Quantum tomography is a method to learn an unknown quantum state or process by a series of measurements and a reconstruction procedure. Full tomographic reconstruction has a large computational cost both in memory and time and hence is infeasible for systems beyond even just ten qubits. During my practicum at Los Alamos National Laboratory (LANL), my collaborators and I developed an alternative variational quantum algorithm (VQA) for both state and process tomography. By construction, our algorithm uses substantially less memory and generally less time than standard tomography. We achieve this reduction by encoding quantum states and unitary processes as low-depth quantum circuits on a quantum device rather than exponentially large matrices on a classical device. Of course, a direct encoding requires knowing the unknown state or unitary.
The trick, as with all VQAs, is to begin with a good guess — aka ansätz. A good ansätz should enforce known properties of the system such as symmetries and contain adjustable parameters that span over the symmetry sector. The objective, then, is to optimize the choice of parameters so that the low-depth quantum circuit can prepare the unknown quantum state or emulate the unknown quantum process. The details of optimizing these parameters comprise our variational quantum algorithm which consists of (i) a cost function to minimize, (ii) an optimizer and parameter update rule, (iii) performance guarantees, and (iv) avoiding known pitfalls like the vanishing gradient problem (aka barren plateaus). In hashing out these details, we also formulated and solved the "quantum low-rank approximation problem" and discovered the non-trivial fact that learning the action of a unitary on product states is sufficient to predict its effect on highly entangled states. Our work shows the value in using a quantum computer for quantum primitives rather than brute-forcing them with classical supercomputers.