We explore in this paper incoherent processes, which can be modelled as a distribution of unitary processes corresponding to spatially or slowly varying control parameters. Incoherent processes are an important source of errors in quantum information processing. These processes also cause problems and ambiguities for quantum process tomography for which we give examples to illustrate the issues. In an effort to characterize these processes, we develop methods of analysis based on the eigenvalue spectra of superoperators to extract information about the noise source. Examples from nuclear magnetic resonance are provided to illustrate the utility of such information in designing control sequences that directly counteract incoherent noise.
We explore in this paper incoherent processes, which can be modelled as a distribution of unitary processes corresponding to spatially or slowly varying control parameters. Incoherent processes are an important source of errors in quantum information processing. These processes also cause problems and ambiguities for quantum process tomography for which we give examples to illustrate the issues. In an effort to characterize these processes, we develop methods of analysis based on the eigenvalue spectra of superoperators to extract information about the noise source. Examples from nuclear magnetic resonance are provided to illustrate the utility of such information in designing control sequences that directly counteract incoherent noise.
Incoherence in the controlled Hamiltonian is an important limitation on the precision of coherent control in quantum information processing. Incoherence can typically be modeled as a distribution of unitary processes arising from slowly varying experimental parameters. We show how it introduces artifacts in quantum process tomography and we explain how the resulting estimate of the superoperator may not be completely positive. We then go on to attack the inverse problem of extracting an effective distribution of unitaries that characterizes the incoherence via a perturbation theory analysis of the superoperator eigenvalue spectra.
Errors in the control of quantum systems may be classified as unitary, decoherent, and incoherent. Unitary errors are systematic, and result in a density matrix that differs from the desired one by a unitary operation. Decoherent errors correspond to general completely positive superoperators, and can only be corrected using methods such as quantum error correction. Incoherent errors can also be described, on average, by completely positive superoperators, but can nevertheless be corrected by the application of a locally unitary operation that “refocuses” them. They are due to reproducible spatial or temporal variations in the system’s Hamiltonian, so that information on the variations is encoded in the system’s spatiotemporal state and can be used to correct them. In this paper liquid-state nuclear magnetic resonance is used to demonstrate that such refocusing effects can be built directly into the control fields, where the incoherence arises from spatial inhomogeneities in the quantizing static magnetic field as well as the radio-frequency control fields themselves. Using perturbation theory, it is further shown that the eigenvalue spectrum of the completely positive superoperator exhibits a characteristic spread that contains information on the Hamiltonians’ underlying distribution.
The results of quantum process tomography on a three-qubit nuclear magnetic resonance quantum information processor are presented and shown to be consistent with a detailed model of the system-plus-apparatus used for the experiments. The quantum operation studied was the quantum Fourier transform, which is important in several quantum algorithms and poses a rigorous test for the precision of our recently developed strongly modulating control fields. The results were analyzed in an attempt to decompose the implementation errors into coherent (overall systematic), incoherent (microscopically deterministic), and decoherent (microscopically random) components. This analysis yielded a superoperator consisting of a unitary part that was strongly correlated with the theoretically expected unitary superoperator of the quantum Fourier transform, an overall attenuation consistent with decoherence, and a residual portion that was not completely positive-although complete positivity is required for any quantum operation. By comparison with the results of computer simulations, the lack of complete positivity was shown to be largely a consequence of the incoherent errors which occurred over the full quantum process tomography procedure. These simulations further showed that coherent, incoherent, and decoherent errors can often be identified by their distinctive effects on the spectrum of the overall superoperator. The gate fidelity of the experimentally determined superoperator was 0.64, while the correlation coefficient between experimentally determined superoperator and the simulated superoperator was 0.79; most of the discrepancies with the simulations could be explained by the cumulative effect of small errors in the single qubit gates.
Quantum information processing has been the subject of countless discoveries since the early 1990's. It is believed to be the way of the future for computation: using quantum systems permits one to perform computation exponentially faster than on a regular classical computer. Unfortunately, quantum systems that not isolated do not behave well. They tend to lose their quantum nature due to the presence of the environment. If key information is known about the noise present in the system, methods such as quantum error correction have been developed in order to reduce the errors introduced by the environment during a given quantum computation. In order to harness the quantum world and implement the theoretical ideas of quantum information processing and quantum error correction, it is imperative to understand and quantify the noise present in the quantum processor and benchmark the quality of the control over the qubits. Usual techniques to estimate the noise or the control are based on quantum process tomography (QPT), which, unfortunately, demands an exponential amount of resources. \n \nThis thesis presents work towards the characterization of noisy processes in an efficient manner. The protocols are developed from a purely abstract setting with no system-dependent variables. To circumvent the exponential nature of quantum process tomography, three different efficient protocols are proposed and experimentally verified. The first protocol uses the idea of quantum error correction to extract relevant parameters about a given noise model, namely the correlation between the dephasing of two qubits. Following that is a protocol using randomization and symmetrization to extract the probability that a given number of qubits are simultaneously corrupted in a quantum memory, regardless of the specifics of the error and which qubits are affected. Finally, a last protocol, still using randomization ideas, is developed to estimate the average fidelity per computational gates for single and multi qubit systems. \n \nEven though liquid state NMR is argued to be unsuitable for scalable quantum information processing, it remains the best test-bed system to experimentally implement, verify and develop protocols aimed at increasing the control over general quantum information processors. For this reason, all the protocols described in this thesis have been implemented in liquid state NMR, which then led to further development of control and analysis techniques.
For several decades it has been appreciated that quantum computers hold incredible promise to perform calculations intractable to classical computation. However, this promise has be slow to realize. Dozens of quantum systems are currently being investigated for use in quantum information processing - none of which have yet demonstrated algorithms involving more than a handful of qubits and it remains unclear which, if any, of these systems will ultimately compose a scalable, robust quantum information processing architecture. In this thesis we employ analytical, optimal and algebraic control techniques to evaluate various quantum systems for their potential use in quantum information processing. In doing so, we have additionally identified several novel characterization procedures capable of probing both the coherent and incoherent dynamics of quantum systems. The first part of this thesis discusses work motivated by attempts to utilize donor qubits in silicon as quantum bits. We first propose a measurement of the state of a single donor electron spin using two-dimensional electron gas of a field-effect transistor and electrically detected magnetic resonance. We analyze the potential sensitivity of this measurement and show that it is a quantum nondemolition measurement of an electron-encoded state. We then present the first of two novel qubit characterization procedures. We consider the problem of rapidly characterizing a large number of similarly prepared qubits using techniques from optimal experiment design. All qubits are assumed to evolve according to the same physical processes, though the Hamiltonian parameters may vary from device to device - an inevitability in solid state qubits. We use the Cram\\'er-Rao bound on the variance of a point estimator to construct the optimal series of experiments to estimate these free parameters, and present a complete analysis of the optimal experimental configuration. Though applied to dipole- and exchange-coupled qubits, this technique is widely applicable to other systems. The second part of the thesis discusses the role that control can play in measuring and mitigating noise in qubit systems. Our first result describes a method for quickly simulating the effects of arbitrary markovian noise on qubit systems through the use of a numerically optimized, multi-state Markovian fluctuator. This ability to rapidly simulate the noisy qubit evolution allows us to compute control sequences capable of maximally decoupling the qubit from the noise source. We then introduce the second characterization procedure of the these, showing that a single measurable and controllable qubit may act as a spectrometer of dephasing noise. We show that the formalism of dynamical decoupling can be used to estimate the short-time correlation function of the noise source, while long time correlations may be estimated by a very simple series of free evolution experiments. This technique is applicable to the wide range of physical implementations which suffer from dephasing noise.The final part of this thesis demonstrates that trapped neutral atoms may be utilized for the robust simulation of complex systems exhibiting a topological phase. We present a method to simulate the toric code Hamiltonian stroboscopically, and demonstrate that our technique preserves the ground state degeneracy . Furthermore, we introduce a dissipative mechanism allowing for thermalization of the system to a finite temperature or direct cooling to the ground state manifold.