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Learning Unitary Operators with Help From u(n)
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Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Use the full n²-parameter Lie algebra exponential parametrization rather than the Arjovsky composition | Unitary operator learning · Synthetic unitary recovery, n=6, 8, 14, 20 | test loss: 0.09 \pm 0.01 -> \mathbf{0.03 \pm 0.01}; 1.17 \pm 0.06 -> \mathbf{0.014 \pm 0.006}; 10.8 \pm 0.3 -> \mathbf{0.07 \pm 0.0 | Helped |
| Use the Lie algebra exponential parametrization rather than projection after gradient updates | Unitary operator learning · Synthetic unitary recovery, n=3, 6, 8, 14, 20 | test loss: 8 \pm 1 -> 6.003 \pm 0.003 \times 10^{-4}; 15 \pm 1 -> \mathbf{0.03 \pm 0.01}; 14 \pm 1 -> \mathbf{0.014 \pm 0.006}; 24 | Helped |
| Restrict the Lie algebra coefficients to 7n learnable parameters (ablation) | Lie algebra unitary parametrization · Synthetic unitary recovery, n=8, 14, 20 | test loss: 0.04 \pm 0.01 -> 1.0 \pm 0.2; 0.25 \pm 0.03 -> 12.6 \pm 0.4; 0.19 \pm 0.03 -> 28.0 \pm 0.6 | Hurt |
| Generate the target unitary matrix using different generation methods | Lie algebra and Arjovsky learning approaches · Synthetic unitary recovery | normalized test loss | No clear effect |
| Add a scaling factor β before the unitary transition operator (ablation) | General unitary RNN · Adding problem, T=100 | task performance | Helped |
| Add a scaling factor β before the unitary transition operator (ablation) | General unitary RNN · Memory problem, T=100 | task performance | Helped |
| Use the general unitary RNN rather than the LSTM | Long-memory recurrent modeling · Memory problem, T=100 | task performance | Helped |
| Use the general unitary RNN rather than the LSTM | Long-memory recurrent modeling · Adding problem, T=100 | task performance | Mixed |
| Use the general unitary RNN rather than the restricted unitary RNN | Long-memory recurrent modeling · Adding and memory problems, T=100 | speedup: 4.8 and 2.6 | Helped |
| Change the Lie algebra basis (ablation) | Lie algebra unitary parametrization · Synthetic unitary recovery, n=6 | validation loss | Hurt |
| Adjust the learning rate after changing the Lie algebra basis (ablation) | Lie algebra unitary parametrization · Synthetic unitary recovery, n=6 | validation loss | Helped |
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