Quantum advantage using quantum circuit for gradient estimation

A method comprising:

• receiving a function ƒ(x) describing a quantity of a resource, x representing a set of k parameters of the quantity;
• preparing a quantum circuit configured to implement a quantum gradient algorithm when executed on four qubit registers of a quantum computing system, the quantum gradient algorithm scaling as (√{square root over (k)}/ϵ) and including a phase oracle OSƒm defined by a finite difference approximation with an order greater than zero, wherein, to implement the phase oracle OSƒm, the quantum circuit includes (1) a first quantum unitary operator to be applied to qubits in the first and second registers and (2) a second quantum unitary operator to be applied to qubits in the third and fourth registers;
• and
• repeatedly executing the quantum circuit on qubits of the quantum computing system to determine a k-dimensional gradient of the function ƒ(x) within an error ϵ at point x0.