Normalizer Circuits and Quantum Computation

In this thesis, we introduce new models of quantum computation to study the potential and limitations of quantum computer algorithms. Our models are based on algebraic extensions of the qubit Clifford gates (CNOT, Hadamard and π/4-phase gates) and Gottesman’s stabilizer formalism of quantum codes. We give two main kinds of technical contributions with applications in quantum algorithm design, classical simulations and for the description of generalized stabilizer states and codes. Our first main contribution is a formalism of restricted quantum operations, which we name the normalizer circuit formalism, wherein the allowed gates are quantum Fourier transforms (QFTs), automorphism gates and quadratic phase gates associated to a set G, which is either an abelian group or an abelian hypergroup. These gates extend the qubit Clifford gates, which only have non-universal quantum computational power and can be efficiently simulated classically, to comprise additional powerful gates such as QFTs, which are central in Shor’s celebrated factoring algorithm. Using our formalism, we show that normalizer circuit models with different choices of G encompass famous quantum algorithms, including Shor’s and those that solve abelian Hidden Subgroup Problems (HSP). Exploiting self-developed classical-simulation techniques, we further characterize under which scenarios normalizer circuits succeed or fail to provide a quantum speed-up. In particular, we derive several no-go results for finding new quantum algorithms with the standard abelian Fourier sampling techniques. We also devise new quantum algorithms (with exponential speedups) for finding hidden commutative hyperstructures. These results offer new insights into the source of the quantum speed-up of the quantum algorithms for abelian and normal HSPs. Our second main contribution is a framework for describing quantum many-body states, quantum codes and for the classical simulation of quantum circuits. Our framework comprises algebraic extensions of Gottesman’s Pauli Stabilizer Formalism (PSF) [1], in which quantum states/codes are written as joint eigenspaces of stabilizer groups of commuting Pauli operators. We use our framework to obtain various generalizations of the seminal Gottesman-Knill theorem [2, 3], which asserts the classical simulability of Clifford operations. Specifically, we use group and hypergroup theoretic methods to manipulate novel types of stabilizer groups and hypergroups, from infinite continuous ones, to others that contain non-monomial non-unitary stabilizers and mimic reactions of physical particles. While the PSF is only valid for qubit and (low dimensional) qudit systems, our formalism can be applied both to discrete and continuousvariable systems, hybrid settings, and anyonic systems. These results enlarge the known families of quantum states/codes that can be efficiently described with classical methods. This thesis also establishes the existence of a precise connection between the quantum algorithm of Shor and the stabilizer formalism, revealing a common mathematical structure in several quantum speed-ups and error-correcting codes. This connection permits a beautiful transfer of ideas between the fields of quantum algorithms and codes, which lies at the roots of our methods and results.