Thermodynamics is one the most important tools we have for understanding the behavior of large physical systems. However, it is very important to realize when thermodynamics is applicable and when it is not applicable. For example, try to apply thermodynamics to the Intel processor inside the laptop I am writing this entry on. Certainly the silicon crystal is in thermal equilbrium, but then how am I able to make this system compute: if states are occupied with probabilities proportional to a Boltzman factor, then how can my computer operate with all sorts of internal states corresponding to, say, it’s memory? Let’s say that all of these internal states, states of my computing machine, are all energetically about the same energy (which is, to a decent approximation, true.) Then, according to thermodynamics, each of these states should be occupied with the same probability. But the last time I checked, the sentence I am typing is not white noise (some of you may object, )
Today, Robert Alicki, Daniel Lidar, and Paolo Zanardi have posted a paper in which they question the threshold theorem for fault-tolerant quantum computation and claim that the normal assumptions for this theorem cannot be fullfilled by physical systems. I have a lot of objections to the objections in this paper, but let me focus on one line of dispute.
The main argument put forth in this paper is that if we want to have a quantum computer with Markovian noise as is assumed in many (but not all) of the threshold calculations, then this leads to the following contradictions. If the system has fast gates, as required by the theorem, then the bath must be hot and this contradicts the condition that we need cold ancillas. If, on the other hand, the bath is cold, then the gates can be shown to be necessarily slow in order to get a valid Markovian approximation. Both of these conditions come from standard derivations of Markovian dynamics. The authors make the bold claim:
These conclusions are unavoidable if one accepts thermodyanmics…We take here the reasonable position that fault tolerant [quantum computing] cannot be in violation of thermodynamics.
Pretty strong words, no?
Well, reading the first paragraph of this post, you must surely know what my objection to this argument is going to be. Thermodyanmics is a very touchy subject and cannot and should not be applied adhoc to physical systems.
So lets imagine running the above argument through a quantum computer operating fault-tolerantly. Let’s say we do have a hot environment. We also have our quantum system, which we want to make behave like a quantum computer. Also we have cold ancilla qubits. Now what do we do when we are performing quantum error correction? We bring the cold ancillas into contact with the quantum computer interact the two and throw away the cold ancillas. Now we can ask the question, is the combined state of the cold anicllas and the hot environment in thermal equilbrium? Well, yes, both are in thermal equibrium before we start this process, but they will be in thermal equilbrium with two different temperatures. OK, so now we have an interaction between the system and the cold ancillas. So let’s do this. Now these two systems, the quantum computer and the ancillas clearly couple to the hot bath. Therefore we can assume that the Markovian assumption holds and further that the gate speed for the combined system-ancilla system is fast. No problem there, right. OK, now we throw away the cold ancillas. So we’ve done a cycle of the quantum error correction without violating the conditions set forth by the authors. How did we do this?
We did this by being careful about what we called the “system.” (Or, more directly we have to be careful what we call the “bath.” But really these are symmetric, no?) We started out the cycle with the system being the quantum computer. Then we brought in the cold ancillas. Our system now includes both the quantum computer and the ancillas. Since we are now enacting operations on this combined system, our enviornment is the original bath, which is hot (which may now couple to the ancillas.) We can perform fast gates on this combined system and then we may discard the ancillas.
In order for the authors argument to work, they have to assume that the “system” is always just the quantum computer. But then clearly the assumption of the environment being in thermal equilibrium is violated at the beginning of the error correcting cycle: the ancillas are cold but the bath is hot. Both are independently in thermal equibrium, but the combined system is not in thermal equilbrium at the same temperature. The interactions with the hot bath do imply that we can perform fast gates. The interactions with the cold ancilla do imply that we will have slow gates. But when we bring the cold ancillas and quantum computer together, we can also have fast gates: because our system now consists of the computer plus the ancillas and the remaining environment it hot. The ancillas are not part of the thermal bath which is causes problems for our quantum computer. Certainly the authors are not objecting to the fact that we can prepare cold ancilla states? So I see no contradiction in this paper with the threshold theorem. (A further note is that there is also a threshold for fault-tolerance when the noise is non-Markovian. I’m still trying to parse what the authors have to say about these theorems. I’m not sure I understand their arguments yet.)
Thermodynamics is, basically, a method for reasoning about large collections of physical systems when certain assumptions are made about this system. Often we cannot make these assumptions. (A classical case of this, which is not relevant to our discussion, but which is interesting is the case of the thermodynamics of a system of many point particles interacting via gravity: here thermodynamics can fail spectacularly, and indeed, things like the internal energy of the system are no long extensive quantities!) In the above argument, we cannot talk about two systems being at the same temperature: we have two separate systems with different tempatures. Certainly if we bring them together and they interact, under certain conditions, the two will equilibriate. But this is explicitly what doesn’t happen in the fault-tolerant constructions. This is, indeed, exactly what we mean by cold ancillas!
Understanding the limits of the threshold for fault-tolerant quantum computation is one of the most interesting areas of quantum information science. I’ve bashed my head up against the theorem many times trying to find a hole in it. I think that this process, of attempting to poke holes in the theorem, is extremely valuable. Because even if the theorem still holds, what we learn by bashing our heads against it is well worth the effort.
Updated Update: Daniel Lidar, Robert Alicki and other have posted responses and comments below. I highly recommend that you read them if you found this entry interesting!