Figure 1.
Structure of subsystems required for performing the quantum part of variational SVD. It includes five n-qubit subsystems for encoding the matrix M(subsystems R, C), orthonormal eigenvectors 〈ψj| and |ψj〉 (subsystems χ, ψ) and weights qi (subsystem q). In addition, the one-qubit subsystem K serves as a controlling qubit in controlled operators of the algorithm and two one-qubit ancillae B and serve for the controlled measurement.
Figure 2.
The circuit for the quantum part of the variational SVD algorithm. The depth of this circuit can be estimated as O(Qlog(N)/ε). (a) The circuit for creating the state |ψout〉K given in (27); the operatorsW(j), j = 0, …,6 are presented without subscripts for brevity. (b) The operators applied to the state |ψout〉KB to probabilistically obtain the normalization G and the objective function L(α, β).
Figure 3.
The circuit for the operators (and ). Here the set of parameters γ is either α (for ) or β (for ), Z ≡ σ(z).
Figure 4.
The hybrid algorithm for calculating SVD. The matrices Û, D and are defined in terms of U(α*) and U(β*) according to (4), ΔL = |L[k+1] – L[k]|. For simplicity, we indicate only the function L and first derivatives ∂γL to be transferred from the output of the quantum algorithm to the input of the classical algorithm. However, the higher order derivatives might be required as well.