Barbara Maria Terhal

We introduce the fermionic satisfiability problem, Fermionic k-SAT: this is the problem of deciding whether there is a fermionic state in the null-space of a collection of fermionic, parity-conserving, projectors on n fermionic modes, where each fermionic projector involves at most k fermionic modes. We prove that this problem can be solved efficiently classically for k = 2. In addition, we show that deciding whether there exists a satisfying assignment with a given fixed particle number parity can also be done efficiently classically for Fermionic 2-SAT: this problem is a quantum-fermionic extension of asking whether a classical 2-SAT problem has a solution with a given Hamming weight parity. We also prove that deciding whether there exists a satisfying assignment for particle-number-conserving Fermionic 2-SAT for some given particle number is NP-complete. Complementary to this, we show that Fermionic 9-SAT is QMA_1-hard.

We consider the optimization problem (ground energy search) for fermionic Hamiltonians with classical interactions. This QMA-hard problem is motivated by the Coulomb electron-electron interaction being diagonal in the position basis, a fundamental fact that underpins electronic-structure Hamiltonians in quantum chemistry and condensed matter. We prove that fermionic Gaussian states achieve an approximation ratio of at least 1/3 for such Hamiltonians, independent of sparsity. This shows that classical interactions are sufficient to prevent the vanishing Gaussian approximation ratio observed in SYK-type models. We also give efficient semi-definite programming algorithms for Gaussian approximations to several families of traceless and positive-semidefinite classically interacting Hamiltonians, with the ability to enforce a fixed particle number. The technical core of our results is the concept of a Gaussian blend, a construction for Gaussian states via mixtures of covariance matrices.

We introduce spectral quantum tomography, a simple method to extract the eigenvalues of a noisy few-qubit gate, represented by a trace-preserving superoperator, in a SPAM-resistant fashion, using low resources in terms of gate sequence length. The eigenvalues provide detailed gate information, supplementary to known gate-quality measures such as the gate fidelity, and can be used as a gate diagnostic tool. We apply our method to one- and two-qubit gates on two different superconducting systems available in the cloud, namely the QuTech Quantum Infinity and the IBM Quantum Experience. We discuss how cross-talk, leakage and non-Markovian errors affect the eigenvalue data.

I address the question of how a quantum computer can be used to simulate experiments on quantum systems in thermal equilibrium. I present two approaches for the preparation of the equilibrium state on a quantum computer. For both approaches, one can show that the output state of the algorithm, after long enough time, is the desired equilibrium.