Computational limitations of small-depth circuits
Computational limitations of small-depth circuits
PP is as hard as the polynomial-time hierarchy
SIAM Journal on Computing
Fault-tolerant quantum computation with constant error
STOC '97 Proceedings of the twenty-ninth annual ACM symposium on Theory of computing
SIAM Journal on Computing
Polynomial-Time Algorithms for Prime Factorization and Discrete Logarithms on a Quantum Computer
SIAM Journal on Computing
Complexity limitations on Quantum computation
Journal of Computer and System Sciences
Quantum computation and quantum information
Quantum computation and quantum information
Quantum Circuits That Can Be Simulated Classically in Polynomial Time
SIAM Journal on Computing
Fast parallel circuits for the quantum Fourier transform
FOCS '00 Proceedings of the 41st Annual Symposium on Foundations of Computer Science
A polynomial-time approximation algorithm for the permanent of a matrix with nonnegative entries
Journal of the ACM (JACM)
Classical complexity and quantum entanglement
Journal of Computer and System Sciences - Special issue: STOC 2003
On the power of quantum computation
SFCS '94 Proceedings of the 35th Annual Symposium on Foundations of Computer Science
BQP and the polynomial hierarchy
Proceedings of the forty-second ACM symposium on Theory of computing
Permutational quantum computing
Quantum Information & Computation
Adptive quantum computation, constant depth quantum circuits and arthur-merlin games
Quantum Information & Computation
The equivalence of sampling and searching
CSR'11 Proceedings of the 6th international conference on Computer science: theory and applications
On the complexity of mixed discriminants and related problems
MFCS'05 Proceedings of the 30th international conference on Mathematical Foundations of Computer Science
The equivalence of sampling and searching
CSR'11 Proceedings of the 6th international conference on Computer science: theory and applications
Concentration and moment inequalities for polynomials of independent random variables
Proceedings of the twenty-third annual ACM-SIAM symposium on Discrete Algorithms
International Journal of Adaptive, Resilient and Autonomic Systems
Commuting quantum circuits: efficient classical simulations versus hardness results
Quantum Information & Computation
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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 P#P=BPPNP, 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 #P-hard to approximate the permanent of a matrix A of independent N(0,1) Gaussian entries, with high probability over A; and the Permanent Anti-Concentration Conjecture, which says that |Per(A)|=√(n!)poly(n) 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.