Fast nonadiabatic two-qubit gates for the Kane quantum computer

In this paper, we apply the canonical decomposition of two-qubit unitaries to find pulse schemes to control the proposed Kane quantum computer. We explicitly find pulse sequences for the controlled-NOT, swap, square root of swap, and controlled Z rotations. We analyze the speed and fidelity of these gates, both of which compare favorably to existing schemes. The pulse sequences presented in this paper are theoretically faster, with higher fidelity, and simpler. Any two-qubit gate may be easily found and implemented using similar pulse sequences. Numerical simulation is used to verify the accuracy of each pulse scheme.

The effects of J-gate potential and interfaces on donor exchange coupling in the Kane quantum computer architecture

We calculate the electron exchange coupling for a phosphorus donor pair in silicon perturbed by a J-gate potential and the boundary effects of the silicon host geometry. In addition to the electron-electron exchange interaction we also calculate the contact hyperfine interaction between the donor nucleus and electron as a function of the varying experimental conditions. Donor separation, depth of the P nuclei below the silicon oxide layer and J-gate voltage become decisive factors in determining the strength of both the exchange coupling and hyperfine interaction-both crucial components for qubit operations in the Kane quantum computer. These calculations were performed using an anisotropic effective-mass Hamiltonian approach. The behaviour of the donor exchange coupling as a function of the parameters varied in this work provides relevant information for the experimental design of these devices.

Gates for the kane quantum computer in the presence of dephasing

In this paper we investigate the effect of dephasing on proposed quantum gates for the solid-state Kane quantum computing architecture. Using a simple model of the decoherence, we find that the typical error in a controlled-NOT gate is 8.3x10(-5). We also compute the fidelities of Z, X, swap, and controlled Z operations under a variety of dephasing rates. We show that these numerical results are comparable with the error threshold required for fault tolerant quantum computation.

Numerical method for determination of the NMR frequency of the single-qubit operation in a silicon-based solid-state quantum computer

A numerical method is introduced to determine the nuclear magnetic resonance frequency of a donor (P-31) doped inside a silicon substrate under the influence of an applied electric field. This phosphorus donor has been suggested for operation as a qubit for the realization of a solid-state scalable quantum computer. The operation of the qubit is achieved by a combination of the rotation of the phosphorus nuclear spin through a globally applied magnetic field and the selection of the phosphorus nucleus through a locally applied electric field. To realize the selection function, it is required to know the relationship between the applied electric field and the change of the nuclear magnetic resonance frequency of phosphorus. In this study, based on the wave functions obtained by the effective-mass theory, we introduce an empirical correction factor to the wave functions at the donor nucleus. Using the corrected wave functions, we formulate a first-order perturbation theory for the perturbed system under the influence of an electric field. In order to calculate the potential distributions inside the silicon and the silicon dioxide layers due to the applied electric field, we use the multilayered Green's functions and solve an integral equation by the moment method. This enables us to consider more realistic, arbitrary shape, and three-dimensional qubit structures. With the calculation of the potential distributions, we have investigated the effects of the thicknesses of silicon and silicon dioxide layers, the relative position of the donor, and the applied electric field on the nuclear magnetic resonance frequency of the donor.

Topological Quantum Computation - An Analysis of an Anyon Model Based on Quantum Double Symmetries

There exists various suggestions for building a functional and a fault-tolerant large-scale quantum computer. Topological quantum computation is a more exotic suggestion, which makes use of the properties of quasiparticles manifest only in certain two-dimensional systems. These so called anyons exhibit topological degrees of freedom, which, in principle, can be used to execute quantum computation with intrinsic fault-tolerance. This feature is the main incentive to study topological quantum computation. The objective of this thesis is to provide an accessible introduction to the theory. In this thesis one has considered the theory of anyons arising in two-dimensional quantum mechanical systems, which are described by gauge theories based on so called quantum double symmetries. The quasiparticles are shown to exhibit interactions and carry quantum numbers, which are both of topological nature. Particularly, it is found that the addition of the quantum numbers is not unique, but that the fusion of the quasiparticles is described by a non-trivial fusion algebra. It is discussed how this property can be used to encode quantum information in a manner which is intrinsically protected from decoherence and how one could, in principle, perform quantum computation by braiding the quasiparticles. As an example of the presented general discussion, the particle spectrum and the fusion algebra of an anyon model based on the gauge group S_3 are explicitly derived. The fusion algebra is found to branch into multiple proper subalgebras and the simplest one of them is chosen as a model for an illustrative demonstration. The different steps of a topological quantum computation are outlined and the computational power of the model is assessed. It turns out that the chosen model is not universal for quantum computation. However, because the objective was a demonstration of the theory with explicit calculations, none of the other more complicated fusion subalgebras were considered. Studying their applicability for quantum computation could be a topic of further research.

Quantum error correction with biased noise

Quantum computing offers powerful new techniques for speeding up the calculation of many classically intractable problems. Quantum algorithms can allow for the efficient simulation of physical systems, with applications to basic research, chemical modeling, and drug discovery; other algorithms have important implications for cryptography and internet security.

At the same time, building a quantum computer is a daunting task, requiring the coherent manipulation of systems with many quantum degrees of freedom while preventing environmental noise from interacting too strongly with the system. Fortunately, we know that, under reasonable assumptions, we can use the techniques of quantum error correction and fault tolerance to achieve an arbitrary reduction in the noise level.

In this thesis, we look at how additional information about the structure of noise, or "noise bias," can improve or alter the performance of techniques in quantum error correction and fault tolerance. In Chapter 2, we explore the possibility of designing certain quantum gates to be extremely robust with respect to errors in their operation. This naturally leads to structured noise where certain gates can be implemented in a protected manner, allowing the user to focus their protection on the noisier unprotected operations.

In Chapter 3, we examine how to tailor error-correcting codes and fault-tolerant quantum circuits in the presence of dephasing biased noise, where dephasing errors are far more common than bit-flip errors. By using an appropriately asymmetric code, we demonstrate the ability to improve the amount of error reduction and decrease the physical resources required for error correction.

In Chapter 4, we analyze a variety of protocols for distilling magic states, which enable universal quantum computation, in the presence of faulty Clifford operations. Here again there is a hierarchy of noise levels, with a fixed error rate for faulty gates, and a second rate for errors in the distilled states which decreases as the states are distilled to better quality. The interplay of of these different rates sets limits on the achievable distillation and how quickly states converge to that limit.

The power of quantum Fourier sampling

How powerful are Quantum Computers? Despite the prevailing belief that Quantum Computers are more powerful than their classical counterparts, this remains a conjecture backed by little formal evidence. Shor's famous factoring algorithm [Shor97] gives an example of a problem that can be solved efficiently on a quantum computer with no known efficient classical algorithm. Factoring, however, is unlikely to be NP-Hard, meaning that few unexpected formal consequences would arise, should such a classical algorithm be discovered. Could it then be the case that any quantum algorithm can be simulated efficiently classically? Likewise, could it be the case that Quantum Computers can quickly solve problems much harder than factoring? If so, where does this power come from, and what classical computational resources do we need to solve the hardest problems for which there exist efficient quantum algorithms?

We make progress toward understanding these questions through studying the relationship between classical nondeterminism and quantum computing. In particular, is there a problem that can be solved efficiently on a Quantum Computer that cannot be efficiently solved using nondeterminism? In this thesis we address this problem from the perspective of sampling problems. Namely, we give evidence that approximately sampling the Quantum Fourier Transform of an efficiently computable function, while easy quantumly, is hard for any classical machine in the Polynomial Time Hierarchy. In particular, we prove the existence of a class of distributions that can be sampled efficiently by a Quantum Computer, that likely cannot be approximately sampled in randomized polynomial time with an oracle for the Polynomial Time Hierarchy.

Our work complements and generalizes the evidence given in Aaronson and Arkhipov's work [AA2013] where a different distribution with the same computational properties was given. Our result is more general than theirs, but requires a more powerful quantum sampler.

Quantum computation using artificial molecules

Two kinds of quantum computation systems using artificial molecules: quantum computer and quantum analog computer are described. The artificial molecule consists of two or three coupled quantum dots stacked along z direction and one single electron, In quantum computer, one-qubit and two-qubit gates are constructed by one molecule and two molecules, respectively. The coupling between two qubits in a quantum gate can be controlled by thin film electrodes. We also constructed a quantum analog computer by designing a three-dot molecule network and mapping a graph 3-colorability problem onto the network. The ground-state configuration of the single electrons in the network corresponds to one of the problem solutions, We numerically study the operations of the two kinds of the quantum computers and demonstrate that they quantum gates can perform the quantum computation and solve complex problems.

Bound on quantum computation time: Quantum error correction in a critical environment

We obtain an upper bound on the time available for quantum computation for a given quantum computer and decohering environment with quantum error correction implemented. First, we derive an explicit quantum evolution operator for the logical qubits and show that it has the same form as that for the physical qubits but with a reduced coupling strength to the environment. Using this evolution operator, we find the trace distance between the real and ideal states of the logical qubits in two cases. For a super-Ohmic bath, the trace distance saturates, while for Ohmic or sub-Ohmic baths, there is a finite time before the trace distance exceeds a value set by the user. © 2010 The American Physical Society.

Quantum-information processing with noisy cluster states

We provide an analysis of basic quantum-information processing protocols under the effect of intrinsic nonidealities in cluster states. These nonidealities are based on the introduction of randomness in the entangling steps that create the cluster state and are motivated by the unavoidable imperfections faced in creating entanglement using condensed-matter systems. Aided by the use of an alternative and very efficient method to construct cluster-state configurations, which relies on the concatenation of fundamental cluster structures, we address quantum-state transfer and various fundamental gate simulations through noisy cluster states. We find that a winning strategy to limit the effects of noise is the management of small clusters processed via just a few measurements. Our study also reinforces recent ideas related to the optical implementation of a one-way quantum computer.

Optimizing Robust Quantum Gates in Open Quantum Systems

We are currently at the cusp of a revolution in quantum technology that relies not just on the passive use of quantum effects, but on their active control. At the forefront of this revolution is the implementation of a quantum computer. Encoding information in quantum states as “qubits” allows to use entanglement and quantum superposition to perform calculations that are infeasible on classical computers. The fundamental challenge in the realization of quantum computers is to avoid decoherence – the loss of quantum properties – due to unwanted interaction with the environment. This thesis addresses the problem of implementing entangling two-qubit quantum gates that are robust with respect to both decoherence and classical noise. It covers three aspects: the use of efficient numerical tools for the simulation and optimal control of open and closed quantum systems, the role of advanced optimization functionals in facilitating robustness, and the application of these techniques to two of the leading implementations of quantum computation, trapped atoms and superconducting circuits. After a review of the theoretical and numerical foundations, the central part of the thesis starts with the idea of using ensemble optimization to achieve robustness with respect to both classical fluctuations in the system parameters, and decoherence. For the example of a controlled phasegate implemented with trapped Rydberg atoms, this approach is demonstrated to yield a gate that is at least one order of magnitude more robust than the best known analytic scheme. Moreover this robustness is maintained even for gate durations significantly shorter than those obtained in the analytic scheme. Superconducting circuits are a particularly promising architecture for the implementation of a quantum computer. Their flexibility is demonstrated by performing optimizations for both diagonal and non-diagonal quantum gates. In order to achieve robustness with respect to decoherence, it is essential to implement quantum gates in the shortest possible amount of time. This may be facilitated by using an optimization functional that targets an arbitrary perfect entangler, based on a geometric theory of two-qubit gates. For the example of superconducting qubits, it is shown that this approach leads to significantly shorter gate durations, higher fidelities, and faster convergence than the optimization towards specific two-qubit gates. Performing optimization in Liouville space in order to properly take into account decoherence poses significant numerical challenges, as the dimension scales quadratically compared to Hilbert space. However, it can be shown that for a unitary target, the optimization only requires propagation of at most three states, instead of a full basis of Liouville space. Both for the example of trapped Rydberg atoms, and for superconducting qubits, the successful optimization of quantum gates is demonstrated, at a significantly reduced numerical cost than was previously thought possible. Together, the results of this thesis point towards a comprehensive framework for the optimization of robust quantum gates, paving the way for the future realization of quantum computers.

Hybrid optimization schemes for quantum control

Optimal control theory is a powerful tool for solving control problems in quantum mechanics, ranging from the control of chemical reactions to the implementation of gates in a quantum computer. Gradient-based optimization methods are able to find high fidelity controls, but require considerable numerical effort and often yield highly complex solutions. We propose here to employ a two-stage optimization scheme to significantly speed up convergence and achieve simpler controls. The control is initially parametrized using only a few free parameters, such that optimization in this pruned search space can be performed with a simplex method. The result, considered now simply as an arbitrary function on a time grid, is the starting point for further optimization with a gradient-based method that can quickly converge to high fidelities. We illustrate the success of this hybrid technique by optimizing a geometric phase gate for two superconducting transmon qubits coupled with a shared transmission line resonator, showing that a combination of Nelder-Mead simplex and Krotov’s method yields considerably better results than either one of the two methods alone.

Exploring Quantum Speed-up Through Cluster-state Computers

The aim of this thesis is to investigate the nature of quantum computation and the question of the quantum speed-up over classical computation by comparing two different quantum computational frameworks, the traditional quantum circuit model and the cluster-state quantum computer. After an introductory survey of the theoretical and epistemological questions concerning quantum computation, the first part of this thesis provides a presentation of cluster-state computation suitable for a philosophical audience. In spite of the computational equivalence between the two frameworks, their differences can be considered as structural. Entanglement is shown to play a fundamental role in both quantum circuits and cluster-state computers; this supports, from a new perspective, the argument that entanglement can reasonably explain the quantum speed-up over classical computation. However, quantum circuits and cluster-state computers diverge with regard to one of the explanations of quantum computation that actually accords a central role to entanglement, i.e. the Everett interpretation. It is argued that, while cluster-state quantum computation does not show an Everettian failure in accounting for the computational processes, it threatens that interpretation of being not-explanatory. This analysis presented here should be integrated in a more general work in order to include also further frameworks of quantum computation, e.g. topological quantum computation. However, what is revealed by this work is that the speed-up question does not capture all that is at stake: both quantum circuits and cluster-state computers achieve the speed-up, but the challenges that they posit go besides that specific question. Then, the existence of alternative equivalent quantum computational models suggests that the ultimate question should be moved from the speed-up to a sort of “representation theorem” for quantum computation, to be meant as the general goal of identifying the physical features underlying these alternative frameworks that allow for labelling those frameworks as “quantum computation”.