Abstract. While solid-state devices offer naturally reliable hardware for modern classical computers, thus far quantum information processors resemble vacuum tube computers in being neither reliable nor scalable. Strongly correlated many body states stabilized in topologically ordered matter offer the possibility of naturally fault tolerant computing, but are both challenging to engineer and coherently control and cannot be easily adapted to different physical platforms. We propose an architecture which achieves some of the robustness properties of topological models but with a drastically simpler construction. Quantum information is stored in the symmetry-protected degenerate ground states of spin-1 chains, while quantum gates are performed by adiabatic non-Abelian holonomies using only single-site fields and nearest-neighbor couplings. Gate operations respect the symmetry, and so inherit some protection from noise and disorder from the symmetry-protected ground states. ar X iv
Abstract. While solid-state devices offer naturally reliable hardware for modern classical computers, thus far quantum information processors resemble vacuum tube computers in being neither reliable nor scalable. Strongly correlated many body states stabilized in topologically ordered matter offer the possibility of naturally fault tolerant computing, but are both challenging to engineer and coherently control and cannot be easily adapted to different physical platforms. We propose an architecture which achieves some of the robustness properties of topological models but with a drastically simpler construction. Quantum information is stored in the symmetry-protected degenerate ground states of spin-1 chains, while quantum gates are performed by adiabatic non-Abelian holonomies using only single-site fields and nearest-neighbor couplings. Gate operations respect the symmetry, and so inherit some protection from noise and disorder from the symmetry-protected ground states. ar X iv
While solid-state devices offer naturally reliable hardware for modern classical computers, thus far quantum information processors resemble vacuum tube computers in being neither reliable nor scalable. Strongly correlated many body states stabilized in topologically ordered matter offer the possibility of naturally fault tolerant computing, but are both challenging to engineer and coherently control and cannot be easily adapted to different physical platforms. We propose an architecture which achieves some of the robustness properties of topological models but with a drastically simpler construction. Quantum information is stored in the symmetry-protected degenerate ground states of spin-1 chains, while quantum gates are performed by adiabatic non-Abelian holonomies using only single-site fields and nearest-neighbor couplings. Gate operations respect the symmetry, and so inherit some protection from noise and disorder from the symmetry-protected ground states.
Implementing quantum gates as non-Abelian holonomies, a class of topologically protected unitary operators, is a particularly promising paradigm for the design of intrinsically stable quantum computers [1]. In contrast to dynamic phases, the geometric phase accumulated by a quantum system propagating through a Hilbert spaceHdepends exclusively on its path. In general, geometric phases can exhibit arbitrary dimensionality. Wilczek and Zee introduced the idea of multi-dimensional, non-Abelian geometric phases - so called holonomies [2]. Anandan later dropped the requirement of adiabaticity to create holonomies, that are truly time-independent [3]. Non-adiabatic holonomies rely on a subspaceHgeoof the Hilbert-space that is spanned by states{∣Φk⟩}kthat fulfill(Φk∣H^∣Φj⟩=0, whereH^is the system's Hamiltonian. Restricting the propagation toHgeoensures parallel transport and, thus, a purely geometric phase (see Fig. 1a) [4], [5]. Quantum optics constitutes a particularly versatile platform for quantum information processing, and in particular for the construction of non-adiabatic holonomic quantum computers: In addition to integration and miniaturization provided by the platform, the bosonic nature of photons also conveniently allows for multiple excitations of the same mode, readily expandingHgeoand enabling the synthesis of holonomies from higher symmetry groupsU(N)as larger and more capable computational units [6], [7].
In recent years, topological phases of matter have presented exciting new avenues to achieve scalable quantum computation. In this thesis, we investigate a class of quantum many-body spin models known as symmetry-protected topological (SPT) phases for use in quantum information processing and storage. We explore the fault-tolerant properties of SPT phases, and how they can be utilized in the design of a quantum computer. Of central importance in this thesis is the concept of quantum error-correction, which in addition to its importance in fault-tolerant quantum computation, is used to characterise the stability of topological phases at finite temperature. We begin with an introduction to quantum computation, quantum error correction, and topological phases of matter. We then focus on the fundamental question of whether symmetry-protected topological phases of matter can exist in thermal equilibrium; we prove that systems protected by global onsite symmetries cannot be ordered at nonzero temperature. Subsequently, we show that certain three-dimensional models with generalised higher-form symmetries can be thermally SPT ordered, and we relate this order to the ability to perform fault-tolerant measurement-based quantum computation. Following this, we assess feasibility of these phases as quantum memories, motivated by the fact that SPT phases in three dimensions can possess protected topological degrees of freedom on their boundary. We find that certain SPT ordered systems can be self-correcting, allowing quantum information to be stored for arbitrarily long times without requiring active error correction. Finally, we develop a framework to construct new schemes of fault-tolerant measurement-based quantum computation. As a notable example, we develop a cluster-state scheme that simulates the braiding and fusion of surface-code defects, offering novel alternative methods to achieve fault-tolerant universal quantum computation.
Adiabatic quantum transistors (AQT) allow quantum logic gates to be performed by applying a large field to a quantum many-body system prepared in its ground state, without the need for local control. The basic operation of such a device can be viewed as driving a spin chain from a symmetry-protected (SP) phase to a trivial phase. This perspective offers an avenue to generalize the AQT and to design several improvements. The performance of quantum logic gates is shown to depend only on universal symmetry properties of a SP phase rather than any fine tuning of the Hamiltonian, and it is possible to implement a universal set of logic gates in this way by combining several different types of SP matter. Such SP AQTs are argued to be robust to a range of relevant noise processes.
We show that universal holonomic quantum computation can be achieved fault tolerantly by adiabatically deforming the gapped stabilizer Hamiltonian of the surface code, where quantum information is encoded in the degenerate ground space of the system Hamiltonian. We explicitly propose procedures to perform each logical operation, including logical state initialization, logical state measurement, logical controlled-not (cnot), state injection, distillation, etc. In particular, adiabatic braiding of different types of holes on the surface leads to a topologically protected, non-Abelian geometric logical cnot. Throughout the computation, quantum information is protected from both small perturbations and low-weight thermal excitations by a constant energy gap and is independent of the system size. Also, the Hamiltonian terms have weight at most four during the whole process. The effect of thermal error propagation is considered during the adiabatic code deformation. With the help of active error correction, this scheme is fault tolerant, in the sense that the computation time can be arbitrarily long for large-enough lattice size. It is shown that the frequency of error correction and the physical resources needed can be greatly reduced by the constant energy gap.
Quantum computers have shown great promise as a resource providing exponential speedup over classical computers for certain problems.Akin to their classical counterparts, quantum computers will be prone to errors.There are two major sources of errors.First, the most formidable source of error is decoherence.To perform quantum computation, quantum information must be stored, processed and read out while protecting it from the debilitating effects of decoherence.Second, even if the system is protected from decoherence, it is almost certain that all the operations performed on the quantum information during its processing will be imperfect.These errors will accumulate over the duration of the computation, eventually causing failure.Thus, for a quantum computing scheme to be feasible, it needs to be fault-tolerant.What this means is that the quantum computer can still perform its task effectively while its components are imperfect.A major breakthrough in this field is the threshold theorem, which asserts that ideal quantum circuits can be simulated efficiently by noisy ones provided the error rates of individual gates are below a certain threshold [1,2,3].Topological quantum computing is an approach to fault-tolerant quantum computation, where the protection from errors occurs not from active intervention, but at a hardware level.The advantage of this approach is that it is robust to localized imperfections.First envisioned by Alexei Kitaev [4], this scheme makes use of a non-abelian exchange statistics of elementary excitations of some two-dimensional quantum many-body system.These excitations form the degenerate ground-space of this topologically ordered system, and all other states are separated from this space by a finite excitation gap.In order to perform computation in this scheme, one needs to create pairs of these non-abelian excitations (called non-abelian anyons) from vacuum, separating them spatially, transporting them around each other to implement the logical gates and finally, fusing them together for measurements.In order for this scheme to be viable, there are two conditions that must be fulfilled.First, there could be errors due to quantum tunneling between the non-abelian anyons.These processes can take place even at zero temperature.But, the amplitude of these processes are exponentially suppressed with the spatial separation of these particles and goes as e -L/l 0 , where L is the spatial separation and l 0 is some characteristic length scale of topological ordering of the system.Thus, to prevent such tunneling during quantum information processing and storage, L has to be much larger than l 0 .Second, at finite temperature, undesired thermally excited quasiparticles can lead to errors.The generation rate of these particles are exponentially suppressed by a Boltzmann factor e -∆/T , where ∆ is the excitation energy and T is the temperature.Therefore, the temperature must be low enough so that there are sufficiently low number of these undesired excitations.There are several physical systems where these excitations could, in principle, be found, manipulated and measured.Non-abelian anyons were proposed as elementary excitations of the ν = 5/2 state in the fractional quantum hall effect [5].The braiding statistics of these particles and their possible use in quantum computing has been investigated thoroughly.For more information on this topic, the reader is invited to consult [6,7].In a pathbreaking work [8], it was proposed that the non-abelian statistics of the excitations of the ν = 5/2 state are shared by the Majorana excitations of a 2D spinless p + ip superconductor.Since then, there have been other theoretical models [9,10] which also support these non-abelian excitations.Moreover, several experimental realizations of these Majorana excitations in solid state systems have also been proposed (for a comprehensive review, see [11]).In this lecture note, we will focus on implementation of topological quantum computation with