Ultimate physical limits to computation

In the paper

“Combining this formula with the formula 2 E/π ¯ h for the number of logical operations that can be performed per second, we see that when it is using all its memory, the number of operations per bit per second that our ultimate laptop can perform is k B 2 ln 2 E/π ¯ hS ∝ k B T/ ¯ h , where T = ( ∂S/∂E ) -1 is the temperature of a kilogram of matter in a maximum entropy in a liter volume.”

In the paper

“Accordingly, the ultimate laptop can perform a maximimum of 5 . 4258 × 10 50 operations per second.”

• Also stated in: § 1.3 Ultimate limits to speed of computation · p. 5 · plain

  “Applying this result to a one kilogram computer with energy E = mc 2 = 8 . 9874 × 10 16 joules show that our ultimate laptop can perform a maximum of 5 . 4258 × 10 50 operations per second.”

In the paper

“Note that in this parallel case, the overall spread in energy of the computer as a whole is considerably smaller than the average energy: in general ∆ E = √∑ ℓ ∆ E 2 ℓ ≈ √ N ∆ E ℓ while E = ∑ E ℓ ≈ NE ℓ .”

In the paper

“Integrating the relationship T = ( ∂S/∂E ) -1 over E yields T = CE/S , where C is a constant of order unity (e.g., C = 4 / 3 for black-body radiation, C = 3 / 2 for an ideal gas, and C = 1 / 2 for a black hole).”

In the paper

“A one kilogram computer has average energy E = mc 2 = 8 . 9874 × 10 16 joules.”

In the paper

“That is, applying the potential effects a NOT operation in a time that attains the limit given by quantum mechanics.”

In the paper

“Conventional laptops operate much more slowly than the ultimate laptop.”

In the paper

“Second, a conventional computer employs many degrees of freedom (billions and billions of electrons) for registering a single bit.”

In the paper

“In a computer that has been compressed to its Schwarzschild radius, the energy per bit is E/I = mc 2 /I = ln2¯ hc 3 / 4 πmG = ln2 k B T/ 2, where T = ( ∂S/∂E ) -1 = ¯ hc/ 4 πk B R S is the temperature of the Hawking radiation emitted by the hole.”

In the paper

“At an input/output rate of 10 12 bits per second, an Avogadro-scale computer with 10 23 bits would take about 10 , 000 years to perform a serial read/write operation on the entire memory.”

In the paper

“When the ultimate laptop is using all its memory space it can perform 2 ln 2 k B E/π ¯ hS = 3ln2 k B T/ 2 π ¯ h ≈ 10 19 operations per bit per second.”

In the paper

“The Schwarzschild radius of a 1 kilogram computer is R S = 2 Gm/c 2 = 1 . 485 × 10 -27 meters.”

In the paper

“The StefanBoltzmann law for black-body radiation then implies that the number of bits per unit area than can be sent out to the environment is B = π 2 k 3 B T 3 / 60 ln 2¯ h 3 c 2 = 7 . 195 × 10 42 bits per meter 2 per second.”

In the paper

“Although less than the ≈ 10 51 operations per second in the ultimate laptop, the maximum information processing rate in 'ordinary matter' is still quite respectable.”

In the paper

“The maximum rate at which information can be rejected to the environment is, up to a geometric factor, ln 2 cS/R (all bits in the computer moving outward at the speed of light).”

In the paper

“The amount of information that can be stored by the 1 kilogram computer in the black-hole limit is 3 . 827 × 10 16 bits.”

In the paper

“In other words, according to a distant observer, the amount of time it takes to flip a bit, tfl ip , is on the same order as the amount of time t com = πR S /c it takes to communicate from one side of the hole to the other by going around the horizon: t com /tfl ip = ln 2 /π .”

In the paper

“Using Page's results on the rate of evaporation of a black hole 88 , we obtain a lifetime for the hole t life = G 2 m 3 / 3 C ¯ hc 4 , where C is a constant that is depends on the number of species of particles with a mass less than k B T , where T is the temperature of the hole.”

What the sources say

• Not checked yetcites [88] Pagels, H. The Cosmic Code: Quantum Physics as the Language of Nature (Simon and Schuster, New York, 1982).

In the paper

“As a result, the time it takes to flip a bit on average is tfl ip = π ¯ hI/ 2 E = π 2 R S /c ln 2.”

In the paper

“The average time it takes a bit to flip in the ultimate laptop is the inverse of the number of operations per bit per second calculated above: tfl ip = π ¯ hS/k B 2 ln 2 E .”

In the paper

“As the computer gets smaller, keeping the energy fixed, the energy density inside the computer goes up.”

In the paper

“Note that S depends only insensitively on the total number of species with mass less than k B T/ 2 c 2 .”

In the paper

“To perform a quantum logic operation between two spins takes a time π ¯ h/ 2 E γ , where E γ is the energy of interaction between the two spins.”

In the paper

“If one collides 100 on 100 nucleons at 200 GeV per nucleon, the operation time is π ¯ h/ 2 E ≈ 10 -29 seconds.”

In the paper

“As the formula for S in terms of T shows, each species contributes (2 π ) 5 r ℓ / 90 ln 2 ≈ 10 2 bits of memory space per cubic thermal wavelength λ 3 T where λ T = 2 π ¯ hc/k B T .”

In the paper

“In fact, contemporary quantum 'microcomputers' such as those constructed using nuclear magnetic resonance 76 -80 do indeed operate at the limits of speed and memory space described above.”

What the sources say

• Not checked yetcites [76] Cory, D. G., Fahmy, A. F. & Havel, T. F. in Proceedings of the Fourth Workshop on Physics and Computation—PhysComp96 (eds Toffoli, T., Biafore, M. & Leão, J.) 87–91 (New England Complex Systems Institute, Boston, MA, 1996).
• Not checked yetcites [77] Gershenfeld, N. A. & Chuang, I. L. Bulk spin-resonance quantum computation. Science 275, 350– 356 (1997). source
• Not checked yetcites [78] Chuang, I. L., Vandersypen, L. M. K., Zhou, X., Leung, D. W. & Lloyd, S. Experimental realization of a quantum algorithm. Nature 393, 143– 146 (1998). source
• Not checked yetcites [79] Jones, J. A., Mosca, M. & Hansen, R. H. Implementation of a quantum search algorithm on a quantum computer. Nature 393, 344– 346 (1998). source
• Not checked yetcites [80] Chuang, I. L., Gershenfeld, N. & Kubinec, M. Experimental implementation of fast quantum searching . Phys. Rev. Lett. 80, 3408– 3411 (1998). source

In the paper

“Suppose that each logical operation performed by the ultimate computer has a probability ǫ of being erroneous. The total number of errors committed by the ultimate computer per second is then 2 ǫE/π ¯ h .”

In the paper

“Accordingly, the amount of information that can be stored in a black hole is I = 4 πGm 2 / ln 2¯ hc = 4 πm 2 / ln 2 m 2 P .”

In the paper

“Since the ultimate laptop has a surface area of 10 -2 square meters and is performing ≈ 10 50 operations per second, it must have an error rate of less than 10 -10 per operation in order to avoid over-heating.”

In the paper

“During the collision, then, there is time to perform approximately 10 4 operations on 10 4 bits a relatively simple computation.”

In the paper

“The strength of the interaction between the particles, e 2 /r , determines the amount of time t flip = π ¯ hr/ 2 e 2 it takes to perform a quantum logic operation such as a ControlledNOT on the two particles.”

In the paper

“Interestingly, the time it takes to perform such an operation divided by the amount of time it takes to send a signal at the speed of light between the bits t com = r/c is a universal constant, t flip /t com = π ¯ hc/ 2 e 2 = π/ 2 α , where α = e 2 / ¯ hc ≈ 1 / 137 is the fine structure constant.”

In the paper

“The collision time is short: in the center of mass frame the two nuclei are Lorentz contracted to D/γ where D = 12 -13 fermi and γ = 100, giving a total collision time of ≈ 10 -25 seconds.”

In the paper

“A one-kilogram computer that has been compressed to the black hole limit of R S = 2 Gm/c 2 = 1 . 485 × 10 -27 meters can perform 5 . 4258 × 10 50 operations per second on its I = 4 πGm 2 / ln 2¯ hc = 3 . 827 × 10 16 bits.”

In the paper

“In particular, the speed with which a physical device can process information is limited by its energy 11 -26 and the amount of information that it can process is limited by the number of degrees of freedom it possesses 5 -40 .”

What the sources say

• Not checked yetcites 5 -40

In the paper

“Although an Avogadro scale computer that can act on 10 23 bits might seem implausible, prototype quantum computers that store and process information on individual atoms have already been demonstrated 64 -65 , 76 -80 .”

What the sources say

• Not checked yetcites [64] Turchette, Q. A., Hood, C. J., Lange, W., Mabuchi, H. & Kimble, H. J. Measurement of conditional phase-shifts for quantum logic. Phys. Rev. Lett. 75, 4710– 4713 (1995). source
• Not checked yetcites [65] Monroe, C., Meekhof, D. M., King, B. E., Itano, W. M. & Wineland, D. J. Demonstration of a fundamental quantum logic gate. Phys. Rev. Lett. 75, 4714–4717 (1995). source
• Not checked yetcites [76] Cory, D. G., Fahmy, A. F. & Havel, T. F. in Proceedings of the Fourth Workshop on Physics and Computation—PhysComp96 (eds Toffoli, T., Biafore, M. & Leão, J.) 87–91 (New England Complex Systems Institute, Boston, MA, 1996).
• Not checked yetcites [77] Gershenfeld, N. A. & Chuang, I. L. Bulk spin-resonance quantum computation. Science 275, 350– 356 (1997). source
• Not checked yetcites [78] Chuang, I. L., Vandersypen, L. M. K., Zhou, X., Leung, D. W. & Lloyd, S. Experimental realization of a quantum algorithm. Nature 393, 143– 146 (1998). source
• Not checked yetcites [79] Jones, J. A., Mosca, M. & Hansen, R. H. Implementation of a quantum search algorithm on a quantum computer. Nature 393, 344– 346 (1998). source
• Not checked yetcites [80] Chuang, I. L., Gershenfeld, N. & Kubinec, M. Experimental implementation of fast quantum searching . Phys. Rev. Lett. 80, 3408– 3411 (1998). source

In the paper

“In particular, the speed with which a physical device can process information is limited by its energy 11 -26 and the amount of information that it can process is limited by the number of degrees of freedom it possesses 5 -40 .”

What the sources say

• Not checked yetcites [11] Lebedev, D. S. & Levitin, L. B. Information transmission by electromagnetic field. Inform. Control 9, 1–22 (1966). source
• Not checked yetcites [12] Levitin, L. B. in Proceedings of the 3rd International Symposium on Radio Electronics part 3, 1–15 (Varna, Bulgaria, 1970).
• Not checked yetcites [13] Levitin, L. B. Physical limitations of rate, depth, and minimum energy in information processing . Int. J. Theor. Phys. 21, 299– 309 (1982). source
• Not checked yetcites [14] Levitin, L. B. Energy cost of information transmission (along the path to understanding) . Physica D 120, 162–167 (1998). source
• Not checked yetcites [15] Margolus, N. & Levitin, L. B. in Proceedings of the Fourth Workshop on Physics and Computation—PhysComp96 (eds Toffoli, T., Biafore, M. & Leão, J.) (New England Complex Systems Institute, Boston, MA, 1996).
• Not checked yetcites [16] Margolus, N. & Levitin, L. B. The maximum speed of dynamical evolution. Physica D 120, 188– 195 (1998). source
• Not checked yetcites [17] Bremermann, H. J. in Self-Organizing Systems (eds Yovits, M. C., Jacobi, G. T. & Goldstein, G. D.) 93–106 (Spartan Books, Washington DC, 1962).
• Not checked yetcites [18] Bremermann, H. J. in Proceedings of the Fifth Berkeley Symposium on Mathematical Statistics and Probability (eds LeCam, L. M. & Neymen, J.) Vol. 4, 15–20 (Univ. California Press, Berkeley, CA, 1967).
• Not checked yetcites [19] Bremermann, H. J. Minimum energy requirements of information transfer and computing. Int. J. Theor. Phys. 21, 203–217 (1982). source
• Not checked yetcites [20] Bekenstein, J. D. Universal upper bound on the entropy-to-energy ration for bounded systems . Phys. Rev. D 23, 287– 298 (1981). source

In the paper

“In fact, as will now be shown, a great deal can be determined concerning the ultimate physical limits of computation simply from knowledge of the speed of light, c = 2 . 9979 × 10 8 meters per second, Planck's reduced constant, ¯ h = 1 . 0545 × 10 -34 joule seconds, and the gravitational constant, G = 6 . 673 × 10 -11 meters cubed per kilogram second squared.”

In the paper

“Although existing quantum computers are small and simple, able to perform a few hundred operations on fewer than ten quantum bits or 'qubits,' the fact that they work at all indicates that there is nothing in the laws of physics that forbids the construction of an Avogadro-scale computer.”

In the paper

“Energy limits speed. Entropy limits memory.”

In the paper

“The energy used to perform a logical operation can be 'borrowed' from a store of free energy such as a battery, 'invested' in the logic gate that performs the operation, and returned to storage after the operation has been performed, with a net 'profit' in the form of processed information.”

In the paper

“An interesting feature of this limit is that it is independent of computer architecture.”

In the paper

“As a consequence, a system with average energy E can perform a maximum of 2 E/π ¯ h logical operations per second.”

In the paper

“As will be seen below, the degree of parallelizability of the computation to be performed determines the most efficient distribution of energy among the parts of the computer.”

In the paper

“As will now be shown, to perform an elementary logical operation in time ∆ t requires an average amount of energy E ≥ π ¯ h/ 2∆ t .”

In the paper

“In other words, the rate at which a computer can compute is limited by its energy.”

In the paper

“In general, a system with N accessible states can register log 2 N bits of information.”

In the paper

“More complicated logic operations that cycle through a larger number of quantum states (such as those on non-binary or continuous quantum variables) can be performed at a rate E/π ¯ h - half as fast as the simpler operations 15 -16 .”

What the sources say

• Not checked yetcites [15] Margolus, N. & Levitin, L. B. in Proceedings of the Fourth Workshop on Physics and Computation—PhysComp96 (eds Toffoli, T., Biafore, M. & Leão, J.) (New England Complex Systems Institute, Boston, MA, 1996).
• Not checked yetcites [16] Margolus, N. & Levitin, L. B. The maximum speed of dynamical evolution. Physica D 120, 188– 195 (1998). source

In the paper

“Accordingly, the total number of bits of memory space available to the computer is S ( E,V ) /k B ln 2, where S ( E,V ) is the thermodynamic entropy of a system with expectation value of the energy E confined to volume V .”

In the paper

“The amount of information that can be registered by a physical system is I = S ( E ) /k B ln 2, where S ( E ) is the thermodynamic entropy of a system with expectation value for the energy E .”

In the paper

“The canonical ensemble tells how many bits of memory are available for all possible ways of programming the computer while keeping its average energy equal to E .”

In the paper

“By embedding a Controlled-ControlledNOT gate in a quantum context, it is straightforward to see that AND and FANOUT , like NOT , can be performed at a rate 2 E/π ¯ h times per second, where E is the average energy of the logic gate that performs the operation.”

In the paper

“More recently, Margolus and Levitin 15 -16 extended this result to show that a quantum system with average energy E takes time at least ∆ t = π ¯ h/ 2 E to evolve to an orthogonal state.”

What the sources say

• Not checked yetcites [15] Margolus, N. & Levitin, L. B. in Proceedings of the Fourth Workshop on Physics and Computation—PhysComp96 (eds Toffoli, T., Biafore, M. & Leão, J.) (New England Complex Systems Institute, Boston, MA, 1996).
• Not checked yetcites [16] Margolus, N. & Levitin, L. B. The maximum speed of dynamical evolution. Physica D 120, 188– 195 (1998). source

In the paper

“In 1973, Bennett 28 -30 showed that all computations could be performed using reversible logical operations only.”

What the sources say

• Not checked yetcites [28] Bennett, C. H. Logical reversibility of computation. IBM J. Res. Dev. 17, 525–532 (1973). source
• Not checked yetcites [29] Bennett, C.H. Thermodynamics of computation—a review. Int. J. Theor. Phys. 21, 905–940 ( 1982). source
• Not checked yetcites [30] Bennett, C. H. Demons, engines and the second law. Sci. Am. 257, 108 (1987). source

In the paper

“The laws of physics do not require redundancy to perform logical operations: recently constructed quantum microcomputers use one quantum degree of freedom for each bit and operate at the Heisenberg limit ∆ t = π ¯ h/ 2∆ E for the time needed to flip a bit 64 -65 , 76 -80 .”

In the paper

“The amount of information that a physical system can store and process is related to the number of distinct physical states accessible to the system.”

In the paper

“But it has been known for more than a century that the number of accessible states of a physical system, W , is related to its thermodynamic entropy by the formula: S = k B ln W , where k B is Boltzmann's constant.”

In the paper

“Accordingly, the temperature governs the number of operations per bit per second, k B ln 2 E/ ¯ hS ≈ k B T/ ¯ h , that a system can perform.”

In the paper

“It is straightforward to show 15 -16 that no quantum system with average energy E can move to an orthogonal state in a time less than ∆ t . That is, the speed with which a logical operation can be performed is limited not only by the spread in energy, but by the average energy.”

What the sources say

• Not checked yetcites [15] Margolus, N. & Levitin, L. B. in Proceedings of the Fourth Workshop on Physics and Computation—PhysComp96 (eds Toffoli, T., Biafore, M. & Leão, J.) (New England Complex Systems Institute, Boston, MA, 1996).
• Not checked yetcites [16] Margolus, N. & Levitin, L. B. The maximum speed of dynamical evolution. Physica D 120, 188– 195 (1998). source

In the paper

“Consequently, by Landauer's principle, computation does not require dissipation.”

In the paper

“Essentially, the one-to-one dynamics of Hamiltonian systems implies that when a bit is erased the information that it contains has to go somewhere.”

In the paper

“In particular, the correct interpretation of the time-energy Heisenberg uncertainty principle ∆ E ∆ t ≥ ¯ h is not that it takes time ∆ t to measure energy to an accuracy ∆ E (a fallacy that was put to rest by Aharonov and Bohm 23 -24 ) but rather that that a quantum state with spread in energy ∆ E takes time at least ∆ t = π ¯ h/ 2∆ E to evolve to an orthogonal (and hence distinguishable) state 2”

What the sources say

• Not checked yetcites [23] Aharonov, Y. & Bohm, D. Time in the quantum theory and the uncertainty relation for the time and energy domain. Phys. Rev. 122, 1649–1658 ( 1961). source
• Not checked yetcites [24] Aharonov, Y. & Bohm, D. Answer to Fock concerning the time-energy indeterminancy relation. Phys. Rev. B 134, 1417–1418 (1964). source
• Not checked yetcites [25] Anandan, J. & Aharonov, Y. Geometry of quantum evolution. Phys. Rev. Lett. 65, 1697–1700 (1990). source
• Not checked yetcites [26] Peres, A. Quantum Theory: Concepts and Methods (Kluwer, Hingham, 1995).

In the paper

“Human ingenuity has proved great in the past, however, and before writing off physical limits as unattainable, one should realize that certain of these limits have already been attained within a circumscribed context in the construction of working quantum computers.”

In the paper

“The problem of how much energy is required for information processing was first investigated in the context of communications theory by Levitin 11 -16 , Bremermann 17 -19 , Beckenstein 20 -22 and others, who showed that the laws of quantum mechanics determine the maximum rate at which a system with spread in energy ∆ E can move from one distinguishable state to another.”

What the sources say

• Not checked yetcites [11] Lebedev, D. S. & Levitin, L. B. Information transmission by electromagnetic field. Inform. Control 9, 1–22 (1966). source
• Not checked yetcites [12] Levitin, L. B. in Proceedings of the 3rd International Symposium on Radio Electronics part 3, 1–15 (Varna, Bulgaria, 1970).
• Not checked yetcites [13] Levitin, L. B. Physical limitations of rate, depth, and minimum energy in information processing . Int. J. Theor. Phys. 21, 299– 309 (1982). source
• Not checked yetcites [14] Levitin, L. B. Energy cost of information transmission (along the path to understanding) . Physica D 120, 162–167 (1998). source
• Not checked yetcites [15] Margolus, N. & Levitin, L. B. in Proceedings of the Fourth Workshop on Physics and Computation—PhysComp96 (eds Toffoli, T., Biafore, M. & Leão, J.) (New England Complex Systems Institute, Boston, MA, 1996).
• Not checked yetcites [16] Margolus, N. & Levitin, L. B. The maximum speed of dynamical evolution. Physica D 120, 188– 195 (1998). source
• Not checked yetcites [17] Bremermann, H. J. in Self-Organizing Systems (eds Yovits, M. C., Jacobi, G. T. & Goldstein, G. D.) 93–106 (Spartan Books, Washington DC, 1962).
• Not checked yetcites [18] Bremermann, H. J. in Proceedings of the Fifth Berkeley Symposium on Mathematical Statistics and Probability (eds LeCam, L. M. & Neymen, J.) Vol. 4, 15–20 (Univ. California Press, Berkeley, CA, 1967).
• Not checked yetcites [19] Bremermann, H. J. Minimum energy requirements of information transfer and computing. Int. J. Theor. Phys. 21, 203–217 (1982). source
• Not checked yetcites [20] Bekenstein, J. D. Universal upper bound on the entropy-to-energy ration for bounded systems . Phys. Rev. D 23, 287– 298 (1981). source

In the paper

“Interestingly, although this hypothetical computation is performed at ultra-high densities and speeds, the total number of bits available to be processed is not far from the number available to current computers operating in more familiar surroundings.”

In the paper

“In other words, to describe behavior at length scales of the size ℓ P = √ ¯ hG/c 3 = 1 . 616 × 10 -35 meter, time scales t P = √ ¯ hG/c 5 = 5 . 391 × 10 -44 second, and mass scales of m P = √ ¯ hc/G = 2 . 177 × 10 -8 kilograms, a unified theory of quantum gravity is required.”

In the paper

“In particular, there is no physical reason why it should not be possible to store one bit of information per atom.”

In the paper

“in order to take full advantage of the memory space available, the ultimate laptop must turn all its matter into energy.”

In the paper

“Clearly, packaging issues alone make it unlikely that this limit can be obtained, even setting aside the difficulties of stability and control.”

In the paper

“That is, the amount of time it takes to communicate from one side of the computer to the other, divided by the amount of time it takes to flip a bit, is approximately equal to the ratio between the size of the system and its thermal wavelength.”

In the paper

“Generally speaking, if the computation to be performed is highly parallelizable or requires many bits of memory, the volume of the computer should be greater and the energy available to perform the computation should be spread out evenly amongst the different parts of the computer.”

In the paper

“In practice, however, any computer - even our ultimate laptop - will dissipate energy.”

In the paper

“Conversely, if the computation to be performed is highly serial and requires fewer bits of memory, the energy should be concentrated in particular parts of the computer.”

In the paper

“In contrast to computation at lesser densities, which is highly parallel as noted above, computation at the horizon of a black hole is highly serial: every bit is essentially connected to every other bit over the course of a single logic operation.”

In the paper

“Only at the ultimate limit of compression - a black hole - is the computation entirely serial.”

In the paper

“As the discussion of error correction above indicates, the rate at which errors can be detected and rejected to the environment by error correction routines puts a fundamental limit on the rate at which errors can be committed.”

In the paper

“A greater degree of serial computation can be obtained at the cost of decreasing memory space by compressing the size of the computer or making the distribution of energy more uneven.”

In the paper

“In other words, the ultimate limits to memory space may prove easier to approach than the ultimate limits to speed.”

In the paper

“If our computer is subject to an error rate of ǫ bits per second, for example, then error-correcting codes can be used to detect those errors and reject them to the environment at a dissipative cost of ǫk B T E ln 2 joules per second, where T E is the temperature of the environment.”

In the paper

“At a particular temperature T , the entropy is dominated by the contributions from particles with mass less than k B T/ 2 c 2 .”

In the paper

“A typical state of the ultimate laptop's memory looks like a plasma at a billion degrees Kelvin: the laptop's memory looks like a thermonuclear explosion or a little piece of the Big Bang!”

In the paper

“Such error-correcting routines in our ultimate computer function as working analogs of Maxwell's demon, getting information and using it to reduce entropy at an exchange rate of k B T ln 2 joules per bit.”

In the paper

“( k B T ln 2 is the minimal amount of energy required to send a bit down an information channel with noise temperature T . 14 )”

What the sources say

• Not checked yetcites [14] Levitin, L. B. Energy cost of information transmission (along the path to understanding) . Physica D 120, 162–167 (1998). source

In the paper

“The ultimate laptop is highly parallel.”

In the paper

“This is because the entropy of a black hole has a well-defined value.”

What the sources say

• Not checked yetcites [90] Preskill, J. Quantum hair. Phys. Scr. T 36, 258– 264 (1991). source

In the paper

“The thermal load of correcting large numbers of errors clearly suggests the necessity of operating at a slower speed than the maximum allowed by the laws of physics.”

In the paper

“As noted above, the serial nature of computation at the black-hole limit can be deduced from the fact that black holes attain the Beckenstein bound 20 -22 , k B RE/ ¯ hcS = 1 / 2 π .”

What the sources say

• Not checked yetcites [20] Bekenstein, J. D. Universal upper bound on the entropy-to-energy ration for bounded systems . Phys. Rev. D 23, 287– 298 (1981). source
• Not checked yetcites [21] Bekenstein, J. D. Energy cost of information transfer. Phys. Rev. Lett. 46, 623–626 (1981). source
• Not checked yetcites [22] Bekenstein, J. D. Entropy content and information flow in systems with limited energy. Phys. Rev. D 30, 1669–1679 (1984). source

In the paper

“In principle, computation does not require dissipation.”

In the paper

“The ultimate laptop must reject errors to the environment at a high rate to maintain reliable operation.”

In the paper

“If a substantial fraction of these nuclei can be made to register a bit, then one can get quite close to the ultimate physical limit of memory without having to resort to thermonuclear explosions.”

In the paper

“Although its computational machinery is in fact in a highly specified physical state with zero entropy, while it performs a computation that uses all its resources of energy and memory space it appears to an outside observer to be in a thermal state at ≈ 10 9 degrees Kelvin.”

In the paper

“At the black-hole limit, computation is fully serial: the time it takes to flip a bit and the time it takes a signal to communicate around the horizon of the hole are the same.”

In the paper

“Although computational universality might at first seem to be a stringent demand on a physical system, a wide variety of physical systems - ranging from nearest neighbor Ising models 52 to quantum electrodynamics 84 and conformal field theories 86 - are known to be computationally universal 51 -53 , 55 -65 .”

What the sources say

• Not checked yetcites [51] Lloyd, S. Quantum-mechanical computers and uncomputability. Phys. Rev. Lett. 71, 943–946 ( 1993). source
• Not checked yetcites [52] Lloyd, S. A potentially realizable quantum computer. Science 261, 1569–1571 (1993). source
• Not checked yetcites [53] Lloyd, S. Necessary and sufficient conditions for quantum computation. J. Mod. Opt. 41, 2503–2520 (1994). source
• Not checked yetcites [55] Lloyd, S. Quantum-mechanical computers. Sci. Am. 273, 140–145 (1995). source
• Not checked yetcites [56] DiVincenzo, D. Quantum computation. Science 270, 255– 261 (1995). source
• Not checked yetcites [57] DiVincenzo, D. P. 2-Bit gates are universal for quantum computation. Phys. Rev. A 51, 1015–1022 ( 1995). source
• Not checked yetcites [58] Sleator, T. & Weinfurter, H. Realizable universal quantum logic gates. Phys. Rev. Lett. 74, 4087– 4090 (1995). source
• Not checked yetcites [59] Barenco, A. et al. Elementary gates for quantum computation. Phys. Rev. A 52, 3457–3467 ( 1995). source
• Not checked yetcites [60] Lloyd, S. Almost any quantum logic gate is universal. Phys. Rev. Lett. 75, 346–349 (1995). source
• Not checked yetcites [61] Deutsch, D., Barenco, A. & Ekert, A. Universality in quantum computation. Proc. R. Soc. Lond. A 449, 669–677 (1995). source

In the paper

“The rate at which the components of a computer can communicate is limited by the speed of light.”

In the paper

“Essentially any quantum system that admits controllable nonlinear interactions can be shown to be computationally universal 60 -61 .”

What the sources say

• Not checked yetcites [60] Lloyd, S. Almost any quantum logic gate is universal. Phys. Rev. Lett. 75, 346–349 (1995). source
• Not checked yetcites [61] Deutsch, D., Barenco, A. & Ekert, A. Universality in quantum computation. Proc. R. Soc. Lond. A 449, 669–677 (1995). source

In the paper

“But no computer can be compressed to smaller than its Schwarzschild radius without becoming a black hole.”

In the paper

“Indeed, computational universality seems to be the rule rather than the exception.”

In the paper

“For example, the ordinary electrostatic interaction between two charged particles can be used to perform universal quantum logic operations between two quantum bits.”