The question of how much heat is given off when erasing a single bit of information has excited scientists for decades, given its fundamental implications for thermodynamics and computation. Physicists in Ireland and the UK have now asked this question of quantum-mechanical bits (qubits) and found that the qubits’ coherence can lead to surprisingly high heat dissipation. They say that the result has important implications for protecting thermally sensitive quantum hardware and also shines further light on the paradox of Maxwell’s demon.

James Clerk Maxwell proposed his famous demon in 1867 to show how it might be possible to violate the second law of thermodynamics, which states that the entropy of a closed system tends only to increase over time. Maxwell envisaged a tiny intelligent being controlling a door in a partition dividing up a box of gas, which is initially at a uniform temperature. He argued that by opening and closing the door at just the right times, the demon could sort the gas molecules according to their speeds so that one half of the box contains faster molecules and is therefore hotter than the other half. This would decrease entropy without directly transferring energy to the particles.

In 1961, Rolf Landauer at IBM put forward a principle stating that any logically irreversbile computation produces entropy. In particular, he found that erasing a single bit of information necessitates the release of a certain minimum amount of heat – kTln(2), where k is Boltzmann’s constant and T is the temperature. Drawing on this idea, Charles Bennett, also at IBM, then argued in 1982 that this principle explains Maxwell’s paradox. The demon relies on a memory to sort the molecules, with that memory needing to be continually updated and scrubbed – thereby boosting entropy.

### Is qubit erasure more expensive?

In the latest work, John Goold, Giacomo Guarnieri and Mark Mitchison at Trinity College Dublin and Harry Miller at the University of Manchester, investigated whether qubit erasure dissipates more heat than does scrubbing a classical bit. Qubits can be in superposition states, allowing them to exist as a one and a zero at the same time and so forming the basis of quantum computers.

As Goold and colleagues point out, heat dissipation can be minimized by erasing information over an extended time in order to keep the system close to thermal equilibrium. Nevertheless, there will always be some thermal fluctuation, meaning that in practice there is a chance that any given bit erasure dissipates significantly more heat than the Landauer limit.

Physicists have done several experiments looking at how low this dissipation can be pushed – typically using particles undergoing Brownian motion and confined in double-well potentials. This research showed that when erasing information repeatedly, the amount of heat dissipated forms a Gaussian distribution with an average close to the Landauer limit.

To see whether the statistical distribution of dissipations looks any different in the quantum case, Goold and colleagues extended the theoretical analysis of double-well potentials. In these two-level systems, the stored bit value simply corresponds to which of the wells the particle occupies. A demon can raise the energy level of either well so that its potential is nearly flat and then allow thermal fluctuations to tip the particle into the other well – if it is not there in the first place. This re-sets the system to a pre-defined state, so erasing the bit value and releasing some heat to the environment.

### Tunnelling option

In the case of a qubit that is a coherent superposition of the two well states, however, there are two ways to overcome the remaining energy barrier – one is to hop over it via thermal fluctuations and the other is to tunnel underneath it. This latter mechanism can generate large quantum fluctuations in the dissipated heat, meaning that on average some small fraction of erasure cycles will release a large amount of heat to the environment.

Doing their sums, Goold and colleagues found, as expected, that the average amount of heat dissipated when erasing a qubit is like that given off in the case of a classical bit. However, they discovered that the energies involved in the most wasteful erasures in the two cases are very different. In one simulation they found that the most heat given off in any single classical cycle was about four times the Landauer limit, whereas in erasures that involved tunneling – roughly one in every thousand for a qubit – heat dissipation could exceed the Landauer limit by more than a factor of 30.

Quantum coherence turns up the heat on Maxwell’s demon

They reckon that even this small fraction of erasure cycles could pose a serious problem of potential overheating in future quantum-scale devices, given that modern computers irreversibly process many billions of bits each second. They also say that it provides further perspective on the paradox of Maxwell’s demon, given that a demon with a quantum memory will end up dissipating even more heat than one with a classical memory.

The results for the moment remain purely theoretical, but the group says that they provide markers for experimentalists to distinguish between quantum and thermal fluctuations – the fact that only the former can generate two consecutive events involving emission of energy quanta. State-of-the-art quantum technologies such as superconducting circuits could be used to look for such dual events, they add.

The research is described in *Physical Review Letters*.