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Cambridge University Press eBooks

Fault-tolerant measurement-based quantum computing

Debbie Leung

Published Sep 5, 2013• 0 citations
Publisher

Abstract

Introduction In the standard model of quantum computation [D95], some simple quantum state is prepared and evolved by a partially ordered set of unitary gates, and finally is measured in the computation basis. Up until the late 1990s, transformations that deviate from unitarity were commonly associated with irreversibility, and thus incoherence and harm induced in the quantum data of interest. However, this viewpoint was challenged by three notable results: (1) teleportation [BBC+93], (2) syndrome measurements [S95, S96e], and (3) the fault-tolerant Toffoli and π/8-gate constructions [S96b, BMP+99]. In these schemes, well-designed measurements project the quantum data of interest onto a subspace of the original ambient space for each outcome, up to some known unitary transformation that is either reversible (and harmless) or intended. Subsequently, researchers found many useful error-correction procedures and unitary gates that rely heavily on measurements. Two “measurement models” of quantum computation were proposed in the early 2000s that make extreme use of measurements. In the cluster model [RB01], any quantum computation can be performed by applying a partially time-ordered set of single-qubit measurements on a certain entangled state. The state must be of appropriate size, but is otherwise independent of the computation. The related graph state model [RBB03] uses an initial state that depends on the computation itself. In the seemingly different teleportation-based model [N03, L01a, L04], any quantum computation can be implemented by applying one- or two-qubit nondemolition measurements on an arbitrary initial state, so as to “teleport” each gate in the simulated circuit.

DOI
10.1017/cbo9781139034807.020

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The last two decades have witnessed the rapid growth of quantum information processing as an active field of multidisciplinary science and technology. Tremendous theoretical and experimental progress has been made in quantum communication and quantum computation, and it seems clear that some of this progress will soon result in practical applications outside the laboratory. Quantum computation aims to build a quantum computer and to explore the origins and limits of quantum computing power. The original circuit model of a quantum computer, due to Deutsch, has a strong formal similarity with the circuit model of a classical computer. Both models depict a computation as processing via the wiring of a number of fiducial gates, each of which dynamically couples a small number of input (qu)bits in a controlled manner. In other words, unitary evolution is the basic mechanism of information processing in the circuit model of quantum computers. 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New Journal of Physics•Jun 29, 2007•2 citations

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Eduardo R. Mucciolo

The approach we considered so far for computing using quantum mechanics relies on preparing a suitable initial state, applying single and two-qubit quantum gates, and performing measurements. It is known as the circuit model of quantum computing. Howeveer, this is not the only approach. In this chapter, we will cover a few alternatives. Some still rely on circuits to some extend, but they either explore alternative paradigms of fault tolerance, utilize protocols that do not require entangling gates, or avoid multi-shot computations. One of these alternative approaches is entirely based on measurements. Another requires fine tunning gates and repeating measurments.

CRC Press eBooks•Aug 6, 2025•0 citations
Introduction In the standard model of quantum computation [D95], some simple quantum state is prepared and evolved by a partially ordered set of unitary gates, and finally is measured in the computation basis. Up until the late 1990s, transformations that deviate from unitarity were commonly associated with irreversibility, and thus incoherence and harm induced in the quantum data of interest. However, this viewpoint was challenged by three notable results: (1) teleportation [BBC+93], (2) syndrome measurements [S95, S96e], and (3) the fault-tolerant Toffoli and π/8-gate constructions [S96b, BMP+99]. In these schemes, well-designed measurements project the quantum data of interest onto a subspace of the original ambient space for each outcome, up to some known unitary transformation that is either reversible (and harmless) or intended. Subsequently, researchers found many useful error-correction procedures and unitary gates that rely heavily on measurements. Two “measurement models” of quantum computation were proposed in the early 2000s that make extreme use of measurements. In the cluster model [RB01], any quantum computation can be performed by applying a partially time-ordered set of single-qubit measurements on a certain entangled state. The state must be of appropriate size, but is otherwise independent of the computation. The related graph state model [RBB03] uses an initial state that depends on the computation itself. In the seemingly different teleportation-based model [N03, L01a, L04], any quantum computation can be implemented by applying one- or two-qubit nondemolition measurements on an arbitrary initial state, so as to “teleport” each gate in the simulated circuit.
We discuss a simple variant of the one-way quantum computing model [R. Raussendorf and H.-J. Briegel, PRL 86, 5188, 2001], called the Pauli measurement model, where measurements are restricted to be along the eigenbases of the Pauli X and Y operators, while auxiliary qubits can be prepared both in the ∣+π4⟩:=1/2(∣0⟩+eiπ4∣1⟩)\ket{+_{\pi\over 4}}:={1/\sqrt{2}}(\ket{0}+e^{i{\pi\over 4}}\ket{1})∣+4π​​⟩:=1/2​(∣0⟩+ei4π​∣1⟩) state, and the usual ∣+⟩:=1/2(∣0⟩+∣1⟩)\ket{+}:={1/ \sqrt{2}}(\ket{0}+\ket{1})∣+⟩:=1/2​(∣0⟩+∣1⟩) state. We prove the universality of this quantum computation model, and establish a standardization procedure which permits all entanglement and state preparation to be performed at the beginning of computation. This leads us to develop a direct approach to fault-tolerance by simple transformations of the entanglement graph and preparation operations, while error correction is performed naturally via syndrome-extracting teleportations.
In the formalism of measurement based quantum computation we start with a given fixed entangled state of many qubits and perform computation by applying a sequence of measurements to designated qubits in designated bases. The choice of basis for later measurements may depend on earlier measurement outcomes and the final result of the computation is determined from the classical data of all the measurement outcomes. This is in contrast to the more familiar gate array model in which computational steps are unitary operations, developing a large entangled state prior to some final measurements for the output. Two principal schemes of measurement based computation are teleportation quantum computation (TQC) and the so-called cluster model or one-way quantum computer (1WQC). We will describe these schemes and show how they are able to perform universal quantum computation. We will outline various possible relationships between the models which serve to clarify their workings. We will also discuss possible novel computational benefits of the measurement based models compared to the gate array model, especially issues of parallelisability of algorithms.
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 discuss a simple variant of the one-way quantum computing model (Raussendorf R and Briegel H-J 2001 Phys. Rev. Lett. 86 5188), called the Pauli measurement model, where measurements are restricted to be along the eigenbases of the Pauli X and Y operators, while qubits can be initially prepared both in the state and the usual state. We prove the universality of this quantum computation model, and establish a standardization procedure which permits all entanglement and state preparation to be performed at the beginning of computation. This leads us to develop a direct approach to fault-tolerance by simple transformations of the entanglement graph and preparation operations, while error correction is performed naturally via syndrome-extracting teleportations.
The last two decades have witnessed the rapid growth of quantum information processing as an active field of multidisciplinary science and technology. Tremendous theoretical and experimental progress has been made in quantum communication and quantum computation, and it seems clear that some of this progress will soon result in practical applications outside the laboratory. Quantum computation aims to build a quantum computer and to explore the origins and limits of quantum computing power. The original circuit model of a quantum computer, due to Deutsch, has a strong formal similarity with the circuit model of a classical computer. Both models depict a computation as processing via the wiring of a number of fiducial gates, each of which dynamically couples a small number of input (qu)bits in a controlled manner. In other words, unitary evolution is the basic mechanism of information processing in the circuit model of quantum computers. Starting with a given fixed state of many qubits, measurement based quantum computation (MBQC) processes information by applying a sequence of measurements to designated qubits in designated bases. This is remarkable, given the probabilistic nature of quantum measurement and the murky physical status of the dynamics of 'wavefunction collapse'. The final result of the computation is determined from the classical data of all the measurement outcomes. There exist two principle models: the teleportation based model and the 'one-way quantum computer' of 'cluster state computation'. The formalism of cluster states has proven to be a powerful way of describing the essential entanglement resources needed to perform quantum information processing. Unlike the circuit model, the MBQC models have no natural classical counterpart. There are a number of physical architectures for which measurements on the relevant systems are more easily performed than controlled unitary evolution. Moreover, it appears the MBQC models can offer novel computational benefits, especially regarding issues of fault-tolerance and parallelizability of algorithms. As a result, the theory of MBQC is now shaping the latest experimental proposals across the full spectrum of quantum information processing technologies. Naturally, understanding the scope of measurement based quantum computation is also yielding many insights into what is necessary (as opposed to merely sufficient) for us to harness the power of quantum computers. Quantum communication aims to offer efficient and secure ways of information exchange, directly based on quantum mechanics. Photons serve as natural information carriers, owing to their robustness against decoherence. A major problem for photonic communication systems is due to photon loss. It turns out that this difficulty can be overcome by the quantum repeater, a measurement-based technique. The basic idea is to divide the transmission channel into many segments and perform some quantum measurement on each segment. Additional measurement-based protocols and technologies have also been demonstrated to improve the quality of quantum communication, such as entanglement swapping and purification. The articles in this invited Focus Issue cover topics both in measurement-based quantum communication and computation, but the emphasis is given to quantum computation. A significant number of articles deal with the preparation of cluster states, the experimental realization of a one-way quantum computer, and the associated efficiency and benefits. We hope that you will find that some of these new results are of interest to you, and further hope that the issue will increase your interest in this interdisciplinary research field if you are a non-expert. Focus on Measurement-Based Quantum Information Processing Contents A proof-of-principle experiment of eliminating photon-loss errors in cluster states Wei-Bo Gao, Xiao-Qi Zhou, Jin Zhang, Tao Yang and Jian-Wei Pan Continuous variable multipartite entanglement and optical implementations of quantum communication networks Yimin Lian, Changde Xie and Kunchi Peng Generalized flow and determinism in measurement-based quantum computation Daniel E Browne, Elham Kashefi, Mehdi Mhalla and Simon Perdrix The entanglement of the four-photon cluster state C Schmid, N Kiesel, W Wieczorek and H Weinfurter Optical zeno gate: bounds for fault tolerant operation Patrick M Leung and Timothy C Ralph Towards experimental entanglement connection with atomic ensembles in the single excitation regime Julien Laurat, Chin-wen Chou, Hui Deng, Kyung Soo Choi, Daniel Felinto, Hugues de Riedmatten and H J Kimble Towards minimal resources of measurement-based quantum computation Simon Perdrix Experimental realization of a quantum game on a one-way quantum computer Robert Prevedel, Andre Stefanov, Philip Walther and Anton Zeilinger Fundamentals of universality in one-way quantum computation M Van den Nest, W Dur, A Miyake and H J Briegel Loss tolerant linear optical quantum memory by measurement-based quantum computing Michael Varnava, Daniel E Browne and Terry Rudolph Graph state generation with noisy mirror-inverting spin chains Stephen R Clark, Alexander Klein, Martin Bruderer and Dieter Jaksch One-way quantum computing in adecoherence-free subspace M S Tame, M Paternostro and M S Kim Cluster state preparation using gates operating at arbitrary success probabilities K Kieling, D Gross and J Eisert Topological fault-tolerance in cluster state quantum computation R Raussendorf, J Harrington and K Goyal Strategies for the preparation of large cluster states using non-deterministic gates Peter P Rohde and Sean D Barrett A repeat-until-success quantum computing scheme A Beige, Y L Lim and L C Kwek Efficient growth of complex graph states via imperfect path erasure Earl T Campbell, Joseph Fitzsimons, Simon C Benjamin and Pieter Kok Investigation of a single-photon source based on quantum interference T B Pittman, J D Franson and B C Jacobs Entanglement and local information access for graph states Damian Markham, Akimasa Miyake and Shashank Virmani The efficiencies of generating cluster states with weak nonlinearities Sebastien G R Louis, Kae Nemoto, W J Munro and T P Spiller A direct approach to fault-tolerance in measurement-based quantum computation via teleportation Marcus Silva, Vincent Danos, Elham Kashefi and Harold Ollivier Experimental ancilla-assisted qubit transmission against correlated noise T Yamamoto, R Nagase, J Shimamura, S K Ozdemir, M Koashi and N Imoto Terry Rudolph, Imperial College, London, UK Jian-Wei Pan, University of Science and Technology of China, Hefei, People's Republic of China and Universitat Heidelberg, Germany
The approach we considered so far for computing using quantum mechanics relies on preparing a suitable initial state, applying single and two-qubit quantum gates, and performing measurements. It is known as the circuit model of quantum computing. Howeveer, this is not the only approach. In this chapter, we will cover a few alternatives. Some still rely on circuits to some extend, but they either explore alternative paradigms of fault tolerance, utilize protocols that do not require entangling gates, or avoid multi-shot computations. One of these alternative approaches is entirely based on measurements. Another requires fine tunning gates and repeating measurments.