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Measurement-Based Quantum Computation with Cluster States

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Measurement-Based Quantum Computation with Cluster States Measurement-based quantum computation with cluster states Robert Raußendorf M¨unchen2003 Messungsbasiertes Quantenrechnen mit Clusterzust¨anden Dissertation an der Fakult¨atf¨urPhysik der Ludwig–Maximilians–Universit¨at M¨unchen vorgelegt von Robert Raußendorf aus Dresden M¨unchen, den 20. Juni 2003 First referee: Dr. Hans-J¨urgenBriegel Second referee: Prof. Dr. Herbert Wagner Day of exam: 06.10.2003 Contents Zusammenfassung/ Abstract viii 1 Introduction 1 2 The one-way quantum computer 9 2.1 General picture of the QCC ........................... 9 2.2 Universality of the QCC ............................ 11 2.2.1 Cluster states and their quantum correlations ............ 11 2.2.2 A universal set of quantum gates ................... 16 2.2.3 Removing the redundant cluster qubits ................ 19 2.2.4 Concatenation of gate simulations ................... 21 2.2.5 Randomness of the measurement results ............... 24 2.2.6 Using quantum correlations for quantum computation ........ 28 2.2.7 Functioning of the CNOT gate and general one-qubit rotations ... 33 2.2.8 Upper bounds on resource consumption ................ 41 2.2.9 Quantum circuits in the Clifford group can be realized in one step . 43 2.3 Examples of practical interest ......................... 44 2.3.1 Multi-qubit swap gate ......................... 44 2.3.2 Simulating multi-qubit Hamiltonians ................. 46 2.3.3 CNOT between distant qubits ..................... 48 2.3.4 Controlled phase gate .......................... 50 2.3.5 Quantum Fourier transformation ................... 52 2.3.6 Multi-qubit controlled gates ...................... 53 2.3.7 Circuit for addition ........................... 58 2.4 Computation with limited spatial resources .................. 62 2.5 Discussion .................................... 63 3 Computational model underlying the one-way quantum computer 65 3.1 Motivation for a non-network model of the QCC ............... 65 3.2 Beyond the network picture .......................... 67 3.2.1 The sets Qt of simultaneously measurable qubits ........... 67 3.2.2 The forward- and backward cones ................... 67 3.2.3 The algorithm- and measurement angles ............... 70 vi Inhaltsverzeichnis 3.2.4 Quantities for the processing of the measurement results ...... 71 3.2.5 To what a quantum logic network condenses ............. 80 3.3 Symmetry considerations ............................ 80 3.4 Computational model for the QCC ...................... 87 3.4.1 Obtaining the computational result from the measurement outcomes 88 3.4.2 Description of the model ........................ 94 3.4.3 Proof of the model ........................... 96 3.5 Logical depth and temporal complexity .................... 99 3.5.1 D = 2 for circuits of CNOT gates and U(1)-rotations ........ 99 3.5.2 The logical depth D is a good measure for temporal complexity .. 102 3.5.3 Temporal complexity of computing the circuit layout ........ 104 3.6 Quantum algorithms and graphs ........................ 106 3.7 Discussion .................................... 108 4 Fault-tolerant quantum computation with the QCC 111 4.1 Fault-tolerant quantum computation in the network model ......... 111 4.2 The error model for the QCC .......................... 124 4.3 Fault-tolerance of the QCC ........................... 129 4.4 Checksums .................................... 138 4.4.1 A first example ............................. 138 4.4.2 The encoded CNOT gate on the Steane code ............. 140 5 Conclusion and outlook 143 A QCC-computation in the presence of classically correlated noise 147 Bibliography 153 Danksagung/ Acknowledgements 159 Curriculum Vitae 161 List of Figures 2.1 Network simulation ............................... 10 2.2 Universal set of quantum gates on the QCC .................. 16 2.3 Gate concatenation ............................... 23 2.4 Vertical cuts ................................... 27 2.5 Simulation of the conjugated gate HUH ................... 37 2.6 Euler-decomposition of a rotation. ....................... 38 2.7 Pattern of correlation centers. ......................... 41 2.8 The multi-qubit swap gate. ........................... 45 2.9 Simulation of the Hamiltonian H4. ...................... 48 2.10 Measurement pattern for a CNOT gate. ................... 49 2.11 Two equivalent networks for the distant CNOT gate. ............ 49 2.12 Controlled phase gate with additional swap. ................. 50 2.13 Quantum Fourier transformation. ....................... 52 2.14 QCC-realization of a quantum Fourier transformation. ............ 53 2.15 QCC-realization of the Toffoli phase gate. ................... 54 2.16 The three qubit controlled gate CARRY . ................... 55 2.17 Quantum correlations for the gate CARRY , first part. ........... 56 2.18 Quantum correlations for the gate CARRY , second part. .......... 57 2.19 Interchangeability between target input and output in CARRY . ...... 58 2.20 Quantum adder network. ............................ 58 2.21 Quantum adder network, bent. ........................ 59 2.22 QCC-circuit for combination of CNOT gates. ................. 60 2.23 QCC-circuit for the quantum adder. ...................... 61 3.1 Forward and backward cones .......................... 68 3.2 General scheme of the QCC .......................... 88 3.3 Network for a diagonal gate .......................... 101 4.1 Stabilizer measurement circuit for the Steane code .............. 118 4.2 Measurement pattern for gates exposed to decoherence ........... 130 4.3 Avoiding two-qubit errors to leading order .................. 132 4.4 Checksums for error identification. ...................... 139 4.5 Encoded CNOT gate .............................. 142 viii Zusammenfassung/ Abstract Zusammenfassung In dieser Dissertation beschreiben wir den Einweg-Quantenrechner (QCC), ein Schema zum universellen Quantenrechnen, das allein aus Einteilchenmessungen an einem hochgradig verschr¨anktenVielteilchenzustand, dem Clusterzustand, besteht. Wir beweisen die Uni- versalit¨atdes QCC, beschreiben das zugrunde liegende Rechnermodell und zeigen, dass der QCC fehlertolerantes Quantenrechnen erlaubt. In Kapitel 2 zeigen wir, dass der QCC als ein Simulator quantenlogischer Netzwerke aufgefasst werden kann. Damit beweisen wir dessen Universalit¨atund stellen den Zusam- menhang zum Netzwerkmodel her, welches das verbreitete Model eines Quantenrechners darstellt. Wir weisen auch darauf hin, dass die Beschreibung des QCC als Netzwerksimu- lator nicht in jeder Hinsicht passend ist. In Kapitel 3 leiten wir das dem Einweg-Quantenrechner zugrunde liegende Rechnermo- dell her. Es ist sehr verschieden vom Netzwerkmodell des Quantenrechners. Der QCC be- sitzt keinen Quanten-Input, keinen Quanten-Output und kein Quantenregister. Unit¨are Quantengatter aus einem universellen Satz sind nicht die elementaren Bestandteile von QCC-Quantenalgorithmen. Dar¨uber hinaus sind die Messergebnisse aus den Einteilchen- messungen die einzige Information, die vom QCC verarbeitet wird, und somit existiert Informationsverarbeitung beim QCC nur auf klassischem Niveau. Dennoch arbeitet der QCC fundamental quantenmechanisch, da er den hochverschr¨anktenClusterzustand als zentrale physikalische Resource nutzt. In Kapitel 4 zeigen wir, dass positive Fehlerschranken f¨urdas fehlertolerante Quanten- rechnen mit dem QCC existieren. Desweiteren skizzieren wir das Konzept der Pr¨ufsummen im Zusammenhang mit dem QCC, das ein Element zuk¨unfitigerpraktikabler und zweck- m¨aßigerMethoden f¨urfehlertolerantes QCC-Quantenrechnen werden kann. x Zusammenfassung/ Abstract Zusammenfassung/ Abstract xi Abstract In this thesis we describe the one-way quantum computer (QCC), a scheme of universal quantum computation that consists entirely of one-qubit measurements on a highly entan- gled multi-particle state, the cluster state. We prove universality of the QCC, describe the underlying computational model and demonstrate that the QCC can be operated fault- tolerantly. In Chapter 2 we show that the QCC can be regarded as a simulator of quantum logic networks. In this way, we give the universality proof and establish the link to the network model, the common model of quantum computation. We also indicate that the description of the QCC as a network simulator is not adequate in every respect. In Chapter 3 we derive the computational model underlying the one-way quantum computer, which is very different from the quantum logic network model. The QCC has no quantum input, no quantum output and no quantum register, and the unitary gates from some universal set are not the elementary building blocks of QCC-quantum algorithms. Further, all information that is processed with the QCC are the outcomes of one-qubit measurements and thus processing of information exists only at the classical level. The QCC is nevertheless quantum mechanical as it uses a highly entangled cluster state as the central physical resource. In Chapter 4 we show that there exist nonzero error thresholds for fault-tolerant quan- tum computation with the QCC. Further, we outline the concept of checksums in the context of the QCC which may become an element in future practicable and adequate methods for fault-tolerant QCC-computation. xii Zusammenfassung/ Abstract Chapter 1 Introduction Quantum computation has come into the focus of physics and information science largely due to Peter Shor’s discovery of a quantum algorithm for factoring large numbers [1]. This algorithm demonstrated that a quantum computer is capable of solving a problem for
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