The Riemann Conjecture

Mein Lieber Herr Riemann,
All night I will dream on,
'Bout how you deserve a lecture.
But of course I allude
To your famous and shrewd
Outstanding and unsolved conjecture.

Oh, I owe you my life,
My 3 kids and my wife,
For the proof of the Prime Number Theorem.
Your zeta function trick
Made the proof really slick,
And those primes — no more do I fear 'em.

But I just stop to think,
How I've taken (hic) to drink,
And evolved this hysterical laugh —
Because still I don't know
If ζ's roots all go
On the line Re z = 1/2!

So I don't sleep at night,
And I'm losing my sight
In search of this darn thing's solution.
As my mind starts to go
My calculations grow
In a flood of "complex" confusion.

I bought a computer;
Not any astuter,
It ran for nearly 10 years — no jive!
But still it doesn't know
If zeta's roots all go
On that line Re z = .5

Now I sit in my room —
I feel doomed in the gloom —
And entombed by mountains of paper.
Still, I pray that some night
My "ol' lightbulb" will light
With the clue that could wrap-up this caper!


— "The Riemann Conjecture," Jonathan P. Dowling, Mathematics Magazine 62 (June 1989) 197; Reprinted in Gamma: Exploring Euler's Constant, by Julian Havil (Princeton University Press, 2003); see also the German-Language Edition, "Die Riemannsche Vermutung," in GAMMA - Euler's Konstante, PrimzahlstrŠnde und die Riemannsche Vermutung (Springer Heidelberg 2007) .

Postdoc, Ergo Proper Doc!

Postdoc in Quantum Information and/or applications to complexity  theory, foundations, quantum gravity, quantum many-body physics,  and thermodynamics

Limit of tenure: up to 3 years
Closing date: Sept 7, or until the positions are filled

The quantum information group at University College London (UCL) invite applications for two post-doctoral research positions in the groups of Fernando Brandao and Jonathan Oppenheim. We are especially interested in applications of quantum information theory to research areas such as complexity theory, foundations, quantum gravity, quantum many-body physics, and thermodynamics. Other faculty in Quantum Information Theory include Dan Browne, Simone Severini, Alessio Serafini and Sougato Bose.

The start date of the appointment is flexible, and both junior and senior research associate positions are available. The salary is £32,375-£39,132 p.a.

For any further queries on the scientific details and requirements of the post, please contact Jonathan Oppenheim (j.oppenheim[at]ucl.ac.uk) and/or Fernando Brandao (fgslbrandao[at]gmail.com).

Applications should include a full CV, publications list, a summary of research interests and previous contributions (two pages), and the names of three referees. Applications should be emailed to qi-jobs@ucl.ac.uk, and the candidate should arrange for the three reference letters to be sent to qi-references@ucl.ac.uk by the closing date with the subject line being the applicants name.

UCL values diversity and is committed to equality of opportunity.

Estimation of Phase and Diffusion: Combining Quantum Statistics and Classical Noise

Sergey I. Knysh, Gabriel A. Durkin

http://arxiv.org/abs/1307.0470

Coherent ensembles of $N$ qubits present an advantage in quantum phase estimation over separable mixtures, but coherence decay due to classical phase diffusion reduces overall precision. In some contexts, the strength of diffusion may be the parameter of interest. We examine estimation of both phase and diffusion in large spin systems using a novel mathematical formulation. For the first time, we show a closed form expression for the quantum Fisher information for estimation of a unitary parameter in a noisy environment. The optimal probe state has a non-Gaussian profile and differs also from the canonical phase state; it saturates a new tight precision bound. For noise below a critical threshold, entanglement always leads to enhanced precision, but the shot-noise limit is beaten only by a constant factor, independent of $N$. We provide upper and lower bounds to this factor, valid in low and high noise regimes. Unlike other noise types, it is shown for $N \gg 1$ that phase and diffusion can be measured simultaneously and optimally.

✭✭✭✭✭ Historical, Scientific, Hysterical View of Quantum Information

Second Amazon five-star review of my book, Schrödinger's Killer App, and this one is not written by my sister or any other close relative!

"Quantum computing is one of the most popular and well-funded ideas in modern physics research. This book explains, in a lively way, the ideas behind this field, its history, and colorful characters who are playing a role in making QC a reality. It starts with an overview of quantum mechanics so you can appreciate why people are going through so much trouble to make a quantum computer. Before you know it you're learning (and understanding) advanced quantum information, and how science is done in the government and academic worlds.

If you are a high school student thinking of becoming a scientist, a teacher who wants to liven up the standard curriculum, a science lover, a fan of sci-fi, considering investing in a quantum computer "company," or just love good writing, you'll want to read this book. However, if you like dry, long-winded, opaque academic writing, you're going to have to find another author.

Physicists in the field should attend a lecture by Prof. Dowling before reading this book for full effect. Knowing his personality amplifies the hilarity of his stories."

— Ellie Arroway, 29 June 2013

On The Computational Power of Linear Optics II

"Classical Computers Very Likely Can Not Efficiently Simulate Multimode Linear Optical Interferometers with Arbitrary Fock-State Inputs-An Elementary Argument," Version 3.0.

With thanks to Scott Aaronson and a referee for suggestions for improvement, we have expanded our argument, originally based on the exponential blowup in the Hilbert space for a bosonic, Fock-state interferometer, to analyze the similar exponential blowup for a fermionic, Fock state interferometer. We then examine both interferometers using an input-output formalism that allows us to treat them both as a boundary value problem. In the limit that the particle coherence length is much larger than the interferometer we then, sticking very close to the physics, argue that in order to access these exponentially large Hilbert spaces, the particles must arrive at the beamsplitters in a way that they are indistinguishable there, and thence their multi-particle  wavefunctions must be properly symmetrized everywhere in space.

In this model, then, to compute an arbitrary output state (our goal) we must simply prepare an arbitrary input state and then use the efficient matrix transfer to transform the input to the output. Therein lies the rub. For fermions there is an efficient protocol for constructing an arbitrary input state but for bosons there is not. In other words, for a large interferometer, we cannot even setup the initial conditions in the general case, much less compute the corresponding output. 

For fermions the appropriate anti-symmetrization of the total input wavefunction (the product of the spatial and spin wavefunctions) may be carried out efficiently (in polynomial time) by constructing and evaluating the Slater determinant of the basis functions. For bosons the equivalent procedure would be to symmetrize the total input wavefunction by using the 'Slater' permanent. (Slater never wrote down any such thing.) In the case of the bosons the most efficient known algorithm for computing the permanent is the Ryser algorithm, discovered 50 years ago, which scales exponentially in the size of the matrix.

Thus we conclude our elementary argument that linear optical interferometers cannot likely be efficiently simulated on a classical computer. We do this using simple arguments from quantum optics and undergraduate quantum physics and without resorting to complexity theoretic notions such as 'collapse of the polynomial hierarchy.'  The small amount of quantum computer complexity that remains in our argument can now discuss in language that can be understood by the ordinary quantum physicist on the street (or in the gutter).

An amusing and informative lecture on these results, given this past week at the Quantum Information and Measurement Conference in Rochester, NY,  may be found here [PDF, PPT].

Entagled-Photon Holes!

Entangled-photon-hole-boosted quantum teleportation of Schrödinger-cat states

Kaushik P. Seshadreesan, Jonathan P. Dowling, Girish S. Agarwal

We study the teleportation of non-Gaussian, non-classical Schrodinger-cat states of light using two-mode squeezed vacuum light embedded with Franson's entangled photon holes. We discuss two figures of merit a) the fidelity of teleportation and b) the maximum negativity of the Wigner function at the output. We show that the addition of entangled photon holes to two-mode squeezed vacuum light lowers the requirements on the amount of squeezing necessary to achieve any given fidelity of teleportation, or to achieve negative values of the Wigner function at the output.

arXiv:1306.3168

What IS a Quantum Computer?

Having a delightful time in Beijing visiting the Computational Science Research Center, which last week hosted the Fifth International Workshop on Quantum Optics and New Materials — this time to celebrate the 60th birthday of my old friend and collaborator, M. Suhail Zubairy.

On the last day, Wednesday, I sat at lunch at the vegetarian table with Zubairy, Barry Sandars, Selim Shahriar, Wolfgang Schleich, Girish Agarwal, and Jörg Evers. The discussion turned to the computational complexity of linear optical interferometers (the topic of my talk that morning) and then more broadly to D-Wave's machine and then somebody (might have been me but I had so many vegetables who can tell) asked each person in turn to define a quantum computer.

To us quantum opticians the fact that there was no agreement at all came as a bit of a surprise. Shahriar held out for a universal machine capable of running Shor's algorithm. I pointed out that, as per a 1998 article by Seth Lloyd, that coherence and strong projective measurements were necessary and sufficient to run Grover's algorithm with a quadratic speedup — shouldn't such a machine be called a type of quantum computer?

Shahriar rejected that as it was not universal. Then came the D-Wave discussion. If the D-Wave machine can do something useful, also exploiting quantum coherence, should it not also be classified as a type of quantum computer? Sanders seemed to side against this. I was more positive.

And so the discussion continued as the vegetables arrived in an never-ending sea of serving platters. In the end Zubairy thought it was quite surprising that we all could not agree on the definition of a quantum computer. I rejoined that people still argue over what was the first classical computer. The Babbage difference engine? The ENIAC? The Atanasoff–Berry Computer? How about that ancient Greek computer the Antikythera Mechanism? If historians of computer science can't even agree on what is a a classical computer what hope do we have of now agreeing on what is a quantum computer.

My own definition? Any computer that exploits some of the weirder features of quantum mechanics to attain a computational advantage over the best classical machine, no matter how slight, should be at least entertained as a type of quantum computer.