Dear John, this is very interesting and inspirational question. (A) If I remember correctly, the lower bound on energy kTln2 (~2.8E-21 @ 20°C) you mention is associated with minimum heat generated during logic operation due to the erase of one bit of information. In general heat is generated only in the case of "irreversible logic operations" (e.g. AND, OR). However, "reversible logic operations" can dissipate arbitrarily little heat (e.g. inverter, Toffoli, Fredkin), hence, theoretically it is not necessary to consume energy for well selected logic operations. (B) I'm not sure if the CMOS is candidate for achieving limit above. In the case of CMOS the minimum energy operation is achieved (currently) in sub threshold regime by optimizing supply voltage Vdd. [S. HANSON ET AL, Ultralow-voltage, minimum-energy CMOS. IBM J. RES. & DEV. VOL. 50 NO. 4/5 JULY/SEPTEMBER 2006]. Minimum energy point is trade-off of dynamic and static energy (e.g. Minimum energy of inverter is ~10^-15J at 130nm) and leakage is the main limiting factor for energy consumption minimization. Hence, in order to reduce leakage the methods improving channel control in MOS transistor at lower supply voltage are necessary i.e. transistor geometry, materials. (C) Delay at minimum energy point for CMOS is usually much higher as minimum achievable delay in max. performance regime, hence, the number of operations per time unit is lower as would be possible. For processing constant amount of work we need more gates in minimum energy regime against maximum performance regime, and hence, we have to consider additional (overhead) energy for information exchange between gates. Is the energy/logic operation representative metric in this case?

Dear John, this is very interesting and inspirational question. (A) If I remember correctly, the lower bound on energy kTln2 (~2.8E-21 @ 20°C) you mention is associated with minimum heat generated during logic operation due to the erase of one bit of information. In general heat is generated only in the case of "irreversible logic operations" (e.g. AND, OR). However, "reversible logic operations" can dissipate arbitrarily little heat (e.g. inverter, Toffoli, Fredkin), hence, theoretically it is not necessary to consume energy for well selected logic operations. (B) I'm not sure if the CMOS is candidate for achieving limit above. In the case of CMOS the minimum energy operation is achieved (currently) in sub threshold regime by optimizing supply voltage Vdd. [S. HANSON ET AL, Ultralow-voltage, minimum-energy CMOS. IBM J. RES. & DEV. VOL. 50 NO. 4/5 JULY/SEPTEMBER 2006]. Minimum energy point is trade-off of dynamic and static energy (e.g. Minimum energy of inverter is ~10^-15J at 130nm) and leakage is the main limiting factor for energy consumption minimization. Hence, in order to reduce leakage the methods improving channel control in MOS transistor at lower supply voltage are necessary i.e. transistor geometry, materials. (C) Delay at minimum energy point for CMOS is usually much higher as minimum achievable delay in max. performance regime, hence, the number of operations per time unit is lower as would be possible. For processing constant amount of work we need more gates in minimum energy regime against maximum performance regime, and hence, we have to consider additional (overhead) energy for information exchange between gates. Is the energy/logic operation representative metric in this case?

Thanks, that is partly my understanding as well. The arguments appear to connect the destruction of information to an increase in entropy. So if an operation is not reversible then entropy has to increase. I think this makes a distinction between the logic element itself and the logic operations. My understanding is that for CMOS there are dissipative processes inherent in the devices, so those processes would result in additional energy consumption, even if the logic constructed out of CMOS was reversible. I think that people have constructed Fredkin and other types of logic out of CMOS, but I don't know it achieves any energy consumption advantage. My guess is that any advantage might be offset by the additional number of elements used to implement the logic, and the non-reversible energy consumption of those elements.

With sub-threshold CMOS, I think that because of gate leakage, and perhaps transistor to transistor variability that older technologies are usually used to minimize the energy consumption.

The trade off between performance, e.g. computations per unit time, which I think almost directly translates into power, and larger number of gates which might operate at a slower rate is also interesting. My guess is that there would also be issues in framing the problems in a format that type of computation would be optimized for. That probably confuses the issues some bringing the energy efficiency of algorithms for example serial versus parallel processing into play. So perhaps you are right, about energy/operation not being the right/only metric. Perhaps total energy per specific problem solution or Entropy increase per problem being solved is a better metric, but would change depending on the problem being benchmarked.

Hi

constructing reversible circuits using CMOS itself is wrong, as the operations cannot be reversible. So comparing reversible circuit(CMOS) with conventional CMOS circuit itself is a wrong idea according to my knowledge

Hi everyone, this really IS a very inspirational question. kTln2 seems to be related to (somehow) moving ONE electron from here to there to CHANGE the information. IF we could TALK to the electron for free, and tell him to move from point A to point B, and he wasted that much energy, Ok , so, that's the limit.

Wait a minute. this is if you are using ELECTRONs to REPRESENT information. What about photons ? In a quantum computer, very non-traditional things happen. First of all,

QUANTUM ENTANGLEMENT blows away everything we imagine. Once you get one photon going, another one follows. Hmmm. Buy one get one free. Well, that's a way to reduce your kTln2 . So, the question kind of turns into this:

What is the best medium to use to represent information ? Will there be others in the future ? If you know that medium, how much energy does it take to CHANGE the information in that new medium ? Photon, electron, whatever-on ?

Ok, let me go one step further: In this new medium (say, photon) , is there a phenomenon, like Quantum Entanglement that makes CHANGING THE INFORMATION a) much less expensive, b) completely DIFFERENT, so, changing information is less necessary to STORE or COMPUTE INFORMATION, c) IRRELEVANT to take it to the extreme, since there is no need to change the information.

Dear John,

I think that your question has to be supported by description of the conditions of measurement of the NEW state that defines that the "logical opration" is performed, even if only because, as we know from physics, when measuring something, we can change it. Firthermore, you can consider that our eye can register in darkness ONE photon, and this registration means, of course, a "logical operation" in the brain. HOWEVER, is really only the energy of the photon involved in this process, from its very beginning? The same question ma be relevant to electronic registraion. You of course already thought about that, but it seems to me that formulation of your very question requires some work.

It is good that you try to connect the question with entropy. The fields of application of entropy are being widen, and will be more. Shrodinger (I ask your pardon for a step aside) spoke about entropy in his "What is life?", and I tried to extend his outlook, from physiology to our thinking (i.e. from usual digestion to the "digestion" of information that one recieves), -- quite sociologically (pardon again!) an important topic, -- in my works found at http://vixra.org/author/emanuel_gluskin. Especially see there the set of some conference slides, in the title of which the name "Shrodinger" aroses, but not only.

I am sorry that I do not answer you in terms of electronic scheme only, but the topic really requires some wide outlook.

Yours, Emanuel.

I think some of the new magnetic memories developed at Purdue are at a femto joule per operation (read or write)

Here is a link to one of their papers on arxiv:

http://arxiv.org/ftp/arxiv/papers/1308/1308.4169.pdf

Reduce the swwitching activities to minimize energy

to Tolga Soyata:

Dear Tolga,

kT ln 2 - minimum possible amount of energy required to change one bit of information - Landauer limit is consequence of second law of thermodynamics for the closed system, and does depend on the way bit of information is represented and changed: electron or photon or anything else moved from here to there or any other way changed one state to an other … ·

Correction:

I meant to say "does not depend on the way" instead of "does depend on the way" in the previous post - sorry for the missing "not"

Intel 22nm CMOS transistors have a gate charging energy of roughly CV^2=10 fJ=62500 eV (Intel, Proc. IEDM 2012). Landauer limit is 17 meV at room temperature. The Landauer limit was experimentally confirmed last year (Berut et al., Nature 2012). Note that the Landauer limit assumes operation under thermal equilibrium conditions. For faster operation the energy per irreversible operation will exceed the Landauer limit.

Vladimir, thank you for clearing that up. If anybody is from a deep Physics background, I would love to learn about the root of the ktln2 formula. Also, how Quantum Entanglement gets one particle to mimic the other particle, and if ktln2 incorporates this phenomenon

Tolga,

To learn about the roots of kTln2 formula I would recommend to start with "Maxwell's demon and the thermodynamics of computation" by Jeffrey Bub - http://arxiv.org/pdf/quant-ph/0203017v1.pdf. This article reviews thermodynamics of computation by following the discussion in Richard Feynman's "Feynman Lectures On Computation" (Ch5 - "Reversible Computation and Thermodynamics of Computing") - absolutely incredible reading - and has references to major papers including references seminal papers of R. Landauer, C. H. Bennett, E, Fredkin, T. Toffoli. You can Google all those works in e-form.

It’s important to stress that kTln2 limit is associated only with any irreversible operation. Information erasure is irreversible operation, merge of 2 or more computation branches is another example of irreversible operation.

kTLn2 limit does not apply to conservative logic=reversible computing. By definition conservative system (system of conservative field(s) - for example: gravitational or electrostatic fields) is the system where mechanical energy is conserved - in other words no work is done along the closed path. Copy is example of reversible operation - theoretically zero work is required to carry copy operation of any length.

It’s possible to make irreversible operations reversible by expense of additional storage …

Pardon for one more "outside" comment. Since computers made both "A" and "NOT-A" similarly well physically realizable (say, for "1" this transistor is open, and for "0"--that), and intuitionistic logic by Brouwer, who was concerned with the possibility of returning, by rejection, from "NOT-A" to "A" (since one cannot reject what is not given, not realized), was defeated (especially after Shannon's contribution), in the sense of the applicational power, by Boolean logic, -- we consider UNIT logical operations, and one can speak about the energy needed for one such. Well, it is a bit funny to call the transfer from 1 to 0 a logical operation. It has no more logic than when a glass falls from the table and is found broken on the floor. Logic appears only when go back in a synthesis, considering combinations of the unit operations. Since, finally, you all have to face that you speak about PHYSICS, I would ask something related both the scopes of logic and energy. You obviously think (and the mention reason of the realization for the "technical success" Boolean logic seems to support this) that EACH unit operation required similar ("minimal") energy. However, if absorption of a photon means "1 to 0", and emission of the (a) photon means "0 to 1", -- couldn't it be that after the logical operation composed of these TWO sequential unit operation, we would have the same energy state without no energy lost, instead of double loss of the minimal energy, that all of you mention. As in my previous comment, I again ask about physical realizations. I believe that Feynman would give a deep physical analysis of the TWO- UNITS situation, but pardon again for the not purely technical commenting.

I think that the current connection you may be looking for is the connection between Entropy and Information content. This appears to be something that has emerged as a way to not have to talk about discrete bits of information. For example, one can compare the entropy of different languages and find out that they have about the same amount of entropy. There is of course a connection between the amount of energy expended and the change in entropy. One paradox, apparently connected to entropy is what happens when matter falls into a black hole? The information contained in the matter is destroyed, but apparently the entropy of the black hole increases....

As for other physical realizations, perhaps the photon case can be considered as a reversible computing operation? I suppose you could have a chain of absorption emission to some number of time, but perhaps until you read the value you have not really obtained any information.

This question seems to be physically-motivated, but, it has been stuck in my mind. I am from a VLSI background and I have been trying to make a sense out of this in the intuitive VLSI sense ... Let me take a shot at this: ktln2 = ~18 meV. So, what this is saying to me as a VLSI person:

"even if you could change information (or, perform a logic operation) by controlling a single electron (I assume you can't use 0.5 electrons !) , you still have to elevate (up/down) the electron at least across an 18 mV potential , say, from 0V to 18mV "

So, thinking in terms of VLSI (and, more specifically CMOS), this corresponds to operating with a MOSFET transistor threshold of somewhere around 10 mV (or, even less). Considering that, we are at about 600-800mV right now with 22nm, this shows that, we have an 80x to go. So, this is almost pointing to sub-nm technologies ...

Am I way off base ? or, is this kind of analogy meaningless ???

I think that is the right track. Ultimately, if the single electron is being considered, I think that the idea is that one state is in a potential well, and to transition to the other state would require energy to overcome the potential barrier. If the barrier is less than KT, the two states can't be distinguished. One loophole in this may be how the electron is transitioned from one state to the other. If the process is very slow, some may make the argument that an adiabatic process might use less energy.

By that argument we are a few orders of magnitude away. Given the difficulty in fabrication of every smaller devices considering other state variables than the charge of the electron, such as the spin of the electron might get closer to the limit.

Realistically, improvements in sub-threshold logic circuits and innovation in design of circuits can probably squeeze out a lot more performance even given current device fabrication techniques.

0 to 1 or 1 to 0 or in other words logical NOT operation is example of the reversible - one to one -- operation, and as such does not change entropy and requires as the minimum 0 - zero - energy for execution, regardless of how 0 or 1 state of the particle (electron or photon or phonon or exciting or magnon or polariton or plazmon, etc) or anything else is presented and regardless of the energy height separating 0 and 1 states. This is clearly explained in the works of Landauer, Bennett, Feynman ...

Although this question is might initially seem like a theoretical curiosity, it is very important in the sense that, it sheds light into certain CONSTRAINTS in computation. I am doing research in energy aware computation, and this question really intrigued me. Ever since John asked it, I have been trying to do a "practical" mapping of it to VLSI. I kept going back to Dennard scaling and see if it is possible to take this original theoretical question and perform any kind of a projection from where we are (22 nm) and may be arrive at a technology node this Landauer [rinciple is prescribing (e.g., 1nm ?? ?). Of course, you will hit many road blocks in this SCALING technologically, but, this is a theoretical question anyway ,,, It is Ok to find the THEORETICAL BEST CASE ...

It seems like, there are a few "components" in the Landauer principle that need to be clarified for any kind of a practical mapping. First, from what I read, a lot of the roots go back to thermodynamics (if not ALL): Let me take another shot at this:

--1-- Per Vladimir's - the PARTICLE doesn't matter, but, rather, the state

of the particle ...

--2-- Per John -- you have to expand sufficient energy to move a particle from

one state to another ... otherwise, the states are not distinguishable

--3-- the calculation of this state according to Landauer's principle is completely

thermodynamically motivated ... It is saying that : "changing from one state to

another distinguishable state will generate x amount of heat ...

--4-- this assumes NON-ADIABATIC process, as if it is, you are not generating heat

to change the state. So, Landauer's principle doesn't apply ...

Is this on track so far ? If so, what is the definition of VERY SLOW (per John) for the

adiabatic - non-adiabatic boundary ?