Figure 15.1. Illustrating the causal wedge puzzle.
The source code of spacetime
In Figure 15.1, we show a slice through AdS spacetime with an entangled quantum theory on the boundary; the Poincaré disk once more. The boundary has been split into three parts, labelled A, B and C. Let’s first focus on region A. Ryu–Takayanagi tells us that the entanglement entropy of region A with regions B and C is given by the length of the shortest line that can divide the two regions. In this AdS spacetime, the shortest line is a curve. Holography tells us that if we know what is happening on the boundary (A, B and C), we know everything about the interior. It also tells us, and this is not obvious but has been proved, that if we know what is happening on A, we also know what is happening in the shaded region. In the jargon, the shaded region is known as the entanglement wedge of A, because the quantum theory on A entirely determines what happens in the shaded ‘wedge’. The same is true for regions B and C, as illustrated in the remainder of the figure. Now consider a point somewhere near to the centre of the disk (the black dot). The left disk tells us that it is encoded on boundary regions B and C. The middle disk says it is encoded on A and C and the right disk says it is encoded on A and B. The only way all three statements can be true is if the information is encoded redundantly. This means that we could erase region A and still know what is happening at the black dot; or we could erase region B or region C. What we can’t get away with is erasing two of the three boundary regions – that would be too much. This is intriguing. It means that the answer to the question, ‘where on the boundary is the information associated with the region around the black dot encoded?’ is ‘it isn’t in any single region (A, B or C), but the information can be determined from knowledge of any two regions’. It is quantum entanglement that makes this robust distribution of information possible.
According to holography, the information needed to encode the interior space is scrambled up and distributed across the boundary, which makes it hard to read but very robust against destruction. This is very similar to a technique computer scientists have discovered that is central to the construction of working quantum computers. At the time of writing, the largest quantum computers are networks of around 100 entangled qubits. The potential of these computers is vast because the ‘space’ in which calculations can be performed grows exponentially with the number of qubits, exploiting quantum entanglement as an information resource. These 100-qubit quantum computers can perform calculations in minutes that would take a conventional supercomputer longer than the current age of the Universe to complete.
One of the biggest challenges in building large-scale quantum computers is preventing the qubits from becoming entangled with their environment. Given what we know about entanglement and quantum information, it should be clear that this would be a bad thing because information would ‘leak out’ of the computer into the surroundings and the computer wouldn’t work. Perfect isolation isn’t practicable, so what is needed is a way to protect the important qubits that are needed to program the computer: a way to encode information that makes it hard to destroy. This can be done by exploiting quantum entanglement to encode the information in a robust way. This is quantum error correction.
Classical error correction is a routine part of our everyday technology. A QR code, for example, encodes multiple copies of information so that a sizeable part of it can be destroyed while still allowing the information to be decoded. Quantum computers can’t rely on storing multiple copies of the information because, as we’ve seen, the quantum no-cloning theorem prevents quantum information from being copied. The solution is to devise a quantum circuit that encodes the important information in a redundant way without copying, but also in a way that is robust against interactions with the environment. It turns out that the latter is equivalent to requiring that the information should be scrambled up such that, in a sense, it is kept secret from the environment. It is rather like the environment can destroy the precious information only if it understands how we have encoded it. If we scramble things up sufficiently then the environment can’t crack the code. We give a non-quantum example of redundant, non-local information encoding in Box 15.1.
BOX 15.1. Encoding information
Suppose we want to encode the three-digit combination to a safe (abc). One way to do it is to make use of the function f(x) = ax2 + bx + c. To crack the code, one needs to know the values of a, b and c. It is possible to hide this information among a large group of people by giving each person a pair of numbers; a particular value of x and the corresponding value of f(x). To crack the code, we must interrogate any three people in the room for their pairs of numbers x and f(x). This is sufficient to determine a, b and c. This secret sharing scheme is a means to redundantly encode information in a non-local way. The method is robust against losing people: so long as we have at least three people we can get the code.
The challenge for those wanting to build a quantum computer is to invent a compact device for encoding a qubit (or a bunch of qubits) inside a bigger block of qubits, so that the qubits we want are safe even if there is damage to the exterior qubits due to their interactions with the environment. Error correction is all about trying to achieve that using an optimal combination of redundancy and secrecy. We can now appreciate the connection with holography, because the coding we’ve discussed in the context of AdS/CFT is an impressive combination of redundancy and secrecy.*52 In holography, the boundary codes for the interior space, and it does so in a redundant way because we can erase part of the boundary without losing the information in the interior. It also stores information in a way that is hard to decode, since the information is scrambled up and encoded non-locally by quantum entanglement. To destroy the interior space (as Van Raamsdonk imagined) we need to destroy the entanglement over a substantial part of the boundary and not just a small part of it.
Figure 15.2. The HaPPY holographic pentagon code.
In 2015, Fernando Pastawski, Beni Yoshida, Daniel Harlow and John Preskill53 devised an arrangement of networked qubits that redundantly encodes information about the interior of the network on the boundary. This is precisely the situation we’ve been discussing in the context of holography. The coding is known as the HaPPY code (after the authors’ initials) and is shown schematically in Figure 15.2. The open circles around the outside are qubits, as are the circles inside the pentagons. In a quantum computer, the boundary qubits are those most in danger from the environment. The qubits inside the pentagons are the ones the computer will use for its operations, and these are safer because of the structure of the network. The pentagons are devices that entangle the six qubits that feed into them. They operate such that any three qubits are maximally entangled with the other three. This means that the information encoded by the central qubit is robust against the erasure of up to three of the surrounding qubits.
The diagram shows a network with just a few layers of pentagons. You can see (from the underlying shaded pattern) that the pentagons are linked together in a manner that matches the hyperbolic tiling of the Poincaré disk. We could add more layers by moving the external qubits out by another layer. The pentagons would look very small on the diagram but that does not mean they are very small in real life. As physical devices, the pentagons could all be the same size. What matters is the way they are networked together and that is governed by the underlying hyperbolic geometry. This hyperbolic linking is an important feature, as we will now see.