Quantum Computation as Geometry

TL;DR

It is shown that finding optimal quantum circuits is essentially equivalent to finding the shortest path between two points in a certain curved geometry, recasting the problem of finding quantum circuits as a geometric problem.

Abstract

Quantum computers hold great promise, but it remains a challenge to find efficient quantum circuits that solve interesting computational problems. We show that finding optimal quantum circuits is essentially equivalent to finding the shortest path between two points in a certain curved geometry. By recasting the problem of finding quantum circuits as a geometric problem, we open up the possibility of using the mathematical techniques of Riemannian geometry to suggest new quantum algorithms, or to prove limitations on the power of quantum computers.

Figures (1)

• Figure 1: Schematic of the three steps used to construct a quantum circuit approximating the unitary operation $U$. The circuit is of size polynomial in the distance $d(I,U)$ between the identity and $U$. First we project the Hamiltonian $H(t)$ for the minimal geodesic path onto one- and two-qubit terms, giving $H_P(t)$. By choosing the penalty $p$ large enough ($p=4^n$) we ensure the error in this approximation is small, $\epsilon_1 \leq d(I,U)/2^n$. Next we break up the evolution according to $H_P(t)$ into $N$ small time steps of size $\Delta=d(I,U)/N$, and approximate with a constant mean Hamiltonian $\bar{H}^j_P$ over each step. Finally we approximate evolution according to the constant mean Hamiltonian over each step by a sequence of one- and two-qubit quantum gates. The total errors, $\epsilon_2$ and $\epsilon_3$, introduced by these approximations can be made smaller than any desired constant by choosing the step size $\Delta$ sufficiently small, $\Delta=O(1/(n^2 d(I,U)))$. In total we need $O(n^6 d(I,U)^3)$ quantum gates to approximate $U$ to within some constant error which can be made arbitrarily small.