Discrete Wigner Function Derivation of the Aaronson-Gottesman Tableau Algorithm

The Gottesman-Knill theorem established that stabilizer states and operations can be efficiently simulated classically. For qudits with dimension three and greater, stabilizer states and Clifford operations have been found to correspond to positive discrete Wigner functions and dynamics. We present a discrete Wigner function-based simulation algorithm for odd-$d$ qudits that has the same time and space complexity as the Aaronson-Gottesman algorithm. We show that the efficiency of both algorithms is due to the harmonic evolution in the symplectic structure of discrete phase space. The differences between the Wigner function algorithm and Aaronson-Gottesman are likely due only to the fact that the Weyl-Heisenberg group is not in $SU(d)$ for $d=2$ and that qubits have state-independent contextuality. This may provide a guide for extending the discrete Wigner function approach to qubits

Exploring Quantum Computation Through the Lens of Classical Simulation

It is widely believed that quantum computation has the potential to offer an ex- ponential speedup over classical devices. However, there is currently no definitive proof of this separation in computational power. Such a separation would in turn imply that quantum circuits cannot be efficiently simulated classically. However, it is well known that certain classes of quantum computations nonetheless admit an efficient classical description. Recent work has also argued that efficient classical simulation of quantum circuits would imply the collapse of the Polynomial Hierarchy, something which is commonly invoked in clas- sical complexity theory as a no-go theorem. This suggests a route for studying this ‘quantum advantage’ through classical simulations. This project looks at the problem of classically simulating quantum circuits through decompositions into stabilizer circuits. These are a restricted class of quantum computation which can be efficiently simulated classically. In this picture, the rank of the decomposition determines the temporal and spatial complexity of the simulation. We approach the problem by considering classical simulations of stabilizer circuits, introducing two new representations with novel features compared to previous meth- ods. We then examine techniques for building these so-called ‘stabilizer rank’ decom- positions, both exact and approximate. Finally, we combine these two ingredients to introduce an improved method for classically simulating broad classes of circuits using the stabilizer rank method

Advancing classical simulators by measuring the magic of quantum computation

Stabiliser operations and state preparations are efficiently simulable by classical computers. Stabiliser circuits play a key role in quantum error correction and fault-tolerance, and can be promoted to universal quantum computation by the addition of "magic" resource states or non-Clifford gates. It is believed that classically simulating stabiliser circuits supplemented by magic must incur a performance overhead scaling exponentially with the amount of magic. Early simulation methods were limited to circuits with very few Clifford gates, but the need to simulate larger quantum circuits has motivated the development of new methods with reduced overhead. A common theme is that algorithm performance can often be linked to quantifiers of computational resource known as magic monotones. Previous methods have typically been restricted to specific types of circuit, such as unitary or gadgetised circuits. In this thesis we develop a framework for quantifying the resourcefulness of general qubit quantum circuits, and present improved classical simulation methods. We first introduce a family of magic state monotones that reveal a previously unknown formal connection between stabiliser rank and quasiprobability methods. We extend this family by presenting channel monotones that measure the magic of general qubit quantum operations. Next, we introduce a suite of classical algorithms for simulating quantum circuits, which improve on and extend previous methods. Each classical simulator has performance quantified by a related resource measure. We extend the stabiliser rank simulation method to admit mixed states and noisy operations, and refine a previously known sparsification method to yield improved performance. We present a generalisation of quasiprobability sampling techniques with significantly reduced exponential scaling. Finally, we evaluate the simulation cost per use for practically relevant quantum operations, and illustrate how to use our framework to realistically estimate resource costs for particular ideal or noisy quantum circuit instances

Contextuality and Ontological Models: A Tale of Desire and Disappointment

Since being defined by Kochen and Specker, and separately by Bell, contextuality has been proposed as one of the key phenomena that distinguishing quantum theory from classical theories. However, with the rise of quantum information contextuality's position as the leading definition of the quantum/classical boundary has been called into question. This is due to the fact that a contextual explanation is required by subtheories that offer no exponential quantum computational advantage over classical computation. In this thesis, submitted in requirement for a PhD in physics with quantum information, we shall explore this unwanted contextuality, and show that generalized contextuality is more prevalent than was previously thought. First we will show that the single-qubit stabilizer subtheory, which was previously thought of as a non-contextual subtheory, requires a generalized contextual ontological model, when transformations are included in the operational description. In addition to this we show that the even smaller single-rebit stabilizer subtheory, a strict subtheory of the single-qubit stabilizer subtheory, must also admit a generalized contextual ontological model, again when transformations are included in the operational description. We then show that both these subtheories require negatively represented quasi-probability representations, re-establishing the link between contextuality and negativity for transformations. We also investigate the representation of transformations in generic quasi-probability representations, showing that under a reasonable assumption almost-all unitaries must be negatively represented by a finite quasi-probability representation. Second we will investigate the efficiently simulable n-qubit stabilizer subtheory, which exhibits all forms of contextuality and thus represents the main obstacle to identifying contextuality as a resource for quantum computation. To this end we present an attempt at constructing a model based on a frame-switching Wigner function. This leads us to constructing a contextual ψ-epistemic ontological model of the n-qubit stabilizer formalism. We shall see that this model is outcome deterministic, which is one of the core assumptions in the definition of traditional non-contextuality. This model then will lead us to a result that proves that any ontological model of the n-qubit stabilizer formalism requires at least n-1 generators to be encoded in the ontology of the model. As n-1 generators represents almost full knowledge about the stabilizer state, we therefore posit that ψ-onicity, a more controversial notion of non-classicality, is actually the resource for quantum computation

Wigner negativity on the sphere

The rise of quantum information theory has largely vindicated the long-held belief that Wigner negativity is an indicator of genuine nonclassicality in quantum systems. This thesis explores its manifestation in spin-j systems using the spherical Wigner function. Common symmetric multi-qubit states are studied and compared. Spin coherent states are shown to never have vanishing Wigner negativity. Pure states that maximize negativity are determined and analyzed using the Majorana stellar representation. The relationship between negativity and state mixedness is discussed, and polytopes characterizing unitary orbits of lower-bounded Wigner functions are studied. Results throughout are contrasted with similar works on symmetric state entanglement and other forms of phase-space nonclassicality

Predicting Many Properties of a Quantum System from Very Few Measurements

Predicting the properties of complex, large-scale quantum systems is essential for developing quantum technologies. We present an efficient method for constructing an approximate classical description of a quantum state using very few measurements of the state. This description, called a ‘classical shadow’, can be used to predict many different properties; order log(M) measurements suffice to accurately predict M different functions of the state with high success probability. The number of measurements is independent of the system size and saturates information-theoretic lower bounds. Moreover, target properties to predict can be selected after the measurements are completed. We support our theoretical findings with extensive numerical experiments. We apply classical shadows to predict quantum fidelities, entanglement entropies, two-point correlation functions, expectation values of local observables and the energy variance of many-body local Hamiltonians. The numerical results highlight the advantages of classical shadows relative to previously known methods