David Gosset

Prior work of Beverland et al. has shown that any exact Clifford+T implementation of the n-qubit Toffoli gate must use at least n T gates. Here we show how to get away with exponentially fewer T gates, at the cost of incurring a tiny 1/poly(n) error that can be neglected in most practical situations. More precisely, the n-qubit Toffoli gate can be implemented to within error ϵ in the diamond distance by a randomly chosen Clifford+T circuit with at most O(log(1/ϵ)) T gates. We also give a matching Ω(log(1/ϵ)) lower bound that establishes optimality, and we show that any purely unitary implementation achieving even constant error must use Ω(n) T gates. We also extend our sampling technique to implement other Boolean functions. Finally, we describe upper and lower bounds on the T-count of Boolean functions in terms of non-adaptive parity decision tree complexity and its randomized analogue.

Abstract A general quantum circuit can be simulated in exponential time on a classical computer. If it has a planar layout, then a tensor-network contraction algorithm due to Markov and Shi has a runtime exponential in the square root of its size, or more generally exponential in the treewidth of the underlying graph. Separately, Gottesman and Knill showed that if all gates are restricted to be Clifford, then there is a polynomial time simulation. We combine these two ideas and show that treewidth and planarity can be exploited to improve Clifford circuit simulation. Our main result is a classical algorithm with runtime scaling asymptotically as ~$ n^{\omega/2}<n^{1.19}$ which samples from the output distribution obtained by measuring all $n$ qubits of a planar graph state in given Pauli bases. Here $\omega$ is the matrix multiplication exponent. We also provide a classical algorithm with the same asymptotic runtime which samples from the output distribution of any constant-depth Clifford circuit in a planar geometry. Our work improves known classical algorithms with cubic runtime. A key ingredient is a mapping which, given a tree decomposition of some graph $G$, produces a Clifford circuit with a structure that mirrors the tree decomposition and which emulates measurement of the quantum graph state corresponding to $G$. We provide a classical simulation of this circuit with the runtime stated above for planar graphs and otherwise $n t^{\omega-1}$ where $t$ is the width of the tree decomposition. Our algorithm incorporates two subroutines which may be of independent interest. The first is a matrix-multiplication-time version of the Gottesman-Knill simulation of multi-qubit measurement on stabilizer states. The second is a new classical algorithm for solving symmetric linear systems over $\mathbb{F}_2$ in a planar geometry.

We consider a computational problem where the goal is to approximate the maximum eigenvalue of a two-local Hamiltonian that describes antiferromagnetic Heisenberg interactions between qubits located at the vertices of the graph. Previous work has shed light on this problem’s approximability by \textit{product states}. For any instance of this problem the maximum energy attained by a product state is lower bounded by the Max Cut of the graph and upper bounded by the standard Goemans-Williamson semidefinite programming relaxation of it. Gharibian and Parekh described an efficient classical approximation algorithm for this problem which outputs a product state with energy at least $0.498$ times the maximum eigenvalue in the worst case, and observe that there exist instances where the best product state has energy $1/2$ of optimal. We investigate approximation algorithms with performance exceeding this limitation which are based on optimizing over tensor products of few-qubit states and shallow quantum circuits. We provide an efficient classical algorithm which achieves an approximation ratio of at least $0.53$ in the worst case. We also show that for any instance defined by a $3$ or $4$-regular triangle-free graph, there is an efficiently computable shallow quantum circuit that prepares a state with energy larger than the best product state (larger even than its semidefinite programming relaxation).

We generalize an efficient exact synthesis algorithm for single-qubit unitaries over the Clifford+T gate set which was presented by Kliuchnikov, Maslov and Mosca. Their algorithm takes as input an exactly synthesizable single-qubit unitary--one which can be expressed without error as a product of Clifford and T gates--and outputs a sequence of gates which implements it. The algorithm is optimal in the sense that the length of the sequence, measured by the number of T gates, is smallest possible. In this paper, for each positive even integer n we consider the ``Clifford-cyclotomic'' gate set consisting of the Clifford group plus a z-rotation by pi/n. We present an efficient exact synthesis algorithm which outputs a decomposition using the minimum number of pi/n z-rotations. For the Clifford+T case n=4 the group of exactly synthesizable unitaries was shown to be equal to the group of unitaries with entries over the ring Z[e^{i*pi/n},1/2]. We prove that this characterization holds for a handful of other small values of n but the fraction of positive even integers for which it fails to hold is 100\%.