1. Travelling Salesman… Let the home city = the halt state, let every other city = an atomic instruction of a Turing machine’s program.
2. If an instruction can be executed, assign a weight of 1.
3. If an instruction cannot be executed, assign a weight of 2.
4. If and only if the salesman can complete a circuit that visits every city exactly once for a cost of the number of cities, there is some Turing machine that will halt when run with some input.
5. If the number of cities is the same as the number of instructions in the program of the Turing machine evaluating the TSP instance, the evaluation is a self-evaluation.
6. And if we assign the weight 2, rather than 1, then this Turing machine is required to determine that an input can only be run by a more complex program than itself.
7. This is because -unlike polynomial-time problems- NP-complete problems have the same order of complexity as their means of solution – they are self-inputs.
8. The arithmetic hierarchy implied by these limits means that there is a corresponding limit on the size of factoring problems that doesn’t exist in the case of multiplication.
9. And yet Shor’s algorithm implies that quantum computers can factor integers in polynomial time…
10. Verdict: Scalable quantum computers cannot be built – quantum computation is the sole province of non-classical objects.