Topological quantum computing (QC) is a way of allowing precise quantum computations to run on noisy and imperfect hardware. One implementation uses surface codes created by forming defects in a highly-entangled cluster state. Such a method of computing is a leading candidate for large-scale QC. However, there has been a lack of sufficiently powerful high-level languages to describe computing in this form without resorting to single-qubit operations, which quickly become prohibitively complex as the system size increases. In this paper, we apply the category-theoretic work of Abramsky and Coecke to the topological cluster-state model of QC to give a high-level graphical language that enables direct translation between quantum processes and physical patterns of measurement in a computer—a ‘compiler language’. We give the equivalence between the graphical and topological information flows, and show the applicable rewrite algebra for this computing model. We show that this gives us a native graphical language for the design and analysis of topological quantum algorithms, and finish by discussing the possibilities for automating this process on a large scale.
Topological quantum computing (QC) is a way of allowing precise quantum computations to run on noisy and imperfect hardware. One implementation uses surface codes created by forming defects in a highly-entangled cluster state. Such a method of computing is a leading candidate for large-scale QC. However, there has been a lack of sufficiently powerful high-level languages to describe computing in this form without resorting to single-qubit operations, which quickly become prohibitively complex as the system size increases. In this paper, we apply the category-theoretic work of Abramsky and Coecke to the topological cluster-state model of QC to give a high-level graphical language that enables direct translation between quantum processes and physical patterns of measurement in a computer—a ‘compiler language’. We give the equivalence between the graphical and topological information flows, and show the applicable rewrite algebra for this computing model. We show that this gives us a native graphical language for the design and analysis of topological quantum algorithms, and finish by discussing the possibilities for automating this process on a large scale.
Topological quantum computing has recently proven itself to be a powerful computational model when constructing viable architectures for large scale computation. The topological model is constructed from the foundation of a error correction code, required to correct for inevitable hardware faults that will exist for a large scale quantum device. It is also a measurement based model of quantum computation, meaning that the quantum hardware is responsible only for the construction of a large, computationally universal quantum state. This quantum state is then strategically consumed, allowing for the realisation of a fully error corrected quantum algorithm. The number of physical qubits needed by the quantum hardware and the amount of time required to implement an algorithm is dictated by the manner in which this universal quantum state is consumed. In this paper we examine the problem of algorithmic optimisation in the topological lattice and introduce the required elements that will be needed when designing a classical software package to compile and implement a large scale algorithm on a topological quantum computer.
Topological quantum computing has recently proven itself to be a powerful computational model when constructing viable architectures for large scale computation. The topological model is constructed from the foundation of a error correction code, required to correct for inevitable hardware faults that will exist for a large scale quantum device. It is also a measurement based model of quantum computation, meaning that the quantum hardware is responsible only for the construction of a large, computationally universal quantum state. This quantum state is then strategically consumed, allowing for the realisation of a fully error corrected quantum algorithm. The number of physical qubits needed by the quantum hardware and the amount of time required to implement an algorithm is dictated by the manner in which this universal quantum state is consumed. In this paper we examine the problem of algorithmic optimisation in the topological lattice and introduce the required elements that will be needed when designing a classical software package to compile and implement a large scale algorithm on a topological quantum computer.
Quantum computers are devices that can solve certain problems faster than ordinary, classical computers. The fundamental units of quantum information are qubits, superpositions of two states, a "zero" state and a "one" state. There are various approaches to construct such two-level systems, among others, using superconducting circuits, trapped ions or photons. A common feature of these physical systems is that their coherence times are relatively short compared to the length of useful computations. Superconducting qubits, for instance, are currently the most advanced solid-state qubits, but they decohere after around 100 microseconds, and any information stored in these qubits is lost. On the other hand, useful quantum computations may require quantum information to survive on time scales that are many orders of magnitude longer, as their runtimes can reach several hours or even days. Topological quantum computing is an approach to construct qubits that survive for the entire duration of such a long computation. Topological quantum computing comes in two flavors. The condensed matter approach is to build error-resilient qubits using exotic quasiparticles in topological materials, most prominently Majorana zero modes in topological superconductors. Even though no such qubit has been built to date, the hope is that their coherence times may be significantly longer than the coherence times of currently available solid-state qubits, but are still expected to be too short for large-scale quantum computing. The quantum information approach is to combine many error-prone qubits to build more robust logical qubits using topological error-correcting codes, e.g., surface codes. Even though the first approach is hardware-based and the second approach is software-based, they are deeply related. With Majorana-based qubits, the main logical operations are Majorana fermion parity measurements. By replacing Majorana-based qubits with surface-code patches and parity measurements with lattice-surgery operations, schemes for quantum computation with Majorana-based qubits or with surface codes can be identical. In this thesis, we explore how to construct a large-scale topological fault-tolerant quantum computer that can perform useful quantum computations. Here, topological refers to the nature of the quantum error-correcting code, while the underlying hardware may be based on non-topological qubits, but could also be composed of Majorana-based qubits. We provide a complete picture of such a large-scale device, breaking down large quantum computations into logical qubits and logical operations, describing how these logical operations are performed on the level of physical qubits and physical gates, and finally discussing how these physical qubits can be pieced together in a Majorana-based system using topological superconducting nanowires.
Abstract—Topological quantum computing has recently proven itself to be a very powerful model when considering large-scale, fully error corrected quantum architectures. In addition to its robust nature under hardware errors, it is a software driven method of error corrected computation, with the hardware responsible for only creating a generic quantum resource (the topological lattice). Computation in this scheme is achieved by the geometric manipulation of holes (defects) within the lattice. Interactions between logical qubits (quantum gate operations) are implemented by using particular arrangements of the defects, such as braids and junctions. We demonstrate that junction-based topological quantum gates allow highly regular and structured implementation of large CNOT (controlled-not) gate networks, which ultimately form the basis of the error corrected primitives that must be used for an error corrected algorithm. We present a number of heuristics to optimise the area of the resulting structures and therefore the number of the required hardware resources. I.
Topological quantum computing has recently proven itself to be a very powerful model when considering large-scale, fully error corrected quantum architectures. In addition to its robust nature under hardware errors, it is a software driven method of error corrected computation, with the hardware responsible for only creating a generic quantum resource (the topological lattice). Computation in this scheme is achieved by the geometric manipulation of holes (defects) within the lattice. Interactions between logical qubits (quantum gate operations) are implemented by using particular arrangements of the defects, such as braids and junctions. We demonstrate that junction-based topological quantum gates allow highly regular and structured implementation of large CNOT (controlled-not) gate networks, which ultimately form the basis of the error corrected primitives that must be used for an error corrected algorithm. We present a number of heuristics to optimise the area of the resulting structures and therefore the number of the required hardware resources.
Topological measurement-based quantum computation (MBQC) enables one to carry out universal fault-tolerant quantum computation via single-qubit measurements with a family of large entangled states called cluster states as resources. Raussendorf's three-dimensional cluster states (RTCSs) based on the surface codes are mainly considered for topological MBQC. In such schemes, however, the logical Hadamard, phase (Z1/2), and T(Z1/4) gates which are essential for building up arbitrary logical gates are not implemented natively without using state distillation or lattice dislocations, to the best of our knowledge. In particular, state distillation generally consumes many ancillary logical qubits; thus it is a severe obstacle against practical quantum computing. To solve this problem, we suggest an MBQC scheme via a family of cluster states called color-code-based cluster states (CCCSs) based on the two-dimensional color codes instead of the surface codes. We define logical qubits, construct elementary logical gates, and describe error correction schemes. We show that all logical Clifford gates generated by the cnot, Hadamard, and phase gates can be implemented natively in a fault-tolerant manner, although the T gate still requires state distillation to be fault-tolerant. The error thresholds of MBQC via CCCSs for logical-Z errors are calculated to be 2.7%--2.8%, which are comparable to the values for RTCSs, assuming a simple error model where physical qubits have nontrivial errors independently with the same probability. We analyze and compare the resource overheads of both the schemes. In particular, we show that the number of physical qubits required for implementing a phase gate with CCCSs is at least about 26 times smaller than with RTCSs using state distillation, for the same code distance.