qrisp.cks.cks_params

cks_params(eps: float, kappa: float, max_beta: int = None) → Tuple[int, int]

Computes the complexity parameter \(\beta\) and the truncation order \(j_0\) for the truncated Chebyshev approximation of \(1/x\), as described in the Childs–Kothari–Somma paper.

Given the condition number \(\kappa\) and the target precision \(\epsilon\), the parameters are computed as:

\[\beta = \kappa^2 \log\!\left(\frac{\kappa}{\epsilon}\right), \quad j_0 = \sqrt{\beta \log\!\left(\frac{4\beta}{\epsilon}\right)}.\]

If max_beta is provided, \(\beta\) is capped to \(\min(\beta, \beta_{\max})\). The returned values are cast to integers via the floor operation.

Parameters:

epsfloat

Target precision \(\epsilon\).

kappafloat

An upper bound for the condition number \(\kappa\) of \(A\). This value defines the “gap” around zero where the function \(1/x\) is not approximated.

max_betafloat, optional

Optional upper bound on the complexity parameter \(\beta\).

Returns:

j0int

Truncation order of the Chebyshev expansion \(j_0 = \lfloor\sqrt{\beta \log(4\beta/\epsilon)}\rfloor\).

betaint

Complexity parameter \(\beta = \lfloor\kappa^2 \log(\kappa/\epsilon)\rfloor\).