Summary of quantum computing work

Read on for some background, or jump ahead to the discussion of my contributions.

Background

The discovery of Shor’s factoring algorithm (along with schemes for quantum error correction) led to an explosion of research in quantum computing in the last 25 years. The subtlety of designing quantum algorithms to achieve meaningful speedup attracted many theoreticians; the application to public-key codebreaking attracted many funding agencies. Another important factor was that qubits, the fundamental building block of quantum computers, are just quantum systems with two accessible energy levels.

These turn out to be ubiquitous, and so a huge array of experimental fields were each able to put forth their own proposals for how to realize qubits. Experimentally, quantum computing is a profound engineering challenge. To make the magic work, you need to keep your system coherent, which requires decoupling it from the rest of the world. However, to use your computer to do anything, the qubits need to be able to communicate with each other and the experimenter… which requires coupling them to the rest of the world. Balancing the tradeoff between these two concerns is the fundamental issue in experimental quantum computing.

Superconducting qubits — the area I studied — are a popular choice. The fabrication technologies used in building them are largely the same as in semiconductors, and interconnections consist of patterned electrical circuits. Superconductivity provides a low loss electrical environment; since dissipation unavoidably leads to decoherence, this is essential for a qubit. Even so, superconducting qubits typically suffer from worse coherence times relative to other technologies. Researchers proposed several different styles of superconducting qubit. All of them were based on Josephson junctions, usually based on a thin insulating layer between two superconductors. These provided the necessary nonlinearity to the system to allow restriction to two energy levels.

I worked in a group at the University of Maryland (led by Fred Wellstood, Chris Lobb, and Bob Anderson) that studied dc SQUID phase qubits. Unpacking that, dc SQUID indicates that it was a ring with two Josephson junctions, in contrast to a single junction phase qubit. This SQUID design, originally proposed by Martinis et al., conceived of one junction as the qubit junction and the other as a filter. By using a junction in the filter, you maintain enough degrees of freedom to bias the system to appropriate operating conditions. In Josephson junctions, the dc current through the junction corresponds to the phase difference in the superconducting wavefunction in the two sides of the junction. When you include the capacitance of the junction, you get an equation of motion for the phase difference that looks like motion in a “tilted washboard” potential. By appropriately biasing the dc SQUID with current and flux, you can operate in a regime where this potential is strongly anharmonic, allowing you to isolate the two lowest resonances of the well as qubit states.

We handled state readout of the qubit by measuring quantum tunneling out of this potential well. When the qubit tunneled, this produced a measurable voltage pulse because of the change in phase. Tunneling is exponentially enhanced with increasing energy, which gives a big difference (and thus a mechanism for distinguishing between) in the 0 and 1 states. Because of the statistical nature of individual quantum measurements, experiments to monitor the quantum state of the qubit involve repeating the same conditions many times.

My work

This tunneling readout method was a key weakness of the phase qubit for at least two reasons. In the “voltage state”, the junction dissipates energy, requiring a relatively lengthy cooling period afterward to return to base temperature. This puts a ceiling on the experimental repetition rate of a few hundred measurements per second. This made building up enough counts for statistically clean measurements slow, reducing turnaround time on the devices we were characterizing. A more fundamental issue is that tunneling based readout is not projective: following the tunneling measurement, the system is generally not in the same state that you just measured. While this doesn’t represent a problem for device characterization, it’s a dealbreaker for general fault tolerant quantum computation. I decided introducing a better readout to resolve these issues would be a good research direction.

We took inspiration from the Josephson bifurcation amplifier. A Josephson junction can be viewed as a nonlinear inductance, and oscillators built using this nonlinearity can exhibit bistable response to rf excitation. By biasing near the bifurcation where bistability appears, small changes in the frequency arising from modulation of junction parameters yield large changes in a reflected rf signal. We believed that it would be possible to use the filter junction of the dc SQUID in a fashion similar to this. A theoretical study of the dc SQUID phase qubit predicted that when the qubit junction was excited, it would lead to a shift in circulating current in the SQUID loop. This would modify the current through the filter junction, shifting the Josephson inductance enough to enable a rf reflectometry readout that would be tunneling free and projective.

I started designing circuits to test this readout approach. The major modification from earlier dc SQUID qubit designs in our lab was the addition of a large shunt capacitor before the SQUID loop. The capacitor was effectively in parallel with the capacitance of the filter junction, depressing the resonant frequency of that system. This moved the frequency of the signal used for readout far from the transition frequency for the qubit to minimize unwanted transitions during readout. After designing these new devices, I began fabricating them. The circuits were patterned on silicon (and later sapphire) wafers using photolithography, and I deposited aluminum using a double-angle thermal evaporation process. A controlled oxidation step between the two evaporation angles formed a thin insulating layer between two layers of aluminum, defining the Josephson junctions of each sample.

The superconducting transition temperature of aluminum is close to 1 K; to achieve those temperatures we mounted our qubits in a dilution refrigerator. We had already installed heavily filtered low frequency lines for qubit measurement and control, and we installed some new microwave lines (along with cryogenic microwave circulators for isolation) for the reflectometry measurement.

When we measured the first sample, we found that the microwave setup (including the on-chip lumped element microwave resonator we were measuring) looked good, but we didn’t succeed in measuring the qubit this way. However, while characterizing the device, we realized that by operating with rf power slightly below the bifurcation regime, we could use the resonator as a sensitive probe to measure 1/f flux noise in the device. This type of noise is ubiquitous in solid state systems and is an important source of decoherence in many superconducting qubit designs. We found values for our system consistent with older results showing weak dependence on most circuit parameters.

After testing a few more devices and ruling out various aspects of the experimental setup as explanations of why the measurement failed, I became suspicious about how well the theory of the projective microwave measurement was fitting our qubits. When I reexamined the theory, a central approximation was that the two junctions of the dc SQUID were well isolated by the large inductance of the SQUID loop. Early phase qubit designs used relatively large area Josephson junctions. One reason for this was to reduce the importance of critical current fluctuations in limiting coherence times. However, other researchers showed that two-level systems in the Josephson junction dielectric played a more significant role in decoherence. The natural way to minimize these unwanted two-level fluctuators is to use smaller junctions. As junctions grow smaller, their Josephson inductance increases. In moving to smaller junctions, we now had Josephson inductances in the filter arm that were non-trivial compared to the combined inductance of the SQUID loop and qubit junction.

The original theory viewed the dynamics as a nonlinear perturbation to two harmonic oscillators given by each junction. My revised theory instead considered a basis where the appropriate harmonic oscillators are the two normal modes of the circuit. With appropriate parameters, this normal modes approach returned to the independent junction model. After working out the new theory, we found it was substantially better at explaining the behavior of other qubits (in a similar parameter regime) being studied by other researchers in the lab.

I spent a while longer working on new designs with stronger nonlinearities and more tunability, with the goal of demonstrating microwave readout in our systems. Eventually, I switched over to the main research direction of our group: trying to understand and improve the qubit lifetime $T_1$ in our systems. With Rangga Budoyo, another graduate student in the lab, I designed and built a new qubit. This device used a large interdigitated capacitor on the smaller junction to improve dielectric quality factor, and handled microwave excitation of the qubit using a coplanar waveguide modeled to have more controlled coupling to the qubit.

As we characterized the new qubit, we found it showed unusual spectroscopic features. In particular, the peak shape was highly broadened on one side. We found that we could explain this well as a consequence of the large shunt capacitor in front of the SQUID. I showed that when the capacitor was included in my quantum circuit theory, you could extract an effective Jaynes-Cummings model for the system. This model, which had become central in superconducting quantum computing from its use in the circuit QED framework, describes the coupling between a two-level system (e.g. a qubit) and a harmonic oscillator (e.g. a microwave cavity). In the dispersive limit of the model, the frequency of the qubit is shifted proportional to the number of excitations in the cavity. Since we had chosen a large capacitance (and thus low frequency filter) to maximize isolation to the bias leads, thermal excitation of the filter modes was likely substantial. This led to many relatively narrow individual peaks being superposed to give the broadened measured peak.