Intermediate models of quantum computation are those who do not provide universal quantum computation, however, are capable of efficiently solving particular problems which exhibit an exponential-time speedup compared to classical computers. BosonSampling is one such a model. As defined by Aaronson and Arkhipov in Ref. [22], this model has two types of im-plementations. First is the case of exact sampling, in which every element of the model is assumed to be ideal. Indeed, this is not a physically fair assumption, but is useful for defining the underlying computational model. Moreover, from the fundamental perspective of quantum correlations concerned in the present dissertation, we are not really worried about physical
feasibility of our protocol; we are only interested in the implications of quantum computational supremacy regarding the correlations inherent within quantum states.
Consider a lossless passive linear-optical network (PLON), where n single photons are in-jected into m input modes of the PLON. The initial state is given by
|ni = action of the unitary transformation corresponding to the PLON is linear8, it can be equivalently described by a linear transformation of the form (4.4) between the mode operators given by a matrix U ,
As a result, it preserves the positivity of the global P-function, that is, SGQOC remains invariant under this transformation.
Assuming that m n (typically, m∼O(n2)), a particular sample from output distribution can be represented by the vector s = (s1, . . . , sm) with si∈{0, 1} so that Pm
i=1si = n, and occurs with the probability [23]
p(s| ˆU , n) = |hs| ˆU |ni|2 = |PerA(s; n)|2. (4.8) Here, PerA(s; n) is the permanent of a particular submatrix A(s; n) of the unitary U deter-mined by the input and sample vectors n and s. An exact BosonSampling device produces samples from the probability distribution px, which is exactly the one describing the output arrangements of the photons generated by the unitary describing the PLON (as the input to the problem) from a fixed arrangement of input single photons. The subscript x represents the events sampled and, in the case of BosonSampling, it is the space of all photon counting events of n photons in m modes.
According to Aaronson and Arkipov [22], generating samples of the output distribution of the BosonSampling device described above is a computationally #P-hard problem, meaning that it cannot be efficiently done on a classical computer. The exact sampling hardness argu-ment relates the complexity class of estimating matrix permanents to the complexity of the BosonSampling problem which encodes these same matrix permanents into the probability distribution of the detection events as appeared in Eq. (4.8). If the BosonSampling prob-lem could be simulated efficiently on a classical computer, then one must accept a so called
“polynomial hierarchy collapse” of the classical complexity classes [24] which is believed to be implausible.
Furthermore, Aaronson and Arkipov showed, up to some very feasible conjectures, that even
8Note that, the generator of a PLON transformation is at most quandratic in the creation and annihilation operators of all modes, and at most linear with respect to the creation or annihilation operators of each mode.
sampling from a distribution close to the exact output distribution of a BosonSampler cannot be efficiently simulated on a classical computer [22]. Specifically, an approximate BosonSam-pling device samples from a probability distribution qx which is a deviation from the exact distribution px in the sense that
X
x
|px− qx| 6 ε, (4.9)
where ε is an additional input parameter. This restriction constrains the total variation distance between the ideal and actual distributions. In other words, ε determines how much deviation is allowed from the expected probability distribution. The origin of the deviation is arbitrary; it could be due to unavoidable physical imperfections of the device or a deliberate intention of an eavesdropper to deceive the user of the samples. Anyhow, a device which claims implementing the approximate BosonSampling must satisfy Eq. (4.9).
Note that, in any finitely constrained computational device there are limits on the size of the inputs. Hence, a device implementing approximate BosonSampling will necessarily have some restrictions on the allowed values of ε. An analogy here would be that of “machine epsilon”
for floating point operations. For every floating point operation, there exists an underlying precision for that operation determined by an implicit input parameter. In the case of floating point numbers, this precision depends on the number of bits used to represent a real number.
This error can be reduced by allowing more resources (bits) to be used, up to some bound which is determined by the details of how the algorithm is implemented. Note that the algorithm used does not need to change when the error parameter is changed, as it merely accepts the level of precision requested. The implementation determines the bound of machine epsilon. A similar situation will occur in implementing approximate BosonSampling algorithms and the input parameter ε.
The result of Aaronson and Arkhipov implies, given some plausible conjectures hold, that the approximate BosonSampling algorithm cannot be implemented efficiently using classical computational resources alone, just like the exact model. In other words, although the physical device might be sampling qx rather than px, yet classically producing the samples from qx is not efficient. Consequently, both exact and approximate BosonSampling protocols have a computational power beyond any classical computer, demonstrating the power of (intermediate) quantum computers.
Importantly, to preserve the hardness of the exact BosonSampling in the approximate case, some changes are needed over that of the exact model. The size of the space of detec-tion events is exponentially large and thus, for randomly chosen PLONs, the probability of any particular event is going to be exponentially small. As a result, in this case, it is not possible to follow the procedure of proving the hardness of exact BosonSampling via matrix permanent estimation through an event’s probability without any changes. This is because, in the worst case, a nefarious implementation would adapt the output distribution depending on which event’s probability has encoded the matrix permanent to be estimated, and would apply the entire error budget ε on that one event. This would increase the error of the matrix permanent estimation drastically simplifying the complexity of the estimation. Aaronson and
Figure 4.1: The schematic of a nonlocal BosonSampling protocol. Charlie uses m SPDC sources and a series of dephasing channels (DC) to produce fully dephased two-mode squeezed vacuum states, and shares the final state between two spatially separated agents. Alice performs BosonSampling using a passive linear-optical network (PLON) and {0, 1} Fock basis measurements, while Bob only performs {0, 1} Fock basis measurements. We showed that, Alice and Bob can simulate their local sample statistics classically efficiently. However, they cannot efficiently simulate the correlations between their outcomes using classical computers and an infinite amount of classical communication, although there is no entanglement or discord between agents at any time.
Arkhipov showed that if the estimation was performed on input matrices whose elements were close to Gaussianly distributed, then it is possible to hide the matrix whose permanent is to be estimated in a distribution that is close to that of Haar random unitary matrices (with di-mensions at least equal to the square of the input matrix) but with the required matrix hidden in a submatrix of each sample [22]. Under these conditions, the errors in applying the exact sampling algorithm will manifest themselves as the average case error for approximate sampling rather than the worst case. Hence, the hardness argument can be restated for the approximate sampling scenario.