The problem considered in this thesis is the following: We are given a Hamiltonian H and time t, and our goal is to approximately implement the unitary operator e^{-iHt} with an efficient quantum algorithm. We present an efficient algorithm for simulating sparse Hamiltonians. In terms of the maximum degree d and dimension N of the space on which the Hamiltonian acts, this algorithm uses (d^2(d+log^* N)||Ht||)^{1+o(1)} queries. This improves the complexity of the sparse Hamiltonian simulation algorithm of Berry, Ahokas, Cleve, and Sanders, which scales like (d^4(log^* N)||Ht||)^{1+o(1)}. In terms of the parameter t, these algorithms are essentially optimal due to a no--fast-forwarding theorem. In the second part of this thesis, we consider non-sparse Hamiltonians and show significant limitations on their simulation. We generalize the no--fast-forwarding theorem to dense Hamiltonians, and rule out generic simulations taking time o(||Ht||), even though ||H|| is not a unique measure of the size of a dense Hamiltonian H. We also present a stronger limitation ruling out the possibility of generic simulations taking time poly(||Ht||,log N), showing that known simulations based on discrete-time quantum walks cannot be dramatically improved in general. We also show some positive results about simulating structured Hamiltonians efficiently.

The problem considered in this thesis is the following: We are given a Hamiltonian H and time t, and our goal is to approximately implement the unitary operator e^{-iHt} with an efficient quantum algorithm. We present an efficient algorithm for simulating sparse Hamiltonians. In terms of the maximum degree d and dimension N of the space on which the Hamiltonian acts, this algorithm uses (d^2(d+log^* N)||Ht||)^{1+o(1)} queries. This improves the complexity of the sparse Hamiltonian simulation algorithm of Berry, Ahokas, Cleve, and Sanders, which scales like (d^4(log^* N)||Ht||)^{1+o(1)}. In terms of the parameter t, these algorithms are essentially optimal due to a no--fast-forwarding theorem. In the second part of this thesis, we consider non-sparse Hamiltonians and show significant limitations on their simulation. We generalize the no--fast-forwarding theorem to dense Hamiltonians, and rule out generic simulations taking time o(||Ht||), even though ||H|| is not a unique measure of the size of a dense Hamiltonian H. We also present a stronger limitation ruling out the possibility of generic simulations taking time poly(||Ht||,log N), showing that known simulations based on discrete-time quantum walks cannot be dramatically improved in general. We also show some positive results about simulating structured Hamiltonians efficiently.

This project was motivated by the following question: what information do the properties of a random graph contain about the performance of a quantum search acting on it? To investigate this problem, we define a notion of search time to quantify the behavior of a quantum search, and find strong evidence of a relation between its distribution and the model of random graph on which the search was performed. Surprisingly, we also find strong evidence that the return time of a classical random walk initialized at the marked vertex is closely related to its search time, and that the distribution of degrees over the graph vertices may play a significant role in this relation. / Mathematics

This thesis is on quantum algorithms. It has three main themes: (1) quantum walk based search algorithms, (2) quantum rejection sampling, and (3) the Boolean function hidden shift problem. The first two parts deal with generic techniques for constructing quantum algorithms, and the last part is on quantum algorithms for a specific algebraic problem. In the first part of this thesis we show how certain types of random walk search algorithms can be transformed into quantum algorithms that search quadratically faster. More formally, given a random walk on a graph with an unknown set of marked vertices, we construct a quantum walk that finds a marked vertex in a number of steps that is quadratically smaller than the hitting time of the random walk. The main idea of our approach is to interpolate the random walk from one that does not stop when a marked vertex is found to one that stops. The quantum equivalent of this procedure drives the initial superposition over all vertices to a superposition over marked vertices. We present an adiabatic as well as a circuit version of our algorithm, and apply it to the spatial search problem on the 2D grid. In the second part we study a quantum version of the problem of resampling one probability distribution to another. More formally, given query access to a black box that produces a coherent superposition of unknown quantum states with given amplitudes, the problem is to prepare a coherent superposition of the same states with different specified amplitudes. Our main result is a tight characterization of the number of queries needed for this transformation. By utilizing the symmetries of the problem, we prove a lower bound using a hybrid argument and semidefinite programming. For the matching upper bound we construct a quantum algorithm that generalizes the rejection sampling method first formalized by von~Neumann in~1951. We describe quantum algorithms for the linear equations problem and quantum Metropolis sampling as applications of quantum rejection sampling. In the third part we consider a hidden shift problem for Boolean functions: given oracle access to f(x+s), where f(x) is a known Boolean function, determine the hidden shift s. We construct quantum algorithms for this problem using the "pretty good measurement" and quantum rejection sampling. Both algorithms use the Fourier transform and their complexity can be expressed in terms of the Fourier spectrum of f (in particular, in the second case it relates to "water-filling" of the spectrum). We also construct algorithms for variations of this problem where the task is to verify a given shift or extract only a single bit of information about it.

This thesis is on quantum algorithms. It has three main themes: (1) quantum walk based search algorithms, (2) quantum rejection sampling, and (3) the Boolean function hidden shift problem. The first two parts deal with generic techniques for constructing quantum algorithms, and the last part is on quantum algorithms for a specific algebraic problem. In the first part of this thesis we show how certain types of random walk search algorithms can be transformed into quantum algorithms that search quadratically faster. More formally, given a random walk on a graph with an unknown set of marked vertices, we construct a quantum walk that finds a marked vertex in a number of steps that is quadratically smaller than the hitting time of the random walk. The main idea of our approach is to interpolate the random walk from one that does not stop when a marked vertex is found to one that stops. The quantum equivalent of this procedure drives the initial superposition over all vertices to a superposition over marked vertices. We present an adiabatic as well as a circuit version of our algorithm, and apply it to the spatial search problem on the 2D grid. In the second part we study a quantum version of the problem of resampling one probability distribution to another. More formally, given query access to a black box that produces a coherent superposition of unknown quantum states with given amplitudes, the problem is to prepare a coherent superposition of the same states with different specified amplitudes. Our main result is a tight characterization of the number of queries needed for this transformation. By utilizing the symmetries of the problem, we prove a lower bound using a hybrid argument and semidefinite programming. For the matching upper bound we construct a quantum algorithm that generalizes the rejection sampling method first formalized by von~Neumann in~1951. We describe quantum algorithms for the linear equations problem and quantum Metropolis sampling as applications of quantum rejection sampling. In the third part we consider a hidden shift problem for Boolean functions: given oracle access to f(x+s), where f(x) is a known Boolean function, determine the hidden shift s. We construct quantum algorithms for this problem using the "pretty good measurement" and quantum rejection sampling. Both algorithms use the Fourier transform and their complexity can be expressed in terms of the Fourier spectrum of f (in particular, in the second case it relates to "water-filling" of the spectrum). We also construct algorithms for variations of this problem where the task is to verify a given shift or extract only a single bit of information about it.

The last forty years have seen an astounding level of progress in the field of quantum computing. Rapidly-improving techniques for fabricating and controlling devices, increasingly refined theoretical models, and innovative quantum computing algorithms have allowed us to pass a number of important milestones on the path towards fault-tolerant general purpose quantum computing. There remains, however, uncertainty regarding the feasibility and logistics of scaling quantum computing platforms to useful sizes. A great deal of work remains to be done in developing sophisticated control techniques, designing scalable quantum information processing architectures, and creating resource-efficient algorithms. This dissertation is a collection of seven manuscripts organized into three sections which aim to contribute to these efforts. In the first section, we explore quantum control techniques for exchange-coupled solid-state electronic spin qubits in arrays of gate-defined quantum dots. We start by demonstrating theoretically the existence of a discrete time crystal phase in finite Heisenberg spin chains. We present driving pulses that can be used to induce time crystalline behavior and probe the conditions under which this behavior can exist, finding that it should be realizable with current experimental capabilities. Next, we use a correspondence between quantum time evolution geometric space curves to design fast, high-fidelity entangling gates in two-spin double quantum dots. In the second section, we study systems of quantum dot spin qubits coupled to one another via mutual coupling to superconducting microwave resonators. We start with two qubits, developing and refining an effective model of resonator-mediated entangling interactions, and then use that model to ultimately design fast, long-distance, high-fidelity entangling gates which are robust to environmental noise. We then take the model further, extending our model to a system of three qubits coupled by a combination of short-range exchange interactions and long-range resonator-mediated interactions, and numerically demonstrate that previously-developed protocols can be used to realize both short- and long-range entangling operations. The final section investigates adaptive variational algorithms for efficient preparation of thermal Gibbs states on a quantum computer, a difficult task with a number of important applications. We suggest a novel objective function which can be used for variational Gibbs state preparation, but which requires fewer resources to measure than the often-used Gibbs free energy. We then introduce and characterize two variational algorithms using this objective function which adaptively construct variational ansätze for Gibbs state preparation. / Doctor of Philosophy / The computers we have now are able to perform computations by storing information in bits (units of memory which can take on either of two values e.g. 0 or 1) and then comparing and modifying the values of these bits according to a simple set of logical rules. The logic these computers use is suited to a universe that obeys the laws of classical mechanics, which was our best theory of physics prior to the 20th century, but the last 120 years have seen a radical shift in our understanding of nature. We now know that nature is much better described by the laws of quantum mechanics, which includes a great deal of surprising and unintuitive non-classical phenomena. The aim of quantum computing is to use our improved understanding of nature to design and build a new kind of computer which stores information in the states of quantum bits ("qubits") and then compares and modifies the combined state of these qubits using a logic adapted to the laws of quantum mechanics. By leveraging the quantum nature of reality, these quantum computers are capable of performing certain computations faster and more efficiently than is possible using classical computers. The prospect of faster computing has inspired a massive effort to develop useful quantum computers, and the last forty years have seen impressive progress towards this goal, but there is a great deal left to do. Current quantum computing devices are too sensitive to their surroundings and far too error-prone to do useful computations. To reach tolerable error rates, we need to develop better devices and better methods for controlling those devices. Meanwhile, although several different device platforms are being continually developed, none of them currently operates with a collection of qubits anywhere near as large as the billions of bits our classical computers are able to use. It is not yet clear that practical scaling of these platforms up to that level is even possible, let alone how we can do so. Furthermore, only a handful of promising quantum algorithms have been discovered, and the efficiency of many is questionable at best. We have much that we still need to learn about what quantum computers can do and how best to use them. This dissertation is a collection of seven papers arranged into three sections, all attempting to help address some of these issues. In the first two sections, we focus on one promising type of quantum computing platform -- solid-state electronic spin qubits. We introduce new methods for quickly performing quantum logic operations in these platforms, we suggest protocols for making these systems exhibit novel and potentially useful behavior, and we characterize and design control methods for a device design which might facilitate scaling up to large numbers of qubits. In the final section, we turn our attention to quantum software, and present two algorithms for using quantum computers to efficiently simulate physical systems at a fixed temperature.

Heterogeneous computing (HC) systems are essential parts of modern-day computing architectures such as cloud, cluster, grid, and edge computing. Many algorithms exist within the classical environment for mapping computational tasks to the HC system’s nodes, but this problem is not well explored in the quantum area. In this work, the practicality, accuracy, and computation time of quantum mapping algorithms are compared against eleven classical mapping algorithms. The classical algorithms used for comparison include A-star (A*), Genetic Algorithm (GA), Simulated Annealing (SA), Genetic Simulated Annealing (GSA), Opportunistic Load Balancing (OLB), Minimum Completion Time (MCT), Minimum Execution Time (MET), Tabu, Min-min, Max-min, and Duplex. These algorithms are benchmarked using several different test cases to account for varying system parameters and task characteristics. This study reveals that a quantum mapping algorithm is feasible and can produce results similar to classical algorithms.

Recent advances in quantum computational science promise substantial improvements in the speed with which certain classes of problems can be computed. Various algorithms that utilize the distinctively non-classical characteristics of quantum mechanics have been formulated to take advantage of this promising new approach to computation. One such algorithm was formulated by David Deutsch and Richard Jozsa. By measuring the output of a quantum network that implements this algorithm, it is possible to determine with N 1 measurements certain global properties of a function f(x), where N is the number of network inputs. Classically, it may not be possible to determine these same properties without evaluating f(x) a number of times that rises exponentially as N increases. Hitherto, the potential power of this algorithm has been explored in the context of qubits, the quantum computational analogue of classical bits. However, just as one can conceive of classical computation in the context of non-binary logic, such as ternary or quaternary logic, so also can one conceive of corresponding higher-order quantum computational equivalents.<p>This thesis investigates the behaviour of the Deutsch-Jozsa algorithm in the context of these higher-order quantum computational forms of logic and explores potential applications for this algorithm. An important conclusion reached is that, not only can the Deutsch-Jozsa algorithms known computational advantages be formulated in more general terms, but also a new algorithmic property is revealed with potential practical applications.

Recent advances in quantum computational science promise substantial improvements in the speed with which certain classes of problems can be computed. Various algorithms that utilize the distinctively non-classical characteristics of quantum mechanics have been formulated to take advantage of this promising new approach to computation. One such algorithm was formulated by David Deutsch and Richard Jozsa. By measuring the output of a quantum network that implements this algorithm, it is possible to determine with N 1 measurements certain global properties of a function f(x), where N is the number of network inputs. Classically, it may not be possible to determine these same properties without evaluating f(x) a number of times that rises exponentially as N increases. Hitherto, the potential power of this algorithm has been explored in the context of qubits, the quantum computational analogue of classical bits. However, just as one can conceive of classical computation in the context of non-binary logic, such as ternary or quaternary logic, so also can one conceive of corresponding higher-order quantum computational equivalents.<p>This thesis investigates the behaviour of the Deutsch-Jozsa algorithm in the context of these higher-order quantum computational forms of logic and explores potential applications for this algorithm. An important conclusion reached is that, not only can the Deutsch-Jozsa algorithms known computational advantages be formulated in more general terms, but also a new algorithmic property is revealed with potential practical applications.

Since their original introduction, strongly complementary observables have been a fundamental ingredient of the ZX calculus, one of the most successful fragments of Categorical Quantum Mechanics (CQM). In this thesis, we show that strong complementarity plays a vastly greater role in quantum theory. Firstly, we use strong complementarity to introduce dynamics and symmetries within the framework of CQM, which we also extend to infinite-dimensional separable Hilbert spaces: these were long-missing features, which open the way to a wealth of new applications. The coherent treatment presented in this work also provides a variety of novel insights into the dynamics and symmetries of quantum systems: examples include the extremely simple characterisation of symmetry-observable duality, the connection of strong complementarity with the Weyl Canonical Commutation Relations, the generalisations of Feynman's clock construction, the existence of time observables and the emergence of quantum clocks. Secondly, we show that strong complementarity is a key resource for quantum algorithms and protocols. We provide the first fully diagrammatic, theory-independent proof of correctness for the quantum algorithm solving the Hidden Subgroup Problem, and show that strong complementarity is the feature providing the quantum advantage. In quantum foundations, we use strong complementarity to derive the exact conditions relating non-locality to the structure of phase groups, within the context of Mermin-type non-locality arguments. Our non-locality results find further application to quantum cryptography, where we use them to define a quantum-classical secret sharing scheme with provable device-independent security guarantees. All in all, we argue that strong complementarity is a truly powerful and versatile building block for quantum theory and its applications, and one that should draw a lot more attention in the future.