Richard Feynman proposed a quantum computer that is autonomous, that is, a computer that operates without external control. See Quantum Mechanical Computers (pdf) for more information. This computer is composed of 2 registers, a computation register and a counter register. Given a list of unitaries $U_{1}, U_{2}... U_{k}$, the quantum computer can be described by the hamiltonian $$H = \sum_{i=1}^{k-1} a_{i+1}^{\dagger} a_{i} U_{i+1} + a_{i}^{\dagger} a_{i+1} U_{i+1}^{\dagger}$$
The computation goes as follows: The clock register of $k + 1$ qubits is initialized to $|100..00\rangle$ and the computation register initialized to $|\psi_{0}\rangle$. The registers are then evolved according to the hamiltonian. When the clock register reaches the state $|000..01\rangle$, the state of the computation register is $U_{1}, U_{2}... U_{k} |\psi_{0}\rangle$.
How can one perform error correction such a device? One can use external control to reset or replace ancillary qubits. The problem is that this and standard methods of error correction break the autonomy of the Hamiltonian quantum computer.
Relatedly, dabacon came up with a hamiltonian that is self-error-correcting. A hamiltonian $$H = -\lambda \sum_{i,j=1}^n \sum_{k=1}^{n-1} (X_{k,i,j}X_{k+1,i,j} + X_{i,k,j}X_{i,k+1,j} + Z_{i,k,j}Z_{i,k+1,j} + Z_{i,j,k}Z_{i,j,k+1})$$ was argued to be inherently robust to error without the need of active error correction. The problem with this self-error-correcting hamiltonian is that, while it is an autonomomous memory, performing logical operations on the memory requires external control which breaks its autonomy.
How can we combine Feynman's and Bacon's hamiltonians to get autonomous computation and autonomous error correction? Given that Bacon's hamiltonian doesn't allow for universal computation, the new hamiltonian need not be universal, it just needs to perform it's logical operations and error correction autonomously.
