Quantum signal processing

Quantum Signal Processing is a Hamiltonian simulation algorithm with optimal lower bounds in query complexity. It linearizes the operator of a quantum walk using eigenvalue transformation. The quantum walk takes a constant number of queries. So quantum signal processing's cost depends on the constant number of calls to the quantum walk operator, number of single qubit quantum gates that aid in the eigenvalue transformation and an ancilla qubit.[1]

## Eigenvalue transformation

Given a unitary W|ui⟩=eiθi|ui⟩, calculate A=f(W)=∑ieif(θi)|ui⟩⟨ui|. For example, if W=[eiθ1000eiθ2000eiθ3], A=[eif(θ1)000eif(θ2)000eif(θ3)]. [1]

## Algorithm

Input: Given a Hamiltonian H, define a quantum walk operator W using 2 d-sparse oracles OH and OF. OH accepts inputs j and k (j is the row of the Hamiltonian and k is the column) and outputs ⟨j|H|k⟩, so querying OH=|j⟩|k⟩|l⟩=|j⟩|k⟩|l⊕Hj,k⟩. OF accepts inputs j and l and computes the lth non-zero element in the jth row of H. [2]
Output: eiHt

1. Create an input state θ
2. Define a controlled-gate, c−W
3. Repeatedly apply single qubit gates to the ancilla followed applications of c−W to the register that contains θ O(td||H||max+log⁡1ϵlog⁡log⁡1ϵ) times.

## References

1. ↑ 1.0 1.1 Low, Guang Hao; Chuang, Isaac (2017). "Optimal Hamiltonian Simulation by Quantum Signal Processing". Physical Review Letters 118 (1): 010501. doi:10.1103/PhysRevLett.118.010501. PMID 28106413. Bibcode: 2017PhRvL.118a0501L.
2. ↑ Guan Hao Low (January 17, 2017). Optimal Hamiltonian simulation by quantum signal processing (YouTube). Retrieved September 9, 2019.

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