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10⁵⁰ Operations Per Second: Why Physics Says Computers Have a Hard Limit

10⁵⁰ Operations Per Second: Why Physics Says Computers Have a Hard Limit

PhysicsComputingScienceFundamental Theory

Sources：Lobsters + web research

The phone in your pocket performs roughly 5 billion operations per second. Fifty years ago, that number required an entire room full of machinery. This kind of exponential progress makes it feel like computers can just keep getting faster forever, with no end in sight.

But physics disagrees.

In 1962, a mathematician named Hans-Joachim Bremermann took two pens — Einstein’s mass-energy equivalence and quantum mechanics’ uncertainty principle — and calculated an iron gate: one kilogram of matter, no matter what form of computer you shape it into, can perform at most roughly 1.36 × 10⁵⁰ elementary operations per second. Not one more. The laws of physics don’t allow it.

That number is staggeringly large, but it is “hard” — it doesn’t come from engineering bottlenecks, material limitations, or heat dissipation problems. It is a direct consequence of the fundamental constants of the universe. It’s like the physical maximum speed on a highway: determined by the friction between tire rubber and the road, not by a speed limit sign erected by a traffic cop. You can install a better engine, a lighter chassis, a smarter driver — but you cannot bypass the friction coefficient.

How a Counterintuitive Number Came to Be

To understand Bremermann’s limit, you only need three things. All three are in any high school physics textbook.

The first is E = mc². It tells us that mass and energy are two sides of the same coin. Locked inside one kilogram of matter is 9 × 10¹⁶ joules of energy — roughly twice the energy released by the Hiroshima atomic bomb. If you could use one kilogram of matter “entirely” for computation, that energy is your total budget.

The second is the Heisenberg uncertainty principle. It has a less frequently cited formulation: energy and time cannot both be precisely determined. In mathematical language, ΔE·Δt ≥ h/4π, where h is Planck’s constant. In plain English: the minimum time a system needs to complete a “state transition” — i.e., perform one computation — depends on how much energy it has available. More energy means each operation can be faster.

The third is combining the first two. Since one kilogram of matter provides at most mc² of energy, and each operation requires at least h/(4π·mc²) of time, take the reciprocal: maximum operations per second = mc² / (h/4π) ≈ mc²/h. Set aside the constant factors — the order of magnitude is this: c² divided by h, about 10⁵⁰.

What I find most beautiful about this is: it’s not an empirical formula. Not a fitted curve. Not a trend line drawn through lab data points. It follows from two iron laws of physics, verified by countless experiments. As long as you accept that E=mc² is correct, and as long as you accept the uncertainty principle is correct, this ceiling necessarily exists — regardless of technology, material, or architecture.

Moore's Law: exponential growth in microprocessor transistor count, 1970–2020 Source: Wikimedia Commons, Moore’s Law Transistor Count 1970-2020

The “Fuel Cost” of Computation: Erasing a Bit Isn’t Free

If Bremermann’s limit governs “how fast you can go,” then a principle discovered in 1961 by another physicist, Rolf Landauer, governs “how much it costs you.”

Landauer was working at IBM. He asked a deceptively simple question: where does the heat in a computer come from? Circuits have resistance and generate heat — that much is obvious. But is there a kind of heat that is generated by the act of computation itself — independent of the data content, independent of the circuit material, independent of manufacturing sophistication?

The answer is yes.

Landauer proved a result that has been repeatedly experimentally verified ever since: every time you erase 1 bit of information, you must release at least kT·ln 2 of energy into the environment as heat. Here, k is Boltzmann’s constant and T is the ambient absolute temperature. At room temperature (~27°C), this comes to about 2.85 × 10⁻²¹ joules — unimaginably tiny, but absolutely not zero.

Why must erasing information generate heat? It follows from the second law of thermodynamics: the entropy of an isolated system cannot decrease. When you merge two bit paths into one — say, writing 0 regardless of whether the original was 0 or 1 — information decreases, entropy increases, and heat must be expelled in some form. Physicists like to say: information is not free. Like fuel, using it leaves “waste heat.”

Interestingly, if computation were completely reversible — every operation’s input could be deduced from its output — then, in theory, no heat would be generated at all. This has spawned a research direction called “reversible computing.” But in practice, the vast majority of computational operations (addition, comparison, conditional branching) discard information, so Landauer’s principle is nearly unavoidable.

Chips are the core vehicle of computation — and the battleground of Bremermann's limit and Landauer's principle Source: Unsplash, photo by Louis Reed

The End of Moore’s Law Isn’t the Destination — It’s Just the First Toll Booth

Many people’s first reaction upon hearing Bremermann’s limit is: “10⁵⁰ operations? The best chips today do about 10¹⁰ — that’s 40 orders of magnitude away. Why worry?”

That reaction isn’t wrong in itself. But here’s the problem: on the road to Bremermann’s limit, the first roadblock we hit turns out to be Landauer’s principle and its engineering cousin — the heat problem.

Moore’s Law has performed spectacularly over the last sixty years: transistor counts on chips doubled every two years. But after 2005, processor clock speeds stopped rising. The best desktop CPU you can buy today still hovers between 3 and 5 GHz — not much different from fifteen years ago. Heat dissipation can’t keep up — and that is the core reason engineers can’t push frequencies higher. Higher frequency means higher power consumption, and higher power consumption means higher heat density. If all the transistors on a modern CPU ran at full speed simultaneously, the heat per unit area would exceed that of an electric stove burner.

This is the phenomenon known as “dark silicon”: chips have vast numbers of transistors, but you can’t light them all up at once — otherwise the chip would burn through itself.

Bremermann’s limit assumes “turn one kilogram of matter into a perfect computer.” In reality, the few hundred grams of silicon, copper wiring, and plastic packaging inside your computer — the transistors that actually do the computation — account for only a tiny fraction of that total mass. The vast majority of mass and energy is either idle or dissipated as heat. We are very far from the Bremermann ceiling, but we are right up against the Landauer floor.

Can Quantum Computers Break These Rules?

Whenever physical limits come up, someone always asks: what about quantum computers? Can they bypass these constraints?

The answer is: no — at least not in the Bremermann and Landauer sense.

Quantum computers are genuinely impressive. By exploiting superposition and entanglement, they can achieve exponential speedups on certain specific problems (like large-number factorization and quantum chemistry simulation). But that doesn’t mean they can ignore physical laws. A qubit is still a piece of matter, and it still obeys E=mc², the uncertainty principle, and the second law of thermodynamics. Bremermann’s limit governs the maximum computational rate of any self-contained physical system — quantum systems are no exception.

However, at the Landauer level, quantum computing has an intriguing possibility. Because quantum logic gate operations can in theory be reversible (the fundamental evolution equations of quantum mechanics are invariant under time reversal), some researchers believe quantum computation could far surpass classical computation in energy efficiency. But this remains an unverified engineering hypothesis, with a long road to practicality.

To put it bluntly: a quantum computer might let you solve certain problems in fewer steps, but it won’t let you perform more than 10⁵⁰ elementary operations per second in the same kilogram of matter.

Engineering Ambition vs. Iron Laws of Physics: A Race You’re Destined to Lose

To me, the most compelling tension in this whole story lies in the asymmetry between human engineering ambition and the iron laws of physics.

We are conditioned to the narrative that “if you try hard enough, you can break through any limit.” The four-minute mile was once considered impossible — then it was broken. The sound barrier was once deemed unbreakable — then it was broken. Stories like these, repeated over and over, create an illusion: that any “limit” is merely temporary.

But Bremermann’s limit and Landauer’s principle are not that kind of limit.

They don’t exist because your materials aren’t good enough, your design isn’t clever enough, or your fabrication isn’t advanced enough. They arise from the structure of the universe itself. The speed of light c, Planck’s constant h, Boltzmann’s constant k — these numbers were not invented by humans, nor can humans modify them. Like gravity, they are the factory settings of the universe we inhabit.

When Bremermann wrote down that formula in 1962, the integrated circuit was only four years old. IBM’s most advanced computer, the System/360, hadn’t even been released yet. He could not possibly have foreseen what today’s chips look like. Yet the upper bound he derived applies to every chip ever built — and to every chip that will be built a hundred years from now.

That is the “brutality” of physical law: it doesn’t negotiate. It doesn’t compromise. It gives you no appeals process.

Flip it around, though, and this is also a form of liberation. Knowing where the ceiling is, you don’t have to lie awake wondering “will we ever catch up?” The ceiling is there. You can focus your energy on more meaningful questions: what fascinating things can we still do before we reach it? How many technologies we haven’t yet invented lie hidden in those 40 orders of magnitude of headroom?

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