On The Dowling-"Neven" Law

In a recent article in the magazine Quanta, much ado was made about the new so-called Neven's law: 

"That rapid improvement has led to what’s being called 'Neven’s law,' a new kind of rule to describe how quickly quantum computers are gaining on classical ones. The rule began as an in-house observation before Neven mentioned it in May at the Google Quantum Spring Symposium. There, he said that quantum computers are gaining computational power relative to classical ones at a 'doubly exponential' rate — a staggeringly fast clip."

Hartmut Neven is the director of the Google Quantum Artificial Intelligence Lab. In some sense — at least for the quantum cognoscenti! — this is a trivial observation. We have known for well over 30 years that the processing power of a quantum computer scales exponentially in the number of qubits. Hence, when the point came that the number of qubits on a chip began to grow exponentially, it is simple to deduce that the processing power will grow doubly exponentially. 

Nevertheless, the general public has difficulty grasping ordinary exponential growth — much less doubly exponential growth! So in my 2013 book, Schrödinger's Killer App: Race to Build the World's First Quantum Computer, I wrote an entire short section on this double-exponential growth in processing power. At the time I called it "Moore's Law for Quantum Computing." 

This double exponential growth has reached a stage where it is a reality, as Neven noted, since the number of superconducting qubits on a wide variety of platforms is doubling every six months. I thought it would be useful to set the historical record straight by posting here the excerpt from my book that discusses the double-exponential scaling, a feature of quantum computers that I propose we call the Dowling-Neven Law, given that I cooked it up six years before Neven did. 

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Excerpt from: Schrödinger's Killer App: Race to Build the World's First Quantum Computer, by Jonathan P. Dowling (Taylor and Francis CRC Press, 2013) pp. 391–392. 

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"I will take [Bill] Phillips up on his wager and bet that we have a quantum computer capable of running Shor’s factoring algorithm in 50 years or so. I will also take the science fiction writer point of view and extrapolate wildly beyond known physics to get us there. (Remember, the science fiction writers are more often right than the too conservative scientists.) There is consensus in the community that such a practical factoring engine will require around a trillion (10¹²) qubits. Conjecturing something like Moore’s law for quantum computing I will further conjecture that new technologies are around the corner that will allow us to double the size of our quantum processors every few years. So if we think of the new NIST ion trap working this in a few years (2020) with 1000 entangled ions we then just scale this up, doubling the number of qubits every few years, until we get to a billion entangled qubits. In figure 6.4, I plot this trajectory with the number of qubits on the red horizontal scale, the size of the corresponding Hilbert space on the green vertical scale, and the year along the diagonal on the orange scale. I show a few qubit ion trap photograph around the year 2000, a 1000 qubit ion trap around the year 2020, a fanciful rendition of a 100000 qubit machine made from a carbon graphene lattice in 2040, and then I run out of ideas for the hardware. But you can bet that scientists in 2040 will not run out of ideas (they have thirty years of new discoveries in hand that I don’t) and they will build the million qubit machine by 2060 and finally the Internet hacking billion qubit machine running Shor’s algorithm (for which I display the quantum circuit) by 2080. The growth in the number of qubits, as per Moore’s law, is exponential by year. Since the size of the Hilbert space, vertical green scale, is exponential in the number of qubits it is therefore super exponential. Following this trajectory, if I hedge William Phillips bet and we have a billion qubit machine in 70 years by 2080 then we humans will have explored 3000000000000 (three-trillion) orders of magnitude in size in Hilbert space. That is to be compared to the rather paltry 60 orders of magnitude in three-dimensional space humans have explored in all of recorded history, tens of thousands of years. The exploration of all of three-dimensional space in all of human history is nothing compared to the exploration of Hilbert space in the next 70 years. That is the promise of quantum technology.

Taking that position that Hilbert space is real, just as real as three-space, and then this is not just a mathematical exercise. If Hilbert space is a physical resource then by 2080 we’ll have a 10^{3000000000000} dimensional resource at our disposal. It is difficult to wrap your brain around the size of this number.[i] This is a one followed by three trillion zeros. This book you are reading has around a million characters in it. Hence to print out the number 10^{3000000000000} in full on paper would require a printing of 3000000 (three-million) books this size filled with all zeros; that is about a tenth of all the books in the US Library of Congress, one of the largest book repositories in the world. The number of particles in the entire universe is only about 10⁸⁰ or one followed by only 80 zeros. In the Church of the Larger Hilbert space we believe that not only are these huge numbers attainable but that they are attainable in a generation and that they correspond to the size of a real thing, Hilbert space, that we can build things with using it as a resource. What is in Hilbert space? Well nothing I’m sure at the moment. That is whenever we opened a new window in three-dimensional space, by inventing the telescope or microscope; we found things that were already there, new planets in the former case and microbes in the latter case. This is why the exploration of three-dimensional space is a science. As our observing tools see bigger or smaller things we find stuff that has been there all along but that we just could not see before. Not so in Hilbert space. Hilbert space is not independent of us, there for us to find stuff in, but rather we create Hilbert space and then use it in turn to power new types of machines. The exploration of Hilbert space is much less a science and much more of a technology. I rather doubt that when we build the billion-qubit quantum computer that it will open a portal into an exponentially large Hilbert space that contains Hilbert space creatures that will leap out at us and gobble us all up. Rather then Hilbert space is empty until we make it and begin to manipulate our quantum mechanical states inside of it and fill it with our technologies. It is because we make it that it is a technology. What will we make in a 10^{3000000000000} dimensional Hilbert space? Well there is plenty of room in there so I would be surprised if all we came up with was a quantum computer.

Figure 6.4: Exponentially large Hilbert space over the next fifty years. The red horizontal scale assumes a type of Moore’s law for quantum processors with the number of qubits in the processor following an exponential growth doubling in size every few years. (This is the type of scaling we’ll have to have if a universal quantum computer with a billion qubits is to be built in fifty years.) The dimension of the Hilbert space scales exponentially again with the number of qubits and so it scales ‘super’ exponentially along the vertical green axis. The diagonal orange arrow indicates approximate year. The figures show an ion trap quantum computer with one, two, three six, and ten qubit (lower left) where ten qubits gives a Hilbert space dimension of 2¹⁰ or about 1000. The second graphic is the NIST ion trap that may have 1000 entangled qubits in 2020 with a Hilbert space dimension of 2¹⁰⁰⁰ or 10³⁰⁰. The third graphic is a schematic of the carbon graphene structure that might have 1000000 entangled qubits or a Hilbert space of 2^1000000 that is about 10³⁰⁰⁰⁰⁰. The final graphic (upper right) is a circuit for Shor’s algorithm running on a billion-qubit machine (of unknown technology) that has a Hilbert space dimension of 2^100000000000 or 10^300000000000. This chart implies that we will cover 300000000000 orders of magnitude of Hilbert space in the next fifty years compared to 60 orders of magnitude (figure 6.3) in three-dimensional space covered in the past several thousand years.[ii]

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[i] See, “The Biggest Numbers in the Universe,” by Bryan Clair in Strange Horizons (02 April 2001) <http://www.strangehorizons.com/2001/20010402/biggest_numbers.shtml>.

[ii] The figure is a composite. The two ion trap photos are courtesy of NIST and as work of the US Government are not subject to copyright. The graphene molecule, a two-dimensional crystalline form of carbon that has been proposed as platform for quantum computing, is a schematic that is the work of Krapnik <http://en.wikipedia.org/wiki/File:CF_1.png> and the circuit diagram of Shor’s algorithm was created by Bender2k14 <http://en.wikipedia.org/wiki/File:Shore_code.svg> and both are and licensed under the Creative Commons Attribution-Share Alike 3.0 Unported license.