Most quantum processors requires pulse sequences for controlling quantum states. Here we present an alternative algorithm for computing an optimal pulse sequence so as to perform a specific task, being an implementation of a quantum gate or a quantum-state preparation. In our method, we reduce greatly the number of parameters to be fitted by using a limited number of functions as the modulations for the amplitude and phase of the radio-frequency pulses. Unlike other algorithms, ours uses some approximations to make it fast and scalable, and uses some efficient methods to obtain pulses that are robust to common types of experimental errors. We demonstrate the success of the proposed algorithm by performing several real experiments for 4-, 7-, and 12-quantum-bit systems using NMR. In addition, we show the efficiency of the algorithm, finding pulses for controlling with good fidelity the quantum states of spins in fictional square two-dimensional lattices containing 16, 36, and 100 qubits.
Most quantum processors requires pulse sequences for controlling quantum states. Here we present an alternative algorithm for computing an optimal pulse sequence so as to perform a specific task, being an implementation of a quantum gate or a quantum-state preparation. In our method, we reduce greatly the number of parameters to be fitted by using a limited number of functions as the modulations for the amplitude and phase of the radio-frequency pulses. Unlike other algorithms, ours uses some approximations to make it fast and scalable, and uses some efficient methods to obtain pulses that are robust to common types of experimental errors. We demonstrate the success of the proposed algorithm by performing several real experiments for 4-, 7-, and 12-quantum-bit systems using NMR. In addition, we show the efficiency of the algorithm, finding pulses for controlling with good fidelity the quantum states of spins in fictional square two-dimensional lattices containing 16, 36, and 100 qubits.
We present a versatile rf pulse control system that has been designed for multi-qubit quantum experiments. One instrument can be scaled to provide 32 channels of rf between 10 - 450 MHz. Synchronization can be achieved across multiple instruments. By using direct digital synthesis and custom control circuitry contained within a field-programmable gate array, sequences of transform-limited pulses can be produced. These have been used to carry out quantum gates that are able to meet fault-tolerant thresholds for single- and two-qubit gate fidelities, as published elsewhere. We have also extended the frequency to the gigahertz regime using additional mixers to address hyperfine transitions in atomic systems. The system uses an efficient memory management scheme and a low-latency communications protocol that allows pulse sequences to be updated in real-time. Together these can enable outcome-based algorithms such as quantum error correction to be executed. The system is fully programmable in C++, and other languages such as Python can be supported by the on-board CPU, offering a highly flexible platform for a wide variety of experimental systems, and has been proven in trapped-ion quantum information experiments.
Quantum Fourier transform (QFT) is a quantum analogue of the classical discrete Fourier transform. It is a fundamental quantum gate in quantum algorithms which has an exponential advantage over the classical computation and has been excessively studied. Normally, an n-qubit quantum Fourier transform could be resolved into the tensor product of n single-qubit operations, and each operation could be implemented by a Hadamard gate and a controlled phase gate. Then the complexity of an n-qubit QFT is of order O(n2). To reduce the complexity of quantum operations, optimal control (OC) method has recently been used successfully to find the minimum time for implementing a quantum operation. Up to now, two types of quantum optimal control methods have been presented, i.e. analytical and numerical methods. The analytical approach is to change the problem of efficient synthesis of unitary transformations into the geometrical one of finding the shortest paths. Numerical optimal control procedures are based on the gradient methods (GRAPE, Gradient Ascent Pulse Engineering) and Krotov methods. Notable application mainly focus on nuclear magnetic resonance fields, including imaging, liquid-state NMR, solid-state NMR, and NMR quantum computation. One obvious advantage of optimal control NMR quantum computation is that the OC unitary evolution transformation pulse sequences are normally shorter than the conventional corresponding ones. Here we use the optimal control method to find the minimum duration for implementing QFT quantum gate. A linear spin chain with nearest-neighbor Ising interaction is used to find the optimization. And the optimized pulse sequence is experimentally demonstrated on an NMR quantum information processor. By using optimal control method with numerical calculation, a three-qubit QFT in an indirect-linear-coupling chain system is optimized. The duration of the OC QFT is obviously shorter than that of conventional approaches. The OC pulse sequence has been experimentally implemented on a liquid-state NMR spectrometer. To verify the optimally controlled pulse sequence for the three-qubit QFT, different initial states are assumed. The accuracy of the OC pulse sequence could be demonstrated by the consistency of theoretical simulation spectra with the experimental results. The good consistency between the simulation and the experimental spectra demonstrates that the OC QFT is of high fidelity.
The intent of this study is to determine the feasibility of diatomics as molecular quantum computing candidates and shed insight into the use of such experimental laser pulse shaping methods to represent quantum logic gates. Four appropriate rovibrational states of model diatomic molecules are encoded as the qubit states. A set of 2-qubit quantum logic gates (ACNOT, CNOT, NOT, Hadamard) are represented by amplitude and phase shaped laser pulses. The combinations of amplitudes and phases that produce the optimal laser pulse representation, for each quantum logic gate, are determined by a Genetic Algorithm optimization routine. The theoretical laser pulse shaping is analogous to current experimental frequency-domain pulse shaping apparatus with amplitude and phase control at individual frequencies. A model set of diatomics is sampled in order to determine a relationship between optimal laser pulse shaping and the choice of diatomic molecule. We show that the choice of diatomic molecule greatly influences the ability to produce optimal laser pulse shapes to represent quantum logic gates. Tuneable parameters specific to laser pulse shaping instruments are varied to determine their effect on optimal pulse production. They include varying the number of amplitude and phase components, adjusting the number of frequency components, and altering the frequency resolution which is synonymous with altering the laser pulse duration. A time domain analytic form of the original frequency domain laser pulse function is derived, providing a useful means to infer the laser pulse dependencies on these parameters. Initially, we show that the appropriate choice of rovibrational state qubits of carbon monoxide (12C16O) and the use of simple shaped binary pulses, 2 amplitude and 2 phase components, can provide significant control for specific quantum gates. Further amplitude variation at each frequency component is shown to be a crucial requirement for optimal laser pulse shaping, whereas phase variation provides minimal contribution. We show that the generation of optimal laser pulse shapes is highly dependent upon the frequency resolution and increasing the number of frequency components provides incremental improvements to optimal laser pulses.
This thesis details work performed on the improvement of experimental control of quantum systems through the use of optimal control theory. We pursue this objective through both conceptual and practical routes. We first devise improved methods of optimal control for quantum systems. We then apply them, as well as more established methods, to experimental situations where robustness to inaccuracy and error is sought. We develop a method for improving the computational efficiency of optimal control theory in the context of quantum control. This approach makes use of the iterative nature of optimisation algorithms, by using perturbation theory to minimise the effort needed to calculate the time propagator associated with a controlled Hamiltonian. We also create a new way of defining a figure of merit when optimising for robustness against error in an experiment. This method specifically applies to errors that have a relatively small effect on the controlled system's Hamiltonian. We compare this with an ensemble method based around randomly sampling several points on the control landscape, and find these methods to be largely equivalent for small errors, with the ensemble method being superior for larger error. On the applied side, we design radio frequency magnetic pulses that create robustness in spin ensembles to inhomogeneous broadening and variations of coupling strength to the control field for two applications. In the first, we create reliable gates in a situation where limitations are imposed on the control that can be applied. In the second application, we design controls for radio frequency magnetic pulses to create a superradiant state in a spin ensemble. Finally, we collaborate with the experiment to measure the electrons electric dipole moment that is being performed at Imperial College London. For this collaboration, we shape the envelope of the radio frequency control pulses used in the experiment to create superpositions of Zeeman and Stark-shifted excited states in YbF molecules. This shaping allows us to impart pre-determined phases to these superpositions, in a manner that is robust to errors in the control pulse envelope. We obtain sets of robust and sensitive pulses that have been sent to the experiment for testing purposes.
Working on quantum systems entail different interests, for example, working on fundamental \nunderstanding of quantum systems also lay foundation for better quantum computation \ntechniques. A test for whether a system is behaving quantum mechanically or \nclassically is devised by Leggett and Garg in form of inequalities, called Leggett-Garg Inequalities \n(LGI). Such Inequalities are violated by a system whose evolution is governed \nby quantum mechanics. A precise experiment to violate LGIs require a guarantee that the \nmeasurement does not affect the system or its future dynamics. These Inequalities were \nproposed for dichotomic systems,systems which can have two outcomes. Here we present \nan LG experiment on a three-level quantum system, which theoretically have larger quantum \nupper bound than that of a two-level quantum system. This larger violation also \nprovides a bigger buffer to taking in account of the various experimental imperfections. \nPerforming a quantum computing task requires precise level of control to initialize, \nperform and measure the quantum system. With increasing size of the quantum processor \nthe challenge is to maintain optimal control. Nuclear Magnetic Resonance (NMR) has \nalways been a very faithful test-bed for quantum processing ideas. In NMR, we perform \nRadio Frequency (RF) pulses to control and steer the system to the desired state. Most \nused method to derive the exact frequency and amplitude of these pulses for a given \ntask is based on gradient. Although systematic, one have to simulate these pulses on \na classical computer first, which makes the task very inefficient. We report a a way of \nperforming optimization with a hybrid quantum-classical scheme. This scheme helps us \nperform classically harder computational tasks on the quantum processor. We optimize \npulses which drive our system from 7-coherence state to 12-coherence state on a 12-qubit \nNMR processor. \nElectron Spin Resonance (ESR) employs the same techniques as of NMR but having \nadvantage in larger polarization compared to later. Although this does not imply better \ncontrol, cause the frequency at which pulses are required to control an ESR system fall into \nmicrowave region. Microwave frequency are harder to control electronically, thus making \nit harder for performing ESR quantum computing. The hybrid scheme used in NMR \nexperiment relies on some ideal pulses which are needed to be optimized classically. We \nalleviate this requirement by using finite difference method of calculating gradient. We \ncompare these methods with the earlier methods to show the superiority of such a scheme. \nState-to-state transfer pulses are sufficient for most of the quantum computing task, \nbut, an universal quantum information implementation requires state independent pulses. \nThe techniques used in optimizing state-to-state pulses can be modified to optimize for a \nstate independent pulse. We show that this methods scale polynomially with the number \nof qubits and is general in terms of its implementation. We further reduce the resource \nrequirement by using a NMR related implementation.
In the context of molecular quantum computation the optimal control theory (OCT) is used to obtain shaped laser pulses for high-fidelity control of vibrational qubits. Optimization is done in time domain and the OCT algorithm varies values of electric field in each time step independently, tuning hundreds of thousands of parameters to find one optimal solution. Such flexibility is not available in experiments, where pulse shaping is done in frequency domain and the number of "tuning knobs" is much smaller. The question of possible experimental interpretations of theoretically found OCT solutions arises. In this work we analyze very accurate optimal pulse that we obtained for implementing quantum gate CNOT for the two-qubit system encoded into the exited vibrational states of thiophosgene molecule. Next, we try to alter this pulse by reducing the number of available frequency channels and intentionally introducing systematic and random errors (in frequency domain, by modifying the values of amplitudes and phases of different frequency components). We conclude that a very limited number of frequency components (only 32 in the model of thiophosgene) are really necessary for accurate control of the vibrational two-qubit system, and such pulses can be readily constructed using OCT. If the amplitude and phase errors of different frequency components do not exceed ±3% of the optimal values, one can still achieve accurate transformations of the vibrational two-qubit system, with gate fidelity of CNOT exceeding 0.99.