There is a fundamental trade-off between making a clock accurate and counting ever shorter ticks, due to the second law of thermodynamics. This trade-off could one day be important for clocks in quantum computers, where computing operations occur on extremely short timescales with high accuracy.
A common way of expressing the second law of thermodynamics is that the amount of disorder in a system (its entropy) must always increase over time. As a consequence of this increasing disorder, there is an arrow of time. All physical laws we know would be the same if time was reversed, says at the Vienna University of Technology in Austria, “so we wouldn’t be able to tell time, but there’s one exception and this is the second law of thermodynamics.”
He and his colleagues realised that an idealised clock would also be subject to this second law and would need to generate disorder to tick. “Entropy needs to be produced for time to be measured, but does this mean the more I know about the time passed, or the more accurately I can tell time, the more entropy I need to produce?” says Huber.
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In an earlier study, they proved that more accurate clocks must produce more entropy, but they didn’t yet know how this related to how often the clock ticks, known as its resolution.
To answer this question and see how accuracy, entropy and resolution were related, Huber and his team realised they needed a better definition of what a clock is, in thermodynamic terms. They concluded that a clock needs a system that can “tick” in an irreversible and random way, but it also requires another system to count this ticking.
One way of illustrating this is with a sand-filled hourglass that randomly drops grains of sand through a hole – this is a kind of ticking. The ticks are counted by watching for when there are no grains left in the top half. A 10-minute sand timer is very accurate at measuring a 10-minute period, but you can’t use it to measure shorter intervals, so it has a very low resolution. If you could count each individual grain that goes through the hole, you would get a higher resolution, but it wouldn’t be very accurate because the randomness in the system means the grains don’t go through at regular intervals.
This isn’t unique to the sand timer – it is an inherent property of all clocks, Huber and his team found. By examining this relationship using quantum mechanics and thermodynamics, they proved that the resolution, defined as how frequently a clock can tick, is equal to the inverse of its accuracy squared, where accuracy is the number of times it can tick before being off by one tick.
This limit isn’t relevant for the most accurate clocks today, such as atomic clocks, because they only measure things on the resolution of around a second, but it could apply to future quantum computers because they will need extremely fine timescales and high accuracy, says Huber.
It could also have particular relevance in quantum computers that can control themselves, says at the National Institute of Standards and Technology in Maryland.
Quantum computers are often controlled manually with wires, which introduce heat to the system and errors, but if the computer can be controlled autonomously, then these errors could be reduced, she says. “These machines would need to have their own clocks to tell them what to do at the right time, and those clocks might eventually be able to operate close to the proven fundamental bound.”
While it is important to understand the fundamental trade-off limit between accuracy and resolution, it doesn’t provide an obvious path forward to help us design better quantum clocks, says Huber.
arXiv
