Draft:Parameter shift rule

The parameter shift rule (PSR) is a method used in quantum computing, specifically for variational quantum algorithms (VQAs), to compute the exact gradient of an expectation value with respect to a parameter in a quantum circuit. PSRs enabl efficient and exact optimization crucial for the advancement of quantum algorithms in chemistry, optimization, and machine learning.

A variational quantum circuit (VQC) is a parameterized quantum circuit (PQC) where certain gates depend on continuous parameters. These parameters are adjusted to optimize a cost function, typically the expectation value of an observable, by training the circuit in a way analogous to training weights in a neural network. To optimize the parameters, one needs to compute the gradient of the cost function with respect to these parameters. However, directly computing gradients on quantum hardware is non-trivial due to the probabilistic nature of quantum measurements and the inability to directly access the quantum state.

Known Parameter Shift Rules

Let $|\psi \rangle$ denote the quantum state in the Hilbert space. Consider the unitary operator $U(t)$, defined by a Hamiltonian $H$ and a parameter $t$. The eigenvalues of $U(t)$ are expressed as $\{\exp(i\lambda _{j}t)\}_{j=1}^{n}$ with real-valued $\{\lambda _{j}\}_{j=1}^{n}$. We aim to determine the mean value of a measurable observable $C$ defined as follows:

$\displaystyle f(t)\equiv \langle \psi |U(t)^{\dagger }CU(t)|\psi \rangle$

A parameterized quantum circuit (PQC) generates probabilistic results, using the expectation value of an observable as an estimate. While mean values of simple variables are obtained by averaging measurement outcomes, there are several methods to optimize the estimation of expectation values for observables involving multiple qubits .

The PSRs establish connections between derivatives of a quantum function and the function's evaluations at distinct points. For the two eigenvalue Hamiltonian with $\mu _{12}=|\lambda _{1}-\lambda _{2}|$ the PSR is

${\frac {\partial f(t)}{\partial t}}{\Big |}_{t=0}={\frac {\mu _{12}}{2\sin {(\mu _{12}\phi _{1})}}}(f(\phi _{1})-f(-\phi _{1})).$

where $\phi _{1}\in (0,\pi )$. In, the latter rule is generalized to gates with eigenvalues $\{-1,0,1\}$, resulting in $m=2$ frequencies:

$\left.{\frac {\partial f(t)}{\partial t}}\right|_{t=0}=y_{1}\left[f(\phi _{1})-f(-\phi _{1})\right]-y_{2}\left[f(\phi _{2})-f(-\phi _{2})\right],$

where $\phi _{1},\phi _{2}\in (0,\pi )$, and $y_{1},y_{2}$ are the corresponding coefficients.

In, the general parameter-shift rules are introduced for the scenario of evenly spaced parameter shifts $\phi _{j}={\frac {(2j-1)\pi }{2n}}$ (or $\phi _{j}={\frac {j\pi }{n}}$), where $j\in [1,n]$ is considered to reconstruct odd (even) functions:

$f'(0)=\sum _{j=1}^{2n}f\left({\frac {(2j-1)\pi }{2n}}\right){\frac {(-1)^{j-1}}{4n\sin ^{2}\left({\dfrac {(2j-1)\pi }{4n}}\right)}},$

$f''(0)=-f(0){\frac {2n^{2}+1}{6}}+\sum _{j=1}^{2n-1}f\left({\frac {j\pi }{n}}\right){\frac {(-1)^{j-1}}{2\sin ^{2}\left({\dfrac {j\pi }{2n}}\right)}}.$