# Quantum Computation in Qudit Space and Applications in Open Quantum Dynamics

#### Abstract

Qudit, a multi-level computational unit for quantum computing, provides a larger state space for information processing, and thus can reduce the circuit complexity, simplify the experimental setup. We promote the qudit-based quantum computing by providing an overview that covers a variety of qudit topics ranging from gate universality, circuit building, algorithm design, to physical realization methods. Among all the important qudit algorithms, we perform the first experimental realization of a qudit-based phase estimation algorithm(PEA) on a photonic platform, utilizing the high dimensionality in time and frequency degrees of freedom (DoFs) in a single photon. In our scheme the controlled-unitary gates can be realized in a deterministic fashion, as the control and target registers are now represented by two DoFs in a single photon. Next we improve the PEA by introducing a new statistical and variational approach to the PEA that we called SPEA. The SPEA can determine any unknown eigenstate-eigenphase pair from a given unitary matrix by treating the probabilistic output of an Iterative PEA (IPEA)-like circuit as an eigenstate-eigenphase proximity metric, using this metric to estimate the proximity of the input state and input phase to the nearest eigenstate-eigenphase pair and approaching this pair via a variational process on the input state and phase. The SPEA can search over the entire computational space as well as some specified given range efficiently and thus outperforms the original PEA. The simulation of open quantum dynamics has attracted wide interests recently with a variety of quantum algorithms developed and demonstrated. The second half of the thesis focus on the simulation of the open quantum dynamics which is a useful application for quantum computer based on qudit as well as qubit. We perform the first quantum simulations of the radical pair mechanism(RPM) in the avian compass with a Sz.-Nagy dilation theorem-based quantum algorithm to demonstrate the generality of the quantum algorithm and to open new opportunities for studying the avian compass with quantum computing devices. Next we apply the same quantum algorithm to simulate open quantum dynamics based on the Generalized Quantum Master Equation (GQME). This approach overcomes the limitations of the Lindblad equation by providing a rigorous derivation of the equations of motion for any subset of elements of the reduced density matrix. We validate our quantum algorithm as applied to the spin-boson benchmark model by analyzing the impact of the quantum circuit depth on the accuracy of the results when the subset is limited to the diagonal elements of the reduced density matrix. Our findings demonstrate that our approach yields reliable results on noisy intermediate-scale quantum (NISQ) computers.

#### Degree

Ph.D.

#### Advisors

Kais, Purdue University.

#### Subject Area

Computer science|Quantum physics|Computational physics

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