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Quantum Protocols for Few-Qubit Devices Quantum protocols for few-qubit devices Koen Groenland Quantum protocols for few-qubit devices ILLC Dissertation Series DS-2020-02 For further information about ILLC publications, please contact Institute for Logic, Language and Computation Universiteit van Amsterdam Science Park 107 1098 XG Amsterdam phone: +31-20-525 6051 e-mail: [email protected] homepage: http://www.illc.uva.nl/ This research was supported by the EU SIQS project (grant agreement ID: 600645), the Quantum Software Consortium, and by a grant of the University of Amsterdam, supporting the Research Priority Area Quantum Matter & Quantum Information and QuSoft. Copyright © by Koen Groenland. Cover design by Sarah Brands. Printed and bound by NBD Biblion. ISBN: 978–90–903–2694–8. Quantum protocols for few-qubit devices ACADEMISCH PROEFSCHRIFT ter verkrijging van de graad van doctor aan de Universiteit van Amsterdam op gezag van de Rector Magnificus prof. dr. ir. K.I.J. Maex ten overstaan van een door het College voor Promoties ingestelde commissie, in het openbaar te verdedigen in de Agnietenkapel op vrijdag 17 januari 2020, te 10.00 uur door Koen Laurentius Groenland geboren te Amsterdam iv Promotiecommissie Promotores: prof. dr. H.M. Buhrman Universiteit van Amsterdam prof. dr. C.J.M. Schoutens Universiteit van Amsterdam Overige leden: prof. dr. S. Bose University College London prof. dr. M. Christandl University of Copenhagen prof. dr. J.-S. Caux Universiteit van Amsterdam prof. dr. M.S. Golden Universiteit van Amsterdam dr. V. Gritsev Universiteit van Amsterdam dr. M. Walter Universiteit van Amsterdam dr. J. van Wezel Universiteit van Amsterdam Faculteit der Natuurwetenschappen, Wiskunde en Informatica v List of publications This dissertation is based on the following papers (in chronological order). In each work, all authors contributed equally unless stated otherwise. [GS18] Many-body strategies for multiqubit gates: Quantum control through Krawtchouk- chain dynamics Koen Groenland, Kareljan Schoutens, Phys. Rev. A 97, 042321 (2018). Chapter5 is based on this paper. [Gro19] Adiabatic state distribution using anti-ferromagnetic spin systems Koen Groenland, SciPost Phys. 6, 011 (2019). Chapter9 is based on this paper. [GS19] Quantum gates by resonantly driving many-body eigenstates, with a focus on Polychronakos’ model Koen Groenland, Kareljan Schoutens, JSTAT 2019, 7, 073103 (2019). Chapters4 and6 are based on this paper. [GGK19] Adiabatic transfer of amplitude using STIRAP-like protocols generalizes to many bipartite graphs Koen Groenland, Carla Groenland, Reinier Kramer, Submitted to J. Math. Phys.. Chapter8 is based on this paper. [RGG+19] Single-step implementation of high fidelity n-bit Toffoli gates Stig E. Rasmussen, Koen Groenland, Rene Gerritsma, Kareljan Schoutens, Nikolaj T. Zinner, Submitted to Phys. Rev. A. Chapter7 is based on this work. The first two authors are the main contributors. vi Abstract Quantum protocols for few-qubit devices With the advent of scalable quantum information technology, accurate control of quan- tum systems is becoming increasingly important. We study theoretical control proto- cols tailored for near-term experiments. The main results can be grouped in two cat- egories. PartII deals with the construction of multiqubit logic gates through resonant driving, whereas in Part III we focus on adiabatic transfer of quantum states. The pre- ceding chapters in PartI contain preliminaries to these subjects. In Part II, we consider multiqubit gates, such as the ToffoliN . These are essential for most quantum algorithms, but turn out to be hard to perform on current quantum computer experiments in practice. As opposed to the standard approach of compil- ing larger unitaries into sequences of elementary gates that act on at most two qubits, we propose a continuously evolving Hamiltonian which implements highly selective multiqubit gates in a strongly-coupled many-body quantum system. We exploit the selectiveness of resonant driving to exchange only 2 out of 2N eigenstates of some background Hamiltonian, leading to a unitary time evolution we call iSWAPt1,t2 . The basis in which these states are exchanged is the eigenbasis of the background Hamil- tonian, and to make this operation relevant to the computational basis, we introduce the concept of eigengates, operations that map between the two bases. We analyze and simulate such gates in three concrete systems: • the Krawtchouk chain, an example of an open XX chain that maps to free fermions, • Polychronakos’ model, an example of an XXX or Heisenberg chain with long- ranged interactions, • and the Ising model, in which one special qubit must be coupled to all others. In Part III, we consider adiabatic protocols that transfer a quantum state through a spatially extended system. Such protocols find widespread application in current physics and chemistry experiments, and may be necessary in future quantum infor- mation technologies. Still, most adiabatic protocols are only defined on linear chains. We extend two types of protocols to work on more general bipartite graphs, under cer- tain restrictions. The first type deals with a single quantum excitation hopping on a graph. The con- ventional protocols called STImulated Raman Adiabatic Passage (STIRAP) and Coher- ent Tunnelling by Adiabatic Passage (CTAP) allow the excitation to be transported be- tween the ends of a linear chains, in a way that is highly resilient to decoherence. We prove that similar protocols can be applied on any (semi-)bipartite graph which allows a perfect matching both when the sender is removed and when the receiver is removed. Secondly, we consider an anti-ferromagnetic XXX spin system laid out on a graph. We find that spin states can be transferred and entanglement between distant sites can be created, as long as the graph is bipartite and obeys a certain balance between the maximum spin on both parts. Contents I Preliminaries 1 A gentle introduction3 2 Condensed matter models9 2.1 Qubits . .9 2.2 Modeling interacting qubits . 10 2.3 The Ising model . 14 2.4 The XX model and free fermions . 15 2.5 The XXX model, or Heisenberg model . 17 3 Manipulating quantum states 25 3.1 How to form a quantum gate . 25 3.2 How to transport a quantum state . 33 II Resonantly driven multiqubit gates 4 Introduction to resonant multiqubit gates 45 4.1 Introduction . 45 4.2 Mapping the computational basis to the eigenbasis . 48 4.3 Resonant driving . 50 5 The Krawtchouk chain 57 5.1 Analysing the model . 57 5.2 Mapping between eigenbases . 59 5.3 Resonant driving on Krawtchouk eigenstates . 60 5.4 Numerical results . 62 5.5 Experimental implementations . 63 5.6 Matrix elements of driving operators in free models . 64 5.7 Conclusion . 67 6 Polychronakos’ model 69 6.1 Analysing the model . 69 6.2 Mapping between eigenbases . 70 6.3 Resonant driving on Polychronakos eigenstates . 71 6.4 Numerical results . 72 6.5 Conclusion . 75 7 The Ising star model 77 7.1 Analysing the model . 78 viii CONTENTS 7.2 Resonant driving on Ising eigenstates . 79 7.3 Turning the driven evolution into a Toffoli . 80 7.4 Discussion of asymptotic scaling . 84 7.5 Experimental implementations . 85 7.6 Conclusion . 86 III Adiabatic transport of quantum states 8 STIRAP and CTAP on semi-bipartite graphs 89 8.1 Introduction .................................... 89 8.2 Conventional STIRAP . 90 8.3 Generalizing STIRAP . 91 8.4 Applications . 96 8.5 Examples of viable graphs . 97 8.6 Numerics . 100 8.7 Conclusion . 102 9 State transfer in bipartite anti-ferromagnetic spin networks 105 9.1 Introduction .................................... 105 9.2 Ground states of symmetry-protected subspaces . 107 9.3 Applications . 111 9.4 Errors and scaling . 114 9.5 Numerics on star graphs . 115 9.6 Experimental implementations . 118 9.7 Discussion and outlook . 118 IV The Closing Matters Bibliography 123 List of symbols and notation 137 Nederlandse samenvatting 138 Acknowledgements 140 Part I Preliminaries CHAPTER 1 A gentle introduction “My metaphor for a quantum computer is that it’s like a coastline. [...] I can manipulate how these waves behave, [...] and the waves crashing on the beach are like the output of the algorithm.” Shelby Kimmel The Schrödinger Sessions II, 7/29/2016 The front cover illustrates this metaphor. This thesis is about quantum computers: devices just like our everyday PCs, except that they exploit the theory of microscopic particles to perform tasks that current comput- ers may be unable to do. The reason to study such machines probably deserves some justification. One way to look at quantum computers is from the perspective of Moore’s Law: with transistors reaching atomic scales, we are clearly reaching a limit to shrinking structures on computer chips, calling for other alternatives. However, this is not the proper motivation for quantum computers. Contrary to what might be naively ex- pected, quantum computers will probably not have a higher clock speed, nor allow more bits to be stored. Their advantage stems from the ability to perform certain op- erations that are classically impossible. Imagine traveling from Morocco to Spain. If your technology limits you to travel by land, you would have to traverse North-Africa, all the way past the Arabic peninsula, and through Europe, before you can reach your destination. This represents the classical algorithm. In the same analogy, a quantum computer endows you with the ability to traverse the water at the Strait of Gibraltar. The new set of operations allows you to travel through previously inaccessible routes, in a fundamentally different way. Whether this extended set of operations is useful, depends on the precise problem that the computer is supposed to solve. Some problems are already solved in a virtu- ally optimal way (from an algorithmic viewpoint at least) by our current computers - problems such as sorting a list, or running a simple text editor. Other problems can greatly benefit from the additional possibilities that quantum offers.
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