Investigating Quantum Computing via Algebraic and Logical Tools

Chaowen Guan

Abstract: Quantum supremacy" or "quantum advantage" means demonstrating a quantum computer's ability to compute a task that no classical device can emulate in a comparable amount of time. This raises the question: How can one determine such an advantage? Much work on studying quantum supremacy has been done. In pursuing this question, this thesis takes a different angle from the great recent efforts at achieving quantum supremacy. In particular, it develops logical and algebraic tools for investigating how well classical computers can emulate/simulate quantum computers. The first part studies a logical approach to classically emulate general quantum circuits. Specifically, we give a new explicit conversion from a quantum circuit C into a small set of Boolean formulas such that the acceptance amplitude of the circuit (on a given input x) can be computed from the numbers of satisfying assignments to the formulas. The fact of this has been known for two decades, but our compact constructions promote the use of heuristic #SAT solvers to perform emulation of quantum circuits. We implement a prototype of our simulator in which #SAT solver can be utilized to compute the acceptance probabilities. The second part's main discovery is the tight connection between the strong simulation of quantum stabilizer circuits and two bedrock mathematical tasks: computing matrix rank and counting solutions to quadratic polynomials (both over the field F2). Precisely, it uses quadratic forms to obtain a strong simulation (i.e. computing the probability for any input and output) of stabilizer circuits.Our results improve the asymptotic running time from O(n3) to O(n omega), where omega = 2.372...is the known exponent of matrix multiplication, as well as show a near-tight relationship to the task of computing matrix rank that was not known before. They also improve the O(n3)-time algorithm for solution counting of quadratic forms over F2 to O(n omega). Besides, we also find further connections to graph theory and matroid theory. The third part builds on the second part to launch a direct attack on computing matrix rank over F2. Although rank reduces to matrix multiplication, they are not known to be equivalent. Getting any time better than O(n omega) would be a major breakthrough. At a high-level, it combines quadratic forms and Fourier analysis to improve the time in some very special cases. In the conclusion we close with a brief discussion of future research directions and then speculate about further applications of algebraic geometry in search of measures of the effort required to operate a quantum circuit that might explain the sustained difficult obstacles to maintaining quantum coherence that have been encountered.