with the number of qubits $m.$ We will therefore stick to the GR states or to Eq. , unless otherwise noted. However, if we need to save some qubits, we can try variations, such as a probability state that implements the trapezoidal or the Simpson rule $$\label{eq:Simpson}
\ket{p_S^{(m)}} = \frac{1}{S_m^{1/2}}\sum_s \sqrt{\theta_sp(x_s)}\ket{s},\;\theta_s=\left\{\begin{array}{ll}1,&s=0,2^{m}-1\\ 4-(s\,\mathrm{mod}\,2),&\mathrm{else}. \end{array}\right.$$ The discretization error of the Simpson state decreases faster with the interval size, $\varepsilon_\text{sample}\sim \mathcal{O}(\delta_m^2).$ This scaling means that the qubits required to achieve a given precision $m\sim \log_2(\varepsilon_\text{sample})$ can be half those required by the uniform ansatz. These savings may be interesting in resource-starved architectures, such as NISQ computers, and also in the algorithms to be considered later in Sect. \[sec:quantum-inspired-algorithms\].

Existence of efficient constructs {#sec:bounds}
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We can show that adding $k$ new qubits to a GR probability state with $m$ qubits demands a vanishingly small amount of entanglement, that decreases exponentially with $2^m.$ This small amount of entanglement suggests that the GR unitary $U^{(m)}_p$ can be replaced with simpler decomposition in terms of quasi-local gates. The reasoning proceeds as follows. In App. \[app:reduced-dty\] we study the reduced density matrix $\rho^{(m,k)}$ of the $k$ new qubits in a GR state with $m+k$ bits of resolution. 