Clifford Simulation: Techniques and Applications
| dc.contributor.author | Kerzner, Alexander | |
| dc.date.accessioned | 2021-05-28T13:09:17Z | |
| dc.date.available | 2021-05-28T13:09:17Z | |
| dc.date.issued | 2021-05-28 | |
| dc.date.submitted | 2021-05-24 | |
| dc.description.abstract | 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. | en |
| dc.identifier.uri | http://hdl.handle.net/10012/17038 | |
| dc.language.iso | en | en |
| dc.pending | false | |
| dc.publisher | University of Waterloo | en |
| dc.subject | quantum computing | en |
| dc.subject | clifford circuits | en |
| dc.subject | classical simulation | en |
| dc.subject | treewidth | en |
| dc.subject | stabilizer formalism | en |
| dc.subject | graph states | en |
| dc.title | Clifford Simulation: Techniques and Applications | en |
| dc.type | Master Thesis | en |
| uws-etd.degree | Master of Mathematics | en |
| uws-etd.degree.department | Combinatorics and Optimization | en |
| uws-etd.degree.discipline | Combinatorics and Optimization (Quantum Information) | en |
| uws-etd.degree.grantor | University of Waterloo | en |
| uws-etd.embargo.terms | 0 | en |
| uws.contributor.advisor | Gosset, David | |
| uws.contributor.affiliation1 | Faculty of Mathematics | en |
| uws.peerReviewStatus | Unreviewed | en |
| uws.published.city | Waterloo | en |
| uws.published.country | Canada | en |
| uws.published.province | Ontario | en |
| uws.scholarLevel | Graduate | en |
| uws.typeOfResource | Text | en |