Implementing the Operator $U|x\rangle \mapsto e^{-it 2cos(2\pi \frac{x}{N})}|x\rangle$

Implementing the Operator U|x⟩↦ e −it2cos(2π x N ) |x⟩ 𝑈 | 𝑥 ⟩ ↦ 𝑒 − 𝑖 𝑡 2 𝑐 𝑜 𝑠 ( 2 𝜋 𝑥 𝑁 ) | 𝑥 ⟩ Ask Question Asked today Modified today Viewed 10 times 2 Recently, I've become interested in the paper: https://arxiv.org/pdf/1510.08657 and how to implement a diagonal gate required for continuous time quantum walks. While the paper states this is possible using a fixed-point approximation of 2cos(2π x N ) 2 𝑐 𝑜 𝑠 ( 2 𝜋 𝑥 𝑁 ) for a cycle graph (which I can verify is true), e.g. we could compute it via truncated Taylor approximation. I am wondering if there is a better way of implementing the appromxation of the diagonal gate by using controlled incrementers ( S|x⟩↦|x+1⟩ 𝑆 | 𝑥 ⟩ ↦ | 𝑥 + 1 ⟩ ) [open to other suggestions]? I just haven't been able to work out the correct way of piecing it together. Would highly value some insight on the topic, not looking to be spoon-fed, but would appreciate showing how to decompose it into a circuit / reference to papers. quantum-algorithmsgate-synthesisquantum-walks Share Improve this question Follow asked 4 hours ago Ramezzez 3861 1 silver badge 7 7 bronze badges Does this answer or this one answer your question? You would still have to build an oracle for |x,y⟩↦ ∣ ∣ x,y⊕2cos(2π x N )⟩ | 𝑥 , 𝑦 ⟩ ↦ | 𝑥 , 𝑦 ⊕ 2 cos ⁡ ( 2 𝜋 𝑥 𝑁 ) ⟩ though. –  Tristan Nemoz ♦ Commented 3 hours ago Add a comment Know someone who can answer? Share a link to this question via email, Twitter, or Facebook. Your Answer Sign up or log in Sign up using Google Sign up using Email and Password Post as a guest Name Email Required, but never shown Post Your Answer By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy. Start asking to get answers Find the answer to your question by asking. Ask question Explore related questions quantum-algorithmsgate-synthesisquantum-walks See similar questions with these tags. 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