What role does Landauer's principle play in quantum reversibility?

Question

In section 3.2.5 of Nielsen and Chuang (starting page 153) they talk about Landauer’s principle, where they discuss the lower bound on the thermodynamic cost of erasing information.

In irreversible computing (i.e. classical computers), we can build any algorithm from a series of NAND gates. Since NAND gates take 2 inputs and give 1 output, a bit of information is erased (i.e. lost) and therefore Landauer’s principle is in effect.

Nielson and Chuang discuss reversible computing and how it would not incur the cost of Landauer’s principle.

QUESTION

1. Irreversible computing incurs the thermodynamic cost from Landauer’s principle and reversible computing does not. Heat and noise can ruin quantum states within quantum computers, is this related to Landauer’s principle?

2. I don’t see any direct statements connecting Landauer’s principle to why quantum computers must be reversible. Can someone clarify the exact role Landauer’s principle plays in the reversibility of quantum computers and its relation to the unitary nature of quantum gates?

Answer 1

1. That is not directly related. Heat and noise can ruin classical information in a classical computer too. In practice that's not such a problem because classical information, unlike quantum information, has the nice property that it can be copied. From the point of view of fundamental physics, the ones and zeros in your computer, or written down on a piece of paper, or whatever, are very redundantly encoded. In other words, there are many degrees of freedom all separately telling you "this is a one" or "this is a zero". That is not the case for a qubit. A qubit is, in a very real sense, a smallest unit of information. It occupies the smallest amount of degrees of freedom possible.

2. Quantum computing can be irreversible. Let me explain what I mean by that. The formalism of state vectors acted on by unitaries can be neatly extended to include probabilistic mixtures of states as well as operations that are irreversible. See Wikipedia or chapter 8 of Nielsen and Chuang. These more general operations describe the evolution of open quantum systems. Closed quantum systems (as well as closed classical systems) are of course reversible. The reversible/irreversible distinction is not a quantum/classical distinction, but rather a closed system/open system distinction.

   In practice quantum computers must do irreversible operations, for example when re-initializing a qubit to $\left|0\right>$.

   As for your question, I don't see why there shouldn't be a corresponding increase in entropy whenever an irreversible operation is performed on a quantum computer, just like there is when an irreversible operation is performed on a classical computer. I don't think Landauer's principle fails to hold in any meaningful way for a quantum computer.

Answer 2

Actually, classical physics is also reversible, whether you're considering classical dynamics or considering it as the limit of quantum physics (which is reversible).

The means for neither classical nor quantum computers, there is no operation which, in isolation, deletes a bit (that is, takes a bit in state $0$ or $1$ and sends it to $0$ in either case). Instead, you need some environmental system, and you modify the environmental system reversibly.

For example, suppose your 0s and 1s are stored as different voltage levels. If you map $0\rightarrow 0$, nothing needs to happen; if you map $1\rightarrow 0$, then the extra voltage (i.e., electrical energy) needs to go somewhere. In practice that "somewhere" is the environment, as heat.

All of this implies that the environment basically "learns" the bits of your memory when you try to delete them. This fact has two consequences: (1) Landauer's law, and (2) quantum circuits (generally) need to be reversible.

The reason this implies Landauer's law is that if you're uncertain about your computer's memory (is it a $1$ or a $0$ at the address you're deleting?) then that uncertainty gets pushed to the environment: after deleting, you know the computer has a $0$ but the environment either gained a bit of heat or not. Any increase in uncertainty about the environment translates to an increase in entropy, which translates to heat.

Since that's a complicated way to think about things, rather than model classical operations as reversible operations on the computer+environment, we model irreversible operations on just the computer and lump all environmental interactions as "heat".

(this is described in a slightly unusually way because I firmly believe probabilities are subjective, and thus so is entropy; consult other sources if you want an explanation in a different (and therefore wrong :P) perspective.)

The reason this forces quantum circuits to be reversible is that if the environment learns something about a quantum state, it "collapses" the quantum state (or in better interpretations, the environment becomes entangled with the computational state). This changes the quantum state and ruins your computation, therefore, you cannot let this happen. You need to ensure that the environment learns nothing about your quantum state, which means the quantum computer must genuinely stay isolated, and thus the operations you perform on it are reversible.

Why doesn't classical data worry about this? You can't collapse classical data!

As John Gardiner points out, many quantum computations do contain irreversible operations. In an error-correcting code, we are supposed to measure error syndromes all the time. Error-correcting codes are carefully designed so that this measurement tells us only about the errors that happened, and nothing about the computational state itself. Here, as in classical computations, Landauer's law would apply.