AND takes two bits in and gives one out — the other bit is thrown away, and throwing a bit away, Landauer proved, must release at least kT·ln2 of heat. But that floor is only for forgetting. Build logic that never forgets — bijective gates that keep every bit, so the whole computation can run backward — and the floor vanishes. This body computes by refusing to erase, and recovers the energy on the way out. It is also, not by accident, the classical shadow of the cubi: every quantum gate is reversible too.Landauer & reversible logic. The kT·ln2 erasure cost (Landauer 1961, measured 2012) and universal reversible gates (Toffoli, Fredkin; Bennett 1973) are proven; adiabatic CMOS recovers energy in real chips.
Practical reversible chips. Adiabatic and reversible processors are built — but slow, and burdened with garbage bits and uncomputation. The energy win shows only at low clock rates.
Sub-Landauer at speed. Fast, general computing below the erasure floor is open — and largely lives in the quantum world, which is reversible by nature.
Landauer's principle ties information to heat: erase one bit of information and you must dissipate at least kT·ln2 ≈ 2.87 zJ at room temperature. A normal logic gate erases constantly — two inputs collapse to one output, and the lost input is paid for in heat. Toggle the irreversible gate below and watch the discarded bit leave as a puff of warmth.
The fix is to never discard. The Toffoli gate (controlled-controlled-NOT) takes three bits to three bits: it flips the third only if the first two are both 1, and passes the first two through untouched. Same count in and out, every input recoverable — so it's bijective, runs backward (apply it twice and you're home), and is universal: set the third input to 0 and the output is a AND b, with a and b still in hand. A reversible perceptron is built from these — the threshold computed without throwing the inputs away.
| a b c | → a b c′ |
|---|---|
| 1 1 0 | 1 1 1 |
| 1 0 0 | 1 0 0 |
| 0 1 0 | 0 1 0 |
If you also switch slowly — charge each capacitor gently instead of slamming it — you recover most of the ½CV² you'd normally burn. That's adiabatic logic: dissipation falls as you slow down, trading speed for near-free computation. And here's the quiet endnote of the whole series: because reversible logic and quantum logic are the same demand — keep all the information, run unitary — this body is the classical doorstep of the cubi.
Reversibility doesn't make computing free. You pay in garbage bits (the records you keep so nothing is erased) and in speed (adiabatic means slow). The energy win is real but only cashes out at low clock rates, which is why your laptop isn't reversible. Its true home is where erasure is forbidden anyway — quantum hardware — and at the ultimate-efficiency edge of classical design. A beautiful floor, rarely the fast floor.
And so the road closes where it has to: fourteen bodies, from a branch predictor guessing one bit to a gate that refuses to forget one. The gate kept open the whole way — say what's true, mark what's hoped — and the last word is the cleanest: the only way to compute for free is to never throw anything away.