College of Liberal Arts & Sciences

# Math Physics Seminar

**Abstract:**

Quantum computers are quantum systems with a finite number of degrees of freedom. The general problem in quantum computations is to compute real-time evolution of a state in a finite dimensional Hilbert space. In this talk I discuss a path integral treatment of this problem motivated by a paper by Palle Jorgensen and Ekatrina Nathanson that gave a reinterpretation of Feynman's path integral in terms of complex probabilities. I start with a general observable that is chosen to prepare initial and final states. Following a construction by Schwinger I use that observable to construct an irreducible pair of operators. A general Hamiltonian can be expressed as a finite degree polynomial in this irreducible pair. Mixed matrix elements of the Hamiltonian in eigenstates of the irreducible pair of operators are analogous to a classical ``phase-space Hamiltonian''. In this framework time evolution is expressed as the expectation value of a complex probability on cylinder sets of ``phase space paths''. The complex probability is constructed from the completeness relation of non-commuting irreducible operators. In this picture smooth time evolution results from the expectation value of cylinder sets of nowhere continuous paths. In this finite dimensional case there are no Fresnel integrals and the complex probabilities have an exact factorization into products of one time step probabilities.

Applications to scattering theory and interacting quantum field theory are discussed.

**UICapture link to recordings:**