Gradients with the parameter-shift rule

Complete the TypeScript function

parameterShiftGradient(evaluate: (params: number[]) => number, params: number[]): number[]

What evaluate does. evaluate(p) runs a fixed unitary circuit on $\lvert0\cdots0\rangle$ with the parameter values p and returns the exact expectation value $f(p)=\langle\psi(p)\vert H\vert\psi(p)\rangle$ of a fixed observable $H$ (a real combination of Pauli strings). p must be an array with one finite number per parameter; entry p[j] is bound to the circuit's $j$-th declared parameter. A wrong length or a non-finite entry throws. Every call is counted, as each call would be a separate circuit execution on hardware.

What to return. An array grad of length $P=$ params.length with grad[j] $=\partial f/\partial\theta_j$ at $\theta=$ params. Every parameter-dependent gate is a Pauli rotation $R_P(\theta_j)=e^{-i\theta_jP/2}$ with $P\in\{X,Y,Z\}$, and each parameter appears in exactly one gate, so

$\frac{\partial f}{\partial\theta_j}(\theta)=\frac12\Big[f\big(\theta+\tfrac\pi2e_j\big)-f\big(\theta-\tfrac\pi2e_j\big)\Big],$

where $e_j$ is the $j$-th unit vector: only entry $j$ is shifted.

Call budget. Exactly two calls of evaluate per parameter, i.e. exactly $2P$ calls in total. The unshifted value $f(\theta)$ is not needed. Do not modify params; work on copies.

Tests. Every test calls your function with the arguments (evaluate, params). The values check compares your gradient with the exact one; the calls check runs the same problems again and also requires exactly $2P$ calls; the exact check (tests exact-visible and exact-random-1 to exact-random-4) runs them once more with the absolute tolerance $10^{-12}$. The shift rule is exact up to floating-point rounding, so it passes all three; central finite differences with a small step also use $2P$ calls but fail exact.

• visible and calls-visible: the circuit $R_Y^{(0)}(t_1)$, $R_Y^{(1)}(t_2)$, $\mathrm{CNOT}(0,1)$ with $H=Z_1$, for which $f(t_1,t_2)=\cos t_1\cos t_2$. At params = [0.4, 1.1] the gradient is $\big(-\sin0.4\cos1.1,\,-\cos0.4\sin1.1\big)=(-0.1766386496831817,\,-0.8208563369208728)$ (absolute tolerance $10^{-9}$), using $2P=4$ calls.
• random-1 to random-4 and calls-random-1 to calls-random-4 (hidden): random three-qubit circuits of $R_X$, $R_Y$, $R_Z$ rotations interleaved with CNOTs, with respectively 7, 7, 7, 8 parameters and an observable that is a Pauli string or a sum of two weighted Pauli strings (absolute tolerance $10^{-7}$). Only their arguments are sent to your browser; the expected gradients stay on the server.