1. Energy as Rate of Computing In quantum mechanics, the energy eigenvalues of any system give the rate of phase rotation of its energy eigenstates.In quantum mechanics, the energy eigenvalues of any system give the rate of phase rotation of its energy eigenstates. Through U ∆t |E  =e iE∆t/  |E . But, the average energy E of any quantum system also gives the maximum rate of rotation in Hilbert space of any state vector.But, the average energy E of any quantum system also gives the maximum rate of rotation in Hilbert space of any state vector. dθ/dt ≤ E/  where cos dθ = |  vU dt |v  |. A conditional transition to a “nearby” distinguishable state (“primitive op”) requires some state vectors to rotate to an orthogonal direction, or through a total angle of ∆θ = π/2,A conditional transition to a “nearby” distinguishable state (“primitive op”) requires some state vectors to rotate to an orthogonal direction, or through a total angle of ∆θ = π/2, –Which thus takes time at least ∆t = h/4E. To transition to the next state along a long chain of orthogonal states (a “computational op”) actually requires 2 primitive ops (e.g. annihilate/ create), i.e., rotation angle ∆θ = π,To transition to the next state along a long chain of orthogonal states (a “computational op”) actually requires 2 primitive ops (e.g. annihilate/ create), i.e., rotation angle ∆θ = π, –The time required for this is ∆t = h/2E. Defining 1 “op” as o = π = h/2, the rate at which ops are performed is R o = o/∆t = E.Defining 1 “op” as o = π = h/2, the rate at which ops are performed is R o = o/∆t = E. Energy can thus be considered the rate of computing, or amount of computing activity.Energy can thus be considered the rate of computing, or amount of computing activity. –Taken to include all of the quantum evolution that is physically occurring, continually, at the nanoscale.

2. Generalized Temperature The concept of temperature can be generalized to apply even to non- equilibrium systems.The concept of temperature can be generalized to apply even to non- equilibrium systems. –Where entropy is less than the maximum. Example: Consider an ideal fermi gas.Example: Consider an ideal fermi gas. –Heat capacity/fermion is C = π 2 k 2 T/2μ. μ = Fermi energy; k = log e; T = temperatureμ = Fermi energy; k = log e; T = temperature –Equilibrium temperature turns out to be: T = (2/πk)(E x μ) 1/2, thus C = πk(E x /μ) 1/2 where:T = (2/πk)(E x μ) 1/2, thus C = πk(E x /μ) 1/2 where: E x = E − E 0, avg. energy excess/fermion rel. to T=0 –Equilibirum (max) entropy/fermion is: S max = ∫dS = ∫d′Q/T = ∫dE x /T = πk(E x /μ) 1/2 = CS max = ∫dS = ∫d′Q/T = ∫dE x /T = πk(E x /μ) 1/2 = C Consider this to be the total information content S max = I tot = S + X (entropy plus extropy).Consider this to be the total information content S max = I tot = S + X (entropy plus extropy). –We thus have: T = 2(E x /I tot ) The temperature is simply 2× the excess energy per unit of total information content.The temperature is simply 2× the excess energy per unit of total information content. Note that the expression E x /I tot is well- defined even for non-equilibrium states, where the entropy is S < S max = I tot.Note that the expression E x /I tot is well- defined even for non-equilibrium states, where the entropy is S < S max = I tot. –Thus, we can validly ascribe a (generalized) temperature to such states.

3. Generalized Temperature as “Clock Speed” Consider systems such as the Fermi gas, where T = cE/I.Consider systems such as the Fermi gas, where T = cE/I. –Where c is a constant of integration. –E is excess energy above the ground state. –I is total physical info. content For such systems, we can say that the generalized temperature gives a measure of the energy content, per bit of physical information content.For such systems, we can say that the generalized temperature gives a measure of the energy content, per bit of physical information content. E b = c -1 Tb = c -1 k B T ln 2 E b = c -1 Tb = c -1 k B T ln 2 Since energy (we saw) gives the rate of computing, the temperature therefore gives the rate of computing per bit.Since energy (we saw) gives the rate of computing, the temperature therefore gives the rate of computing per bit. –In other words, the clock frequency! For our case c=2, room temperature corresponds to a max. frequency of:For our case c=2, room temperature corresponds to a max. frequency of: f max = 2c -1 Tb/h = k B (300 K)(ln 2)/h = ~4.3 THz –Comparable to freq. of room-T IR photons A computational subsystem that is at a generalized temperature equal to room temperature can never update its digital state at a higher frequency than this!A computational subsystem that is at a generalized temperature equal to room temperature can never update its digital state at a higher frequency than this!

4. Generic Subsystem Model Models the internal state and the I/O channels for the smallest-scale subsystems in a technology-independent way, based on universal physical/computational concepts.Models the internal state and the I/O channels for the smallest-scale subsystems in a technology-independent way, based on universal physical/computational concepts. –Energy, entropy, temperature, bandwidth… Generic Subsystem Characteristics: Generic Subsystem Characteristics: State space Σ, internal Hamiltonian HState space Σ, internal Hamiltonian H Instantaneous reduced quantum mixed state ρ tInstantaneous reduced quantum mixed state ρ t Internal energy above ground state E x = E − E 0Internal energy above ground state E x = E − E 0 Total physical information content I = log NTotal physical information content I = log N Local entropy content S ≤ I, extropy X = I − SLocal entropy content S ≤ I, extropy X = I − S Internal generalized temp. T i  E x /I.Internal generalized temp. T i  E x /I. I/O Channel interaction energy E io For each direction along the channel: Power transfer P = dE/dt Physical information bandwidth B = dI/dt Generalized channel temperature T c = dE/dI = P/B

5. Generic Subsystems of a CORP Device Coding subsystem: State manipulated to store & process information.Coding subsystem: State manipulated to store & process information. –Logical subsystem: The digital bits being represented. –Redundancy subsystem: Correlated info. for noise immunity & error correction. Non-coding subsystem: The rest of the physical information in the system.Non-coding subsystem: The rest of the physical information in the system. –Power subsystem: Provides extropy and “chill” (clean energy) to replenish losses. –Thermal subsystem: Allowed to vary randomly. Stores and removes entropy and waste heat. –Structural subsystem: Nominally constant for device to function properly. Changes represent degradation/decay of device structure. Device Coding Subsystem Non-coding Subsystem Logical Subsystem Redundancy Subsystem Structural Subsystem Thermal Subsystem Power Subsystem Computing with Optimal, Realistic Physics

6. Thermal Subsystem Subsystem Interactions A CORP device is characterized by interaction energies between its various generic subsystems.A CORP device is characterized by interaction energies between its various generic subsystems. –Some of the most important interactions are shown. Arrows show dominant direction of energy/info. transfer.Arrows show dominant direction of energy/info. transfer. These energies determine the maximum rate of information & entropy transfer between the subsystems.These energies determine the maximum rate of information & entropy transfer between the subsystems. –And also rates of entropy generation, energy dissipation and structural degradation. Each subsystem also interacts with the corresponding subsystems of neighboring devices. (Not shown.)Each subsystem also interacts with the corresponding subsystems of neighboring devices. (Not shown.) –These inter-device interactions are harnessed to propagate energy & information (incl. heat & entropy) through the system in desired flows. For power delivery, communication, synchronization & cooling.For power delivery, communication, synchronization & cooling. Structural Subsystem Coding Subsystem Logical Subsystem Power Subsystem E TS E PC E CT E PT Timing Subsystem E TC E PT