantum linear algebra (e.g., HHL algorithm) could accelerate matrix inversions, gradient descent’s iterative nature and adaptive step sizes prevent direct integration814. Hybrid quantum-classical approaches introduce synchronization overhead that negates speedups12.

5. Complexity Analysis

For a dd-dimensional problem with TT iterations:

• Classical: O(T?d)O(T?d) time.
• Quantum:Gradient estimation: O(d?δ?2)O(d?δ?2) measurements per iteration9.Total: O(T?d?δ?2)O(T?d?δ?2), which is slower unless δ?2?1δ?2?1 (unachievable with NISQ error rates).

Conclusion

Quantum computing does not accelerate classical gradient descent due to:

1. Matching O(d)O(d) scaling in gradient estimation19,
2. Oracle conversion overhead2,
3. Barren plateaus in high-dimensional spaces616,
4. Lower bounds for non-smooth optimization1011,
5. Noise and measurement constraints69.

While quantum methods like variational quantum eigensolvers (VQE) or quantum approximate optimization (QAOA) show promise in quantum-native tasks512, they do not translate to speedups for classical gradient descent. Hybrid quantum-classical algorithms remain limited by classical bottlenecks in parameter updates and error correction14.