Quantum computation in molecular magnets is studied by solving the time-dependent Schr\"odinger equation numerically. Following Leuenberger and Loss [Nature (London) $410,$ 789 (2001)], an external alternating magnetic field is applied to populate and manipulate a superposition of single-spin states in molecular magnet ${\mathrm{Mn}}_{12}.$ The conditions to realize parallel recording and reading databases of Grover algorithms in molecular magnets are discussed in detail. It is found that an accurate time duration of the magnetic pulse as well as the discrete frequency spectrum and the amplitudes are required to design a quantum computing device.
Quantum computation in molecular magnets is studied by solving the time-dependent Schr"odinger equation numerically. Following Leuenberger and Loss [Nature (London) 410, 789 (2001)], an external alternating magnetic field is applied to populate and manipulate a superposition of single-spin states in molecular magnet Mn12. The conditions to realize parallel recording and reading databases of Grover algorithms in molecular magnets are discussed in detail. It is found that an accurate time duration of the magnetic pulse as well as the discrete frequency spectrum and the amplitudes are required to design a quantum computing device.
Quantum algorithms use the principles of quantum mechanics, such as, for example, quantum superposition, in order to solve particular problems outperforming standard computation. They are developed for cryptography, searching, optimization, simulation, and solving large systems of linear equations. Here, we implement Grover's quantum algorithm, proposed to find an element in an unsorted list, using a single nuclear 3/2 spin carried by a Tb ion sitting in a single molecular magnet transistor. The coherent manipulation of this multilevel quantum system (qudit) is achieved by means of electric fields only. Grover's search algorithm is implemented by constructing a quantum database via a multilevel Hadamard gate. The Grover sequence then allows us to select each state. The presented method is of universal character and can be implemented in any multilevel quantum system with nonequal spaced energy levels, opening the way to novel quantum search algorithms.
Standard CMOS circuits are expected to be no longer able to satisfy nowadays demand for higher performances and lower power consumption. This commits to look for new solutions. Quantum computation, whose fundamental quantity of information is the qubit, is a thoroughly new way of thinking to electronics. There are at least two main reasons for which quantum computing is expected to broaden computation horizons. First, specific algorithms can be engineered, manipulating the probability amplitudes of different states, to exploit the quantum superposition principle and achieve high parallelism. Second, it is expected to be spontaneously suitable to simulate complex quantum physical systems, as large molecules. Quantum computing requires the cooperation of several scientific fields. The leading idea has been to highlight the operations and algorithms a quantum computer must be able to perform and, then, to propose a hardware solution, providing the rigorous physical treatment mandatory for a deep understanding and for an appropriate modelling, something which is hard to find in nowadays literature. Thus, the first chapters address the quantum algorithms, focusing on Grover's search algorithm and deriving, from fundamentals, the corresponding circuit. The latter is then described in Quantum Assembly (QASM), which can be seen as the quantum equivalent of VHDL, and simulated on IBM Q Experience. The following chapters deal with a possible physical implementation of quantum processors: the encoding of qubits on nuclear spins in diamagnetic molecules. When a static magnetic field is applied to a spin-1/2 nucleus, two well-defined energy eigenstates arise, as a result of the Zeeman effect. Each of them can be associated with a qubit eigenstate. Thanks to nuclear magnetic resonance principles, the probability with which one of the two eigenstates is measured can be handled by the superposition of a radio frequency field at the resonance frequency of the nucleus. This can be interpreted as a rotation of the spin, and so of the qubit, about the RF field and enables the execution of single qubit quantum gates, since they are simple rotations of the state vector. The two-qubit CNOT gate is obtained through the interaction of spins via J-coupling. Considering that these gates constitute a universal set, every quantum algorithm can theoretically be run on an NMR processor. Different nuclei have different resonant frequencies, but the addressing of a specific spin can be achieved also in homonuclear molecules, since the resonant frequency depends also on the nuclear environment (nuclear shielding). While unwanted J-couplings can be removed thanks to refocusing, spin decoherence and relaxation phenomena force an upper limit to the allowed timescale. A main advantage of NMR is that this timescale is long, even at room temperature. Conversely, scalability is still an issue. A simple heteronuclear MATLAB model, based on the relations derived from first principles in the thesis, is proposed to prove the feasibility of NMR quantum processors. The input technological parameters, as chemical shielding and J-coupling, can be computed resorting on ORCA, or obtained from experimental data. The model, able to run a universal set of quantum gates and simple algorithms, can be useful to find an optimal operating point as a compromise between molecule physical properties and the quantities which can be controlled by nowadays NMR instrumentation.
Following an alternative version of Grover's algorithm, we propose a scheme to read- in information in Mn 12 molecular magnets with the help of multi-frequency magnetic pulses in two separate steps: first to realize an intermediate state |Ψ(T1)> and then to reach the state |Ψ(T2)> to encode information. All controllable parameters are calculated to implement read-in of information, and the durations of magnetic pulses are less than the dephasing time of quantum states in Mn 12. It is also shown that all the states of information can be initialized back to the intermediate state.
Molecular nanomagnets can be considered as serious candidates for the definition of a scalable and reliable quantum computing technology: information is encoded on their spins and they ensure relaxation and decoherence times sufficiently long for the execution of tens of quantum operations. Among these devices, Cr7Ni supramolecular complexes are extremely of interest since they implement a universal set of quantum gates, made of single-qubit gates and the two-qubit Controlled-phase gate. A model for analyzing Cr7Ni molecules and a potential quantum computer architecture are proposed: starting from the energy parameters required for spin manipulations, these systems have been proved to implement elementary quantum algorithms. Within this paper, two, three and four-qubit systems have been analyzed: the simulation of quantum operations and the analysis of dynamical non-idealities have been done in parallel. Our approach enables the definition of an operating point - in terms of magnetic driving fields and latency - in which the execution of elementary quantum algorithms is characterized by negligible errors. The proposed model has been entirely implemented in MATLAB, thus obtaining a software infrastructure for the analysis of Cr7Ni molecules that can be extended to quantum systems with analogous Hamiltonian and behavior.
In this paper, molecular quantum computation is numerically studied with the quantum search algorithm (Grover's algorithm) by means of optimal control simulation. Qubits are implemented in the vibronic states of I 2 , while gate operations are realized by optimally designed laser pulses. The methodological aspects of the simulation are discussed in detail. We show that the algorithm for solving a gate pulse-design problem has the same mathematical form as a state-to-state control problem in the density matrix formalism, which provides monotonically convergent algorithms as an alternative to the Krotov method. The sequential irradiation of separately designed gate pulses leads to the population distribution predicted by Grover's algorithm. The computational accuracy is reduced by the imperfect quality of the pulse design and by the electronic decoherence processes that are modeled by the non-Markovian master equation. However, as long as we focus on the population distribution of the vibronic qubits, we can search a target state with high probability without introducing error-correction processes during the computation. A generalized gate pulse-design scheme to explicitly include decoherence effects is outlined, in which we propose a new objective functional together with its solution algorithm that guarantees monotonic convergence.
Many problems intractable on classical devices could be solved by algorithms explicitly based on quantum mechanical laws, i.e. exploiting quantum information processing. As a result, increasing efforts from different fields are nowadays directed to the actual realisation of quantum devices. Here we provide an introduction to Quantum Information Processing, focusing on a promising setup for its implementation, represented by molecular spin clusters known as Molecular Nanomagnets. We introduce the basic tools to understand and design quantum algorithms, always referring to their actual realisation on a molecular spin architecture. We then examine the most important sources of noise in this class of systems and one of their most peculiar features, i.e. the possibility to exploit many (more than two) available states to encode information and to self-correct it from errors via proper design of quantum error correction codes. Finally, we present some examples of quantum algorithms proposed and implemented on a molecular spin qudit hardware.