Milk prices and spherical elephants

Ulrich Hoff
“A litre of milk - that costs about 10 DKK or so, right?”

A little while ago, my ears vaguely picked up this part of a random conversation going on a bit further down the lively open office space at Kvantify. Not a message conveying many bits of information but enough to spark a chain of associations. And almost like an enactment of the Ising-model I reached out with a sloppy remark to my nearest neighbour colleague (hey, we are quantum physicists working in quantum computing) – “That’s probably as good an approximation as saying elephants are spherical.” And this is where the Kvantify mentality showed its smiling face. Without moving his eyes even a bit from the Jupyter notebook in front of him, he just replied: “Can you quantify that?”

Challenge accepted!

Physics is to many an awe-inspiring discipline because of its intricate equations, stone cold facts, and ability to reveal nature’s inner workings on scales ranging from the subatomic to the intergalactic. But when you get to know physics a little better you soon realise that, if anything, physics is about approximations. It is about rounding off the sharp corners and hard edges, simplifying, and making nature tractable and subjectable to mathematical modelling –

while always keeping as much of the messy complicated stuff needed to ensure the models are consistent with experiments. Classical physics corner stones like Newton’s 2nd law (stating that the mass of an object times its acceleration equals the sum of forces acting on that object) are simple to work with and good enough for us to send humans to the Moon and bring them back. However, they are crude simplifications of how nature works and, strictly speaking, only apply in highly idealised scenarios. On the other hand, it would be an insurmountable task – and an utterly fruitless one – to describe at an atomic level, how every part of a space rocket interacts with its environment. You have to choose your fights wisely. I chose to accept the challenge of quantifying my conjecture.

The devil is in the details

Knowing when enough is enough, that is when you’ve reached the black belt in physics. In some cases that threshold resides at a very deep level of detailing. Many industries are limited in their development or forced to follow time consuming and expensive detours because they cannot simulate nature exactly as it is. Pharma is one example. As a physicist, I’ve always considered chemistry a dirty business (and it is! – I know because my other nearest neighbour in the Ising office space is a quantum chemist). Modelling how electrons in even simple molecules fizz around in their orbitals and form bindings with other molecules requires an exponentially large number of bits. It is a job that can bring even the fittest of supercomputers to their knees, sweating, and gasping for RAM. In other words, way beyond possible... at least for now. Quantum computers might bring about a profound change to what is computationally possible, and even though the full potential is still some way away, the light at the end of the tunnel is steadily growing brighter and brighter. Quantum computers naturally lend themselves to computations of quantum physical problems as their qubits directly provide the exponential Hilbert space required to efficiently embed and crunch such problems. The ability to perform fast and precise physical simulations of molecular processes is intriguing and will make a deep impact on medical industries. It will enable computational chemistry to take over more and more of the laborious wet lab work and thereby cut short the long and expensive drug discovery process. Driving this development forward, constantly squeezing the most bits of information out of quantum computing hardware as it matures, by knowing how to find our way through the land of approximations and forging new powerful algorithms, is what we do. This is what floats our boat at Kvantify!

Wasn’t there an elephant in the room?

There was, and I’ll be coming back to that now. Physics is all about making approximations, and among physicists it is classic garage lingo to say that elephants are spherical, because that is for many practical purposes a pretty good approximation. That is the training behind the chain of associations that spurred the challenge in the first place. To my great fortune and joy, the Jupyter notebook at the neighbouring lattice site was busy crunching tensor network calculations and the process owner volunteered to take up the challenge with me. Together we endeavoured into kvantify’ing whether 'a litre of milk costs about 10 DKK is as good an approximation as taking elephants for being spherical’.


So, what is the average market price of milk? A quick search gave us a ballpark interval for milk prices in Denmark: 10-15 DKK per litre. Next up, how much does an elephant weigh and what’s its volume? Again, Google happily provided a number for the average weight, and well aware that there are different species of elephants with each their characteristic specs, we decided to narrow in the investigation to consider only the African one. Average weight: 6.000kg. And for the dimensions: height 2.6 to 4 metres, body length 3 to 5 metres, and width 1.3 to 2.1 metres. This gave us a characteristic dimension for the system of 3.0 metres, and taking half of that for the radius the spherical approximation yielded a volume of 14 m3. Now, the average density of mammal flesh is just below 1000 kg/m3 and the approximated weight thus ended up somewhere below 14 tons, or about a factor of 2 off the actual value. Hmm... but who cares about factors of 2 in physics anyway... The approximated milk price that spurred it all was also not spot on, but only about 20% off. So, it turns out I was wrong, but most importantly the conjecture was quantified.

But why?

Thanks for asking. Spending time on searching the web for average values on physical properties of elephants – and milk prices for that matter – might seem like a complete waste of time. And from a narrow-minded business or efficiency point of view it probably was. But it was a nice little “Fermi problem” to enjoy together with a colleague for a while, just as an alternative to having a coffee. More importantly, this little story nicely illustrates the mindset here at Kvantify. We really like quantifying statements with numbers. And we apply whatever cutting-edge tools from physics, chemistry, math, or computer science it takes to get us there faster and with more precise results than what was possible yesterday.

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Ulrich Hoff
Quantum Engagement Specialist

Ulrich holds a PhD in physics from the Technical University of Denmark and brings many years of experience with research and innovation in quantum technologies from academia. He is deeply engaged in the strategic work towards the formation of a strong national quantum ecosystem and is also a dedicated science communicator.