Clifford Simulation: Techniques and Applications

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.