# Limits for quantum computers: perfect clocks are impossible

It is becoming increasingly possible to carry out calculations with quantum computers. However, calculations by TU Wien show that there are fundamental limits - namely the quality of the clock used.

There are different ideas on how to build quantum computers. But they all have one thing in common: you use a quantum physical system - for example individual atoms - and change their state by exposing them to very specific forces for a very specific time. However, this means that in order to be able to rely on the quantum computing operation delivering the correct result, you need a clock that is as precise as possible.

But this is where you run into problems: perfect time measurement is impossible. Every clock has two fundamental properties: a certain precision and a certain time resolution. The time resolution indicates how small the time intervals are that can be measured - in other words, how often the clock ticks. The precision indicates the inaccuracy that must be expected with each individual tick. The research team was able to show that Since no clock has an infinite amount of energy available (or generates an infinite amount of entropy), it can never have perfect resolution and perfect precision at the same time. This places fundamental limits on the possibilities of quantum computers. This finding has now been published in two recent papers.

Quantum arithmetic steps are like rotations In our classical world, perfect arithmetic operations are no problem. For example, you can use an abacus in which wooden balls are threaded onto a rod and pushed back and forth. The wooden balls have unique states, each one is in a very specific place, if you do nothing, the ball stays exactly where it was. And it makes no difference to the result whether you move the bead quickly or slowly.

In quantum physics, however, things are more complicated. -Changing a quantum state in a quantum computer is mathematically equivalent to a rotation in higher dimensions-, says Jake Xuereb, the first author of the first paper, who conducts research at the Atomic Institute of TU Wien in Marcus Huber’s team. -In order to achieve the desired state in the end, the rotation must be applied for a very specific period of time. Otherwise you either rotate the state too short or too far.

## Entropy: time makes everything messier and messier

Marcus Huber and his team investigated the general laws that must always apply to every conceivable clock. -Time measurement always has to do with entropy," explains Marcus Huber. Entropy increases in every closed physical system, it becomes more and more disordered. It is precisely this development that determines the direction of time: the future is where the entropy is higher, the past is where the entropy was even lower.

As can be shown, any measurement of time is also inevitably associated with an increase in entropy: A clock, for example, requires a battery whose energy is ultimately converted into frictional heat and audible ticking via the mechanics of the clock - a process in which a fairly ordered state in the battery is converted into a fairly disordered state of thermal radiation and sound.

On this basis, the research team was able to create a mathematical model that every conceivable clock must obey. -For a given increase in entropy, there is a trade-off between time resolution and precision," says Florian Meier, the first author of the second publication. -This means that either the clock works quickly or it works precisely - both are not possible at the same time.

## Limits for quantum computers

This insight now brings with it a natural limit for quantum computers: the resolution and precision that can be achieved with clocks limits the speed and reliability that can be achieved with quantum computers.
-This is not yet a problem, says Marcus Huber. -At the moment, the accuracy of quantum computers is still limited by other factors, such as the precision of the components used or electromagnetic fields. But our calculations also show that we are not far away from the area in which the fundamental limits of time measurement play the decisive role.

So if the technology of quantum information processing is further improved, we will inevitably have to deal with the problem of non-optimal time measurement. But who knows: perhaps this is exactly what will allow us to learn something interesting about the world of quanta.

## Original publications

J. Xuereb et al, The Impact of Imperfect Timekeeping on Quantum Control, Phys. Rev. Lett. 131, 160204. Open access version: https://arxiv.org/abs/2301.10767

F. Meier et al, Fundamental accuracy-resolution trade-off for timekeeping devices, accepted in Physical Review letters (not yet published), freely accessible version:
https://arxiv.org/abs/2301.05173