Figure 15.3. The greedy geodesic has a length defined by the number of network legs it cuts through. Starting from the physical qubits dangling on the outside we can move inwards to reconstruct the interior logical qubits shown as black dots inside the pentagons.
The exciting feature of the HaPPY code is that it reproduces the most important features of AdS/CFT, and in particular the Ryu–Takayanagi result. We illustrate this in Figure 15.3. Each black dot represents a qubit. Let’s suppose that we know the state of the ‘dangling’ qubits around the edge. If lines from three known qubits feed into a pentagon, then we also know the state of the other two qubits, and also the central qubit. Qubits from the outer pentagons link into adjacent interior pentagons. As we head inwards and repeat, we always know the state of all the qubits linking into each pentagon until we encounter a pentagon with fewer than three inputs. At this point we can’t go any deeper; the qubits around the edge no longer encode for that part of the interior. When we reach this stage, the line that we cross is known as the ‘greedy geodesic’, shown as a dashed line. It marks out the part of the interior that is described perfectly by the dangling qubits around the edge. It is also the shortest line that can be drawn through the interior that links the edges of the boundary region containing the dangling qubits. Remarkably, the amount of entanglement between the qubits in the boundary region and the rest of the boundary is equal to the number of links the greedy geodesic cuts through in the network. This is nothing other than the Ryu–Takayanagi result.
The gem here – the key point – is that the HaPPY code is a network of qubits, and yet it exhibits the properties of the physics we’ve been discussing in the context of black holes. Try to imagine the HaPPY code without imagining a space in which the qubits are embedded. No space, just entangled qubits. We know that there is an equivalent description of the code using the language of geometry, which is what we’ve been using to visualise it: it’s the hyperbolic geometry of the Poincaré disk. In other words, the way we have wired up the qubits gives rise to an emergent hyperbolic geometry. The notion of distance emerges as the number of links we cut through in the network, i.e. distance is defined by counting the number of links that are cut. Astonishing as it may seem, we are being invited to imagine that the space we live in is built up from an entangled network of elemental entangled quantum units that are too small for us to detect with current experiments. Instead, we are sensitive to the way these entangled units give rise to the physical phenomena we see, including the very idea of space itself. This is quite a remarkable development and one of which John Archibald Wheeler would surely have approved.
So, what is reality?
Are we living inside a giant quantum computer? The evidence is mounting that it may be so. For years, the study of black holes has been an intellectual endeavour that has pushed theoretical physicists into corners. But in the last decade or so, a flurry of understanding, fuelled in large part by exploiting the rapidly developing field of quantum information, has led to a consensus view that holography is here to stay and that it shares many similarities with quantum error correction.
Does living inside a universe that resembles a giant quantum computer suggest that we are virtual creations living inside the computer game of a super-intellect? Probably not. There is no reason to make that link. Rather, in our pursuit of the quantum theory of gravitation, the bluest of blue skies research, we appear to have glimpsed a deeper level of the world, and understanding this deeper level may well be useful to us when we design quantum computers. This has happened so many times in the history of science. We are constantly discovering techniques that Nature has already exploited. It is not so surprising that those techniques turn out to be useful to us technologically: it seems that Nature is the best teacher.
This unlikely link between quantum computing and quantum gravity raises tantalising new possibilities. The future of quantum gravity research may have an experimental side to it, something thought highly unlikely just a few years ago. Maybe we can explore the physics of black holes in the laboratory using quantum computers. And this deep relationship between the two fields flows both ways. There may in turn be a good deal of overlap between pure black hole research and the development of large-scale quantum computers, devices which will be of enormous benefit to our economy and the long-term future of our civilisation. Perhaps it will not be long before we can no more imagine a world without quantum computers than we could imagine a world without classical computers today.
This is the ultimate vindication of research for research’s sake: two of the biggest problems in science and technology have turned out to be intimately related. The challenge of building a quantum computer is very similar to the challenge of writing down the correct theory of quantum gravity. This is one reason why it is vital that we continue to support the most esoteric scientific endeavours. Nobody could have predicted such a link.
‘Be clearly aware of the stars and the infinity on high. Then life seems almost enchanted after all’, wrote Vincent van Gogh. The study of black holes has attracted many of the greatest physicists of the last 100 years because physics is the search for both understanding and enchantment. That the quest to understand the infinities in the sky has led inexorably to the discovery of a holographic universe enchanting in its strangeness and logical beauty serves to underline Van Gogh’s insight. Perhaps it is inevitable that human beings will encounter enchantment when they commit to exploring the sublime. But it’s bloody useful too.
* The penny dropped first for Ahmed Almheiri, Xi Dong and Daniel Harlow, who pointed out the AdS/CFT link with quantum error correction in 2015.
Acknowledgements
We are very grateful for the help and support of numerous colleagues, family and friends. In particular, for their many helpful comments and discussions, we’d like to thank Bob Dickinson, Jack Holguin, Tim Hollowood, Ross Jenkinson, Mark Lancaster, Geraint Lewis, Chris Maudsley, Peter Millington and Geoff Penington. We are also grateful for the support received from the University of Manchester and the Royal Society.
For helping make the book happen, a big thank you is due to Myles Archibald and the team at HarperCollins, and to Diane Banks, Martin Redfern and Sue Rider.
Most of all, we are indebted to our families – to Marieke, Florence, Isabel, Lenny and Tilly, and to Gia, George and Mo.
Thank you all.
Endnotes
1. Hawking, S. W. and Ellis, G. F. R. (1973), The Large Scale Structure of Space-Time (Cambridge University Press, Cambridge).
2. Einstein, A. (1939), ‘On a Stationary System with Spherical Symmetry Consisting of Many Gravitating Masses’, Ann. Math. Second Series, 40(4):922–936.
3. Montgomery, C., Orchiston, W. and Whittingham, I. (2009), ‘Michell, Laplace and the Origin of the Black Hole Concept’, J. Astron. Hist. Herit., 12(2):90–96.
4. Fowler, R. H. (1926), ‘On Dense Matter’, MNRAS, 87:114–122.
5. Chandrasekhar, S. (1931), ‘The Maximum Mass of Ideal White Dwarfs’, Astrophys. J., 74:81–82.