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In a recent question about quantum speed-up @DaftWullie says:

My research, for example, is very much about "how do we design Hamiltonians $H$ so that their time evolution $e^{-iHt_0}$ creates the operations that we want?", aiming to do everything we can in a language that is "natural" for a given quantum system, rather than having to coerce it into performing a whole weird sequence of quantum gates.

This makes me think of chronons, which are a proposed quantum of time.

"There are physical limits that prevent the distinction of arbitrarily close successive states in the time evolution of a quantum system.

If a discretization is to be introduced in the description of a quantum system, it cannot possess a universal value, since those limitations depend on the characteristics of the particular system under consideration. In other words, the value of the fundamental interval of time has to change a priori from system to system."

Introduction of a Quantum of Time ("chronon"), and its Consequences for Quantum Mechanics

Is universal chrononic computing possible?

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Is universal chrononic computing possible?

There is no consensus that chronons even exist.
See the first line of this, for example.

However time (and space) is quantized in one of the most popular generalizations of quantum mechanics called loop quantum gravity.

If loop quantum gravity is an accurate description of the universe (which is not something we will be able to test for a very long time, until we can observe for example, Hawking radiation), then universal quantum computation with chronons would be possible as long as we can find a way to implement a universal set of gates such as {H,CNOT,R($\pi$/4)}.

It is hard enough to implement a useful number of {H,CNOT,R($\pi$/4)} gates with ordinary quanta that we've been working with for a century (such as spin quanta or atomic energy level quanta or photon quanta), so don't be disappointed if you don't see universal chrononic quantum computers on the market during your lifetime. But it is possible, provided that quanta of time actually do exist, which would be true if loop quantum gravity were to be true.

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    $\begingroup$ I didn't downvote, but I'd be interested in how the elementary gates would be realised using 'quanta of time', in the case e.g. that loop quantum gravity were true. $\endgroup$ – Niel de Beaudrap Jul 11 '18 at 10:33
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    $\begingroup$ @NieldeBeaudrap: In relativity we don't think of "time" or "space" as separate concepts but as part of one thing: "spacetime". Everything has a 4-dimensional position in spacetime given by $\vec{q}=(x,y,z,t)$. Time is just the 4th dimension. In the (ordinary) quantum version, $x\rightarrow \hat{x}$ is a continuous-variable position operator. We can be in a superposition of two x-positions $\frac{1}{\sqrt{2}}(|\rm{left}\rangle + |\rm{right}\rangle$. In 4 dimensions this becomes $\endgroup$ – user1271772 Jul 11 '18 at 11:31
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    $\begingroup$ $\frac{1}{\sqrt{2}}(|{\vec{q}_1}\rangle + |\vec{q}_2\rangle)$, so the Hadamard gate would transform one's spacetime coordinate from $|\vec{q}_1\rangle$ to there. How to physically implement this gate is probably something that loop quantum gravity experts haven't thought about since loop quantum gravity might not even be true, and how to go from $|0\rangle \rightarrow|+\rangle$ is already hard enough for most "ordinary" quanta other than simple things like spins and photonic polarizations. For example we don't really know a good way of doing $|0\rangle\rightarrow|+\rangle$ for anyon qubits. $\endgroup$ – user1271772 Jul 11 '18 at 11:36
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    $\begingroup$ It is certainly something way beyond what most engineers trying to build quantum computers have thought about, since we don't even know whether or not chronons (such as those found in loop quantum gravity) exist in the first place. But does the question (by a relatively new user) still deserve the 2 downvotes? It is still a good question isn't it? Then the answers have also got a total of 7 downvotes. Yes it's a highly esoteric topic, but "assuming chronons exist, can you build a universal quantum computer" is still a good question isn't it? $\endgroup$ – user1271772 Jul 11 '18 at 11:47
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    $\begingroup$ While on the subject of LQG, it is instructive to look at the 3 dimensional case. There we have a decent understanding of how state sum models in the flavor of LQG, usual QFT and string theories give different but often coinciding perspectives (way too many references to list here). This coincidence may continue in 4D or they may diverge. I don't know. $\endgroup$ – AHusain Jul 11 '18 at 18:14

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