On stabiliser techniques and their application to simulation and certification of quantum devices INAUGURAL-DISSERTATION ZUR ERLANGUNG DES AKADEMISCHEN GRADES doctor rerum naturalium (Dr. rer. nat.) IN THEORETISCHER PHYSIK der Mathematisch-Naturwissenschaftlichen Fakultät der Universität zu Köln vorgelegt von MARKUS HEINRICH aus Offenburg Köln, 2021 Gutachter: Prof. Dr. David Gross PD Dr. Rochus Klesse Prof. Dr. Markus Müller Tag der Disputation: 3. Mai 2021 ABSTRACT The stabiliser formalism is a widely used and successful subtheory of quantum mechanics consisting of stabiliser states, Clifford unitaries and Pauli measurements. The power of the formalism comes from the description of its elements via simple group theory. Although the origins of the formalism lie in quantum error correction and fault-tolerant quantum computing, the utility of the formalism goes beyond that. The Gottesman-Knill theorem states that the dynamics of stabiliser states under Clif- ford unitaries and Pauli measurements can be efficiently simulated on a classical com- puter. This algorithm can be extended to arbitrary states and unitaries in multiple ways at the cost of an increased runtime. This runtime can be seen as a quantification of the non- stabiliser resources needed to implement a quantum circuit. Moreover, as non-stabiliser elements are necessary for universal quantum computing, the runtime provides a way to measure the “non-classicality” of a computation. This is particularly pronounced in the magic state model of quantum computing where the only non-stabiliser elements are given by magic states. Hence, in the resource theory of magic, resources are measured through magic monotones which are operationally linked to runtimes of classical simulation algo- rithms. In this thesis, I discuss different aspects of the resource theory of magic. The men- tioned classical simulation algorithms require the computation of magic monotones which is in general a computationally intractable problem. However, I show that the compu- tational complexity can be exponentially reduced for certain classes of symmetric states, such as copies of magic states. To this end, the symmetries of the convex hull of stabiliser states are characterised and linked to their properties as so-called designs. In addition, I study the recently introduced class of completely stabiliser-preserving channels (CSP), which is the class of quantum channels unable to generate magic resources. It is shown that this class is strictly larger than the class of stabiliser operations, composed of Clifford unitaries and Pauli measurements. This finding could have several interesting conse- quences. First, it is likely that CSP is efficiently simulable which would allow classi- cal simulation beyond the Gottesman-Knill theorem. Second, it is possible that optimal magic state distillation rates cannot be achieved via stabiliser operations and this gap is in fact significant. Further applications of the stabiliser formalism come through design theory. A uni- tary t-design is an ensemble of unitaries which reproduce the first t moments of the Haar measure on the unitary group. Randomness in the form of Haar-random unitaries is an essential building block in many quantum information protocols. Implementing such Haar-random unitaries is however often impractical. Here, designs can exhibit consider- ably lower resource requirements while still being random enough for most applications. Some of the most prominent examples of such protocols concern the certification and characterisation of quantum systems, such as randomised benchmarking. Interestingly, the group of Clifford unitaries forms a unitary 3-design and is often the prime choice thanks to efficient group operations. i ii In this thesis, I summarise my recent results obtained with collaborators in construct- ing approximate unitary t-designs from the Clifford group supplemented by only few non- Clifford gates. Intriguingly, this construction uses only O˜ (t4) single qubit non-Clifford gates and is independent of the number of qubits n. Overall, this yields a gate count of O˜ (n2t4) which is a significant improvement over O˜ (n2t10) for the Brandao-Harrow- Horodecki construction based on local random circuits. To provide context for this result, I review the representation theory of the Clifford group and define the Clifford semigroup. In an attempt to generalise approximations results for the unitary group to the Clifford group, the Clifford semigroup is investigated for suitable approximations of the Clifford twirl. CONTENTS Abstract i Contents iii Introduction to this dissertation v I The phase space representation of the stabiliser formalism 1 1 Introduction 3 2 Stabiliser formalism for qubits 5 3 Symplectic structures over finite fields 11 3.1 Finite fields and discrete symplectic vector spaces . 11 3.2 The Weil representation in odd characteristic . 18 3.3 A Weil-like representation in even characteristic . 23 4 Stabiliser formalism in prime-power dimensions 29 4.1 The Heisenberg-Weyl and Clifford groups . 29 4.2 Stabiliser states and codes . 32 4.3 Fq versus Fp structure . 39 4.4 Simulation of stabiliser circuits . 41 5 Applications 45 5.1 Discrete Wigner function . 45 5.2 Mutually unbiased bases . 51 5.3 Algorithms . 52 6 Further topics 57 6.1 Construction of Galois extensions of fields and rings . 57 6.2 On the problems in even characteristic . 60 II Classical simulation and the resource theory of magic 71 7 Introduction 73 8 Robustness of Magic and the stabiliser polytope 77 8.1 Introduction . 78 8.2 Robustness of Magic . 82 8.3 Exploiting stabiliser symmetries . 83 8.4 Computing the robustness of magic . 91 8.5 Conclusion & Outlook . 102 8.A Equivalence of the two robustness measures . 103 iii iv CONTENTS 8.B On the dual RoM problem . 104 8.C Symmetries of 3-designs . 105 8.D Numerical implementation . 108 8.E Symmetry reduction of convex optimisation problems . 109 9 Axiomatic vs. operational approaches to resource theories of magic 115 9.1 Introduction . 116 9.2 Preliminaries . 117 9.3 Results . 120 9.4 Summary and open questions . 125 9.A Phase space formalism in a nutshell . 126 9.B Proof of Corollary 9.1 . 128 9.C Polar form of bipartite stabiliser states . 130 9.D Proof that L is extremal . 133 9.E Proof that SO1 = CSP1 .............................. 142 9.F Analysis of the channel decomposition of L . 144 10 Open questions 147 III Exact and approximate unitary designs from the Clifford group 149 11 Introduction 151 12 Unitary designs and the Clifford group 155 12.1 Definitions . 155 12.2 The Clifford group as a design . 158 12.3 Tensor power representations of the Clifford group . 161 13 Group designs are rare and essentially Clifford 171 14 Approximate t-designs with few non-Clifford gates 175 14.1 Introduction . 175 14.2 Results . 176 14.3 Technical background . 179 15 Approximations of the Clifford projector 187 15.1 Introduction . 187 15.2 The Clifford frame operator . 189 15.3 Approximation of the Clifford projector . 191 Conclusion 197 Acknowledgments 201 Bibliography 203 Formalia 219 Zusammenfassung in deutscher Sprache . 219 Erklärung zur Dissertation . 221 Teilpublikationen . 221 INTRODUCTION TO THIS DISSERTATION The defining goal of quantum information science is to study how information can be manipulated by the quantum-mechanical laws of nature. During the last 30 years, sig- nificant progress has been made in developing ideas and methods to store, transmit or process information using quantum effects. Consequently, quantum information science has matured to a broad and prospering field with diverse research directions that reach from quantum communication to quantum computing and the characterisation of quan- tum processes. Among other things, the research finds that faster and more secure ways of communication as well as more powerful computing is possible with quantum de- vices. Some of these ideas have already been successfully demonstrated in a series of proof-of-principle experiments [1–6]. Quantum key distribution is arguably the most advanced quantum technology as several networks have demonstrated over the years. Furthermore, researchers have recently claimed the first quantum advantage of a quan- tum computer over classical computers [5, 6]. Careful voices, however, expect realistic usecases for quantum computers to lie years ahead of us. Nevertheless, these ideas bear the potential to have a technological impact comparable with the silicon revolution of classical computers. Although the development is still in an early stage and high techno- logical barriers have to be overcome, these prospects have already gained a lot of public and political attention. Due to the sensitivity of quantum systems to their environment, the design of quan- tum devices is delicate and makes a close collaboration between theory and experiment necessary. On the theoretical side, a number of methods have been developed which allow to characterise a quantum system to a varying level of detail [7–16]. This allows to quantify the amount of noise or to determine its form. Furthermore, it can be certified that a quantum device is functioning properly. As quantum computations are especially sensitive to noise, these techniques are of special importance there. Besides the char- acterisation of noise, benchmarking the individual components of a quantum computer has become a common method of quantifying its performance [17–23]. At the current state-of-the art, the results of quantum computation can often be simulated on a classical computer can be used to certify the function of a quantum computer [24–39]. Many of these methods are based on a special subclass of quantum states and op- erations with astonishing properties. This subclass consists of stabiliser states, Clifford unitaries, and Pauli measurements, and allows for an efficient description through the so- called stabiliser formalism. In particular, the dynamics of stabiliser states under Clifford unitaries and the outcomes of Pauli measurements can be efficiently simulated on a clas- sical computer – a result which is known as the Gottesman-Knill theorem [40]. Neverthe- less, this subclass can be used to demonstrate many quantum phenomena.