Calculating the inelastic cross section

The scattering experiment cannot measure the quantum dynamics that we can explore with our quantum processor. Their measurement data is limited to the detection of photons, i.e., the final probabilities. For comparison we next simulate only the final excited state probabilities over the same energy range as the Olsen experiment. This spans over two orders of magnitude in projectile energy, corresponding to an order of magnitude variation of collision velocities,

and a wide distribution of impact parameters. In total we simulate 104 distinct parameter

combinations with 100 linearly spaced steps in both center of mass velocity, v, and impact parameter, b, the results of which are plotted in the left column of Fig. 7.7. The large number of distinct parameter combinations and therefore unique quantum evolutions preclude us from individual optimization, thus necessitating methodical calibrations. For comparison in the right column of Fig. 7.7, we plot the corresponding numerical computation done on a classical computer over the same parameter regime.

For large impact parameters, we expect there to be no interaction or population exchange between the states because the atoms effectively never “collide”. For the range of velocities studied, we find that no population is transferred into the other qubits/molecular channels when

b > 3.0 ˚A. For 1.0 ˚A < b < 3.0 ˚A we measure the greatest population transfer when v/c

∼_{1.4 × 10}−3_{, with roughly equal probability of finding the excitation in qubit q2 or q3. For}

other velocities in this range of impact parameters, we find a decrease in the probability of

exciting Na. When b < 1.0 ˚A the collision results in a substantial (75%) probability of exciting

Figure 7.7: P|1i (false color) as a function of collision impact parameter, b, and velocity, v. The left column are data from simulations performed on our quantum processor after correcting for

readout fidelity and T1decay. The right column are numerical computations of the ideal version

of our quantum processor. We find excellent agreement over the full simulation range.

found in q2. Above this velocity we find the excitation is in q3.

We now have the required simulation results to calculate an inelastic cross section. First we note that the scattering experiment did not have the ability to measure an individual collision’s impact parameter. Inherently they are sampling over all impact parameters by using a beam of many Na atoms colliding with the He gas. For comparison with the experimentally determined

parameter

Cross section = 2π

Z ∞

0

b × (1 − P|1i)db. (7.4)

In Fig. 7.8 we plot the inelastic cross section as a function of the center of mass kinetic energy. The original scattering experimental data has been replotted here in the center of mass frame (red squares). The numerical computation performed on a classical computer represent- ing the ideal version of our quantum processor is shown as the solid black line. Qualitatively, the numerical computation agrees very well with the experimental data, including the position of the peak and roll-off at higher and lower energies. Quantitatively the numerical computation predicts approximately twice the cross section measured by the Olsen experiment over the full measurement range. This discrepancy may be due to incorrect diabatic potentials or systematic errors in the original experimental data.

We find excellent agreement between the inelastic cross section measured using our quan- tum processor and the numerical computation over the full range of the original scattering experiment. The mean absolute percent error is 8.7%, while the maximum percent error is 15.8%. We hypothesis that the discrepancy between the numerical computation and the quan- tum processor is from dephasing during the simulation time. The scattering experimental data measures the peak in the cross section at 2.1 keV, in agreement with the adiabatic criterion discussed earlier. For the numerical computation and quantum processor datasets, we find the

Numerical Computation Quantum Simulation Scattering Experiment

Figure 7.8: Inelastic cross section of Na and He collision as a function of center of mass (COM) kinetic energy. The red squares are the original Olsen collision experiment replotted in the COM frame. The solid black line is the cross section from the numerically computed data of Fig. 7.7 after integration according to Eq. 7.4. We perform the same integration of the data obtained using our quantum processor (blue circles). The deviation between the numeri- cal computation and the original Olsen collision experiment may be due to imperfect diabatic potentials or possible systematic error in the collision experiment. The deviation between the simulation performed on our quantum processor and the numerical computation is due to im- perfect calibration of pulse errors or more likely dephasing of the qubits during the simulation.

7.6

Conclusion

We have simulated the inelastic collision of Na and He using a superconducting quantum pro- cessor containing three fully connected gmon qubits. We simulated the collision over the same range of kinetic energies as a previous scattering experiment. We find that both the original experiment and our quantum simulation produce a peak in the cross section in agreement with the adiabatic criterion. Furthermore the quantum simulation predicts the experimentally mea- sured cross section to within a factor of two over the full range of energies, and within 8.7% of the numerical computation. The accuracy of this quantum dynamics simulation shows the maturity of superconducting qubit control technology and paves the way for potentially solving more complicated problems using superconducting qubits.

Chapter 8

Conclusion and Outlook

8.1

Conclusion

A scattering cross section of the inelastic collision between Na and He atoms has been simu- lated using a superconducting quantum processor. The generated quantum chemical dynamics as well as the final scattering cross section agree very well with numerical computations per- formed on a classical computer and a previous scattering experiment. Generating the cross section required fast arbitrary time-dependent modulations of both the individual qubit fre- quencies and the qubit-qubit coupling strengths. This allows complete control over the single excitation subspace of the three qubits. The vast parameter space associated with generating the cross section essentially eliminated the possibility of fine-tuning the individual pulse se- quences. Instead we developed a series of independent calibrations to minimize the effects from pulse distortions and other non-idealities.

taining controllability. Similar requirements are necessary for testing the fundamentals of quantum error correction. We developed the high level of qubit coherence over several years us- ing both simpler qubit circuits and superconducting resonators. We have measured hundreds of resonators over this time. The results have clearly shown the importance of improving both the metal-substrate interface as well as the exposed interfaces by cleaning with a non-destructive method, for example without physical ion bombardment. These improved interfaces have led to better quality factors on both single crystal sapphire and silicon substrates. Besides dielectric loss, resonator experiments have improved our understanding of other energy loss mechanisms coming from quasiparticles, magnetic vortices, and radiation. We applied this knowledge to improve the qubit design, fabrication, and measurement, leading to a thirty-fold improvement

in T1 times over a two year period.

In this work we have achieved record high quality factors on silicon substrates (QLP_i >

5 × 106_{). These resonators exhibit a previously undiscovered temperature dependence of the}

quality factor which is not predicted by the standard tunneling model of two level systems (TLS). Instead these results are consistent with a model of interacting TLS. Further research is required to better understand and utilize this behavior.