3D tomography of exchange phase in a Si/SiGe quantum dot device

The exchange interaction is a foundational building block for the operation of spin-based quantum processors. Extracting the exchange interaction coefficient $J(\mathbf{V})$, as a function of gate electrode voltages, is important for understanding disorder, faithfully simulating device performance, and operating spin qubits with high fidelity. Typical coherent measurements of exchange in spin qubit devices yield a modulated cosine of an accumulated phase, which in turn is the time integral of exchange. As such, extracting $J(\mathbf{V})$ from experimental data is difficult due to the ambiguity of inverting a cosine, the sensitivity to noise when unwrapping phase, as well as the problem of inverting the integral. As a step toward obtaining $J(\mathbf{V})$, we tackle the first two challenges to reveal the accumulated phase, $φ(\mathbf{V})$. We incorporate techniques from a wide range of fields to robustly extract and model a 3D phase volume for spin qubit devices from a sequence of 2D measurements. In particular, we present a measurement technique to obtain the wrapped phase, as done in phase-shifting digital holography, and utilize the max-flow/min-cut phase unwrapping method (PUMA) to unwrap the phase in 3D voltage space. We show this method is robust to the minimal observed drift in the device, which we confirm by increasing scan resolution. Upon building a model of the extracted phase, we optimize over the model to locate a minimal-gradient $π$ exchange pulse point in voltage space. Our measurement protocol may provide detailed information useful for understanding the origins of device variability governing device yield, enable calibrating device models to specific devices during operation for more sophisticated error attribution, and enable a systematic optimization of qubit control. We anticipate that the methods presented here may be applicable to other qubit platforms.

NoMoPy: Noise Modeling in Python

NoMoPy is a code for fitting, analyzing, and generating noise modeled as a hidden Markov model (HMM) or, more generally, factorial hidden Markov model (FHMM). This code, written in Python, implements approximate and exact expectation maximization (EM) algorithms for performing the parameter estimation process, model selection procedures via cross-validation, and parameter confidence region estimation. Here, we describe in detail the functionality implemented in NoMoPy and provide examples of its use and performance on example problems.