The Hamiltonian here is the Hermitian operator $-i[{\varphi }_{\beta },H]$.

Acting on copies with different entries of $\overrightarrow{c}$ is still a well-defined unitary operation. Such unbalanced gates may be useful as heuristic or variational additions, for example near the end of the circuit, but they are not part of the constructions analyzed here.

We note that ${\varphi }_{\beta }$, defined in Eq. (2), is pure, and therefore the expression for the fidelity simplifies to the trace of the product of the two states.

With our sign convention, the tree circuit described here takes the first copy to the bookkeeping value $-n$. This choice is arbitrary: one could instead follow the opposite output branch at each layer, which would take, for example, the last copy to $+n$. One can also follow both branches simultaneously, as in the four-copy example of Eq. (16), so that the final bookkeeping values of the first and last copies are $(-n,n)$. Although these two output copies are generally correlated, each individual reduced state follows the recurrence of Eq. (19), with the sign of $ϵ$ in $U_{12}$ determined by the branch being followed. No measurement or reinitialization is involved in this deterministic construction; these operations are only introduced later in the post-selected variant.

The hedge circuit requires one small adjustment: within each layer, any copy not acted on by a $V$ gate is evolved for an additional $ϵ$ real time. This padding of the circuit equates the real time of the copies on which the $V$ gates act, such that also the acquired phases agree and the copies are symmetric with respect to swap (at leading order).

This is the optimal gate count regardless of the circuit structure. To see this, let us represent the bookkeeping vector after the $m$th use of $U_{ij}$ as ${\overrightarrow{c}}^{(m)}$. Consider the following function $S({\overrightarrow{c}}^{(m)})=(c_1^{(m)}{)}^2+(c_2^{(m)}{)}^2+\cdots$, where $c_i^{(m)}$ are the elements of ${\overrightarrow{c}}^{(m)}$. As $U_{ij}$ only acts on two elements if they are equal, and it adds 1 to one of them and $-1$ to the other, we understand that $S({\overrightarrow{c}}^{(m+1)})-S({\overrightarrow{c}}^{(m)})=2$. Therefore, $S({\overrightarrow{c}}^{(m)})=2m$, and the minimal number $M$ to reach a state is given by $S({\overrightarrow{c}}^{(m)})/2$. For the state ${\overrightarrow{c}}_f=(-n,\dots ,-1,1,\dots ,n)$, we obtain the claimed result.

We note that the sign of $ϵ$ here is opposite to the definition of $U$ in Eq. (11). This is because we (arbitrarily) chose to evolve the first copy from ${\varphi }_0$ to ${\varphi }_{\beta }$, and not to ${\varphi }_{-\beta }$ in the error bound derivation.