Boson-sampling is a simplified model for quantum computing that may hold the key to implementing the first ever post-classical quantum computer. Boson-sampling is a non-universal quantum computer that is significantly more straightforward to build th
an any universal quantum computer proposed so far. We begin this chapter by motivating boson-sampling and discussing the history of linear optics quantum computing. We then summarize the boson-sampling formalism, discuss what a sampling problem is, explain why boson-sampling is easier than linear optics quantum computing, and discuss the Extended Church-Turing thesis. Next, sampling with other classes of quantum optical states is analyzed. Finally, we discuss the feasibility of building a boson-sampling device using existing technology.
Boson sampling is a specific quantum computation, which is likely hard to implement efficiently on a classical computer. The task is to sample the output photon number distribution of a linear optical interferometric network, which is fed with single
-photon Fock state inputs. A question that has been asked is if the sampling problems associated with any other input quantum states of light (other than the Fock states) to a linear optical network and suitable output detection strategies are also of similar computational complexity as boson sampling. We consider the states that differ from the Fock states by a displacement operation, namely the displaced Fock states and the photon-added coherent states. It is easy to show that the sampling problem associated with displaced single-photon Fock states and a displaced photon number detection scheme is in the same complexity class as boson sampling for all values of displacement. On the other hand, we show that the sampling problem associated with single-photon-added coherent states and the same displaced photon number detection scheme demonstrates a computational complexity transition. It transitions from being just as hard as boson sampling when the input coherent amplitudes are sufficiently small, to a classically simulatable problem in the limit of large coherent amplitudes.
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 Schrodinger 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.
