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Quantum Algorithms Quantum Algorithms Student Seminar RWTH Aachen Winter Semester 2020/2021 Contents 1 Gate Set Tomography 6 Lucas Marcogliese (under the supervision of Maarten Wegewijs) 1.1 Introduction....................................6 1.2 Operational representation of quantum processors...............7 1.2.1 Quantum circuits and open quantum systems.............7 1.2.2 Superoperator formalism.........................8 1.2.3 Physical constraints in the canonical gauge..............9 1.2.4 Black box model.............................9 1.3 Breaking self-referentiality in traditional quantum tomography.................................... 10 1.3.1 The self-referentiality problem...................... 11 1.3.2 SPAM errors............................... 12 1.4 Linear gate set tomography........................... 13 1.4.1 Self-consistent approach to quantum tomography........... 13 1.5 Applications of a gate set estimate....................... 15 1.5.1 Predictions for arbitrary sequences................... 16 1.5.2 Figures of merit and the lack of gauge invariance........... 16 1.6 Gauge fixing simulation with pyGSTi ...................... 18 1.6.1 Overview of pyGSTi algorithm..................... 19 1.6.2 Simulation results and analysis..................... 22 1.7 Conclusion..................................... 25 App. 1.A Example: Pauli transfer matrix representation............. 26 App. 1.B Physical constraints on gate set under the blame gauge transformation 26 App. 1.C Overcomplete data with least squares.................. 27 App. 1.D Algorithm scaling analysis......................... 27 App. 1.E Long-sequence GST: commuting errors and amplificational completeness 28 App. 1.F Motivation for maximum likelihood estimation............. 28 App. 1.G Target and estimate gate PTMs..................... 29 App. 1.H Simulation: native state and effect analysis............... 30 2 Variational Quantum Algorithms 32 Bavithra Govintharajah (under the supervision of Fabian Hassler) 2.1 Introduction.................................... 32 2.2 Variational quantum algorithms......................... 34 2.2.1 Outline of the algorithm......................... 34 2.2.2 The variational paradigm and general quantum systems....... 35 2.3 Variational ansatz construction......................... 37 2.3.1 Arbitrary unitary evolution....................... 38 2.3.2 Problem-inspired ansatz construction.................. 39 2.3.3 Hardware-efficient ansatz construction................. 40 CONTENTS 3 2.4 Cost function................................... 41 2.5 Measurement scheme............................... 43 2.5.1 Precision and Chebyshev’s inequality.................. 43 2.5.2 Measurement scheme optimization................... 44 2.6 Optimization routine............................... 44 2.7 Qiskit hands-on example: Molecular quantum simulation using the VQE.. 47 2.7.1 Classical pre-processing......................... 47 2.7.2 Simulating the Hydrogen molecule................... 51 2.8 Outlook and conclusions............................. 58 App. 2.A The variation principle.......................... 58 App. 2.B Pauli basis................................. 60 3 Topological Quantum Error Correction: Codes and Decoders 61 Orkun Şensebat (under the supervision of Eliana Fiorelli) 3.1 Introduction.................................... 61 3.2 Quantum Error Correction............................ 64 3.2.1 The Stabilizer Formalism........................ 64 3.2.2 Comparison of Quantum Codes..................... 66 3.3 Topological Error Correction.......................... 69 3.3.1 The Toric Code.............................. 69 3.3.2 Planar Codes............................... 73 3.3.3 Code Thresholds............................. 75 3.4 Conclusion..................................... 79 3.5 Acknowledgments................................. 80 4 Quantum Walk and its Universality 81 Kiran Adhikari (under the supervision of Thomas Botzung) 4.1 Introduction.................................... 81 4.2 Quick review of Classical Random walk.................... 82 4.2.1 Classical random walk on an unrestricted line............. 82 4.3 Quantum walk.................................. 84 4.3.1 Discrete quantum walk on a line.................... 84 4.3.2 Continuous-time quantum walk (CTQW)............... 87 4.3.3 Universal Computation by Quantum walk............... 90 4.4 Conclusion..................................... 93 App. 4.A Notebook for Discrete quantum walk simulation............ 93 5 Adiabatic Quantum Computing 98 Ananya Singh (under the supervision of Markus Müller) 5.1 Introduction.................................... 98 5.1.1 Overview of the report.......................... 99 5.2 Adiabatic Theorem................................ 99 5.2.1 Adiabatic quantum algorithm...................... 100 5.2.2 Landau-Zener transition......................... 101 5.3 Adiabatic Quantum Computing......................... 103 5.3.1 The ingredients.............................. 103 5.3.2 Some simple examples.......................... 105 5.3.3 A simple application - 2-SAT with 3 qubits.............. 106 5.4 Adiabatic Quantum Algorithm......................... 107 5.4.1 Needle in the haystack.......................... 108 5.4.2 Finding the needle adiabatically..................... 109 4 CONTENTS 5.5 Universality of AQC............................... 111 5.5.1 Gate model can efficiently simulate AQC................ 112 5.6 Summary..................................... 114 5.6.1 NP-Complete problems......................... 114 App. 5.A Adaibatic theorem calculation...................... 115 App. 5.B Codes.................................... 116 6 Topological Quantum Error Correction: Magic State Distillation 117 Dmitrii Dobrynin (under the supervision of Lorenzo Cardarelli) 6.1 Introduction.................................... 117 6.2 Theory of Quantum Error Correction...................... 118 6.2.1 Basic Codes................................ 118 6.2.2 Stabilizer Formalism........................... 119 6.2.3 Fault-tolerance and Universality.................... 121 6.2.4 Topological Error Correction: Surface Code.............. 122 6.3 Magic State Distillation............................. 124 6.3.1 Magic States............................... 124 6.3.2 Distillation Algorithm.......................... 125 6.3.3 Costs of Computation with Magic States................ 128 6.3.4 Implications of Magic States....................... 129 6.4 Conclusion..................................... 129 App. 6.A Program for MSD plots.......................... 130 CONTENTS 5 These notes are the result of a student seminar that took place during the win- ter semester 2020/2021 at RWTH Aachen. The seminar was geared towards beginning master’s students. The goal of the seminar was to introduce the stu- dents to ‘advanced’ quantum algorithms that are not covered in the introductory lecture on quantum information theory. The students were asked to consult the literature, realize some analytical or numerical work on the chosen topic, pre- pare a talk, as well as these notes. The students were assigned a supervisor that offered guidance. The result of the seminar can be found in these notes. Please be aware that the notes are written by the students with little to no corrections. As such, they should be read critically and taken with a grain of salt. Still we hope that the notes might be of use, in particular as a first introduction and as a guide to future literature. The supervisors Chapter 1 Gate Set Tomography Lucas Marcogliese Supervisor: Maarten Wegewijs Gate set tomography (GST) is a quantum algorithm that self-consistently char- acterizes a quantum processor’s set of available operations. By extracting in- formationally complete sets of fiducial states and effects from the original gate set, it successfully overcomes the self-referentiality problem associated with tra- ditional quantum process tomography. In this report, the derivation of the lin- ear GST algorithm is presented. The intrinsic gauge freedom in tomography is shown to emerge naturally from the Born rule and does not prevent one from using the GST estimate to predict experimental observations for arbitrary sequences. Moreover, the dangers of using standard, gauge-variant figures of merit with GST estimates to quantify device performance are considered and a suitable, gauge-invariant alternative is discussed. Advanced techniques tech- niques for higher accuracy, namely long-sequence GST, are introduced and an overview of the software package pyGSTi is given. Finally, the results of a simulation probing the influence of pyGSTi’s gauge fixing procedure on GST’s estimates are analyzed. 1.1 Introduction Modern quantum computing platforms such as ion traps, superconducting qubits and semi- conductor spin qubits, all share a common functionality: the qubit register is initialized in some state, a clever quantum algorithm is performed and the resulting output state is measured yielding a certain outcome. These promising candidates also have the common fragility of interacting with their environment which is on one hand necessary to control the qubits and on the other hand constitutes a channel for noise and decoherence. Luckily, fault-tolerant quantum error correction (FTQEC) schemes predict the feasibility of univer- sal quantum computation with noisy systems as long as their error rates are below certain thresholds. But how do we characterize a quantum processor to evaluate whether it meets FTQEC thresholds and, more generally, to quantify its overall performance with respect
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