10 research outputs found
Holonomic quantum computing in symmetry-protected ground states of spin chains
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.Comment: 19 pages, 4 figures. v2: published versio
Fault tolerant architectures for superconducting qubits
In this short review, I draw attention to new developments in the theory of
fault tolerance in quantum computation that may give concrete direction to
future work in the development of superconducting qubit systems. The basics of
quantum error correction codes, which I will briefly review, have not
significantly changed since their introduction fifteen years ago. But an
interesting picture has emerged of an efficient use of these codes that may put
fault tolerant operation within reach. It is now understood that two
dimensional surface codes, close relatives of the original toric code of
Kitaev, can be adapted to effectively perform logical gate operations in a very
simple planar architecture, with error thresholds for fault tolerant operation
simulated to be 0.75%. This architecture uses topological ideas in its
functioning, but it is not 'topological quantum computation' -- there are no
non-abelian anyons in sight. I offer some speculations on the crucial pieces of
superconducting hardware that could be demonstrated in the next couple of years
that would be clear stepping stones towards this surface-code architecture.Comment: 28 pages, 10 figures. For the Nobel Symposium on Qubits for Quantum
Information, submitted to Physica Scripta. v. 2 Corrections and small changes
to reference
Fault-tolerance in two-dimensional topological systems
This thesis is a collection of ideas with the general goal of building, at least in the abstract, a local fault-tolerant quantum computer. The connection between quantum information and topology has proven to be an active area of research in several fields. The introduction of the toric code by Alexei Kitaev demonstrated the usefulness of topology for quantum memory and quantum computation. Many quantum codes used for quantum memory are modeled by spin systems on a lattice, with operators that extract syndrome information placed on vertices or faces of the lattice. It is natural to wonder whether the useful codes in such systems can be classified. This thesis presents work that leverages ideas from topology and graph theory to explore the space of such codes. Homological stabilizer codes are introduced and it is shown that, under a set of reasonable assumptions, any qubit homological stabilizer code is equivalent to either a toric code or a color code. Additionally, the toric code and the color code correspond to distinct classes of graphs. Many systems have been proposed as candidate quantum computers. It is very desirable to design quantum computing architectures with two-dimensional layouts and low complexity in parity-checking circuitry. Kitaev\u27s surface codes provided the first example of codes satisfying this property. They provided a new route to fault tolerance with more modest overheads and thresholds approaching 1%. The recently discovered color codes share many properties with the surface codes, such as the ability to perform syndrome extraction locally in two dimensions. Some families of color codes admit a transversal implementation of the entire Clifford group. This work investigates color codes on the 4.8.8 lattice known as triangular codes. I develop a fault-tolerant error-correction strategy for these codes in which repeated syndrome measurements on this lattice generate a three-dimensional space-time combinatorial structure. I then develop an integer program that analyzes this structure and determines the most likely set of errors consistent with the observed syndrome values. I implement this integer program to find the threshold for depolarizing noise on small versions of these triangular codes. Because the threshold for magic-state distillation is likely to be higher than this value and because logical CNOT gates can be performed by code deformation in a single block instead of between pairs of blocks, the threshold for fault-tolerant quantum memory for these codes is also the threshold for fault-tolerant quantum computation with them. Since the advent of a threshold theorem for quantum computers much has been improved upon. Thresholds have increased, architectures have become more local, and gate sets have been simplified. The overhead for magic-state distillation has been studied, but not nearly to the extent of the aforementioned topics. A method for greatly reducing this overhead, known as reusable magic states, is studied here. While examples of reusable magic states exist for Clifford gates, I give strong reasons to believe they do not exist for non-Clifford gates
Quantum Algorithms, Architecture, and Error Correction
Quantum algorithms have the potential to provide exponential speedups over some of the best known classical algorithms. These speedups may enable quantum devices to solve currently intractable problems such as those in the fields of optimization, material science, chemistry, and biology. Thus, the realization of large-scale, reliable quantum-computers will likely have a significant impact on the world. For this reason, the focus of this dissertation is on the development of quantum-computing applications and robust, scalable quantum-architectures. I begin by presenting an overview of the language of quantum computation. I then, in joint work with Ojas Parekh, analyze the performance of the quantum approximate optimization algorithm (QAOA) on a graph problem called Max Cut. Next, I present a new stabilizer simulation algorithm that gives improved runtime performance for topological stabilizer codes. After that, in joint work with Andrew Landahl, I present a new set of procedures for performing logical operations called color-code lattice-surgery. Finally, I describe a software package I developed for studying, developing, and evaluating quantum error-correcting codes under realistic noise
Topological Code Architectures for Quantum Computation
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\u27s 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\u27s toric code and Bombin\u27s 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
Resource optimization for fault-tolerant quantum computing
In this thesis we examine a variety of techniques for reducing the resources
required for fault-tolerant quantum computation. First, we show how to simplify
universal encoded computation by using only transversal gates and standard
error correction procedures, circumventing existing no-go theorems. We then
show how to simplify ancilla preparation, reducing the cost of error correction
by more than a factor of four. Using this optimized ancilla preparation, we
develop improved techniques for proving rigorous lower bounds on the noise
threshold.
Additional overhead can be incurred because quantum algorithms must be
translated into sequences of gates that are actually available in the quantum
computer. In particular, arbitrary single-qubit rotations must be decomposed
into a discrete set of fault-tolerant gates. We find that by using a special
class of non-deterministic circuits, the cost of decomposition can be reduced
by as much as a factor of four over state-of-the-art techniques, which
typically use deterministic circuits.
Finally, we examine global optimization of fault-tolerant quantum circuits
under physical connectivity constraints. We adapt techniques from VLSI in order
to minimize time and space usage for computations in the surface code, and we
develop a software prototype to demonstrate the potential savings.Comment: 231 pages, Ph.D. thesis, University of Waterlo
Superconducting Resonator with Composite Film and Circuit Layout Design for Quantum Information
The full manipulation of a quantum system can endow us with the power of computing in exponentially increased state space without exponential growth of physical resources. High-quality, innovative superconducting films plays a key role in the next stage of development of large-scale superconducting quantum computing protected by error correction. The further development of quantum error correction theory should also be tailored to the devices’ limitations brought by modern nanofabrication. Neither hardware nor software shall be absented for realizing efficient quantum information processor. In this thesis, we are dedicated to the development in the material of superconducting devices and its application in future large-scale quantum computation with error correction methods.
For the hardware, we developed composite superconducting films for resonators in the application of low-temperature pulsed-ESR for quantum information processing making use of proximity effect. The superconducting resonator is also the key element in superconducting circuits for quantum storage and quantum bus. We compared the performance of superconducting resonators made from single Nb film with those of composite films, including Nb/NbN/Nb, TiN/tapering/NbN/tapering/TiN, and Al/Nb/Al. We found that the quality factor of resonators from Al/Nb/Al surpasses the previous best results from single Nb film under 0.35 T magnetic field. A formula was established for evaluating the surface impedance of composite superconducting film. It has been proved that adjusting phonon density of states in superconducting films can tailor the recombination time of Cooper pairs in thin films and hence affect the coherence time of qubits and Q values of resonators. For the software, we improved the surface code theory to make it resistant to sparse fabrication defects. A physically faulted qubit can be replaced with a working physical qubit near the defective one. Thereby sparse fabrication errors can be collected into one sacrificial layer and isolated from the working layers by turning off their controllable couplers. We proposed a two-dimensional quantum annealing architecture to solve the 4th order binary optimization problem by encoding four-qubit interactions within the coupled local fields acting on a set of physical qubits. We also designed a layout to implement quantum annealing with error correction through stabilizer codes. The stabilizers are realized with weak measurements for real-time, in-run monitoring of spins, to minimize or control the back-action