This thesis actively focuses on designing, analyzing, and experimentally implementing various QST and QPT protocols using an NMR ensemble quantum processor and superconducting qubit-based IBM cloud quantum processor. Part of the thesis also includes a study of duality quantum simulation algorithms and Sz-Nagy's dilation algorithm on NMR where several 2-qubit non-unitary quantum channels were simulated using only a single ancilla qubit. The work carried out in the thesis mainly addresses several important issues in experimental QST and QPT which include: i) dealing with invalid experimental density (process) matrices using constraint convex optimization (CCO) method, ii) scalable QST and QPT using incomplete measurements via compressed sensing (CS) algorithm and artificial neural network (ANN) technique, iii) selective and direct measurement of unknown quantum states and processes using the concept of quantum 2-design states and weak measurement (WM) approach and iv) quantum simulation and characterization of open quantum dynamics using the dilation technique.
This thesis actively focuses on designing, analyzing, and experimentally implementing various QST and QPT protocols using an NMR ensemble quantum processor and superconducting qubit-based IBM cloud quantum processor. Part of the thesis also includes a study of duality quantum simulation algorithms and Sz-Nagy's dilation algorithm on NMR where several 2-qubit non-unitary quantum channels were simulated using only a single ancilla qubit. The work carried out in the thesis mainly addresses several important issues in experimental QST and QPT which include: i) dealing with invalid experimental density (process) matrices using constraint convex optimization (CCO) method, ii) scalable QST and QPT using incomplete measurements via compressed sensing (CS) algorithm and artificial neural network (ANN) technique, iii) selective and direct measurement of unknown quantum states and processes using the concept of quantum 2-design states and weak measurement (WM) approach and iv) quantum simulation and characterization of open quantum dynamics using the dilation technique.
We experimentally performed complete and optimized quantum process tomography of quantum gates implemented on superconducting qubit-based IBM QX2 quantum processor via two constrained convex optimization (CCO) techniques: least squares optimization and compressed sensing optimization. We studied the performance of these methods by comparing the experimental complexity involved and the experimental fidelities obtained. We experimentally characterized several two-qubit quantum gates: identity gate, a controlled-NOT gate, and a SWAP gate. The general quantum circuit is efficient in the sense that the data needed to perform CCO-based process tomography can be directly acquired by measuring only a single qubit. The quantum circuit can be extended to higher dimensions and is also valid for other experimental platforms.
The experimental implementation of selective quantum process tomography (SQPT) involves computing individual elements of the process matrix with the help of a special set of states called quantum 2-design states. However, the number of experimental settings required to prepare input states from quantum 2-design states to selectively and precisely compute a desired element of the process matrix is still high, and hence constructing the corresponding unitary operations in the lab is a daunting task. In order to reduce the experimental complexity, we mathematically reformulated the standard SQPT problem, which we term the modified SQPT (MSQPT) method. We designed the generalized quantum circuit to prepare the required set of input states and formulated an efficient measurement strategy aimed at minimizing the experimental cost of SQPT. We experimentally demonstrated the MSQPT protocol on the IBM QX2 cloud quantum processor and selectively characterized various two- and three-qubit quantum gates.
The standard procedure for quantum process tomography (QPT) requires a series of experiments. Each experiment involves initialization of the system to a particular basis state, applying the quantum process $\ensuremath{\varepsilon}$ on the system, and finally characterizing the output state by quantum state tomography (QST). The output states collected for a complete set of basis states enable us to calculate the $\ensuremath{\chi}$ matrix characterizing the process $\ensuremath{\varepsilon}$. The standard procedure for QST itself requires independent experiments, each involving measurement of a set of commuting observables. Thus QPT procedure demands a number of independent measurements and, moreover, this number increases rapidly with the size of the system. However, in ensemble systems, the total number of independent measurements can be greatly reduced with the availability of ancilla qubits. Ancilla-assisted process tomography (AAPT) has earlier been shown to require a single QST of system-ancilla space. Ancilla-assisted quantum state tomography (AAQST) has also been shown to perform QST in a single-scan measurement of an ensemble system. Here we combine AAPT with AAQST to realize a single-scan QPT (SSPT), a procedure to characterize a general quantum process in a single ensemble measurement. We demonstrate experimental SSPT by characterizing several single-qubit processes using a three-qubit NMR quantum register. Furthermore, using the SSPT procedure, we experimentally characterize the twirling process and compare the results with theory.
We experimentally implement the Sz.-Nagy dilation algorithm to simulate open quantum dynamics on a nuclear magnetic resonance quantum processor. The Sz.-Nagy algorithm enables the simulation of the dynamics of an n-qubit system using n+1 qubits. We experimentally simulate the action of three nonunitary processes, namely, a phase damping channel acting independently on two qubits, a two-qubit correlated amplitude damping channel, and a magnetic-field-gradient pulse acting on an ensemble of two coupled nuclear spin-21 particles. To evaluate the quality of the experimentally simulated quantum process, we perform convex-optimization-based full quantum process tomography to reconstruct the quantum process from the experimental data and compare it with the target quantum process to be simulated.
This chapter delves deeply into quantum process tomography (QPT) and regression: two critical approaches in the field of processing quantum information. The process of characterizing and estimating unknown quantum processes executed on quantum systems is known as quantum process tomography, while regression aims to find optimal parameters for input states to achieve desired unitary transformations. The chapter begins by introducing the concept of channel-state duality, which establishes a fundamental connection between quantum channels and quantum states. This duality enables the analysis of quantum processes by examining the input and output states, providing insights into their properties and behavior. Quantum process tomography is then discussed in detail, covering various tomographic methods and strategies used to reconstruct unknown quantum processes. Then the mathematical foundations, experimental techniques, and statistical analysis involved in quantum process tomography are explored to provide a comprehensive understanding of the field. Next, the chapter delves into the role of groups, the unitary group, and compact lie groups in the study of quantum processes. These mathematical concepts provide a framework for understanding the structure and symmetries of quantum processes, and representation theory is introduced as a powerful tool for decomposing and analyzing quantum processes into irreducible representations. The concept of storage of unitary and parallel application operations is crucial for the practical implementation of quantum processes insights about techniques such as parallelization and matrix factorization, which optimize the computational resources required for large-scale quantum systems, enabling efficient and scalable execution of quantum processes. The chapter also addresses the concept of finding optimal input states for learning quantum processes, adaptive measurement, and statistical optimization techniques, which enables researchers to identify the input states that provide the most informative data for accurate process estimation. Finally, the chapter concentrates on establishing the unitary operation and defining the parameterization for the initial state to achieve desired transformations. The effective learning criteria and quantum machine learning (QML) for leveraging quantum states provides advantages of QML over classical methods. In summary, this chapter provides a comprehensive exploration of quantum process tomography and regression. By leveraging channel-state duality, representation theory, and optimization techniques, researchers and practitioners can accurately characterize and estimate quantum processes. It also offers insights into the mathematical foundations, experimental methodologies, and practical applications of these techniques, contributing to the advancement of quantum information processing and technology.
Alternatively to the full reconstruction of an unknown quantum process, the so-called selective and efficient quantum process tomography (SEQPT) allows estimating, individually and up to the required accuracy, a given element of the matrix that describes such an operation with a polynomial amount of resources. The implementation of this protocol has been carried out with success to characterize the evolution of a quantum system that is well described by a trace-preserving quantum map. Here, we deal with a more general type of quantum process that does not preserve the trace of the input quantum state, which naturally arises in the presence of imperfect devices and system-environment interactions, in the context of quantum information science or quantum dynamics control. In that case, we show with the aid of a priori information on the losses structure of the quantum channel that the SEQPT reconstruction can be adapted to reconstruct the non-trace-preserving map. We explicitly describe how to implement the reconstruction in an arbitrary Hilbert space of finite dimension d. The method is experimentally verified on a superconducting quantum processor provided by IBM Quantum services, by estimating several non-trace-preserving quantum processes in dimensions up to d=6. Our results show that it is possible to efficiently reconstruct non-trace-preserving processes, with high precision, and with significantly higher fidelity than when the process is assumed to be trace preserving.