Thanks a lot for the blog post! Really appreciate the side-by-side comparison of SQcircuit and scQubits. I totally agree with "Doing the Thing instead of endlessly preparing yourself to Do the Thing"!
To answer your question whether the tunable coupling vs. coupler flux curve is what I expected: No ;-) I would expect that at zero coupler flux, when the coupler is far removed from the qubits in frequency, so it doesn't participate much and the qubits are almost uncoupled. But at zero coupler flux, the plot shows a gap=2*coupling of 500 MHz which is super large.
After some digging: it looks like when you construct a circuit from YAML with scQubits, it uses a default cutoff of |n| = 5 for the non-periodic variables, which is of course too small to model transmons. If you make the cutoff much larger the calculation takes long, so hierarchical diagonalization is probably the way to go. Code below, this works with scqubits==3.0.2 :-)
The "leading order" intuitive explanation of tunable couplers I'm familiar corresponds to eq (2) in Yan et al, https://arxiv.org/abs/1803.09813. It says that a direct capacitance gives a positive coupling, and an indirect coupling via the coupler transmon gives a negative coupling. So if you get the indirect coupling just right by tuning the coupler frequency, the two coupling channels cancel out. I can't really back that explanation up, so I'm looking forward to your coupler blogpost!
That would suggest that the direct qubit-qubit capacitance is necessary to have an off-point. I.e., the stray qubit-qubit capacitance is a feature not a bug here. That turns out to be the case numerically: Plotting the coupling on a logscale, it never actually reaches zero, but stays above 1MHz or so which would be too large in practice. If you add a small capacitance between nodes 1 and 3, with a charging energy of say 48.4 GHz, there is a good off point. Let me know if you agree!
I agree with you that beyond a flux of 0.4 or so, the coupler goes below the qubits in frequency, so getting the coupling as the difference between the second and first excited energies no longer works.
Florian, thanks for the kind words, glad you enjoyed it.
Your comment probably deserves a full post addressing it, since there's a lot of rich detail here. Instead I will just post a few of my thoughts about the points you raised:
1. You're right that I didn't concern myself much with setting the cutoff correctly, this was partially because I only managed to get multiprocessing working very late, and partly because figuring what the appropriate ncut value is, in my view, more art than science. The current documentation describing how to approach it is sorely lacking, IMO, and makes it hard for newbies to understand how and why they're so important. I think if we were doing this 'for real', we'd definitely spend some time increasing ncuts until the spectra across a few values of external flux bias converge to something reasonable. In my heart, I suspect that there is a more elegant way of approaching this problem, but I don't know what it is.
I initially didn't think ncuts would be a problem here, but I checked by increasing all ncuts to 10, which did indeed reduce the coupling to ~15 MHz at zero coupler flux. Raising all ncuts to 20 is takes a pretty long time, but reduces the coupling to 800 kHz at zero coupler flux. Honestly, I would *not* have expected 0 coupling at 0 flux, because of interaction (iii) in the paper:
>the capacitive coupling via the intermediate capacitance network (direct qubit-qubit coupling, indirect connection)
Which I take to mean coupling through C1c, C2c even if the coupler-transmon is just hanging out. So we would expect some small coupling even without the C12 capacitor (which I think is accounted for by interaction (iv)).
2. The effect above dovetails with your second point about the off-state. I think that same positive coupling from mechanism (iii) would still permit there to be an off-state even in the absence of a weak, direct coupling in the form of C12. This appears to still be the case for my simulation with ncuts set to 10. My min coupling value is about 388 kHz using the direct circuit simulation with no C12 coupler and no hierarchical diagonalization. I suspect there is a place in phase space where you can get an off-point without an explicit C12, but it could be hard to find. I didn't really think of it as a feature, but it seems that you are right, it appears to be helpful to include some finite value of direct tmon-tmon coupling to get a nice off-state. Luckily, in real life, nature usually provides this coupling whether we want it or not!
Agreed that at zero coupler flux, the effective coupling won't be zero exactly - there should be a competition between indirect coupling (since the coupler is still there and only finitely removed in frequency) and direct coupling (which can be through the direct or indirect connection). Thanks for pointing out the direct qubit-qubit coupling through the indirect capacitive connection, I hadn't realized that!
If only real life stray couplings were always this nice to us...
Also there’s a new tunable coupler paper that just dropped: https://arxiv.org/pdf/2207.06607.pdf
What if you made a coupler out of two transmons instead of one? Might be fun to simulate
Thanks a lot for the blog post! Really appreciate the side-by-side comparison of SQcircuit and scQubits. I totally agree with "Doing the Thing instead of endlessly preparing yourself to Do the Thing"!
To answer your question whether the tunable coupling vs. coupler flux curve is what I expected: No ;-) I would expect that at zero coupler flux, when the coupler is far removed from the qubits in frequency, so it doesn't participate much and the qubits are almost uncoupled. But at zero coupler flux, the plot shows a gap=2*coupling of 500 MHz which is super large.
After some digging: it looks like when you construct a circuit from YAML with scQubits, it uses a default cutoff of |n| = 5 for the non-periodic variables, which is of course too small to model transmons. If you make the cutoff much larger the calculation takes long, so hierarchical diagonalization is probably the way to go. Code below, this works with scqubits==3.0.2 :-)
The "leading order" intuitive explanation of tunable couplers I'm familiar corresponds to eq (2) in Yan et al, https://arxiv.org/abs/1803.09813. It says that a direct capacitance gives a positive coupling, and an indirect coupling via the coupler transmon gives a negative coupling. So if you get the indirect coupling just right by tuning the coupler frequency, the two coupling channels cancel out. I can't really back that explanation up, so I'm looking forward to your coupler blogpost!
That would suggest that the direct qubit-qubit capacitance is necessary to have an off-point. I.e., the stray qubit-qubit capacitance is a feature not a bug here. That turns out to be the case numerically: Plotting the coupling on a logscale, it never actually reaches zero, but stays above 1MHz or so which would be too large in practice. If you add a small capacitance between nodes 1 and 3, with a charging energy of say 48.4 GHz, there is a good off point. Let me know if you agree!
I agree with you that beyond a flux of 0.4 or so, the coupler goes below the qubits in frequency, so getting the coupling as the difference between the second and first excited energies no longer works.
```
import numpy as np
import scqubits as scqub
import matplotlib.pyplot as plt
tcap_coupled_tmon = """# transmon circuit
branches:
- ["JJ", 0, 1, 12,10]
- ["JJ", 1, 0, 12,10]
- ["C", 1, 0, 0.2]
- ["C", 1, 2, 4.84]
- ["JJ", 0, 2, 36,10]
- ["JJ", 2, 0, 36,10]
- ["C", 2, 0, 0.1]
- ["C", 2, 3, 4.84]
- ["JJ", 0, 3, 12,10]
- ["JJ", 3, 0, 12,10]
- ["C", 3, 0, 0.2]
"""
tcap_tmons = scqub.Circuit(tcap_coupled_tmon, from_file=False, ext_basis="discretized")
tcap_tmons.configure(transformation_matrix=np.eye(3), # work in node variables, rather than automatically choosing variables
system_hierarchy=[[1], [2], [3]], # hierarchical diagonalization, treat each node var as a subsystem
subsystem_trunc_dims=[7, 7, 7]) # diagonalize and truncate each subsystem to 7 dimensions before diagonalizing joint system
for i in [1, 2, 3]:
setattr(tcap_tmons, f'Φ{i}', 0.3) # Set both qubits at flux 0.3
setattr(tcap_tmons, f'cutoff_n_{i}', 50) # To diagonalize each subsystem, go up to |n|=50, rather than the default |n|=5
flux_array = np.linspace(0.0, .4, 201)
# flux_array = np.linspace(0.35, .37, 201) # Zoom in on the off point
fig, ax = tcap_tmons.plot_evals_vs_paramvals('Φ2', flux_array, evals_count=7, subtract_ground=True)
ax.set_title('Spectrum vs. coupler flux')
spec = tcap_tmons.get_spectrum_vs_paramvals('Φ2', flux_array, evals_count=3, subtract_ground=True)
gap = spec.energy_table[:, 2] - spec.energy_table[:, 1]
print(f'Minimum coupling [GHz]: {np.min(gap)/2=}')
plt.figure(figsize=(10, 8))
plt.plot(flux_array, gap / 2)
plt.xlabel('$\Phi$2 ($\Phi_0$)', fontsize=24)
plt.ylabel('Coupling (GHz)', fontsize=24)
plt.yscale('log')
plt.tick_params(labelsize=20)
plt.title('Coupling vs. coupler flux')
plt.tight_layout()
plt.show()
```
Florian, thanks for the kind words, glad you enjoyed it.
Your comment probably deserves a full post addressing it, since there's a lot of rich detail here. Instead I will just post a few of my thoughts about the points you raised:
1. You're right that I didn't concern myself much with setting the cutoff correctly, this was partially because I only managed to get multiprocessing working very late, and partly because figuring what the appropriate ncut value is, in my view, more art than science. The current documentation describing how to approach it is sorely lacking, IMO, and makes it hard for newbies to understand how and why they're so important. I think if we were doing this 'for real', we'd definitely spend some time increasing ncuts until the spectra across a few values of external flux bias converge to something reasonable. In my heart, I suspect that there is a more elegant way of approaching this problem, but I don't know what it is.
I initially didn't think ncuts would be a problem here, but I checked by increasing all ncuts to 10, which did indeed reduce the coupling to ~15 MHz at zero coupler flux. Raising all ncuts to 20 is takes a pretty long time, but reduces the coupling to 800 kHz at zero coupler flux. Honestly, I would *not* have expected 0 coupling at 0 flux, because of interaction (iii) in the paper:
>the capacitive coupling via the intermediate capacitance network (direct qubit-qubit coupling, indirect connection)
Which I take to mean coupling through C1c, C2c even if the coupler-transmon is just hanging out. So we would expect some small coupling even without the C12 capacitor (which I think is accounted for by interaction (iv)).
2. The effect above dovetails with your second point about the off-state. I think that same positive coupling from mechanism (iii) would still permit there to be an off-state even in the absence of a weak, direct coupling in the form of C12. This appears to still be the case for my simulation with ncuts set to 10. My min coupling value is about 388 kHz using the direct circuit simulation with no C12 coupler and no hierarchical diagonalization. I suspect there is a place in phase space where you can get an off-point without an explicit C12, but it could be hard to find. I didn't really think of it as a feature, but it seems that you are right, it appears to be helpful to include some finite value of direct tmon-tmon coupling to get a nice off-state. Luckily, in real life, nature usually provides this coupling whether we want it or not!
Thanks a lot for the detailed reply!
Agreed that at zero coupler flux, the effective coupling won't be zero exactly - there should be a competition between indirect coupling (since the coupler is still there and only finitely removed in frequency) and direct coupling (which can be through the direct or indirect connection). Thanks for pointing out the direct qubit-qubit coupling through the indirect capacitive connection, I hadn't realized that!
If only real life stray couplings were always this nice to us...