I'm wondering if one can (potentially) get any quantum advantage with $R_y$ single-qubit rotations and $CNOT$s only? (Note that I don't care about having a universal quantum computer.)
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$\begingroup$ Some implementation of QMC uses only CNOT and Ry gates. QMC reaches quadratic speed-up. All algorithms with exp. speed-up I saw somehow employed phase - e.g. Shor or HHL. $\endgroup$– Martin VeselyCommented Apr 24, 2022 at 12:36
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It is well known that quantum computing with real numbers (i.e., only states & gates with real entries) is as powerful as quantum computing with complex entries. (While you obviously cannot prepare any quantum state, it is computationally equivalent, i.e. you can carry out all the same computations -- which take classical inputs and return classical outputs -- at almost the same cost.)
See, e.g., https://arxiv.org/abs/quant-ph/0301040
Now whether the gate set you propose is universal, I can't tell on the spot -- you would have to check whether you can use it to implement a universal set of real gates (such as Hadamard and Toffoli).
answered Apr 24, 2022 at 21:04
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1$\begingroup$ Ry + CNOT is universal by reduction to H+Toffoli. $\endgroup$ Commented Apr 24, 2022 at 22:30
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2$\begingroup$ I wouldn't say it's "obvious", but you could give it as an undergrad assignment. Cnot effectively gives you access to X, then H vs sqrt Y differ by X. For toffoli you use the textbook "almost toffoli" that uses 4 Y rotations separated by cnots then fix the phase error. For the phase error I found it easiest to make it work on just |++0> (to make a magic state) and then teleport. The intuition for why it would be possible is that the cnot gives you interactions and the Ry is not constrained enough to stop you from filling the space (except that you can't get a factor of i). $\endgroup$ Commented Apr 25, 2022 at 13:20