To add to the challenge, some Hamiltonians are not static, but change over time. Many of the state of the art quantum simulation algorithms, notably qubitization, only work for time independent Hamiltonians. Thankfully, there is a known trick for eliminating time dependence, which looks a bit like representing time using a timeline. We use a spatial coordinate to represent the flow of time, as in the figure above. This trick had not been properly considered for use in quantum algorithms.

To address this, I conducted research with Alessandro Roggero, Nathan Wiebe, and Dean Lee to apply this approach. We proposed a direct simulation of the resulting "clock space" using qubitization, the current optimal technique for simulation. While we do not prove optimal simulation, this represents the first nontrivial simulation of general time dependence by qubitization. We also introduced a novel algorithm for time dependent simulation using so-called multiproduct formulas. Our method compares well with current state of the art approaches and retains the commutator scaling of Trotter formulas. 

Overall, we make progress to bridge the gap between time dependent and time independent Hamiltonian simulation. 

By far the most popular and well known quantum simulation technique is Trotterization. In this scheme, all the pieces of the Hamiltonian are broken up into simple units that can be simulated one by one. Yet this technique is relatively inaccurate as compared to more sophisticated and recent algorithms. Trotter is cheap and suitable for near term quantum computers, so we would like to improve its performance.

To help, Gumaro Rendon, Nathan Wiebe, and myself investigated the use of polynomial interpolation to improve accuracy. The more Trotter steps in the simulation, the more accurate the result. We use polynomials to extrapolate out to infinite Trotter steps, which is the exact result. We found that we could greatly increase the accuracy for estimates of "observables," which as the name suggests are the detectable outcomes of quantum experiments. However, they may be a cost of quadratically worse performance with respect to the desired simulation time. Nevertheless, this simple technique provides a mechanism by which Trotter may compete more effectively with "post-Trotter" methods in terms of accuracy.

I have been involved in several projects related to the Rodeo Algorithm, a method for measuring eigenvalues and preparing eigenstates on a quantum computer. This is an important step in a number of physics and more general applications. The circuits involved are those seen in iterative phase estimation protocols, but the way they are used is distinct. We compare the eigenvalues of the operator of interest to a selected "target" eigenvalue within a specified selection range. When there are eigenvalues within the selection range, a "signal" will be generated in the form of a high success probability. This allows for searching eigenvalues in a range and location of interest along the line of possible eigenvalues.