Abstract
We give new evidence that quantum computers—moreover, rudimentary quantum computers built entirely out of linear-optical elements—cannot be efficiently simulated by classical computers. In particular, we define a model of computation in which identical photons are generated, sent through a linear-optical network, then nonadaptively measured to count the number of photons in each mode. This model is not known or believed to be universal for quantum computation, and indeed, we discuss the prospects for realizing the model using current technology. On the other hand, we prove that the model is able to solve sampling problems and search problems that are classically intractable under plausible assumptions. Our first result says that, if there exists a polynomial-time classical algorithm that samples from the same probability distribution as a linear-optical network, then ${\mathsf P}^{\mathsf{\#P}}=\mathsf{BPP}^{\mathsf{NP}}$, and hence the polynomial hierarchy collapses to the third level. Unfortunately, this result assumes an extremely accurate simulation. Our main result suggests that even an approximate or noisy classical simulation would already imply a collapse of the polynomial hierarchy. For this, we need two unproven conjectures: the Permanent-of-Gaussians Conjecture, which says that it is $\mathsf{\#P}$-hard to approximate the permanent of a matrix $A$ of independent $\mathcal{N}\left( 0,1\right)$ Gaussian entries, with high probability over $A$; and the Permanent Anti-Concentration Conjecture, which says that $\left\vert \operatorname*{Per}\left( A\right) \right\vert \geq\sqrt{n!}/\operatorname*{poly}\left( n\right)$ with high probability over $A$. We present evidence for these conjectures, both of which seem interesting even apart from our application. This paper does not assume knowledge of quantum optics. Indeed, part of its goal is to develop the beautiful theory of noninteracting bosons underlying our model, and its connection to the permanent function, in a self-contained way accessible to theoretical computer scientists. An extended abstract of this article appeared in the Proceedings of STOC 2011.
Highlights
The Extended Church-Turing Thesis says that all computational problems that are efficiently solvable by realistic physical devices, are efficiently solvable by a probabilistic Turing machine
This provides evidence that quantum computers have capabilities outside the entire polynomial hierarchy, complementing the recent evidence of Aaronson [3] and Fefferman and Umans [24]. Another point worth mentioning is that, even if the exact BOSONSAMPLING problem were solvable by a polynomial-time nonuniform sampling algorithm—that is, by an algorithm that could be different for each boson computer A—we would still get the conclusion P#P ⊆ BPPNP/poly, whence the polynomial hierarchy would collapse
We extend the random self-reducibility of permanents over finite fields proved by Lipton [47], to show that exactly computing the permanent of most Gaussian matrices X ∼ N (0, 1)nC×n is #P-hard
Summary
The Extended Church-Turing Thesis says that all computational problems that are efficiently solvable by realistic physical devices, are efficiently solvable by a probabilistic Turing machine. To run Shor’s algorithm, one needs to be able to perform arithmetic (including modular exponentiation) on a coherent superposition of integers encoded in binary This does not seem much easier than building a universal quantum computer.. Watched the dolphin in its natural habitat, we might see it display equal intelligence with no special training at all Following this analogy, we can ask: are there more “natural” quantum systems that already provide evidence against the Extended Church-Turing Thesis? Perhaps the real question is this: do there exist quantum systems that are “intermediate” between Shor’s algorithm and a Bose-Einstein condensate—in the sense that (1) they are significantly closer to experimental reality than universal quantum computers, but (2) they can be proved, under plausible complexity assumptions (the more “generic” the better), to be intractable to simulate classically?. We will argue that the answer is yes
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