We explore basic necessary protocols to achieve fault tolerance for quantum computation in the cluster state and circuit models. For the cluster state model we simulate the implementation of an arbitrary rotation via only measurement on a decohered cluster state. Fidelity is used to quantify the accuracy of the initial cluster state and a gate fidelity for the arbitrary rotation is determined. In the circuit model we compare the accuracy of two methods that can be used to construct a logical zero state appropriate for the [7, 1, 3] Steane quantum error correction code in a non-equiprobable Pauli operator error environment: a fault tolerant method done by applying error correction on seven qubits all in the state zero, and a non-fault tolerant method done by implementing the encoding gate sequence. We find that the latter construction method, in spite of its lack of "fault tolerance," outputs a seven qubit state with a higher fidelity than the first (fault tolerant) method. However, the fidelity of the single qubit of stored information exhibits almost equivalent values between the two construction methods.
We explore basic necessary protocols to achieve fault tolerance for quantum computation in the cluster state and circuit models. For the cluster state model we simulate the implementation of an arbitrary rotation via only measurement on a decohered cluster state. Fidelity is used to quantify the accuracy of the initial cluster state and a gate fidelity for the arbitrary rotation is determined. In the circuit model we compare the accuracy of two methods that can be used to construct a logical zero state appropriate for the [7, 1, 3] Steane quantum error correction code in a non-equiprobable Pauli operator error environment: a fault tolerant method done by applying error correction on seven qubits all in the state zero, and a non-fault tolerant method done by implementing the encoding gate sequence. We find that the latter construction method, in spite of its lack of "fault tolerance," outputs a seven qubit state with a higher fidelity than the first (fault tolerant) method. However, the fidelity of the single qubit of stored information exhibits almost equivalent values between the two construction methods.
We present a detailed description of an architecture for fault-tolerant quantum computation, which is based on the cluster model of encoded qubits. In this cluster-based architecture, concatenated computation is implemented in a quite different way from the usual circuit-based architecture where physical gates are recursively replaced by logical gates with error-correction gadgets. Instead, some relevant cluster states, say fundamental clusters, are recursively constructed through verification and postselection in advance for the higher-level one-way computation, which namely provides error-precorrection of gate operations. A suitable code such as the Steane seven-qubit code is adopted for transversal operations. This concatenated construction of verified fundamental clusters has a simple transversal structure of logical errors, and achieves a high noise threshold 3 for computation by using appropriate verification procedures. Since the postselection is localized within each fundamental cluster with the help of deterministic bare controlled-Z gates without verification, divergence of resources is restrained, which reconciles postselection with scalability.
Correcting errors is a vital but expensive component of fault tolerant quantum computation. Standard fault tolerant protocol assumes the implementation of error correction, via syndrome measurements and possible recovery operations, after every quantum gate. In fact, this is not necessary. Here we demonstrate that error correction should be applied more sparingly. We simulate encoded single qubit rotations within the [7,1,3] code and show via fidelity measures that applying error correction after every gate is not desirable. The simulations also shed light on what accuracy can be expected for noisy error correction and thus to what accuracy arbitrary single qubit rotations should be implemented.
This thesis is concerned with certain theoretical problems that arise naturally in the context of fault-tolerant quantum computation. Fault-tolerance can be defined as the art of building reliable devices from unreliable components, and is of particular importance for quantum computers that aim to precisely control the dynamics of extremely sensitive quantum systems. A model of quantum computation is a specification of the basic building blocks by which a quantum computation is implemented. The best known model is the quantum circuit model, where computations are implemented by means of unitary quantum gates that are applied to two-level quantum systems known as qubits. In a physical implementation of a quantum circuit, the gates and qubits will inevitably be affected by noise. Fault-tolerant quantum circuits are designed to be resilient against the effects of this noise, provided that it is not too strong. Fault-tolerance in the quantum circuit model is well developed thanks to the theory of quantum error-correcting codes. These codes allow for the correction of small numbers of errors introduced by a variety of noise processes. In a fault-tolerant quantum circuit, qubits are replaced with encoded qubits, and quantum gates with encoded gates that are immediately followed by special quantum circuits for error correction. Provided the rate at which errors occur is below a constant threshold value, the accumulation of errors can be checked so that the correct output of the computation correctly determined. The threshold acts as both a measure of how good the design of a quantum circuit is, and also as a target for experimenters aiming to implement quantum circuits. Much current research in quantum computation is aimed at designing quantum circuits that increase the noise threshold, hopefully to the point where it becomes within the reach of experimenters. A fault-tolerant encoded quantum gate must limit the propagation of errors so that the code’s corrective capabilities are not overwhelmed. It is not so easy to design encoded gates that satisfy this property, and to date only a finite handful of such gates are known. In a faulttolerant quantum circuit, all quantum gates must be decomposed or compiled in terms of those that may be implemented fault-tolerantly. In the first part of this thesis, we present two results that may be applied to this problem of gate compilation. The first is a generic method based on the Solovay-Kitaev theorem that may be applied to all quantum gates, but is most effective for those that act on single qubits. We present the Solovay-Kitaev theorem in its simples known form as an algorithm, together with novel constructions that can be used to implement it. Following this we give a two specialized methods for two-qubit gate decomposition, based on the Cartan decomposition of the Lie group SU(4). The cluster-state model of computation is an alternative to the quantum circuit model, and makes use of quantum measurements and highly entangled cluster-states to implement a quantum computation. The fault-tolerant techniques developed for quantum circuits are not immediately applicable in this model, so in order for it to be a realistic candidate for performing computations we prove that such techniques are possible. In the second part of the thesis we prove that constant fault-tolerance thresholds may be achieved in the cluster-state model, and in particular in an adaption to an optical implementation. Following this we design a complete error correction scheme for optical cluster-state computation, and numerically determine the threshold of this model in the face of the dominant noise models likely to affect such an implementation.
We simulate the implementation of a T gate, or $\frac{\ensuremath{\pi}}{8}$ gate, for a [7,1,3] encoded logical qubit in a nonequiprobable error environment. We demonstrate that the use of certain non-fault-tolerant methods in the implementation may nevertheless enable reliable quantum computation while reducing basic resource consumption. Reliability is determined by calculating gate fidelities for the one-qubit logical gate. Specifically, we show that despite using a non-fault-tolerant procedure in constructing a logical zero ancilla to implement the T gate, the gate fidelity of the logical gate, after perfect error correction, has no first order error terms. Meaning, any errors that may have occurred during implementation are ``correctable'' and fault tolerance may still be achieved.
Introduction Cluster states were introduced in Chapter 18, along with some of the approaches to achieving fault tolerance in measurement-based quantum computing. In this chapter, we describe an extremely promising fault-tolerant cluster state quantum computing scheme [RH07, RHG07] with a threshold error rate of 7.46 × 10 −3 , low overhead arbitrarily long-range logical gates and novel adjustable strength error correction capable of correcting general errors through the correction of Z errors only. Detailed proposed implementations of this scheme exist for ion traps [SJ09] and single photons with cavity mediated interactions [DGI+07]. The discussion is organized as follows. In Section 20.2, we describe the topological cluster state, which is a specific three-dimensional (3D) cluster state, and give a brief overview of what topological cluster state quantum computing involves. Section 20.3 describes logical qubits in more detail and how to initialize them to ∣0 L 〉 and ∣+ L 〉 and measure them in the Z L and X L bases. State injection, the non-fault-tolerant construction of arbitrary logical states, is covered in Section 20.4. Logical gates, namely the logical identity gate and the logical CNOT gate, are discussed in Section 20.5 along with their byproduct operators. Section 20.6 describes the errorcorrection procedure. In Section 20.7, we calculate an estimate for the threshold of this scheme. Section 20.8 presents an analysis of the overhead as a function of both the circuit size and the error rate.
In this work we compare the accuracy of two methods used to construct a logical zero state appropriate for the [7, 1, 3] CSS quantum error correction code in a non-equiprobable Pauli operator error environment. The first method is to apply error correction, via syndrome measurement, on seven physical qubits all in the state zero. To do the syndrome measurements in a fault-tolerant fashion requires the construction of four qubit Shor states. These Shor states are also assumed to be constructed in a non-equiprobable Pauli operator error environment and it is these that are used to implement the syndrome measurement. The second construction method is to implement the [7, 1, 3] encoding gate sequence, also in the nonequiprobable Pauli operator error environment. The fidelity of the output states is calculated for each of these methods. With respect to the Shor state construction we find that the implementation of (noisy) parity based verifications does not necessarily raise the fidelity of the resulting Shor state. We also find that the second logical zero construction method outputs a seven qubit state with a respectfully higher fidelity than the first (fault tolerant) method. However, the fidelity of the single qubit of stored information has almost equivalent fidelity from the two construction methods.