Fixed Depth Hamiltonian Simulation via Cartan Decomposition demo #1261
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| h_0_m = Kc_m @ H_m @ Kc_m.conj().T | ||
| h_0 = qml.pauli_decompose(h_0_m) | ||
| print(len(h_0)) |
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the resulting h_0 is actually not in
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I got it to work.
I did the following tweaks (took some time 😅 )
gammas = [np.pi**i % 2 for i in range(1, len(h)+1)]instead ofgammas = [np.pi**i for i in range(len(h))]. The% 2leads to more uniform gammas, which helps optimization, apparently. This makes sense because we care about numerical stability here and with the previousgammas, some directions had entirely different scale than others. The shift by1in theimight be unnecessary.- Switch from Adam to L-BFGS. This is more expensive, but seems to optimize significantly better.
- Optimize for 2k epochs rather than 500. Together with the optimizer change the optimization then takes 60s instead of 8s.
- Switch
h_0_m = Kc_m @ H_m @ Kc_m.conj().Ttoh_0_m = Kc_m.conj().T @ H_m @ Kc_m(line 314). As we transform thev_minlossbyK_m @ v_m @ K_m.conj().T, we need to do the opposite toH_mwhen applying the optimized parameters. I'd suggest to transformH_minlossas well, to be more consistent (I'm aware that this breaks with how the paper writes it down, but transition from paper to code might be nicer than within code?)
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Let me know whether I should push these changes.
| Us_kak = jax.vmap(Us_kak)(ts) | ||
| Us_trotter50 = jax.vmap(lambda t: qml.matrix(qml.TrotterProduct(H, time=t, n=50, order=4), wire_order=range(n_wires)))(ts) | ||
| Us_trotter500 = jax.vmap(lambda t: qml.matrix(qml.TrotterProduct(H, time=t, n=500, order=4), wire_order=range(n_wires)))(ts) | ||
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| res_kak = 1 - jnp.abs(jnp.einsum("bij,bji->b", Us_exact.conj(), Us_kak)) / 2**n_wires | ||
| res_trotter50 = 1 - jnp.abs(jnp.einsum("bij,bji->b", Us_exact.conj(), Us_trotter50)) / 2**n_wires | ||
| res_trotter500 = 1 - jnp.abs(jnp.einsum("bij,bji->b", Us_exact.conj(), Us_trotter500)) / 2**n_wires | ||
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| plt.plot(ts, res_kak, label="KAK") | ||
| plt.plot(ts, res_trotter50, "x--", label="50 Trotter steps") | ||
| plt.plot(ts, res_trotter500, ".-", label="500 Trotter steps") | ||
| plt.ylabel("empirical error") | ||
| plt.xlabel("t") | ||
| # plt.yscale("log") | ||
| plt.legend() | ||
| plt.show() |
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seems wild to me that both 50 and 500 steps seem to do equally bad 🤔
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I think we want to do time=-t in TrotterProduct because it implements
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Turns out 50 steps is waaay to precise already :D :)
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