Boson-Sampling-Inspired Quantum Metrology

Our group at Louisiana State University has teamed up with researchers at Macquarie University in Sydney and Boise State University in Boise to produce an new publication in Physical Review Letters, entitled, “Linear Optical Quantum Metrology with Single Photons: Exploiting Spontaneously Generated Entanglement to Beat the Shot-Noise Limit.” For regular readers of this blog, you will know that Boson Sampling is a new paradigm in quantum computing whereby single photons, inputted into a linear optical interferometer, can carry out a mathematical sampling problem that would be intractable on classical computer. The buzz surrounding Boson Sampling is that, unlike universal linear optical quantum computing, the experimental implementation requires no special quantum gates, like controlled-NOT gates, nor feed forward nor teleportation or any other fancy stuff. Identical single photons rattle around in the interferometer and they are sampled in the number basis when they come out. Sounds simple, but a classical machine cannot efficiently simulate the sampling output, whereas the linear optical device does this quite easily. For our recent review on Boson Sampling the reader is encouraged to go here.

In spite of all the excitement about Boson Sampling as a new paradigm for quantum information processing, the Boson Sampling problem has no know practical application to any mathematics problem anybody is interested in. In some ways the situation is similar to the late 1980s and early 1990s, before Shor’s invention of his factoring algorithm, when the first quantum algorithm shown to give an exponential speedup was the Deutsch-Jozsa (DJ) algorithm that allowed one to tell if a function was balanced or unbalanced. While a very nice result, nobody really gave a rat’s ass whether a function was balanced or unbalanced. It was however hoped that the DJ algorithm was just the tip of an iceberg and indeed the rest of the iceberg was revealed when Shor’s factoring algorithm was discovered. That was an (apparent) exponential speedup on a problem that people cared deeply about.

So too do we hope that Boson Sampling is just the tip of the iceberg when it comes to the power of linear optical interferometers, with simple single-photon inputs, to carry out tasks that are not only impossible classically but also of practical interest. In that direction our paper makes a frontal attack on the berg with a metrological ice axe. The idea emerged from the understanding that in Boson Sampling, an exponentially large amount of number-path entanglement is generated through the natural evolution of the single photons in the interferometer via repeated implementation of the Hong-Ou-Mandel effect at each beam splitter. It has been known for nearly 30 years the number-path entanglement is a resource for quantum metrology, beating the shot-noise limit, and so it was natural for us to ask if this hidden power in linear optics with single photon inputs might be put to work for a metrological advantage. Our paper shows that this is indeed the case.

To briefly summarize our scheme, we send a sequence of single photons into linear optical interferometer that contains an interferometric implementation of the Quantum Fourier Transform coupled with a bank of phase shifters with an unknown phase that is to be measured. Our signal consists of a sampling of the outputs tuned to the same sequence of single photons emerging from the exit ports. The signal-to-noise analysis was quite challenging as it involves the computation of the permanent of a large square matrix with complex entries. While in general this is classically intractable, to our surprise, something about the structure of the Quantum Fourier Transform seems to allow the permanent to be computed analytically in closed form. As least we conjecture this is so. We were able to eyeball a closed form formula for the permanent of a matrix of any rank and confirm it out to rank 20 or so numerically, but a rigorous mathematical proof of the permanent formula is still wanting.

Once we had the signal and variance analysis carried out, we were able to show (carefully counting resources) that the sensitivity of the device, which we christened the Quantum Fourier Transform Interferometer, is well below the classical shot-noise limit. It has been known for years that exotic number-path entangled states, such as N00N states, can beat the shotnoise limit, but N00N states are resource intensive to create in the first place, requiring either very strong Kerr nonlinearities or non-deterministic heralding. Here in our new paper we get super sensitivity for free from the natural evolution of single photons in a passive optical linear interferometer. This then seems to be the first example of the Boson Sampling paradigm providing a quantum advantage in an arena of importance, which is quantum metrology.

 Who knows what is left on this iceberg still yet unexplored?