The theoretical concepts of quantum computation in the idealized and undisturbed case are well understood. However, in practice, all quantum computation devices do suffer from decoherence effects as well as from operational imprecisions. This work assesses the power of error-prone quantum computation devices using large-scale numerical simulations on parallel supercomput- ers. We present the Juelich Massively Parallel Ideal Quantum Computer Simulator (JUMPIQCS), that simulates a generic quantum computer on gate level. The robustness of various algorithms in the presence of noise has been analyzed. The simulation results show that for large system sizes and long computations it is imperative to actively correct errors by means of fault-tolerant quantum error correction. Fault-tolerant methods require the single qubit error rate to be below a certain threshold. We determined this threshold numerically for Steane’s 7-qubit code. Using the depolar- –6 izing channel as the source of decoherence, we find a threshold error rate of (5.2w 0.2) · 10 . For Gaussian distributed operational over-rotations the threshold lies at a standard deviation of 0.0431 w 0.0002. We can conclude that quantum error correction is especially well suited for the correction of operational imprecisions and systematic over-rotations. For realistic simulations of specific quantum computation devices we extend the generic model to dynamic simulations of realistic hardware models. We focus on today’s most advanced technology, Large-Scale Simulations of Error-Prone QC Devices Simulations of Error-Prone Large-Scale i.e. ion trap quantum computation. We developed the Dynamic Quantum Computer Simulator for Ion Traps (DyQCSI). Starting from a microscopic Hamiltonian, it does not rely on approximations that are usually necessary for an analytical approach. We show that the effects due to these approximations are significant. We present several ways for the visualization of the state of the system during its time evolution and demonstrated the benefit of the simulation approach for parameter optimizations. Doan Binh Trieu Doan Binh Trieu This publication was written at the Jülich Supercomputing Centre (JSC) which is an integral part of the Institute for Advanced Simulation (IAS). The IAS combines the Jülich simulation sciences and the supercomputer facility in one organizational unit. It includes those parts of the scientific institutes at Forschungszentrum Jülich which use simulation on supercomputers as their main research methodology. IAS Series IAS Series Large-Scale Simulations of Error-Prone Quantum Computation Devices Doan Binh Trieu IAS Series Volume 2 ISBN 978-3-89336-601-9 2 der Helmholtz-GemeinschaftMitglied Schriften des Forschungszentrums Jülich IAS Series Volume 2 Forschungszentrum Jülich GmbH Institute for Advanced Simulation (IAS) Jülich Supercomputing Centre (JSC) Large-Scale Simulations of Error-Prone Quantum Computation Devices Doan Binh Trieu Schriften des Forschungszentrums Jülich IAS Series Volume 2 ISSN 1868-8489 ISBN 978-3-89336-601-9 Bibliographic information published by the Deutsche Nationalbibliothek. Die Deutsche Nationalbibliothek lists this publication in the Deutsche Nationalbibliografie; detailed bibliographic data are available in the Internet at <http://dnb.d-nb.de> Publisher Forschungszentrum Jülich GmbH and Distributor: Zentralbibliothek, Verlag D-52425 Jülich phone: +49 2461 61-5368 · fax: +49 2461 61-6103 e-mail: [email protected] Internet: http://www.fz-juelich.de/zb Cover Design: Grafische Medien, Forschungszentrum Jülich GmbH Printer: Grafische Medien, Forschungszentrum Jülich GmbH Copyright: Forschungszentrum Jülich 2009 Schriften des Forschungszentrums Jülich IAS Series Volume 2 D 468 (Diss., Wuppertal, Univ., 2009) ISSN 1868-8489 ISBN 978-3-89336-601-9 The complete volume is freely available on the Internet on the Jülicher Open Access Server (JUWEL) at http://www.fz-juelich.de/zb/juwel Persistent Identifier: urn.nbn:de: 0001-00552 Resolving URL: http://www.persistent-identifier.de/?link=610 Neither this book nor any part may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopying, microfilming, and recording, or by any information storage and retrieval system, without permission in writing from the publisher. Abstract The theoretical concepts of quantum computation in the idealized and undisturbed case are well understood. However, in practice, all quantum computation devices do suffer from decoherence effects as well as from operational imprecisions. This work assesses the power of error-prone quantum computation devices using large- scale numerical simulations on parallel supercomputers. We present the Juelich Massively Parallel Ideal Quantum Computer Simulator (JUMPIQCS), that simulates a generic quan- tum computer on gate level. It comprises an error model for decoherence and operational errors. The robustness of various algorithms in the presence of noise has been analyzed. The simulation results show that for large system sizes and long computations it is imper- ative to actively correct errors by means of quantum error correction. We implemented the 5-, 7-, and 9-qubit quantum error correction codes. Our simulations confirm that using error-prone correction circuits with non-fault-tolerant quantum error correction will always fail, because more errors are introduced than being corrected. Fault-tolerant methods can overcome this problem, provided that the single qubit error rate is below a certain threshold. We incorporated fault-tolerant quantum error correction techniques into JUMPIQCS using Steane’s 7-qubit code and determined this threshold numerically. Using the depolarizing channel as the source of decoherence, we find a threshold error rate of (5.2 ± 0.2) · 10−6. For Gaussian distributed operational over-rotations the threshold lies at a standard deviation of 0.0431±0.0002. We can conclude that quantum error correction is especially well suited for the correction of operational imprecisions and systematic over-rotations. For realistic simulations of specific quantum computation devices we need to extend the generic model to dynamic simulations, i.e. time-dependent Hamiltonian simulations of re- alistic hardware models. We focus on today’s most advanced technology, i.e. ion trap quan- tum computation. We developed the Dynamic Quantum Computer Simulator for Ion Traps (DyQCSI). Starting from a microscopic Hamiltonian, it does not rely on approximations that are usually necessary for an analytical approach. We show that the effects due to these approximations are significant. We present several ways for the visualization of the state of the system during its time evolution and demonstrated the benefit of the simulation ap- proach for parameter optimizations. Contents 1 Introduction and Overview 1 2 Gate Level Simulations 3 2.1 Ideal Quantum Computer Simulations .................... 3 2.1.1 The Circuit Model of Quantum Computation ............ 3 2.1.2 Quantum Algorithms ......................... 8 2.1.2.1 Shor’s Algorithm for Prime Factorization ......... 8 2.1.2.2 Grover’s Search Algorithm ................ 14 2.1.3 Massively Parallel Ideal Quantum Computer Simulator ....... 18 2.2 Error-Prone Quantum Computer Simulations ................ 37 2.2.1 Error Model .............................. 37 2.2.2 H2k-Algorithm ............................ 41 2.2.3 Quantum Fourier Transform ..................... 45 2.2.4 Grover’s Search Algorithm ...................... 50 2.3 Simulation of Quantum Error Correction ................... 56 2.3.1 Quantum Error Correction Codes .................. 57 2.3.1.1 Shor’s 9-Qubit Quantum Error Correction Code ..... 58 2.3.1.2 Steane’s 7-Qubit Quantum Error Correction Code .... 61 2.3.1.3 5-Qubit Quantum Error Correction Code ......... 65 2.3.2 Ideal and Error-Prone Quantum Error Correction .......... 68 2.3.3 Fault-Tolerant Quantum Error Correction .............. 71 2.3.3.1 Theoretical Principles ................... 71 2.3.3.2 Numerical Simulations .................. 81 2.3.3.3 Conclusion .........................102 I CONTENTS 3 Dynamic Simulations of Ion Trap Quantum Computers 105 3.1 Theory of Ion Trap Quantum Computation ..................105 3.2 Dynamic Quantum Computer Simulator for Ion Traps ............120 4 Summary and Outlook 139 Acknowledgements 143 A Quantum Circuit Symbols 145 B Equivalence of Depolarizing Channel and Unitary Over-Rotations 147 C Stabilizer Codes for Quantum Error Correction 153 D Listings 159 II List of Figures 2.1 Bloch sphere representation of a single qubit ................. 4 2.2 Single qubit gates used in quantum computation ............... 7 2.3 Controlled-NOT gate ............................. 7 2.4 Quantum circuit for the quantum Fourier transform ............. 9 2.5 Phase estimation procedure .......................... 10 2.6 Decomposition of Controlled-U j gate.................... 11 2.7 Order-finding algorithm circuit ........................ 12 2.8 Phase inversions in Grover’s algorithm .................... 15 2.9 Geometric visualization of a Grover iteration ................ 17 2.10 Distribution of state vector to MPI tasks ................... 19 2.11 Memory access scheme for a Hadamard operation .............. 21 2.12 Pairs of state vector elements ......................... 22 2.13 Pairs of state vector elements on a distributed memory system ....... 22 2.14 Communication scheme in distributed memory ............... 23 2.15 Even/odd splitting of state vector ....................... 24 2.16