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?