We pose a randomized boson-sampling problem. Strong evidence exists that such a problem becomes intractable on a classical computer as a function of the number of bosons. We describe a quantum optical processor that can solve this problem efficiently based on a Gaussian input state, a linear optical network, and nonadaptive photon counting measurements. All the elements required to build such a processor currently exist. The demonstration of such a device would provide empirical evidence that quantum computers can, indeed, outperform classical computers and could lead to applications.

The idea of signal amplification is ubiquitous in the control of physical systems, and the ultimate performance limit of amplifiers is set by quantum physics. Increasing the amplitude of an unknown quantum optical field, or more generally any harmonic oscillator state, must introduce noise 1 . This linear amplification noise prevents the perfect copying of the quantum state 2 , enforces quantum limits on communications and metrology 3 , and is the physical mechanism that prevents the increase of entanglement via local operations. It is known that non-deterministic versions of ideal cloning 4 and local entanglement increase (distillation) 5 are allowed, suggesting the possibility of non-deterministic noiseless linear amplification. Here we introduce, and experimentally demonstrate, such a noiseless linear amplifier for continuous-variables states of the optical field, and use it to demonstrate entanglement distillation of field-mode entanglement. This simple but powerful circuit can form the basis of practical devices for enhancing quantum technologies. The idea of noiseless amplification unifies approaches to cloning and distillation, and will find applications in quantum metrology and communications.A quantum-noise-free amplifier, if it could be constructed, could aid a wide variety of quantum-enhanced information protocols, primarily through its ability to distill and purify continuous-variable entanglement. This