Programming the Universe

by Seth Lloyd

  The Margolus-Levitin theorem makes calculating the capacity of the ultimate laptop a breeze. The energy that the ultimate laptop has available for computation can be calculated using Einstein’s famous formula E = mc 2, where E is the energy, m is the mass of the laptop, and c is the speed of light. Plugging in our ultimate computer’s mass of one kilogram and the speed of light (300 million meters per second), we find that the ultimate laptop has 100 million billion (1017) joules available with which to perform computations. Measured in a more familiar form of energy, the laptop has about 20 million million (2 × 1013) kilocalories of available energy, equivalent to 100 billion candy bars. That’s a lot of energy.

  Another familiar measure of energy is the amount of energy released in a nuclear explosion. The ultimate laptop has twenty megatons (equivalent to 20 million tons of TNT) of energy available for computation, comparable to the amount of energy released by a large hydrogen bomb. In fact, when it is computing full throttle, using every available calorie to flip bits, the interior of the ultimate computer looks a lot like a nuclear explosion. The elementary particles that register and process information in the ultimate laptop are jiggling around at a temperature of a billion degrees. The ultimate laptop looks like a small piece of the Big Bang. (Packaging technology will have to make considerable progress before anyone will want to put the ultimate laptop on his lap.) Thus, the number of ops per second that our small but powerful computer can perform is a huge number, a million billion billion billion billion billion (1051) ops per second. Intel has a long way to go.

  Just how far does Intel have to go? Recall Moore’s law: over the past half century, the amount of information that computers can process and the rate at which they process it has doubled every eighteen months or so. A variety of technologies—most recently, integrated circuits—have enabled this exponential increase in information-processing power. There is no particular reason that Moore’s law should continue to hold year after year; it is a law of human ingenuity, not of nature. At some point, Moore’s law will break down. In particular, no laptop can calculate faster than the ultimate laptop described above.

  At its current rate of progress, how long will it take the computer industry to produce an ultimate laptop? The power of computers doubles every year and a half or so. Over fifteen years, it doubles ten times, going up by a factor of 1,000. Thus, current computers are a billion times faster than the giant, lumbering electromechanical computers of fifty years ago. Working flat out, current computers perform on the order of a trillion logical operations per second (1012). So (if Moore’s law can be sustained until then) we should be able to buy an ultimate laptop in the store by 2205.

  The amount of energy available for computation limits the speed of the computation. But speed of computation isn’t the only specification you’re concerned with when you’re buying a new laptop. Equally important is the amount of memory space. What is the capacity of the ultimate hard drive?

  The interior of the ultimate laptop is jammed with elementary particles jiggling around like mad at a billion degrees. The same techniques that cosmologists use to measure the amount of information present during the Big Bang can be used to measure the number of bits registered by the ultimate laptop. The jiggling particles of the ultimate laptop register about 10,000 billion billion billion bits (1031). That’s a lot of bits—far more than the amount of information stored on the hard drives of all the computers in the world.

  How long will it take the computer industry to reach the memory specifications of the ultimate laptop? In fact, Moore’s law for memory capacity is currently moving faster than Moore’s law for speed of computation. The capacity of hard drives doubles in a little over a year. At this rate, it will take only about seventy-five years to produce the ultimate hard drive.

  Of course, Moore’s law can be sustained only so long as human ingenuity can continue to overcome obstacles to miniaturization. It’s hard to make wires, transistors, and capacitors smaller, and the smaller you make the components of computers, the harder they are to control. Moore’s law has been declared dead many times in the past, each time because of some such knotty technical problem that seemed unsolvable. But each time it has been declared dead, clever engineers and scientists have found ways to cut the technical knot. Moreover, as we’ve discussed, we have hard experimental evidence that the components of computers can be miniaturized to the size of atoms. Existing quantum computers already store and process information at the atomic scale. At the current rate of miniaturization, Moore’s law is not set to reach the atomic scale for another forty years, so there’s hope for the old law yet.

  The Computational Capacity of the Universe

  Now that we know how much computation can be performed by a piece of matter that can sit on our lap, let’s turn to more powerful computers—like the one envisioned by Isaac Asimov in “The Last Question,” a computer the size of the cosmos itself. Suppose all the matter and energy in the cosmos were put into service to perform computations. How powerful would the resulting computer be? The power of the cosmological computer, consisting of everything in the universe, can be calculated using the same techniques with which we examined the power of the ultimate laptop.

  First of all, energy limits speed. The amount of energy in the universe has been determined to a fairly high degree of accuracy. Much of it is locked up in the mass of atoms. If we count the atoms in all the stars and all the galaxies, adding the matter in interstellar clouds, we find that the overall average density of the universe is about one hydrogen atom per cubic meter.

  There are other forms of energy in the universe, as well. For example, light contains energy (though far less than the energy contained in atoms). The rotation rates of distant galaxies suggest the existence of further, invisible sources of energy. The forms they take are unknown; candidates include such whimsically named constructions as wimps, winos, and machos. The anomalous acceleration of the universe’s expansion suggests the presence of yet another form of energy, currently dubbed “quintessence.” The total amount of energy in these exotic forms appears to be no more than ten times greater than the energy in directly observable matter, which doesn’t make much difference in the total amount of computation the universe can perform.

  Before we calculate the computational power of the universe, let’s be clear about what it is we are calculating. Current observational evidence suggests that the universe is spatially infinite, extending forever in all directions. In a spatially infinite universe, the amount of energy in the universe is also infinite; consequently, the number of ops and number of bits in the universe is infinite, too.

  But observation also tells us that the universe has a finite age: it is slightly less than 14 billion years old. Information can’t travel any faster than the speed of light. Because the universe has a finite age and because the speed of light is finite, the part of the universe about which we can have information is also finite. The part of the universe about which we can have information is said to be “within the horizon.” Beyond the horizon we can only guess as to what is happening. The numbers we will calculate represent the amount of computation that can take place within the horizon. Information processing occurring beyond the horizon cannot affect the result of any computation performed in the observable universe since the Big Bang. So when we calculate “the computational capacity of the universe,” what we are really calculating is “the computational capacity of the universe within the horizon.”

  As time passes, the horizon expands, at three times the speed of light. When we look through a telescope, we also look backward in time, and the most remote objects we can see appear as they were a little under 14 billion years ago. In the intervening time, because of the expansion of the universe, those objects have moved even farther away, and right now they are 42 billion light-years away from us. As the horizon expands, more and more objects swim into view, and the amount of energy available for computation within the horizon increases. The amount of computation that ca
n have been performed within the horizon since the beginning of the universe increases over time.

  The horizon is 42 billion light-years away. On average, every cubic meter of the universe within the horizon contains a mass of about one hydrogen atom. Each hydrogen atom contributes energy E = mc 2. Toting up all the energy in the universe, we find that the universe contains about 100 million billion billion billion billion billion billion billion (1071) joules of energy. Most of this is free energy, available for doing work or performing computation. That’s a lot of calories! To eat that much, you’d have to be the size of the universe itself.

  To get the maximum rate at which the universe can process information, then, apply the Margolus-Levitin theorem: take the amount of energy within the horizon, multiply by 4, and divide by Planck’s constant. The result is that every second, a computer made up of all the energy in the universe could perform 100,000 googol (10105) operations. Over the 14 billion years the universe has been around, this cosmological computer could have performed about 10,000 billion billion googol (10122) ops.

  By comparison, let’s look at the number of ops that have been performed by all the computers on Earth since computers were first constructed. Because of Moore’s law, half this computation has taken place in the last year and a half. (Whenever you have a process that doubles in capacity every year and a half, half that capacity has been generated in the last year and a half.) There are somewhat fewer than a billion computers on Earth. The clock rate of these computers is about a billion cycles per second (i.e., a gigahertz), on average. During each clock cycle, a typical computer performs somewhat fewer than 1,000 elementary operations. A year consists of about 32 million seconds. Over the last year and a half, then, all the computers on Earth have performed somewhat fewer than 10 billion billion billion (1028) ops. Over the entire history of computation on Earth, computers have performed no more than twice this number of ops.

  How many bits of memory space are available to the cosmological computer? Once again, the amount of memory space available is determined by counting the number of bits registered by every atom and every photon. Just as in the calculation of the memory space of the ultimate laptop, this number of bits can be counted using techniques developed by Max Planck a hundred years ago. The result is that the cosmological computer could store 100 billion billion billion billion billion billion billion billion billion billion (1092) bits of information—far greater than the information registered by all of the computers on Earth. The somewhat fewer than a billion earthly computers each have somewhat fewer than 1,000 billion (1012) bits of memory space, on average, so taken together, they register fewer than 1,000 billion billion (1021) bits.

  The cosmological computer can have performed 10122 ops on 1092 bits. Those are big numbers, but I can think of bigger. In fact, when I first calculated the number of ops a computer the size of the universe could have performed, my initial reaction was “Is that all?”

  Yes, that’s all. No computer can have computed more than that, in the whole history of the universe. But it’s also enough. Because of the power of quantum computers to simulate physical systems, a quantum computer that can perform 10122 ops on 1092 bits has enough power to compute everything we can observe. (If you take into account not only the bits that can be stored on elementary particles but those that can be stored using quantum gravity, soon to be described, there might be more bits—10122 bits, to be exact.) These numbers of ops and bits can be interpreted in three ways:

  1. They give upper bounds to the amount of computation that can have been performed by all the matter in the universe since the universe began. As noted, the laws of physics impose fundamental limits on the speed of computation and the number of bits available. The speed of computation is limited by the amount of energy available, and the number of bits is limited by this energy together with the size of the system doing the computation. The size of the universe and the amount of energy in it are known to a fairly high degree of accuracy. No computer that obeys the laws of physics could have computed more.

  2. They give lower bounds to the number of ops and bits required to simulate the universe with a quantum computer. Earlier, we saw that quantum computers are particularly efficient at simulating other quantum systems. To perform such a simulation, a quantum computer needs at least the same number of bits as the system to be simulated. In addition, to simulate each elementary event that occurs in the simulated system—for example, each time an electron moves from here to there—the quantum computer requires at least one op. A quantum computer that simulates the universe as a whole must have at least as many bits as there are in the universe and must perform at least as many ops as the number of elementary events (or ops) that have occurred since the universe began.

  3. The third interpretation is more controversial. If you choose to regard the universe as performing a computation, it could have performed 10122 ops on 1092 bits since its beginning. Whether or not you choose to regard the universe as performing a computation is to some degree a question of taste. To say the universe has performed 10122 ops requires you to define an op in terms of fundamental physical processes. In a computer, an op occurs when the computer flips a bit. (In some logic operations, such as an AND operation, the computer flips a bit or not, depending on the state of several other bits.) Here, we’ll say that a physical system performs an op whenever it applies enough energy for enough time to flip a bit. With this simple physical definition of an op, the number of ops performed by any physical system, including the universe, can be calculated using the Margolus-Levitin theorem.

  As time goes on, our horizon expands and the amount of energy available to register bits of information and perform computation increases. The total number of ops performed and the number of bits grow as a function of the age of the universe. In the Standard Model of cosmology, the total amount of energy within the horizon grows in direct proportion to the age of the universe. Since the rate of information processing is proportional to the energy available, the number of ops per second that the universe can perform within the horizon is also growing in proportion to the age of the universe. The total number of ops the universe has performed since the beginning is proportional to the number of ops per second times the age of the universe; that is, the total number of ops the universe has performed in the entire time since the Big Bang is proportional to the square of that time.

  Similarly, conventional cosmology dictates that the number of bits within the horizon grows as the age of the universe raised to the three-quarters power. The information-processing power of the universe grows steadily with time. The future looks rosy.

  So What?

  We know how the universe is computing. We know how much the universe is computing. “So what?” you may ask. “Just what does this picture of the universe as a quantum computer buy me that I didn’t already have?” After all, we have a perfectly good quantum-mechanical theory of elementary particles. So what if these particles are also processing information and performing computations? Do we really need a whole new paradigm for thinking about how the universe operates?

  These are reasonable questions. Let’s start with the last. The conventional picture of the universe in terms of physics is based on the paradigm of the universe as a machine. Contemporary physics is based on the mechanistic paradigm, in which the world is analyzed in terms of its underlying mechanisms; in fact, the mechanistic paradigm is the basis for all of modern science. A beautiful expression of it can be found in the opening paragraphs of Thomas Hobbes’s Leviathan, his massive treatise on the political state:

  Nature, the art whereby God hath made and governs the world, is by the art of man, as in many other things, so in this also imitated, that it can make an artificial animal. For seeing life is but a motion of limbs, the beginning whereof is in some principal part within; why may we not say, that all automata (engines that move themselves by springs and wheels as doth a watch) have an artificial life? For what is the heart, but a spring; and the nerves, but
so many strings, and the joints, but so many wheels, giving motion to the whole body, such as was intended by the artificer? Art goes yet further, imitating that rational and most excellent work of nature, man.

  Paradigms are highly useful. They allow us to think about the world in a new way, and thinking about the world as a machine has allowed virtually all advances in science, including physics, chemistry, and biology. The primary quantity of interest in the mechanistic paradigm is energy. This book advocates a new paradigm, an extension of the powerful mechanistic paradigm: I suggest thinking about the world not simply as a machine, but as a machine that processes information. In this paradigm, there are two primary quantities, energy and information, standing on an equal footing and playing off each other.

  Just as thinking about the body in terms of clockwork allowed insight into physiology (and in Hobbes’s case, into the inner workings of the body politic), the computational universe paradigm allows new insights into the way the universe works. Perhaps the most important new insight afforded by thinking of the world in terms of information is the resolution of the problem of complexity. The conventional mechanistic paradigm gives no simple answer to the question of why the universe in general, and life on Earth in particular, is so complex. In the computational universe, by contrast, the innate information-processing power of the universe systematically gives rise to all possible types of order, simple and complex.

  A second insight provided by the computational universe pertains to the question of how the universe began in the first place. As noted, one of the great outstanding problems of physics is the problem of quantum gravity. In the beginning of the twentieth century, Albert Einstein proposed a beautiful theory for gravity called general relativity. General relativity, one of the most elegant physical theories of all time, accounts for many of the observed features of the universe at large scales. Quantum mechanics accounts for virtually all observed features of the universe at small scales. But to give a full picture of how the universe began, back when it was new, tiny, and incredibly energetic, requires a theory that unifies general relativity and quantum mechanics, two monumentally useful and undeniably correct theories that are basically incompatible.