The possibility of practical large-scale quantum computation is largely dictated by the error threshold and qubit overhead of the error correcting scheme. To achieve this goal, architectures that perform fault tolerance quantum computation (FTQC) with 3D cluster states have been designed; however, the current benchmarks for such constructions are still below general experimental requirements. In light of this, we propose a new 3D FTQC scheme by exploiting geometric tolerance to erasure errors using cluster states with novel lattice structures, specifically, by replacing the physical qubits with small-sized error detection codes that allow simple applications of logical CZ gates. We are able to locate Pauli errors locally and effectively convert them into loss errors. With a combination of code concatenation and multi-staged decoding, the scheme is able to achieve higher performance than traditional cubic lattices in protecting quantum information. We have surveyed a wide range of possible concatenation schemes and Monte-Carlo simulations have been performed on the decoders to study their respective scaling behavior. The scheme has demonstrated an overall decrement in active qubit overhead and an improvement of the error threshold, which varies from 10 % to 160% depending on the noise model, indicating a relatively higher rate of fault tolerance compared with previous cluster states-based codes.
The possibility of practical large-scale quantum computation is largely dictated by the error threshold and qubit overhead of the error correcting scheme. To achieve this goal, architectures that perform fault tolerance quantum computation (FTQC) with 3D cluster states have been designed; however, the current benchmarks for such constructions are still below general experimental requirements. In light of this, we propose a new 3D FTQC scheme by exploiting geometric tolerance to erasure errors using cluster states with novel lattice structures, specifically, by replacing the physical qubits with small-sized error detection codes that allow simple applications of logical CZ gates. We are able to locate Pauli errors locally and effectively convert them into loss errors. With a combination of code concatenation and multi-staged decoding, the scheme is able to achieve higher performance than traditional cubic lattices in protecting quantum information. We have surveyed a wide range of possible concatenation schemes and Monte-Carlo simulations have been performed on the decoders to study their respective scaling behavior. The scheme has demonstrated an overall decrement in active qubit overhead and an improvement of the error threshold, which varies from 10 % to 160% depending on the noise model, indicating a relatively higher rate of fault tolerance compared with previous cluster states-based codes.
We investigate a family of fault-tolerant quantum error correction schemes based on the concatenation of small error detection or error correction codes with the three-dimensional cluster state. We propose fault-tolerant state preparation and decoding schemes that effectively convert every circuit-level error into an erasure error, leveraging the cluster state's high threshold against such errors. We find a set of codes for which such a conversion is possible, and study their performance against the standard circuit-level depolarizing model. Our best performing scheme, which is based on a concatenation with a classical code, improves the threshold by 16.5% and decreases the spacetime overhead by 32% compared to the scheme without concatenation, with each scheme subject to a physical error rate of 10−3 and achieving a logical error rate of 10−6 .
The threshold theorem indicates that if errors are all local and their rates are below a certain threshold, it is possible to implement large scale quantum computation with arbitrarily small error based on active quantum error correction (QEC). However, most fault-tolerant quantum computation (FTQC) schemes require enormous overhead to achieve this rate by increasing either the concatenation levels for concatenated codes or the distance (hence the size) for topological codes. As a result, a logical qubit is usually encoded in more than thousands of physical qubits, even when the error rate is below the threshold. In this thesis, we try to reduce the resource overhead of FTQC. We first study a quantum computation scheme using clouds of atoms trapped on atomic chips. It suggests that this scheme is robust to noise to some extent and has potential to be scalable. Secondly, we propose a FTQC scheme of encoding many logical qubits into large code blocks. The error performance of such scheme is numerically studied. Preparation of various ancilla states necessary for computation is also explored. Thirdly, we explore the idea of introducing energy gap to suppress thermal noise on each qubit. We show that fault-tolerant holonomic quantum computation (HQC) can be implemented in stabilizer codes with the existence of such energy protection. Especially, we studied fault-tolerant HQC in surface codes in detail. This scheme opens the possibility for self-correcting quantum computation in 2D lattice.
Quantum error correction is the backbone of fault-tolerant quantum computation, a necessary requirement for any large scale quantum computer. The fault-tolerance threshold theorem has long been a target for experimental precision, allowing for the possibility of reducing logical error rates to arbitrarily low levels without excessive overhead. While there are many promising fault-tolerant architectures, the path towards the most practical fault-tolerant scheme is far from decided and may vary for differing physical models. This thesis proposes new schemes for universal fault-tolerant quantum computation in both the concatenated and topological code settings. \n \nThrough the concatenation of two different error correcting codes, a set of universal fault-tolerant gates can be obtained without requiring the need for magic state distillation. A lower bound of 1.28times10−3 for the fault-tolerance threshold under circuit level depolarizing noise is obtained. Additionally, stacked codes are proposed as a means to simulate the action of a 3D topological code in 2D, allowing for the application of a universal set of transversal operations. While fault-tolerant, unfortunately the scheme does not exhibit a threshold due to the decreasing pseudo-threshold with growing code distance, yet points to potential interesting avenues for fault-tolerant computation in 2D without distillation. \n \nOne of the primary avenues to constructing fault-tolerant logical operations is through transversal operations. In this thesis, the set of single qubit logical gates that can be implemented transversally are characterized and determined to all belong to the Clifford hierarchy. Moreover, any diagonal two-qubit operation that can be applied transversally must belong to the same level of the Clifford hierarchy as the set of gates that can be implemented in the single-qubit case. \n \nThe opposite to quantum error correction is privacy, where the output of a channel is disguised from its input. The two are fundamentally related through the complementary channel construction. This thesis presents a new class of private quantum channels, expanding the existing class beyond a seemingly fundamental restriction. This yields interesting insights into the structure of quantum information and the leaking of information to external environments. Additionally, the duality is only recovered when extending the complementary channel to sufficiently high environmental dimension. \n \nFinally, the error properties of bucket brigade quantum Random Access Memory (qRAM) are assessed. It is determined that using the bucket brigade qRAM architecture for the running of Grover's algorithm necessitates reducing the error rate of the individual components to exponentially small levels for an exponential sized memory. As such, fault-tolerant architectures will likely play an essential role in the construction of such computing primitives.
Quantum computers are far more error-prone than their classical counterparts. Therefore, to build a quantum computer capable of running large-scale quantum algorithms, we must use the techniques of quantum error correction to ensure that the computer produces the correct output even when its components are unreliable. However, the resource requirements of building such a fault-tolerant quantum computer are currently prohibitive. Here, we examine the utility of using three-dimensional (3D) surface codes in a fault-tolerant quantum computer. This family of topological error-correcting codes is a generalization of the well-known 2D surface code to three spatial dimensions. We show that certain 3D surface codes have a transversal logical non-Clifford gate. In a quantum computing architecture, a non-Clifford gate is required to achieve computational universality. Transversal gates do not entangle qubits in different codes, so they are naturally fault tolerant because they do not spread errors. Next, we consider the problem of decoding 3D surface codes. In a quantum error-correcting code, we cannot observe the qubits directly, so we measure parity-check operators to gain information about the state of the code. Decoding is the problem of estimating what error has occurred given a list of unsatisfied parity checks. We observe that 3D surface codes offer asymmetric protection against bit-flip and phase-flip errors, but in both cases, we find that a threshold error rate exists below which we can suppress logical errors by increasing the size of the code. We use our results about logical gates and decoding to propose two fault-tolerant quantum computing architectures that utilize 3D surface codes. Finally, we compare the resource requirements of our architectures with the requirements of leading quantum computing architectures based on topological codes. We find that one of our architectures may be competitive with the leading architectures, depending on the properties of the physical systems used to build the qubits.
In this thesis we present three main contributions to the field of topological quantum error correcting codes. We focus on some of the properties of such codes required for fault-tolerant quantum computation. Prior work has concentrated on determining error rate thresholds of particular models, but increasingly other parameters are gaining prominence. One of these is the overhead -- the quantity of a named resource required to achieve a desired level of performance from the code. We characterise the qubit overhead of the toric code in a fault-tolerant setting. These results provide a general framework for determining the overhead for other code constructions with more complicated noise models. Next we introduce a decoding algorithm, applicable to topological codes in a qudit architecture, specifically those where fault-tolerance is achieved through repeated syndrome measurements. It is computationally light and capable of decoding qudits of arbitrarily high dimension with negligible increase in its run time. The threshold of the decoder is limited by the percolation of the syndromes. Using local matching techniques we are able to overcome this limitation, increasing the threshold by almost a factor of two for high qudit dimensions. Finally, we turn our attention to a second family of topological quantum codes: the colour codes. In three and higher spatial dimensions such codes can support transversal non-Clifford gates. We show, using a technique that we call a star-bipartition of the vertices of the lattice, that any existing qubit colour code lattice can be used to define a qudit colour code. By generalising the notion of triorthogonal matrices we derive analogous transversality properties in the qudit codes.
The construction of a large-scale fault-tolerant quantum computer is an outstanding scientific and technological goal. It holds the promise to allow us to solve a variety of complex problems such as factoring large numbers, quick database search, and the quantum simulation of many-body quantum systems in fields as diverse as condensed matter, quantum chemistry, and even high-energy physics. Sophisticated theoretical protocols for reliable quantum information processing under imperfect conditions have been de-veloped, when errors affect and corrupt the fragile quantum states during storage and computations. Arguably, the most realistic and promising ap-proach towards practical fault-tolerant quantum computation are topologi-cal quantum error-correcting codes, where quantum information is stored in interacting, topologically ordered 2D or 3D many-body quantum systems. This approach offers the highest known error thresholds, which are already today within reach of the experimental accuracy in state-of-the-art setups. A combination of theoretical and experimental research is needed to store, protect and process fragile quantum information in logical qubits effectively so that they can outperform their constituting physical qubits. Whereas small-scale quantum error correction codes have been implemented, one of the main theoretical challenges remains to develop new and improve existing efficient strategies (so-called decoders) to derive (near-)optimal error cor-rection operations in the presence of experimentally accessible measurement information and realistic noise sources. One main focus of this project is the development and numerical implementation of scalable, efficient decoders to operate topological color codes. Additionally, we study the feasibility of im-plementing quantum error-correcting codes fault-tolerantly in near-term ion traps. To this end, we use realistic modeling of the different noise sources, computer simulations, and most modern quantum information approaches to quantum circuitry and noise suppression techniques.