The University of Cambridge is a special place. During term time, hordes of students gather, ready to attend formal, a candle-lit dinner in a five-century-old hall overlooked by paintings of academics past. The sight is like an overromanticized still of a bygone era. The men wear suits, the women long dresses, all wear gowns. The simplest gowns belong to the undergrads, with the complexity and length of the gowns growing as one climbs the academic ranks. 

Between the bringing of the bread and the serving of the soup, the conversation at my end of the table typically turns to my topic of research: quantum computers. These theoretical machines harness quantum physics to outperform their classical counterparts. Quantum computers hold great promise for the future, with applications ranging from drug discovery to cybersecurity. Yet, despite their promise, we still do not have a satisfying answer to the fundamental question: what makes quantum computers computationally stronger than classical computers?

To answer this question, imagine that tonight, instead of the soup, the cooks are brewing a happiness potion. This potion causes the drinker to be joyful and content for the rest of the evening. The cooks know how to brew the potion perfectly well; they discovered the list of ingredients in the nineties and have been successfully brewing ever since. But the cooks still cannot solve one mystery: what makes the potion different from the soup? What ingredient sets the potion apart from the soup?

The cooks come up with a simple approach. They leave out one ingredient in each serving, and then carefully observe the formal-goers. They find that almost every seat is filled by a happy student, chatting away to their neighbours, indicating a working potion. However, one student’s spirits have not been lifted, as they were solely served a simple soup. The cooks conclude that the ingredient they withheld for that student was essential. They name this ingredient magic.

In my research, the potion is a quantum computer, the soup is a classical computer, and magic is the actual technical term used for quantum states that promote certain classical computers to quantum computers. Without magic, these quantum computers are computationally no stronger than a normal laptop. 

So magic is necessary for quantum computational advantage, but is it enough? Over a decade ago, researchers found that, for quantum computers built on qutrits, the answer is no. To understand what a qutrit is, consider the regular bit: a switch that’s either zero or one. The qubit, the quantum generalization of the bit, can be both zero and one simultaneously. The qutrit is a roomier qubit that can be zero, one, two, or any of those simultaneously. 

It is possible to build quantum computers using qutrits. However, the vast majority of quantum computers are built on qubits, the quantum generalisation of the everyday bit. My colleagues and I have recently discovered that for qubit-based quantum computers, too, magic is not enough to gain an advantage over classical computers. Picture the cooks dumping jar after jar of magic into a soup, only to find that the soup never gains magical powers. Potions need more than magic. Potions also need Kirkwood-Dirac negativity.

What is Kirkwood-Dirac negativity? The idea developed by John Kirkwood and Paul Dirac builds on probabilities—for example, the odds of obtaining a heads upon flipping a coin. One can describe a quantum computer using numbers that behave similarly to probabilities but come with a twist: they may be negative. These negative “probabilities” mark where quantum departs from classical. The main result of our paper is that this negativity, too, is a necessary ingredient: without it, no amount of magic will turn the soup into a potion.

At this point in the conversation, I am typically cut off by the arrival of the soup. The conversation moves on from quantum computers to different subjects, and I usually look up at the paintings staring down at me. One of them is of Dirac, who spent many dinners discussing quantum theory in the same dining hall. The University of Cambridge is a special place.