Quantum computing has become an increasingly relevant topic in computer science due to the development of functional quantum computers in recent years. Quantum processing units (QPUs) use quantum particles in the form of qubits as information carriers. By using properties of quantum particles, quantum algorithms have been developed that can solve problems in polynomial time, for which only exponentially scaling classical algorithms are known. QPUs will not run on their own in the foreseeable future, due to high error rates and low stability of qubits. Instead, quantum devices will be one component in a quantum-classical (hybrid) computing system (distributed system). Quantum computers embedded in distributed hybrid quantum-classical systems get instructions from a main CPU, execute them and report their results. To fully utilize quantum calculations, a QPU should be able to communicate with a CPU in real-time. This means the qubits information content should stay stable during communication time. In this thesis, we will look at quantum computing systems with real-time feedback between a QPU and a CPU. As quantum computers are going to be used in real-time hybrid systems, we need optimization routines for code running on these systems. Existing works on the optimization of quantum code focus on the optimization of quantum circuits only, and neglect potential classical calculations that are necessary on a hybrid distributed system. We want to examine which kind of optimization routines are possible and sensible for hybrid quantum-classical computations. We will evaluate some of todays quantum programming languages (QPLs), especially with respect to their ability to support and optimize real-time hybrid calculations. We will find that most languages support real-time calculations to some extent, but are more adapted to the programming approach of static circuit creation. Additionally, none of the QPLs offer optimization routines targeted at quantum-classical calculations. Therefore, we will examine options for real-time quantum-classical optimization. For this, we use the low-level QPL Quil. We will introduce optimization operations and apply them to real-time quantum-classical algorithms. We will present metrics to evaluate the performance of hybrid calculations. The results of applying our optimization operations to Quil code will be evaluated against our performance metrics. We will find that we are in principle able to optimize hybrid quantum-classical programs. We encourage more research in this field. QPLs will become more abstract in the future and need more compilation steps until they can be executed on hardware. This will make the optimization of quantum-classical hybrid code more relevant.
Though quantum computing naturally offers an advantage for simulations of real-time quantum systems, implementing Imaginary-Time Evolution (ITE) is comparatively more difficult. Nevertheless, quantum implementations of ITE are useful both in cases where classical implementations have associated sign problems, and also as an exact method of preparing eigenstates on quantum computers. In this work we present an algorithm to obtain ITE of a generic quantum Hamiltonian by performing analytic continuation of measured real-time correlation functions. We present simulations and demonstrations on IBM quantum hardware of 1D Fokker-Planck equations for classical diffusion process and imaginary-time evolution of integrated correlation functions in 1D quantum mechanical scattering as examples to demonstrate the effectiveness of the method.
We outline a strategy to compute deeply inelastic scattering structure functions using a hybrid quantum computer. Our approach takes advantage of the representation of the fermion determinant in the QCD path integral as a quantum mechanical path integral over
0+1-dimensional fermionic and bosonic worldlines. The proper time evolution of these worldlines can be determined on a quantum computer. While extremely challenging in general, the problem simplifies in the Regge limit of QCD, where the interaction of the worldlines with gauge fields is strongly localized in proper time and the corresponding quantum circuits can be written down. As a first application, we employ the color glass condensate effective theory to construct the quantum algorithm for a simple dipole model of the
F2 structure function. We outline further how this computation scales up in complexity and extends in scope to other real-time correlation functions.