Approximation by Quantum Circuits

TL;DR

This paper investigates the limits of approximating arbitrary quantum operations with small circuits, establishing lower bounds and demonstrating the inherent difficulty of solving most problems using quantum circuits.

Contribution

It provides rigorous lower bounds on the approximation capabilities of small quantum circuits and formalizes arguments about their limitations in solving complex problems.

Findings

Almost all problems are hard to solve with quantum circuits.

Strong lower bounds on approximation capabilities of g-qubit operations.

Reduced upper bounds for exact unitary representation.

Abstract

In a recent preprint by Deutsch et al. [1995] the authors suggest the possibility of polynomial approximability of arbitrary unitary operations on $n$ qubits by 2-qubit unitary operations. We address that comment by proving strong lower bounds on the approximation capabilities of g-qubit unitary operations for fixed g. We consider approximation of unitary operations on subspaces as well as approximation of states and of density matrices by quantum circuits in several natural metrics. The ability of quantum circuits to probabilistically solve decision problem and guess checkable functions is discussed. We also address exact unitary representation by reducing the upper bound by a factor of n^2 and by formalizing the argument given by Barenco et al. [1995] for the lower bound. The overall conclusion is that almost all problems are hard to solve with quantum circuits.