#### Paul Herringer

Doctor of Philosophy in Physics (PhD)

**Research Topic**

Classifying computational phases of quantum matter in 2D and 3D

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- Robert Raussendorf

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Dissertations completed in 2010 or later are listed below. Please note that there is a 6-12 month delay to add the latest dissertations.

Hybrid quantum computing: on the role of classical computing in scaling up quantum computing (2021)

In this thesis, we aim to answer one research question: What is the algorithmic role of classical computing in scaling up quantum computing applicability? Quantum computers have promising potential in offering more efficient solutions to some of the most demanding computational problems in the industry. There has been phenomenal progress in the theory and realization of quantum computers in the last few decades. However, quantum computing is a nascent field of technology and scaling up its computing capacity remains impeded by many open engineering/scientific challenges. One emerging solution to alleviate some of the present limitations is finding clever use of classical pre- and post-processing techniques in tandem with quantum algorithms. This helps to advantageously use scarce quantum computing resources for only a portion of a problem that benefits the most from a quantum computer. In other words, it leads to a hybrid framework in which quantum computers are considered special-purpose co-processors for accelerating specific computational tasks. We aim to develop a hybrid theoretical framework for the role of classical algorithms in scaling up quantum computing. We investigate this by focusing on three distinct limitations. We review each limitation in the context of a different application and propose an original hybrid classical-quantum solution to it. First, we focus on classical algorithms' role in a more resource-efficient compilation and embedding of quantum programs. We propose a classical algorithm that helps to embed larger optimization problems in a quantum annealer. Then we explore how a problem decomposition technique and preprocessing of the input problem can help solve significantly larger optimization problems on a quantum computer. Finally, we propose a completely hybrid quantum algorithm for preparing ground states of quantum Hamiltonians. We achieve orders of magnitude faster convergence by combining classical optimization with variational quantum state preparation.

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Reductions in finite-dimensional quantum mechanics: from symmetries to operator algebras and beyond (2021)

The idea that symmetries simplify or reduce the complexity of a system has been remarkably fruitful in physics, and especially in quantum mechanics. On a mathematical level, symmetry groups single out a certain structure in the Hilbert space that leads to a reduction. This structure is given by the irreducible representations of the group, and in general it can be identified with an operator algebra (a.k.a. C*-algebra or von Neumann algebra). The primary focus of this thesis is the extension of the framework of reductions from symmetries to operator algebras, and its applications in finite-dimensional quantum mechanics.Finding the irreducible representations structure is the principal problem when working with operator algebras. We will therefore review the representation theory of finite-dimensional operator algebras and elucidate this problem with the help of two novel concepts: minimal isometries and bipartition tables. One of the main technical results that we present is the Scattering Algorithm for analytical derivations of the irreducible representations structure of operator algebras.For applications, we will introduce a symmetry-agnostic approach to the reduction of dynamics where we circumvent the non-trivial task of identifying symmetries, and directly reduce the dynamics generated by a Hamiltonian. We will also consider quantum state reductions that arise from operational constraints, such as the partial trace or the twirl map, and study how operational constraints lead to decoherence. Apart from our primary focus we will extend the idea of reduction beyond operator algebras to operator systems, and formulate a quantum notion of coarse-graining that so far only existed in classical probability theory. In addition, we will characterize how the uncertainty principle transitions to the classical regime under coarse-grained measurements and discuss the implications in a finite-dimensional setting.

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Theses completed in 2010 or later are listed below. Please note that there is a 6-12 month delay to add the latest theses.

Classification of quantum wire in tensor network states with a local Pauli symmetry (2021)

Certain symmetry-protected topological (SPT) phases also possess the property that every state in the phase is a universal resource for measurement-based quantum computation (MBQC). These phases are called computational phases of quantum matter. Computational phases have been classified in 1D, and several examples are known in 2D. 2D computational phases are of interest because we can simulate any quantum computation using a 2D phase. Therefore, the classification of 2D phases may lead to a better understanding of the relationship between symmetry and quantum computation. Many known 2D phases have a projected entangled pair state (PEPS) representation where the local tensor symmetries define a quantum cellular automaton (QCA) acting on correlation space. However, not every tensor symmetry defines a QCA, which motivates the definition and study of a larger class of PEPS tensors called stabilizer tensors. Stabilizer tensors have symmetries that act as a tensor product of Pauli operators on correlation space but do not necessarily define a QCA. In this thesis, I present a classification of stabilizer tensors by their channel capacity for quantum wire. In particular, I consider the quantum wire capacity of translationally invariant, cylindrical PEPS constructed from stabilizer tensors and find that it falls into one of 13 classes. Furthermore, I prove that no other types of channel capacity exist for stabilizer tensors, regardless of the size of the cylinder. I also discuss the significance of this result for the classification of 2D computational phases along with possible next steps.

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Symmetry protected measurement-based quantum computation in finite spin chains (2021)

We consider ground states of quantum spin chains with symmetry-protected topological (SPT) order and their usefulness as resources for measurement-based quantum computation (MBQC). It is known that SPT phases (applicable to infinite spin chains) protected by a finite abelian symmetry group exhibit uniform computational power. In this work, we extend these ideas to finite spin chains which are more realistic resources for physical computation. Herein, we relate the usefulness of MBQC resource states to a non-vanishing string-order parameter. Furthermore, using the techniques developed, we show that hard-to-think-about regimes of computation are actually more efficient than the textbook prescription. Our results strengthen the connection between condensed matter and quantum computation and also provide the necessary tools to explore such connections in the state-of-the-art noisy small quantum devices. As an outlook, we discuss how this newly developed formalism can potentially be used to identify non-traditional resource states for MBQC and be extended to higher dimensions, leading to universal quantum computation.

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Hidden variable models and classical simulation algorithms for quantum computation with magic states on qubits (2020)

An important problem in quantum computation is to characterize the resources required for a computational speedup over classical computation. Veitch et al. showed that one necessary condition for a computational speedup in the model of quantum computation with magic states is that the discrete Wigner function representing the input state of the quantum circuit must take negative values. The amount of negativity in the discrete Wigner function quantifies the complexity of classical simulation of a quantum computation with simulation being efficient if the Wigner function is nonnegative everywhere. In this sense, negativity of the Wigner quasiprobability representation serves as an indicator of quantumness from a computational perspective. However, this result only holds for systems of qudits where the local Hilbert space dimension is odd.The first main result discussed in this thesis relates to a discrete Wigner function suitable for describing quantum computation with magic states defined for any local Hilbert space dimension. When the local Hilbert space dimension is odd it subsumes the standard discrete Wigner function. When the local Hilbert space dimension is even, as a result of state-independent proofs of contextuality among multiqubit (or multi-even-dimensional-qudit) Pauli observables, the phase space over which the Wigner function is defined becomes much larger. However, for systems of qubits, the properties required for simulation of quantum computation with magic states remain. This simulation method effectively extends the result described above for odd-dimensional qudits to qubits. Although in the even-dimensional case the phase space is much larger, points in multiqubit phase space can be characterized and classified and some structure can be imparted on the phase space.The second main result discussed here is a hidden variable model for quantum computation with magic states on qubits. This model is similar in structure to quasiprobability representations like the discrete Wigner function, but unlike those representations the model is capable of representing all elements of any quantum computation---states, operations, and measurements---using only classical probabilities. No negativity is required. This calls into question the role of negativity in quasiprobability representations as an indicator of quantumness for models of quantum computation.

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Computational power of one-dimensional symmetry-protected topological phases (2017)

We consider ground states of quantum spin chains with symmetry-protected topological (SPT) order as resources for measurement-based quantum computation (MBQC). Using tensor network methods, we show that SPT phases protected by a finite abelian on-site symmetry group exhibit uniform computational power. That is, any state from a given phase leads to the same Lie group of executable gates when used as a resource for MBQC. This Lie group is determined by the same algebraic information that labels the SPT phase itself, and we give a necessary condition on the phase that guarantees a full set of single-qubit gates. To obtain our results, we construct several new techniques in MBQC and refine the structure of quantum states with abelian SPT order. Our results are analogous to similar results relating topological order and topological quantum computation, and we comments on their implications on the general connection between quantum phases of matter and quantum computation.

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An investigation of no-go theorems in hidden variable models of quantum mechanics (2016)

Realism defined in EPR paper as “In a complete theory there is an element corresponding to each element of reality.” Bell showed that there is a forbidden triangle (Realism, Quantum Statistics, and Locality), and we are only allowed to pick two out of three. In this thesis, we investigate other inequalities and no-go theorems that we face. We also discuss possible Hidden Variable Models that are tailored to be consistent with Quantum Mechanics and the specific no-go theorems. In the special case of the Leggett Inequality the proposed hidden variable is novel in the sense that the hidden variable is in the measurement device rather than the wave-function.

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Fast Post Processing Algorithms for Fault Tolerant Quantum Computation Using Surface Codes (2015)

Local architecture of qubits in two dimensional lattices is known as one of the candidates to run fault tolerant quantum computation with high threshold. Topological quantum error correcting codes, such as surface codes, are used to make this architecture robust against various quantum error models. In this research we present a fast, concise, operationally inexpensive and highly parallelizable decoding algorithm for surface codes without using concatenation which has a threshold range of 8.6 % to 10.5 % varying based on a parameter called OMSS. Thanks to parallelization of the proposed algorithm, the time complexity scales logarithmically in the lattice size for small OMSS. This can be compared to the work of \citep{Poulin-2010,Poulin-2010-IEEE} which has a threshold of 8 % for the bit-flip error channel within the same complexity order.

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Wigner Function Negativity and Contextuality in Quantum Computation with Rebits (2015)

We study the resources necessary for quantum computation with rebits (qubit states with real amplitudes in the standard basis). We introduce a scheme for universal quantum computation by state injection, and define a Wigner function appropriate for this scheme. We show that the Wigner function obeys a Hudson’s theorem and transforms covariantly under CSS-ness preserving unitary gates; these results allows us to establish that Wigner function negativity is necessary for quantum computation. Furthermore, we establish contextuality as another necessary computational resource. We show that in contrast with the case of qudits [M. Howard et al., Nature 510, 351 (2014)], negativity does not imply contextuality. We discuss state independent contextuality and why it does not arise in our computational scheme.

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This is a small sample of students and/or alumni that have been supervised by this researcher. It is not meant as a comprehensive list.