Combinatorics and Optimization

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This is the collection for the University of Waterloo's Department of Combinatorics and Optimization.

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Browsing Combinatorics and Optimization by Author "Gosset, David"

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    Clifford Simulation: Techniques and Applications
    (University of Waterloo, 2021-05-28) Kerzner, Alexander; Gosset, David
    Despite the widespread belief that quantum computers cannot be efficiently simulated classically, efficient simulation is known to be possible in certain restricted regimes. In particular, the Gottesman-Knill theorem states that Clifford circuits can be efficiently simulated. We begin this thesis by reviewing and comparing several known techniques for efficient simulation of Clifford circuits: the stabilizer formalism, CH form, affine form, and the graph state formalism. We describe each simulation method and give four different proofs of the Gottesman-Knill theorem. Next we review a recent work [15], which shows that restricting the geometry of Clifford circuits can lead to a further speedup. We give an algorithm for simulating Pauli basis measurements on a planar graph state in time $\widetilde{O}(n^{\omega/2})$, where $\omega < 2.373$ is the matrix multiplication exponent. This algorithm achieves a quadratic speedup over using Clifford simulation methods directly. As an application of this algorithm, we consider a depth-$d$ Clifford circuit whose two-qubit gates act along edges of a planar graph and describe how to sample from its output distribution or compute an output probability in time $\widetilde{O}(n^{\omega/2}d^\omega)$. For $d= O(\log n)$, both of these results are quadratic speedups over using Clifford simulation methods directly. Finally, we extend these simulation algorithms to universal circuits by using stabilizer rank methods. We follow a previously known gadgetization procedure [9] to show that given a depth-$d$ Clifford+$T$ circuit with $t$ $T$ gates and whose two-qubit gates act along edges of a planar graph, we can sample from its output distribution in time $\widetilde{O}(2^{0.7926t}n^{5/2}t^{6} d^3)$ and can compute output probabilities in time $\widetilde{O}(2^{0.3963t}n^{3/2}t^{6} d^3)$. Previous work [9,6], applied to the case $d=O(\log n)$, gives algorithms for sampling in time $O(2^{0.3963t} n^6 t^6)$ and computation of output probabilities in time $O(2^{0.3963t}n^3t^3)$. Our sampling algorithm offers improved scaling in $n$ but poorer scaling in the exponential term, while our algorithm for computing output probabilities offers improved scaling in $n$ with identical scaling in the exponential term.
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    Moments of Random Quantum Circuits and Applications in Random Circuit Sampling
    (University of Waterloo, 2021-12-23) Liu, Yinchen; Gosset, David
    Random quantum circuits and random circuit sampling (RCS) have recently garnered tremendous attention from all sub-fields of the quantum information community, especially after Google’s quantum supremacy announcement in 2019. While the science of RCS draws ideas from diverse disciplines ranging from pure mathematics to electrical engineering, this thesis explores the subject from a theoretical computer science perspective. We begin by offering a rigorous treatment of the t-design and the anti-concentration properties of random quantum circuits in a way that various intermediate lemmas will find further applications in subsequent discussions. In particular, we prove a new upper bound for expressions of the form E_V[<0^n|V\sigma_p|0^n>^2] for 1D random quantum circuits V and n-qubit Pauli operators \sigma_p. Next, we discuss at a high level the RCS supremacy conjecture, which forms the main complexity-theoretic basis supporting the belief that deep random quantum circuits may be just as hard to classically simulate as arbitrary quantum circuits. Finally, we study the performance of quantum and classical spoofing algorithms on the linear cross-entropy benchmark (XEB), a statistical test proposed by Google for the purpose of verifying RCS experiments. We consider an extension of a classical algorithm recently proposed by Barak, Chou, and Gao and try to show that the extended algorithm can achieve higher XEB scores [BCG20]. While we are unable to prove a key conjecture for random quantum circuits with Haar random 2-qubit gates, we do establish the result in other related settings including for Haar random unitaries, random Clifford circuits, and random fermionic Gaussian unitaries.
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    On the Power and Limitations of Shallow Quantum Circuits
    (University of Waterloo, 2022-09-01) Parham, Natalie; Gosset, David; Laflamme, Raymond
    Constant-depth quantum circuits, or shallow quantum circuits, have been shown to exhibit behavior that is uniquely quantum. This thesis explores the power and limitations of constant-depth quantum circuits, in particular as they compare to constant-depth classical circuits. We start with a gentle introduction to shallow quantum and classical circuit complexity, and we review the hardness of sampling from the output distribution of a constant-depth quantum circuit. We then give an overview of the shallow circuit advantage from the 1D Magic Square Problem from [Bravyi, Gosset, Koenig, Tomamichel 2020]. The first novel contribution is an investigation into the limitations of shallow quantum circuits for local optimization problems. We prove that if a shallow quantum circuit's input/output relation is exactly that of a local optimization problem, then we can construct a shallow classical circuit that also solves the optimization problem. We also prove an approximate version of this statement. Finally, we introduce a novel sampling task over an n-bit distribution D_n such that there exists a shallow quantum circuit that takes as input the state \ket{\GHZ_n} = \frac{1}{\sqrt{2}}(\ket{0^n} + \ket{1^n}) and produces a distribution close to D_n whereas, any constant-depth classical circuit with bounded fan-in and n + n^\delta random input bits for some \delta<1, will produce a distribution that is not close to D_n.