How to build a hierarchical memory in physical reality.
| Parameter | Symbol | Description |
|---|---|---|
| Branching factor | $b$ | Sub-containers per container |
| Encoding depth | $D$ | Depth of logical encoding |
| Energy scale | $E_0$ | Barrier at root |
| Barrier exponent | $\alpha$ | $\Delta E_k = E_0 \cdot b^{-\alpha k}$ |
| Noise energy | $\varepsilon$ | $k_B T$ for thermal noise |
Required operations: Initialize, Hold, Read, Write, Reset.
Hamiltonian:
$$H = -\sum_{i=1}^{N-1} J_i \, \sigma_i^z \sigma_{i+1}^z - h \sum_{i=1}^{N} \sigma_i^x$$with $J_i = J_0 \cdot b^{-\alpha i}$ — coupling decays exponentially along the chain.
The domain wall position encodes the state. Moving the wall past spin $k$ costs energy $\propto J_k$. Shallow spins have strong coupling (large barriers); deep spins have weak coupling (small barriers).
$J_0/k_B = 10$ K. $J_0/(k_B T) = 100$. At $D=1$: $\Delta E_1 = 5$ K, error rate $\sim 10^{-10}$ s⁻¹. Mean time between errors: $\sim 300$ years.
An organic molecule with rotatable bonds at different barrier scales. Each bond's orientation is one digit in the tree address. High-barrier bonds encode the logical bit; low-barrier bonds absorb thermal jitter.
Example: Biphenyl derivative with barriers at 1500 K, 1000 K, 500 K, and 250 K — a natural hierarchy. Readout via NMR spectroscopy.
Potential engineering via Fourier synthesis:
$$U(\varphi) = -E_J \sum_{m=0}^{M} b^{-\alpha m} \cos(b^m \varphi)$$Each term creates wells at a different scale. Realized with Josephson junction arrays of varying sizes. At $T = 10$ mK and $E_J/h = 50$ GHz, all levels down to $M = 4$ are well protected.
Series of nested Fabry-Perot cavities. A photon in the innermost cavity is most confined. To escape, it must tunnel through multiple partially reflective mirrors — each a barrier. Mirror reflectivities $0.99 \to 0.999 \to 0.9999 \to 0.99999$ create the hierarchy.
| Aspect | Surface Code | Hierarchical Encoding |
|---|---|---|
| Protection | Active syndrome + feedback | Passive energy barriers |
| Physical/logical ratio | $O(d^2)$ | $O(b^D)$, $D \sim \log(1/p)$ |
| Idle energy | Circuits active | Near zero |
| Error accumulation | Between cycles | Below threshold: none |
| Maturity | Demonstrated | Theoretical proposal |