Standard one-way quantum computers (1WQC) allow for e.g. Shor, Grover algorithms, however, general NP problems seem too difficult for them(?) - bringing an open question if they could be somehow enhanced to be able to attack NP problems?
A natural direction is search for two-way quantum computers (2WQC). For example Ising model allows to attack NP problems (e.g. here or here). Mathematically Ising is similar to quantum mechanics: Boltzmann path ensemble in space (Ising) vs Feynman path ensemble in time (QM). Being in space we can do it "two-way": mounted in left and right ends.
For QM such two-way is problematic due to requirement of "mounting in future", there is missing some kind of T/CPT analogue of state preparation ... but maybe it could be realized? E.g. stimulated emission-absorption equations are CPT analogs - one allows for state preparation, shouldn't the second allow for CPT(state preparation)?
There are T/CPT analogues in optics, e.g. optical heating-cooling, and pushing-pulling: e.g. in optical tweezers, also EM radiation pressure is $\vec{p}=<\vec{E} \times \vec{H}>/c$ vector - there can be negative radiation pressure allowing to pull e.g. solitons (articles).
Could such e.g. stimulated emission/negative radiation pressure as replacement for measurement allow for 2WQC like in diagram below (Section V here)? Specifically, prepare ensemble of $2^n$ inputs, calculate 3-SAT alternatives for them, and enforce output of all alternatives to "true" restricting the ensemble to satisfying 3-SAT instance?
Are 2WQC considered in literature? Are there different approaches to achieve them? Generally to enhance quantum computers to attack NP problems? If possible, could 2WQC solve general NP problems?
Update: we just had a lot of discussion about 2WQC in r/quantum and r/QuantumComputing - a lot of skepticism, I believe I have handled. To summarize:
State preparation allows to enforce boundary values - is more powerful than measurement taking random boundary values. Quantum computation process is unitary, what means time symmetric, so maybe boundary conditions do not have to be treated in asymmetric way?
Imagine pumped with laser state to excited |1> as example of state preparation. Stimulated emission-absorption (equations in diagram) are CPT analogues, hence using stimulated emission to get |0> state could be used as CPT analogue of state preparation.
As 2WQC would replace some measurements, for Born rule we should use time symmetric e.g. as in scattering matrix: $S_{fi} = \lim_{t_2\to+\infty} \lim_{t_1 \to -\infty} \langle \Phi_f|U|\Phi_i\rangle$ - we prepare initial and final states (with state preparation and its CPT analogue), also unitary propagator realized with quantum gates.
Basic approach: prepare <0|s then Hadamard gates for initial ensemble, and enforce some constraint satisfaction for this ensemble using CPT analogues of state preparation for final states. With additional measurements try to extract the remaining ensemble.
Hypothetical 2WQC should be able to do in one run, what postselected 1WQC does in multiple runs. Some measured qubits of 2WQC can be treated as postselected.
Update: Nice "Quantum entanglement on photonic chips: a review" paper. E.g. b) below is the discussed type of quantum computer, there is "pump" on the left - I am asking about adding "CPT(pump)" on the right using e.g. ring laser above, hopefully to simultaneously kind of pull photons of e.g. chosen polarity through such chip.
