A hard-core predicate for all one-way functions
STOC '89 Proceedings of the twenty-first annual ACM symposium on Theory of computing
Learning decision trees using the Fourier spectrum
SIAM Journal on Computing
On the Power of Quantum Computation
SIAM Journal on Computing
Spectral Analysis of Boolean Functions as a Graph Eigenvalue Problem
IEEE Transactions on Computers
Quantum Circuits That Can Be Simulated Classically in Polynomial Time
SIAM Journal on Computing
Simulating Quantum Computation by Contracting Tensor Networks
SIAM Journal on Computing
Quantum Computation and the Evaluation of Tensor Networks
SIAM Journal on Computing
Matrix product state representations
Quantum Information & Computation
Commuting quantum circuits: efficient classical simulations versus hardness results
Quantum Information & Computation
Classical simulations of Abelian-group normalizer circuits with intermediate measurements
Quantum Information & Computation
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We investigate the boundary between classical and quantum computational power. This work consists of two parts. First we develop new classical simulation algorithms that are centered on sampling methods. Using these techniques we generate new classes of classically simulatable quantum circuits where standard techniques relying on the exact computation of measurement probabilities fail to provide efficient simulations. For example, we show how various concatenations of matchgate, Toffoli, Clifford, bounded-depth, Fourier transform and other circuits are classically simulatable. We also prove that sparse quantum circuits as well as circuits composed of CNOT and exp[itheta;X] gates can be simulated classically. In a second part, we apply our results to the simulation of quantum algorithms. It is shown that a recent quantum algorithm, concerned with the estimation of Potts model partition functions, can be simulated efficiently classically. Finally, we show that the exponential speed-ups of Simon's and Shor's algorithms crucially depend on the very last stage in these algorithms, dealing with the classical postprocessing of the measurement outcomes. Specifically, we prove that both algorithms would be classically simulatable if the function classically computed in this step had a sufficiently peaked Fourier spectrum.