In many previous 'always-on' quantum computing (QC) schemes the non-nearest-neighbour couplings are often omitted because of their weak strength compared with the nearest-neighbour coupling. In this paper, we consider the problem of residual long-range interaction. We propose a QC scheme in which by the method of generalized 'barrier spin' encoding, the influence of the long-range coupling can be significantly suppressed. The universal quantum gates could be deterministically established. We also study the application of this scheme in a Josephson junction system.
In many previous 'always-on' quantum computing (QC) schemes the non-nearest-neighbour couplings are often omitted because of their weak strength compared with the nearest-neighbour coupling. In this paper, we consider the problem of residual long-range interaction. We propose a QC scheme in which by the method of generalized 'barrier spin' encoding, the influence of the long-range coupling can be significantly suppressed. The universal quantum gates could be deterministically established. We also study the application of this scheme in a Josephson junction system.
Since the seminal work by Shor who proposed a quantum algorithm factorizing integers into prime factors, it has become manifest that the laws of quantum mechanics provide resources for computation that overpower classical physics. The computational advantages that quantum physics offers have stimulated a tremendous amount of theoretical and experimental research. In this context, spin systems have played a major role, given that the spin degree of freedom -- with the paradigmatic case of the spin-1/2 of electrons -- represents an obvious candidate for the encoding of an elementary bit of quantum information (qubit). \n \nOn the other hand, however, quantum objects are very fragile entities, being very susceptible to the environment they reside in. This fragility of qubits is one of the main obstacles in the realization of a quantum computer. \n \nIn this thesis, we mainly address the two following questions relevant to quantum computation. \n \ni) How is it possible to realize quantum gates both in a reliable and scalable way? \n \nii) How can we store quantum information in a way that is resilient to the errors caused by the thermal environment? \n \nWe focus on spin systems and demonstrate that long-range spin-spin interactions in the models considered can have beneficial effects. \n \nIn their pioneering work, Loss and DiVincenzo proposed a way to perform quantum computation in a semiconductor-based architecture where the spin state of an electron trapped in a quantum dot is chosen to encode the elementary qubit. In this proposal, the spins are required to lie spatially close to each other, and this might complicate the realization of a scalable architecture. \n \nIn the first part of the thesis we thus propose a scheme that allows the constraint on the positioning of the qubits to be relaxed. This is achieved by introducing a ferromagnetic coupler between the distant qubits, to which it is coupled via a dipolar interaction. Most importantly, our proposal is applicable to any type of spin qubits and in particular to the technologically very relevant silicon-based qubits and NV-centers in diamond to which previous coupling schemes do not apply. \n \nAs additional key element, a quantum computer needs a memory capable of reliably storing quantum information in the presence of thermal fluctuations. \n \nThis brings us to the second part of this thesis, where we consider self-correcting memories, for which the protection against thermal noise is built-in at the hardware level. We propose physical models that exhibit these self-correcting properties, using as a starting point the well known topologically ordered toric code. In particular, we investigate how to induce long-range interactions between the spins of the toric code, since such interactions help increase the memory lifetime. \n \nAs a first step, we study a honeycomb quantum spin model coupled to delocalized cavity modes. We investigate the properties of the low-energy toric code Hamiltonian and show that the coupling to cavity modes prolongs the lifetime of the memory and offers a method to detect the presence of excitations. \nWhile the introduction of extended bosonic modes makes the model non-local, we also propose a purely local model consisting of a toric code embedded in a three-dimensional cubic lattice of hopping bosons; the low-energy sector of a toric code coupled to a three-dimensional Heisenberg ferromagnet in a broken-symmetry state realizes this model. Our analysis leads to an energy penalty for the creation of defects that grows linearly with the size L of the memory and thus to a lifetime increasing exponentially with L. \n \nIn the third part of this thesis, we study spin systems that support anyons, i.e., particles with fractional statistics. Similar to the toric code, such systems are topologically ordered: they are immune to local perturbations and quantum gates are implemented by non-local operations, namely the exchange of anyons, whose outcomes depend only on the topology of the exchange. Here again the fault-tolerance is achieved at the level of the hardware and physical systems supporting non-abelian anyons are thus promising platforms for quantum computation. \n \nWe focus on spin systems that exhibit some of these properties and specifically on variations of the honeycomb quantum spin model. We first investigate the exact solution of the honeycomb model in detail and derive an explicit formula for the projector onto the physical subspace. We use this result to study inhomogeneous open spin ladders, related to the honeycomb model, which can be tuned between topological and non-topological phases. We test the robustness of Majorana end states (MES) which emerge at the boundary between sections in different topological phases. Furthermore, we present a trijunction setup where MES can be braided. This is of interest since MES in these spin ladders potentially follow non-abelian braiding statistics. Finally, we study the ground states of the aforementioned ladders and show that they are free of vortices when the signs of the spin couplings are all positive or negative. To prove this, we use exact reflection-positivity-based methods as well as approximate methods. \n \nIn the last part of the thesis, we provide an extension of the Mermin-Wagner theorem to a system of lattice spins that are spin-coupled to itinerant and interacting charge carriers. We prove that neither (anti-) ferromagnetic nor helical long-range order is possible in one and two dimensions at any finite temperature (in the absence of spin-orbit). The fundamental question whether spontaneous ordering of the lattice spins occurs in these systems is of interest in the context of quantum computation; the polarization of nuclear spins coupled to a two-dimensional electron gas is a possible route towards the reduction of decoherence induced by the fluctuating Overhauser field in gate-defined quantum dots. \n \n
We consider a one-dimensional chain of many superconducting quantum interference devices (SQUIDs), serving as charge qubits. Each SQUID is coupled to its nearest neighbor through constant capacitances. We study the quantum logic operations and implementation of entanglement in this system. Arrays with two and three qubits are considered in detail. We show that the creation of entanglement with an arbitrary number of qubits can be implemented, without systematic errors, even when the coupling between qubits is not small. A relatively large coupling constant allows one to increase the clock speed of the quantum computer. We analytically and numerically demonstrate the creation of the entanglement for this case, which can be a good test for the experimental implementation of a relatively simple quantum protocol with many qubits. The logic operations in a system with long-range interaction are not universal. We discuss a possible application of our approach for implementing universal quantum logic for more complex algorithms by decreasing the coupling constant and, correspondingly, decreasing the clock speed. The errors introduced by the long-range interaction for the universal logic gates are estimated analytically and calculated numerically. Our results can be useful for experimental implementation of quantum algorithms using controlled magnetic fluxes and gate voltages applied to the SQUIDs. The algorithms discussed in this paper can be implemented using already existing technologies in superconducting systems with constant interqubit coupling.
We consider the problem of selectively controlling couplings in a practical quantum processor with always-on interactions that are diagonal in the computational basis, using sequences of local not gates. This methodology is well known in nuclear magnetic resonance implementations, but previous approaches do not scale efficiently for the general fully connected Hamiltonian, where the complexity of finding time-optimal solutions makes them only practical up to a few tens of qubits. Given the rapid growth in the number of qubits in cutting-edge quantum processors, it is of interest to investigate the applicability of this control scheme to much larger-scale systems with realistic restrictions on connectivity. Here we present an efficient scheme to find near time-optimal solutions that can be applied to engineered qubit arrays with local connectivity for any number of qubits, indicating the potential for practical quantum computing in such systems.
We consider the problem of selectively controlling couplings in a practical quantum processor with always-on interactions that are diagonal in the computational basis, using sequences of local not gates. This methodology is well known in nuclear magnetic resonance implementations, but previous approaches do not scale efficiently for the general fully connected Hamiltonian, where the complexity of finding time-optimal solutions makes them only practical up to a few tens of qubits. Given the rapid growth in the number of qubits in cutting-edge quantum processors, it is of interest to investigate the applicability of this control scheme to much larger-scale systems with realistic restrictions on connectivity. Here we present an efficient scheme to find near time-optimal solutions that can be applied to engineered qubit arrays with local connectivity for any number of qubits, indicating the potential for practical quantum computing in such systems.
We investigate the design and functionality of a network of loop-shaped charge qubits with switchable nearest-neighbour coupling. The qubit coupling is achieved by placing large Josephson junctions (JJs) at the intersections of the qubit loops and selectively applying bias currents. The network is scalable and makes it possible to perform a universal set of quantum gates. The coupling scheme allows gate operation at the charge degeneracy point of each qubit, and also applies to charge-phase qubits. Additional JJs included in the qubit loops for qubit readout can also be employed for qubit coupling.
We introduce a scheme for realizing universal quantum computing in a linear nearest neighbor architecture with fixed couplings. We first show how to realize a controlled-NOT gate operation between two adjacent qubits without having to isolate the two qubits from qubits adjacent to them. The gate operation is implemented by applying two consecutive pulses of equal duration, but varying amplitudes, on the target qubit. Since only a single control parameter is required in implementing our scheme, it is very efficient. We next show how our scheme can be used to realize single qubit rotations and two-qubit controlled-unitary operations. As most proposals for solid state implementations of a quantum computer use a one-dimensional line of qubits, the schemes presented here will be extremely useful.