Topological quantum computing promises error-resistant quantum computation without active error correction. However, there is a worry that during the process of executing quantum gates by braiding anyons around each other, extra anyonic excitations will be created that will disorder the encoded quantum information. Here, we explore this question in detail by studying adiabatic code deformations on Hamiltonians based on topological codes, notably Kitaev's surface codes and the more recently discovered color codes. We develop protocols that enable universal quantum computing by adiabatic evolution in a way that keeps the energy gap of the system constant with respect to the computation size and introduces only simple local Hamiltonian interactions. This allows one to perform holonomic quantum computing with these topological quantum computing systems. The tools we develop allow one to go beyond numerical simulations and understand these processes analytically.
Topological quantum computing promises error-resistant quantum computation without active error correction. However, there is a worry that during the process of executing quantum gates by braiding anyons around each other, extra anyonic excitations will be created that will disorder the encoded quantum information. Here, we explore this question in detail by studying adiabatic code deformations on Hamiltonians based on topological codes, notably Kitaev's surface codes and the more recently discovered color codes. We develop protocols that enable universal quantum computing by adiabatic evolution in a way that keeps the energy gap of the system constant with respect to the computation size and introduces only simple local Hamiltonian interactions. This allows one to perform holonomic quantum computing with these topological quantum computing systems. The tools we develop allow one to go beyond numerical simulations and understand these processes analytically.
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.
In seminal work (arxiv:quant-ph/9707021) Alexei Kitaev proposed topological quantum computing (arXiv:cond-mat/0010440, arxiv:quant-ph/9707021, arXiv:quant-ph/0001108, arXiv:0707.1889), whereby logic gates of a quantum computer are conducted by creating, braiding and fusing anyonic particles on a two-dimensional plane. Furthermore, he showed the proposal is inherently robust to local perturbations (arXiv:cond-mat/0010440, arxiv:quant-ph/9707021, arXiv:1001.0344, arXiv:1001.4363) when anyons are created as quasiparticle excitations of a topologically ordered lattice model prepared at zero temperature. Over the decades following this proposal there have been considerable technological developments towards the construction of a fault-tolerant quantum computer. Rather than maintaining some target ground state at zero temperature, a modern approach is to actively correct the errors a target state experiences, where we use noisy quantum circuit elements to identify and subsequently correct for deviations from the ideal state. We present an error-correction scheme that enables us to carry out robust universal quantum computation by braiding anyons. We show that our scheme can be carried out on a suitably large device with an arbitrarily small failure rate assuming circuit elements are below some threshold level of local noise. The error-corrected scheme we have developed therefore enables us to carry out fault-tolerant topological quantum computation using modern quantum hardware that is now under development.
This dissertation is concerned with quantum computation using many-body quantum systems encoded in topological codes. The interest in these topological systems has increased in recent years as devices in the lab begin to reach the fidelities required for performing arbitrarily long quantum algorithms. The most well-studied system, Kitaev's toric code, provides both a physical substrate for performing universal fault-tolerant quantum computations and a useful pedagogical tool for explaining the way other topological codes work. In this dissertation, I first review the necessary formalism for quantum information and quantum stabilizer codes, and then I introduce two families of topological codes: Kitaev's toric code and Bombin's color codes. I then present three chapters of original work. First, I explore the distinctness of encoding schemes in the color codes. Second, I introduce a model of quantum computation based on the toric code that uses adiabatic interpolations between static Hamiltonians with gaps constant in the system size. Lastly, I describe novel state distillation protocols that are naturally suited for topological architectures and show that they provide resource savings in terms of the number of required ancilla states when compared to more traditional approaches to quantum gate approximation.
There exists a close relationship between topological quantum error-correcting codes and topological order in condensed matter systems. Indeed, a topological stabilizer code can be regarded as the ground state of a suitable Hamiltonian model, so that "wrong" syndromes correspond to excitations. These excitations are anyons, quasiparticles that carry a topological charge and exhibit exotic statistics. Anyon models can be symmetric under some permutations of their topological charges. One can then conceive topological defects that, under monodromy, transform anyons according to a symmetry. We call these defects twists. Twists give rise to new topological degrees of freedom in the ground state, useful as a quantum memory. Moreover, twists can be braided to perform topologically protected gates on these topological qubits. Thus, twists provide a new way to encode and compute with topological codes through code deformations. Because the properties of twists depend on the anyon model, codes with different anyon content give rise to different computational capabilities. E.g., in the well-known toric code a process where suitable twists are braided and fused has the same outcome as if they were Ising anyons. These are non-abelian anyons: braiding produces non-trivial gates on encoded qubits.
Topological quantum computation encodes quantum information in the internal fusion space of non-Abelian anyonic quasiparticles, whose braiding implements logical gates. This goes beyond Abelian topological order (TO) such as the toric code, as its anyons lack internal structure. However, the simplest non-Abelian generalizations of the toric code do not support universality via braiding alone. Here we demonstrate that such minimally non-Abelian TOs can be made universal by treating anyon fusion as a computational primitive. We prepare a 54-qubit TO wavefunction associated with the smallest non-Abelian group, S3, on Quantinuum's H2 quantum processor. This phase of matter exhibits cyclic anyon fusion rules, known to underpin universality, which we evidence by trapping a single non-Abelian anyon on the torus. We encode logical qutrits in the nonlocal fusion space of non-Abelian fluxes and, by combining an entangling braiding operation with anyon charge measurements, realize a universal topological gate set and read-out, which we further demonstrate by topologically preparing a magic state. This work establishes S3 TO as simple enough to be prepared efficiently, yet rich enough to enable universal topological quantum computation.
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