Locally Connected Quantum Computers Without Control Bits

1 Locally Connected Quantum Computers Without Control Bits 

One of the most severe problems with using control bits is that it is not possible to make loops in a simple way. In particular, it is not possible to let exception criteria depend on the state of the computer. However, this is essential for universal computing. Thus, it is necessary to let the path of the computation depend on the state of the computer rather than on the state of the control bits. This is possible Control bits are not necessary to perform local deterministic computation as will now be shown. 

The trick is to somehow substitute the cursor states by more complicated computational states which nevertheless follow unambiguously one after the other. 

Consider a computational state. .., x,y,z,..where x, y, and z each stand for a certain number of quantum bits. The block of the bits x is adjacent to the block of the bits y, and the block of the bits y is adjacent to the block of the bits z. Suppose that the next computational state depends on the state y). Now, divide the possible states of y) into three sets 5o, Si, and S2. 

A reversible and local update rule for the computer reads as follows. If y) Si, then 

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