Inefficiency of classically simulating linear optical quantum computing with Fock-state inputs

TL;DR

This paper provides an elementary quantum optics-based proof that simulating linear optical quantum computing with Fock-state inputs is computationally infeasible due to exponential Hilbert space growth and permanent calculation complexity.

Contribution

It offers a straightforward, physics-based argument for the classical intractability of simulating such quantum systems, complementing prior complexity-theoretic approaches.

Findings

Hilbert space dimension scales exponentially with resources

Schrödinger and Heisenberg pictures are not computationally equivalent

Bosonic interferometer simulation complexity linked to permanent calculation

Abstract

Aaronson and Arkhipov recently used computational complexity theory to argue that classical computers very likely cannot efficiently simulate linear, multimode, quantum-optical interferometers with arbitrary Fock-state inputs [Aaronson and Arkhipov, Theory Comput. 9, 143 (2013)]. Here we present an elementary argument that utilizes only techniques from quantum optics. We explicitly construct the Hilbert space for such an interferometer and show that its dimension scales exponentially with all the physical resources. We also show in a simple example just how the Schr\"odinger and Heisenberg pictures of quantum theory, while mathematically equivalent, are not in general computationally equivalent. Finally, we conclude our argument by comparing the symmetry requirements of multiparticle bosonic to fermionic interferometers and, using simple physical reasoning, connect the nonsimulatability of the bosonic device to the complexity of computing the permanent of a large matrix.