where $\tilde{\Delta} + \frac{d}{T_0}t := (\Delta_1 + \frac{d}{T_0}t, \Delta_2 + \frac{d}{T_0}t, \dots, \Delta_D + \frac{d}{T_0}t)$, and in the limits $d \to \infty$, $(0, d) \ni \sigma \to \infty$, we have that

$$ \varepsilon_v(t, d) = \begin{cases} |t| \frac{d}{T_0} \left( \mathcal{O} \left( \frac{d^{3/2}}{\bar{v}+1} \right)^{1/2} + \mathcal{O}(d) \right) \exp \left( -\frac{\pi}{4} \frac{\alpha_0^2}{(1+\bar{v})^2} d \right) + \mathcal{O} \left( e^{-\frac{\pi}{2} d} \right) & \text{if } \sigma = \sqrt{d} \\ |t| \frac{d}{T_0} \left( \mathcal{O} \left( \frac{\sigma^3}{\bar{v}\sigma^2/d+1} \right)^{1/2} + \mathcal{O} \left( \frac{d^2}{\sigma^2} \right) \right) \exp \left( -\frac{\pi}{4} \frac{\alpha_0^2}{\frac{d}{\sigma^2} + \bar{v}} \left( \frac{d}{\sigma} \right)^2 \right) \\ + \mathcal{O} \left( |t| \frac{d^2}{\sigma^2} + 1 \right) e^{-\frac{\pi}{4} \frac{d^2}{\sigma^2}} + \mathcal{O} \left( e^{-\frac{\pi}{2} \sigma^2} \right) & \text{otherwise.} \end{cases} $$

It can readily be seen that Theorem C.1 reduces to Theorem 9.1 in [26], in a number of special cases. For example, when $\Delta_1 = \Delta_2 = \dots = \Delta_D$, and we identify $\hat{V}_d$ in [26] with $\sum_{j=1}^D \hat{V}_d^{(j)}$ for any $D \in \mathbb{N}_{>0}$.

*Proof.* The proof follows analogously to the proof of Theorem 9.1 in [26], (which used Lemmas IX.0.1., IX.0.2., IX.0.3., IX.0.4., IX.0.5., as input lemmas to the proof of Theorem 9.1.). One needs to exchange definition 80 in [26], namely

$$ \Theta(\Delta; x) = \int_{x-\Delta}^{x} dy V_{d}(y), \qquad (C.66) $$

with the new definition

$$ \Theta(\Delta; x) := \sum_{j=1}^{D} \Theta_j(\Delta_j; x), \qquad \Theta_j(\Delta_j; x) := \int_{x-\Delta_j}^{x} dy V_d^{(j)}(y). \qquad (C.67) $$

After this change, it is easy to go through the above-mentioned Lemmas line-by-line and the proof of the theorem itself, to verify Theorem C.1. ■

## 5. Lemmas bounding error contributions from control on $C_2$

In this section we will state and prove a lemma which will bound lines eqs. (C.9) and (C.11) in eq. (C.7). However, before doing so, we will need to introduce the following definition and lemma.

$$ \bar{H}_{MC_2}^{[k,r]} := H_{C_2} + \sum_{q \in \mathbf{W}(k,r)} I_M^{(q)} \otimes I_{C_2}^{(q)}, \qquad (C.68) $$

with

$$ \mathbf{W}(k,r) := \begin{cases} \emptyset & \text{if } r=0 \text{ and } k=1. \\ \{1, 2, 3, \dots, k-1\} & \text{if } r=0 \text{ and } k=2, 3, 4, \dots, N_g. \\ \{2, 3, 4, \dots, N_g\} & \text{if } r=1, 2, \dots, L, \text{ and } k=1. \\ \{1, 2, 3, \dots, k-1, k+1, \dots, N_g\} & \text{if } r=1, 2, \dots, L, \text{ and } k=2, 3, 4, \dots, N_g. \end{cases} \qquad (C.69) $$

where $\emptyset$ is the empty set.

In the following lemma, is a technical lemma which will be used in the proofs of lemmas C.4 and C.5.

**Lemma C.3 (Bounding the dynamics on $C_2$).** Consider the following definition.

$$ |\epsilon(t)\rangle_{C_2} := e^{-it(H_{C_2} + \sum_{l \in \mathbf{W}(k,r)} \Omega_{il}^{(l)} I_{C_2}^{(l)})} |\Psi_{i_k}(t_{k-1,r})\rangle_{C_2} - |\Psi_{i_k}(t_{k,r})\rangle_{C_2}, \qquad (C.70) $$

where $k = 1, 2, \dots, N_g$; $r = 0, 1, 2, \dots, L$; $t \in \mathbb{R}$ and $\{\Omega_{il}^{(l)} \in (0, 2\pi]\}_{i_l}$. Assuming $n_2$ to be monotonically increasing with $d$, we have the bound

$$ \| |\epsilon(t)\rangle_{C_2} \|_2 := \varepsilon_v^{(2)}(t, d) \le |t| \frac{d}{T_0} \left( \mathcal{O} \left( d^{1/2+\varepsilon_g} \right) + \mathcal{O}(d) \right) \exp \left( -\frac{1}{8\pi^2 \kappa^2 C_0^2(N_2) N_g n_2^2} \frac{d}{N_g n_2^2} \right) + \mathcal{O} \left( e^{-\frac{\pi}{2} d} \right), \qquad (C.71) $$

as $d \to \infty$.
Proof. Given the definitions, we can readily apply Theorem C.1 with $\bar{\Delta} = (t_{k-2}d/T_0, t_{k-3}d/T_0, t_{k-4}d/T_0, \dots, t_1d/T_0, 0)$ for $r=0$ and $\bar{\Delta} = (rd+t_{k-2}d/T_0, rd+t_{k-3}d/T_0, rd+t_{k-4}d/T_0, \dots, rd, rd+t_{-2}d/T_0, rd+t_{-3}d/T_0, \dots, rd+t_{k-1-N_g}d/T_0)$ for $r=1, 2, \dots, L$. (This amounts to $\bar{\Delta}$ being of dimension $k-1 = |\mathbf{W}(k, 0)|$ and $N_g-1 = |\mathbf{W}(k, r)|$ respectively). In both cases, the Hamiltonian $H_C + \sum_{j=1}^D \hat{V}_d^{(j)}$ is chosen to be $H_{C_2} + \sum_{q \in \mathbf{W}(k,r)} \Omega_{i_q}^{(q)} I_{C_2}^{(q)}$.

Thus, out task is to bound eq. (C.64) for this setting. Thus the summation $\sum_{j=1}^D V_0^{(j)}(x)$ in eq. (C.64) amounts to

$$ \sum_{j=1}^{D} V_{0}^{(j)}(x) = \sum_{q \in \mathbf{W}(k,r)} \Omega_{i_q}^{(q)} \bar{V}_{0}(x) \Big|_{x_0=x_0'^{(q)}}. \quad (C.72) $$

where recall $\bar{V}_0(\cdot)$ is given by eq. (C.45). Therefore for $k=1, 2, 3, \dots$ we have

$$ \max_{x \in [0, 2\pi]} \left| \sum_{j=1}^{D} \frac{d^k}{dx^k} \bar{V}_0(x) \right| \le \pi N_g \max_{x \in [0, 2\pi]} \left| \frac{d^k}{dx^k} V_0(x) \right| \le \pi N_g n_2^{k+1} C_0(N_2)^{k+1} \quad (C.73) $$

where we have used eq. (C.48) in the penultimate inequality and Lemma 28 from [27, page 29] in the last inequality. Recall $C_0^{(2)}(N_2)$ is solely a function on $N_2$, i.e. independent of $k, d$ and $n_2$ (it is denoted $C_0(N)$ in Lemma 28 from [27, page 29]. Here we use the superscript to distinguish it from $C_0(N)$ used for the control on C in the proof of Theorem 1).

For the second term in eq. (C.64) we have for $p=1, 2, 3, \dots$

$$ \left| \sum_{j=1}^{D} \frac{d^p}{dy^p} V_0^{(j)} \left( y - \frac{2\pi \Delta_j}{d} \right) \right| = \left| \sum_{q \in \mathbf{W}(k,r)} \Omega_{i_q}^{(q)} \frac{d^p}{dy^p} \bar{V}_0 \left( y - \frac{2\pi t_{k-1-q,r}}{T_0} \right) \Bigg|_{x_0=x_0'^{(q)}} \right| \quad (C.74) $$

$$ = \left| \sum_{q \in \mathbf{W}(k,r)} \Omega_{i_q}^{(q)} \frac{d^p}{dy^p} \bar{V}_0 \left( y - \frac{2\pi}{N_g} (k-1-q) - x_0'^{(q)} \right) \Bigg|_{x_0=0} \right| \quad (C.75) $$

$$ = \left| \left( \sum_{q \in \mathbf{W}(k,r)} \Omega_{i_q}^{(q)} \right) \frac{d^p}{dy^p} \bar{V}_0 \left( y - \frac{2\pi}{N_g} (k - 3/2) - \pi \right) \Bigg|_{x_0=0} \right| \quad (C.76) $$

$$ \leq \pi N_g \left| \frac{d^p}{dy^p} \bar{V}_0 \left( y - \frac{2\pi}{N_g} (k - 3/2) - \pi \right) \right|_{x_0=0} \quad (C.77) $$

In line eq. (C.75), we have used the $2\pi$ periodicity of $\bar{V}_0$. In line eq. (C.76) we have used definition eq. (C.46). In line eq. (C.77) we have used definition eq. (C.48). Therefore, using Lemma 28 from [27, page 29] we find

$$ \left| \sum_{j=1}^{D} \frac{d^p}{dy^p} V_0^{(j)} \left( y - \frac{2\pi \Delta_j}{d} \right) \right| \le \pi N_g n_2^{p+1} C_0(N_2)^{p+1}. \quad (C.78) $$

Therefore, from definition eq. (C.64) it follows that we can choose $b$ such that

$$ b \ge \max \left\{ 2\pi, \sup_{p \in \mathbb{N}^+} \left( (2\pi N_g)^{1/(p+1)} n_2 C_0(N_2) \right) \right\}. \quad (C.79) $$

$$ = \max \left\{ 2\pi, (2\pi N_g)^{1/2} n_2 C_0(N_2) \right\}. \quad (C.80) $$

Therefore, applying Theorem C.1, we conclude eq. (C.71).

<figure><img src="image_10.png" /></figure>

We will now state and prove a lemma which will bound line eq. (C.9) in eq. (C.7) We will also require the definition of how the switch states change overtime. This was discussed in the main text but we reproduce it here in a more compact form for ease of readability. For $l=0, 1, 2, \dots, N_g$; $r=0, 1, 2, \dots, L$:

$$ |t_{l,r}\rangle_{W_k} = \begin{cases} |\text{on}\rangle_{W_k} & \text{if } r=1, 2, 3, \dots, L \\ |\text{on}\rangle_{W_k} & \text{if } r=0 \text{ and } l \ge k \\ |\text{off}\rangle_{W_k} & \text{if } r=0 \text{ and } l < k \end{cases} \quad (C.81) $$
**Lemma C.4** (Bound for 1st bus-related term). Consider the control states $\{|t_{k,r}\rangle_{C_2}\}_{k,r}$ and Hamiltonian interaction terms described in appendix C3. The following holds for all $N_2 - 2, N_g - 2 \in \mathbb{N}_{\ge 0}$, $n_2 > 0$, $k = 1, 2, 3, \dots, N_g$ and $r \in 0, 1, 2, \dots, L$.

$$ \| |t_{k,r}\rangle_{\bar{M}_{0,k}C_2} |t_{k,r}\rangle_{\bar{W}_k} - e^{-it_1 \bar{H}_{\text{MWC}_2}^{(k)}} |t_{k-1,r}\rangle_{\bar{M}_{0,k}C_2} |t_{k-1,r}\rangle_{\bar{W}_k} \|_2 \quad (C.82) $$

$$ \leq \bar{\delta}_{r,0} A_{\text{nor}}^{(2)} 4\pi^2 \sqrt{2d \left[ \frac{5}{(2N_2-1)^2 8^2} \left( \frac{\pi^2 n_2}{4} \right)^{-4N_2} + 4\pi^2 e^{-2\pi d/8^2} \right]} + \varepsilon_v^{(2)}(t_1, d), \quad (C.83) $$

where

$$ \bar{\delta}_{r,0} = \begin{cases} 0 & \text{if } r = 0 \\ 1 & \text{if } r = 1, 2, \dots, L \end{cases} \quad (C.84) $$

and $\varepsilon_v^{(2)}(t_1, d)$ is upper bounded in lemma C.3.

*Proof.* Due to boundary conditions which would require a change of notation, we only consider $k = 2, 3, \dots, N_g$ here. The case $k = 1$ follows analogously with slight changes in the notation [e.g. eq. (C.87) requires some modification as is readily apparent]. Note that by definition, $\bar{H}_{\text{MWC}_2}^{(k)}$ acts trivially on all memory cells $M_{0,k}, M_{1,k}, M_{2,k}, \dots, M_{L,k}$. What is more, by definition eq. (C.49), $|t_{k,r}\rangle_{\bar{M}_{\#k}} = |t_{k-1,r}\rangle_{\bar{M}_{\#k}}$, where $\bar{M}_{\#k}$ denotes all of $M$ except $M_{\#k}$.

$$ \| |t_{k,r}\rangle_{\bar{M}_{0,k}C_2} |t_{k,r}\rangle_{\bar{W}_k} - e^{-it_1 \bar{H}_{\text{MWC}_2}^{(k)}} |t_{k-1,r}\rangle_{\bar{M}_{0,k}C_2} |t_{k-1,r}\rangle_{\bar{W}_k} \|_2 \quad (C.85) $$

$$ = \| |t_{k,r}\rangle_{\bar{M}_{\#k}C_2} |t_{k,r}\rangle_{\bar{W}_k} - e^{-it_1 \bar{H}_{\text{MWC}_2}^{(k)}} |t_{k-1,r}\rangle_{\bar{M}_{\#k}C_2} |t_{k-1,r}\rangle_{\bar{W}_k} \|_2. \quad (C.86) $$

What is more, $\bar{H}_{\text{MWC}_2}^{(k)}$ is diagonal in the on/off basis of the switch and only has support on the on-switch states $\{|\text{on}\rangle\rangle_{W_l}\}_{l=1}^{N_g}$. What is more, from the definition of $|t_{l,m}\rangle\rangle_W$ [eq. (C.81)] it follows

$$ |t_{k-1,r}\rangle_W = \begin{cases} |\text{on}\rangle_{W_1} |\text{on}\rangle_{W_2} \dots |\text{on}\rangle_{W_{k-1}} |\text{off}\rangle_{W_k} |\text{off}\rangle_{W_{k+1}} \dots |\text{off}\rangle_{W_{N_g}} & \text{if } r=0 \\ |\text{on}\rangle_{W_1} |\text{on}\rangle_{W_2} \dots |\text{on}\rangle_{W_{N_g}} & \text{if } r=1, 2, \dots, N_g \end{cases} \quad (C.87) $$

$$ |t_{k,r}\rangle_W = \begin{cases} |\text{on}\rangle_{W_1} |\text{on}\rangle_{W_2} \dots |\text{on}\rangle_{W_k} |\text{off}\rangle_{W_{k+1}} |\text{off}\rangle_{W_{k+2}} \dots |\text{off}\rangle_{W_{N_g}} & \text{if } r=0 \\ |\text{on}\rangle_{W_1} |\text{on}\rangle_{W_2} \dots |\text{on}\rangle_{W_{N_g}} & \text{if } r=1, 2, \dots, N_g \end{cases} \quad (C.88) $$

Therefore,

$$ \| |t_{k,r}\rangle_{\bar{M}_{0,k}C_2} |t_{k,r}\rangle_{\bar{W}_k} - e^{-it_1 \bar{H}_{\text{MWC}_2}^{(k)}} |t_{k-1,r}\rangle_{\bar{M}_{0,k}C_2} |t_{k-1,r}\rangle_{\bar{W}_k} \|_2 \quad (C.89) $$

$$ = \| |t_{k,r}\rangle_{\bar{M}_{\#k}C_2} - e^{-it_1 \bar{H}_{\text{MWC}_2}^{[k,r]}} |t_{k-1,r}\rangle_{\bar{M}_{\#k}C_2} \|_2, \quad (C.90) $$

where $\bar{H}_{\text{MC}_2}^{[k,r]}$ is defined in eq. (C.68). Let us now compute $e^{-it_1 \bar{H}_{\text{MC}_2}^{[k,r]}} |t_{k-1,r}\rangle_{\bar{M}_{\#k}C_2}$:

$$ e^{-it_1 \bar{H}_{\text{MC}_2}^{[k,r]}} |t_{k-1,r}\rangle_{\bar{M}_{\#k}C_2} \quad (C.91) $$

$$ = e^{-it_1 \bar{H}_{\text{MC}_2}^{[k,r]}} |t_{k-1,r}\rangle_{\bar{M}\setminus\{\bar{M}_{\#k-1}, \bar{M}_{\#k}\}} |t_{k-1,r}\rangle_{\bar{M}_{\#k-1}} \quad (C.92) $$

$$ = \prod_{\substack{l=1 \\ l \notin \{k, k-1\}}}^{N_g} \left[ \sum_{i_l} \bar{M}_{\#l} \langle \Omega_{i_l} | 0 \rangle_{\bar{M}_{\#l}} | \Omega_{i_l} \rangle_{\bar{M}_{\#l}} \right] e^{-it_1 (H_{C_2} + \sum_{q \in \mathbf{w}(k,r)} \Omega_{i_q}^{(q)} I_{C_2}^{(q)})} |\Psi_{\bar{i}_{k-1}, \bar{i}_k}(t_{k-1,r})\rangle_{\bar{C}_2} |t_{k-1,r}\rangle_{\bar{M}_{\#k-1}} \quad (C.93) $$

$$ = \prod_{\substack{l=1 \\ l \neq k}}^{N_g} \left[ \sum_{i_l} \bar{M}_{\#l} \langle \Omega_{i_l} | 0 \rangle_{\bar{M}_{\#l}} | \Omega_{i_l} \rangle_{\bar{M}_{\#l}} \right] e^{-it_1 (H_{C_2} + \sum_{q \in \mathbf{w}(k,r)} \Omega_{i_q}^{(q)} I_{C_2}^{(q)})} |\Psi_{\bar{i}_k}(t_{k-1,r})\rangle_{\bar{C}_2} \quad (C.94) $$

$$ = \prod_{\substack{l=1 \\ l \neq k}}^{N_g} \left[ \sum_{i_l} \bar{M}_{\#l} \langle \Omega_{i_l} | 0 \rangle_{\bar{M}_{\#l}} | \Omega_{i_l} \rangle_{\bar{M}_{\#l}} \right] \left( |\Psi_{\bar{i}_k}(t_{k,r})\rangle_{\bar{C}_2} + |\epsilon\rangle_{\bar{C}_2} \right) \quad (C.95) $$
$$ = |t_{k,r}\rangle_{M\setminus\{M_{\#k}, M_{\#k+1}\}} |t_{k,r}\rangle_{M_{\#k+1}} + |\epsilon_{k+1,r}\rangle_{\bar{M}_{\#k}C_2} + \prod_{\substack{l=1 \\ l \neq k}}^{N_g} \left[ \sum_{i_l} M_{\#l} \langle \Omega_{i_l} | 0 \rangle_{M_{\#l}} |\Omega_{i_l}\rangle_{M_{\#l}} \right] |\epsilon\rangle_{C_2} \quad (C.96) $$

$$ = |t_{k,r}\rangle_{\bar{M}_{\#k}C_2} + |\epsilon_{k+1,r}\rangle_{\bar{M}_{\#k}C_2} + \prod_{\substack{l=1 \\ l \neq k}}^{N_g} \left[ \sum_{i_l} M_{\#l} \langle \Omega_{i_l} | 0 \rangle_{M_{\#l}} |\Omega_{i_l}\rangle_{M_{\#l}} \right] |\epsilon\rangle_{C_2}. \quad (C.97) $$

In line eq. (C.94), we have used that $|\Psi_{\bar{\tau}_{k-1}, \bar{\tau}_k}^{(k-1)}(t_{k-1}, r)\rangle_{C_2} = e^{i\tau \Omega_{\bar{\tau}_{k-1}}^{(k-1)}} |\Psi_{\bar{\tau}_k}^{(k-1)}(t_{k-1}, r)\rangle_{C_2}$. In line eq. (C.95) we have applied lemma C.3. In line eq. (C.96) we have defined $|\epsilon_{k+1,r}\rangle_{\bar{M}_{\#k}C_2}$ as the difference between this line and the previous line.

Therefore, from eqs. (C.90) and (C.96) we conclude:

$$ \| |t_{k,r}\rangle_{\bar{M}_{0,k}C_2} |t_{k,r}\rangle_{\bar{W}_k} - e^{-it_1 \bar{H}_{MWC_2}^{(k)}} |t_{k-1,r}\rangle_{\bar{M}_{0,k}C_2} |t_{k-1,r}\rangle_{\bar{W}_k} \|_2 \le \| |\epsilon_{k+1,r}\rangle_{\bar{M}_{\#k}C_2} \|_2 + \| |\epsilon\rangle_{C_2} \|_2 \quad (C.98) $$

$$ \le \| |\epsilon_{k+1,r}\rangle_{\bar{M}_{\#k}C_2} \|_2 + \epsilon_v^{(2)}(t_1, d), \quad (C.99) $$

where $\epsilon_v^{(2)}(t, d)$ is upper bounded in Lemma C.3. Let us now bound $\| |\epsilon_{k+1,r}\rangle_{\bar{M}_{\#k}C_2} \|_2$:

$$ \| |\epsilon_{k+1,r}\rangle_{\bar{M}_{\#k}C_2} \|_2^2 / 2 \quad (C.100) $$

$$ =1 - \Re \left\{ \left( \prod_{\substack{l=1 \\ l \neq k}}^{N_g} \left[ \sum_{i_l} \left( M_{\#l} \langle \Omega_{i_l} | 0 \rangle_{M_{\#l}} \right)^* M_{\#l} \langle \Omega_{i_l} | \right]_{C_2} \langle \Psi_{\bar{\tau}_k} (t_{k,r}) | \right) |t_{k,r}\rangle_{\bar{M}_{\#k+1}C_2} \right\} \quad (C.101) $$

$$ =1 - \Re \left\{ \prod_{\substack{l=1 \\ l \neq k}}^{N_g} \left[ \sum_{i_l} \left( M_{\#l} \langle \Omega_{i_l} | 0 \rangle_{M_{\#l}} \right)^* M_{\#l} \langle \Omega_{i_l} | \right] \right. \quad (C.102) $$

$$ \left( \sum_{q \in S_d(t_{k,r}d/T_0)} \prod_{l \in \mathbf{W}(k,r)} \left[ e^{i \Omega_{i_l}^{(l)}} \int_{q-rd-t_{k,r}d/T_0}^{q} dy I_{C_2,d}^{(l)}(y) \right] \psi_{\text{nor}}^{(2)*}(t_{k,r}d/T_0, q) \right)_{C_2} \langle \theta_q | \quad (C.103) $$

$$ \prod_{\substack{l'=1 \\ l' \notin \{k, k+1\}}}^{N_g} \left[ \sum_{j_{l'}} M_{\#l', l'} \langle \Omega_{j_{l'}} | 0 \rangle_{M_{\#l', l'}} |\Omega_{j_{l'}}\rangle_{M_{\#l', l'}} \right] |\Psi_{\bar{j}_k, \bar{j}_{k+1}}(t_{k,r})\rangle_{C_2} |t_{k,r}\rangle_{M_{\#k+1}} \right\} \quad (C.104) $$

$$ =1 - \Re \left\{ \prod_{\substack{l=1 \\ l \neq k}}^{N_g} \left[ \sum_{i_l} |M_{\#l, l'} \langle \Omega_{i_l} | 0 \rangle_{M_{\#l, l'}}|^2 e^{-i \tau \Omega_{i_l}^{(k+1)} \delta_{l, k+1}} \left( \sum_{q \in S_d(t_{k,r}d/T_0)} e^{i \bar{\delta}_{r,0} \Omega_{i_l}^{(k+1)} \int_{q-rd-t_{k-l}d/T_0}^{q} dy I_{C_2,d}^{(k+1)}(y)} \right) \right] \right\} \quad (C.105) $$

$$ \left| \psi_{\text{nor}}^{(2)}(t_{k,r}d/T_0, q) \right|^2 \right\} \quad (C.106) $$

$$ =1 - \Re \left\{ \sum_{i_{k+1}} \left| M_{\#k+1} \langle \Omega_{i_{k+1}} | 0 \rangle_{M_{\#k+1}} \right|^2 \left( \sum_{q \in S_d(t_{k,r}d/T_0)} e^{i \bar{\delta}_{r,0} \Omega_{i_{k+1}}^{(k+1)} \int_{q-t-1d/T_0}^{q} dy I_{C_2,d}^{(k+1)}(y)} \right) \left| \psi_{\text{nor}}^{(2)}(t_{k,r}d/T_0, q) \right|^2 \right\} \quad (C.107) $$

$$ = \bar{\delta}_{r,0} \sum_{i_{k+1}} \left| M_{\#k+1} \langle \Omega_{i_{k+1}} | 0 \rangle_{M_{\#k+1}} \right|^2 \sum_{q \in S_d(t_{k,r}d/T_0)} \left( \Omega_{i_{k+1}}^{(k+1)} \right)^2 \left( \int_{q+d/N_g}^{q} dy I_{C_2,d}^{(k+1)}(y) \right)^2 \left| \psi_{\text{nor}}^{(2)}(t_{k,r}d/T_0, q) \right|^2 \quad (C.108) $$

$$ \le \bar{\delta}_{r,0} \sum_{i_{k+1}} \left( \Omega_{i_{k+1}}^{(k+1)} \right)^2 \left| M_{\#k+1} \langle \Omega_{i_{k+1}} | 0 \rangle_{M_{\#k+1}} \right|^2 d \max_{q' \in [-1/2, 1/2]} \left( \frac{2\pi}{d} \right)^2 \left( - \int_{dq'+kd/N_g+rd}^{dq'+(k+1)d/N_g+rd} dy \bar{V}_0 \left( \frac{2\pi}{d} y \right) \right) \quad (C.109) $$

$$ A_{\text{nor}}^{(2)} e^{-\frac{2\pi}{\sigma^2} (dq')^2} \quad (C.109) $$
$$ \le \bar{\delta}_{r,0} dA_{\text{nor}}^{(2)} (2\pi)^4 \left[ \max_{q' \in [-1/8, 1/8]} \left( \left( \int_{q'}^{q'+1/N_g} dz \bar{V}_0 (2\pi(z+k/N_g+r)) \right) \Bigg|_{x_0=x_0^{(k+1)}} \right)^2 \right] \quad (C.110) $$

$$ + \max_{q' \in [-1/2, -1/8] \cup [1/8, 1/2]} \left( \left( \int_{q'}^{q'+1/N_g} dz \bar{V}_0 (2\pi(z+k/N_g+r)) \right) \Bigg|_{x_0=x_0^{(k+1)}} \right)^2 e^{-2\pi \frac{d^2}{\sigma_2^2} (1/8)^2} \quad (C.111) $$

$$ \le \bar{\delta}_{r,0} dA_{\text{nor}}^{(2)} (2\pi)^4 \left[ \max_{q' \in [-1/8, 1/8]} \left( \sum_{p=-\infty}^{\infty} \int_{q'}^{q'+1/N_g} dz V_B \left( 2\pi n_2 \left( z - \frac{1}{2N_g} + \frac{1}{2} + p \right) \right) \right)^2 \right] \quad (C.112) $$

$$ + 4\pi^2 e^{-2\pi \frac{d^2}{\sigma_2^2} (1/8)^2} \quad (C.113) $$

$$ \le \bar{\delta}_{r,0} dA_{\text{nor}}^{(2)} (2\pi)^4 \left[ \frac{(2\pi^2 n_2)^{-4N_2}}{(2N_2-1)^2} \max_{q' \in [-1/8, 1/8]} \left( \sum_{p=-\infty}^{\infty} \left[ \left| q' + \frac{1}{2N_g} + \frac{1}{2} + p \right|^{-2N_2+1} + \left| q' - \frac{1}{2N_g} + \frac{1}{2} + p \right|^{-2N_2+1} \right] \right)^2 \right] \quad (C.114) $$

$$ + 4\pi^2 e^{-2\pi \frac{d^2}{\sigma_2^2} (1/8)^2} \quad (C.115) $$

$$ \le \bar{\delta}_{r,0} dA_{\text{nor}}^{(2)} (2\pi)^4 \left[ \frac{1}{(2N_2-1)^2 8^2} \left( \frac{2\pi^2 n_2}{8} \right)^{-4N_2} \left( \sum_{p=-\infty}^{\infty} \left[ \max_{q' \in [-1/8, 1/8]} \left| 8 \left( q' + \frac{1}{2N_g} + \frac{1}{2} + p \right) \right|^{-2N_2+1} \right] \right) \right] \quad (C.116) $$

$$ + \left[ \max_{q' \in [-1/8, 1/8]} \left| 8 \left( q' - \frac{1}{2N_g} + \frac{1}{2} + p \right) \right|^{-2N_2+1} \right]^2 + 4\pi^2 e^{-2\pi \frac{d^2}{\sigma_2^2} (1/8)^2} \quad (C.117) $$

$$ \le \bar{\delta}_{r,0} dA_{\text{nor}}^{(2)} (2\pi)^4 \left[ \frac{5}{(2N_2-1)^2 8^2} \left( \frac{\pi^2 n_2}{4} \right)^{-4N_2} + 4\pi^2 e^{-2\pi \frac{d^2}{\sigma_2^2} (1/8)^2} \right] \quad (C.118) $$

Where in line eq. (C.106), we have defined $\bar{\delta}_{r,0} := 0$ if $r = 0$, and $\bar{\delta}_{r,0} := 1$ if $r = 1, 2, 3, \dots, L$, while we have defined $\delta_{l,k+1} := 0$ if $l \ne k+1$ and $\delta_{l,k+1} = 1$ if $l = k+1$. In line eq. (C.108), we have taken the real part and used the bound $-\cos\theta \le \theta^2 - 1$, for all $\theta \in \mathbb{R}$. In line eq. (C.109), we have used the fact that $q \in S_d(t_{k,r}d/T_0)$ is equivalent $q \in \{-d/2 + kd/N_g + rd\}, [-d/2 + kd/N_g + rd] + 1, [-d/2 + kd/N_g + rd] + 2, \dots, [-d/2 + kd/N_g + rd] + d-1\}$. Since $[-d/2 + kd/N_g + rd] + d-1 = [-d/2 + d-1 + kd/N_g + rd] \le d/2 + kd/N_g + rd$, we have that $q$ takes on $d$ values in the interval $q \in [-d/2 + kd/N_g + rd, d/2 + kd/N_g + rd]$. We have then performed the change of variable $q' := -(k/N_g + r) + q/d$. In line eq. (C.111), we have recalled that $\Omega_{i_{k+1}}^{(k+1)} \in (0, 2\pi]$ and we have used the change of variable $z := y/d - (k/N_g + r)$. In line eq. (C.112) we used the definition of $V_0$ and its properties to interchange the order of summation and integration while in line eq. (C.113) noted that, due to the properties of $\bar{V}_0, \int_a^b dx \bar{V}_0(2\pi x + x_0) \le 2\pi$ when $0 \le b - a \le 1$, $x_0 \in \mathbb{R}$. In line eq. (C.114), we have used the upper bound $V_B(\cdot) \le (\pi(\cdot))^{-2N_2}$ followed by performing the integrals and upper bounding the outcome. In line eq. (C.118), we have used that $N_2, N_g \ge 2$. In particular, first we have noted that $q' + 1/(2N_g) + 1/2 \in [3/8, 7/8]$, and $q' - 1/(2N_g) + 1/2 \in [1/8, 5/8]$, and used this observation together with $N_2 \ge 2$, to bound each value in the summand individually. Finally, we use $\sigma_2 = \sqrt{d}$ to achieve eq. (C.83).

**Lemma C.5 (Bound for 2nd bus-related term).** Consider the control states $\{|t_{k,r}\rangle\}_{k,r}$ and Hamiltonian interaction
terms described in appendix C3. The following holds for all $N_2 - 2, N_g - 2 \in \mathbb{N}_{\ge 0}$, $d - 4 \in \mathbb{N}_{\ge 0}$, $j = 1, 2, 3, \dots, N_g$ and $r \in \{0, 1, 2, \dots, L\}$.

$$ \max_{y \in [0, t_1]} \left\| \left( I_{\mathrm{MW}}^{(k)} \otimes I_{\mathrm{C}_2}^{(k)} \right) e^{-i y \bar{H}_{\mathrm{MWC}_2}^{(k)}} |t_{k-1,r}\rangle_{\mathrm{MC}_2} |t_{k-1,r}\rangle_{\mathrm{W}} \right\|_2 \quad (C.119) $$

$$ \le \bar{\delta}_{r,0} \frac{4\pi^2}{T_0} n_2 A_0 \left( A_{\mathrm{nor}}^{(2)} \sqrt{d} \left( 3 \left( \frac{\pi^2 n_2}{\sqrt{2}} \right)^{-2N_2} + 2e^{-\frac{\pi}{8}d} \right) + 2\varepsilon_v^{(2)}(t_1, d) \right) \quad (C.120) $$

where

$$ \bar{\delta}_{r,0} = \begin{cases} 0 & \text{if } r = 0 \\ 1 & \text{if } r = 1, 2, \dots, L \end{cases} \quad (C.121) $$

and $\varepsilon_v^{(2)}(t_1, d)$ is upper bounded in Lemma C.3.

*Proof.* Here we prove the result for $k = 2, 3, \dots, N_g$. While the case $k = 1$ follows analogously, it is best treated separately due to the periodic boundary conditions.

$$ \max_{y \in [0, t_1]} \left\| \left( I_{\mathrm{MW}}^{(k)} \otimes I_{\mathrm{C}_2}^{(k)} \right) e^{-i y \bar{H}_{\mathrm{MWC}_2}^{(k)}} |t_{k-1,r}\rangle_{\mathrm{MC}_2} |t_{k-1,r}\rangle_{\mathrm{W}} \right\|_2 \quad (C.122) $$

$$ = \bar{\delta}_{r,0} \max_{y \in [0, t_1]} \left\| \left( I_{\mathrm{M}}^{(k)} \otimes I_{\mathrm{C}_2}^{(k)} \right) e^{-i y \bar{H}_{\mathrm{MC}_2}^{[k,1]}} |t_{k-1,r}\rangle_{\mathrm{MC}_2} \right\|_2 \quad (C.123) $$

$$ = \bar{\delta}_{r,0} \max_{y \in [0,1]} \left\| I_{\mathrm{M}}^{(k)} |t_{k-1,r}\rangle_{\mathrm{M}_{\#k}} \right\|_2 \left\| I_{\mathrm{C}_2}^{(k)} e^{-i t_y \bar{H}_{\mathrm{MC}_2}^{[k,1]}} |t_{k-1,r}\rangle_{\bar{M}_{\#k} \mathrm{C}_2} \right\|_2 \quad (C.124) $$

$$ \le \bar{\delta}_{r,0} \| I_{\mathrm{M}}^{(k)} \|_2 \max_{y \in [0,1]} \left( \left\| I_{\mathrm{C}_2}^{(k)} \prod_{\substack{l=1 \\ l \neq k}}^{N_g} \left[ \sum_{i_l} \mathrm{M}_{\#l} \langle \Omega_{i_l} | 0 \rangle_{\mathrm{M}_{\#l}} | \Omega_{i_l} \rangle_{\mathrm{M}_{\#l}} \right] \right| \Psi_{\bar{i}_k} (t_{k-1+y,r}) \rangle_{\mathrm{C}_2} \|_2 + \| I_{\mathrm{C}_2}^{(k)} | \epsilon \rangle_{\mathrm{C}_2} \|_2 \right) \quad (C.125) $$

$$ \le \bar{\delta}_{r,0} 2\pi \max_{y \in [0,1]} \left( \frac{d}{T_0} \sqrt{\sum_{q \in S_d(t_{k-1+y,r}d/T_0)} \left( I_{C_2,d}^{(k)}(q) \right)^2 \left| \psi_{\mathrm{nor}}^{(2)}(t_{k-1+y,r}d/T_0, q) \right|^2} \right) \quad (C.126) $$

$$ + \frac{d}{T_0} \sqrt{\sum_{q \in [0,1,\dots,d-1]} |I_{C_2,d}^{(k)}(q) C_2 \langle \theta_q | \epsilon \rangle_{C_2}|^2} \quad (C.127) $$

$$ \le \bar{\delta}_{r,0} 2\pi \max_{y \in [0,1]} \left( \frac{2\pi}{T_0} \sqrt{\sum_{q \in S_d(t_{k-1+y,r}d/T_0)} \left( \bar{V}_0 \left( \frac{2\pi}{d}q \right) \bigg|_{x_0=x_0^{(k)}} \right)^2 \left| \psi_{\mathrm{nor}}^{(2)}(t_{k-1+y,r}d/T_0, q) \right|^2} \right) \quad (C.128) $$

$$ + \frac{d}{T_0} \| |\epsilon\rangle_{C_2} \|_2 \max_{q \in [0,1,\dots,d-1]} |I_{C_2,d}^{(k)}(q)| \quad (C.129) $$

$$ \le \bar{\delta}_{r,0} 2\pi \max_{y \in [0,1]} \left( A_{\mathrm{nor}}^{(2)} \frac{2\pi}{T_0} \sqrt{d \max_{q \in [-d/2+[k-1+y]d/N_g+rd, d/2+[k-1+y]d/N_g+rd]} \left( \bar{V}_0 \left( \frac{2\pi}{d}q \right) \bigg|_{x_0=x_0^{(k)}} \right)^2 \left| \psi_{\mathrm{nor}}^{(2)}(t_{k-1+y,r}d/T_0, q) \right|^2} \right) \quad (C.130) $$

$$ + \frac{2\pi}{T_0} \varepsilon_v^{(2)}(t_{k-1+y,r}, d) \max_{x \in [-\pi, \pi]} \bar{V}_0(x) \quad (C.131) $$

$$ \le \bar{\delta}_{r,0} 2\pi \max_{y \in [0,1]} \left( n_2 A_{\mathrm{nor}}^{(2)} A_0 \frac{2\pi}{T_0} \sqrt{d} \max_{q' \in [-1/2, 1/2]} \left[ \sum_{p=-\infty}^{\infty} V_B \left( 2\pi n_2 \left(q' + \left[1/2 + y - 1\right]/N_g + 1/2 + p\right) \right) \right] e^{-\pi \left(\frac{d}{\sigma_2}\right)^2 q'^2} \right) \quad (C.132) $$

$$ + \frac{2\pi}{T_0} \varepsilon_v^{(2)}(t_{k,r}, d) n_2 A_0 \left[ 1 + \sum_{p=0}^{\infty} (\pi + 2\pi p)^{-2N_2} + \sum_{p=0}^{\infty} (-\pi - 2\pi p)^{-2N_2} \right] \quad (C.133) $$
$$
\le \bar{\delta}_{r,0} 2\pi \max_{y \in [0,1]} \left( n_2 A_{\text{nor}}^{(2)} A_0 \frac{2\pi}{T_0} \sqrt{d} \left( \left( \frac{2\pi^2 n_2}{\sqrt{8}} \right)^{-2N_2} \left[ \max_{q' \in [-1/8, 1/8]} \sum_{p=-\infty}^{\infty} \left( \sqrt{8}(q' + [1/2 + y - 1]/N_g + 1/2 + p) \right)^{-2N_2} \right] \right. \\
\left. + \left[ \max_{q' \in [-1/2, -1/8] \cup [1/8, 1/2]} \sum_{p=-\infty}^{\infty} V_B \left( 2\pi n_2 (q' + [1/2 + y - 1]/N_g + 1/2 + p) \right) \right] e^{-\pi \left( \frac{d}{\sigma_2} \right)^2 (1/8)^2} \right) \tag{C.134}
$$

$$
+ \frac{4\pi}{T_0} \varepsilon_v^{(2)}(t_{k,r}, d) n_2 A_0 \tag{C.135}
$$

$$
\le \bar{\delta}_{r,0} 2\pi \max_{y \in [0,1]} \left( n_2 A_{\text{nor}}^{(2)} A_0 \frac{2\pi}{T_0} \sqrt{d} \left( 3 \left( \frac{\pi^2 n_2}{\sqrt{2}} \right)^{-2N_2} + \left( 5 \left( \frac{3\pi^2 n_2}{2} \right)^{-2N_2} + 1 \right) e^{-\pi \left( \frac{d}{\sigma_2} \right)^2 (1/8)^2} \right) \right) \quad (C.136)
$$

$$
+ \frac{4\pi}{T_0} n_2 A_0 \varepsilon_v^{(2)}(t_{k,r}, d) \tag{C.137}
$$

$$
\le \bar{\delta}_{r,0} \frac{4\pi^2}{T_0} n_2 A_0 \left( A_{\text{nor}}^{(2)} \sqrt{d} \left( 3 \left( \frac{\pi^2 n_2}{\sqrt{2}} \right)^{-2N_2} + 2e^{-\pi \left( \frac{d}{\sigma_2} \right)^2 (1/8)^2} \right) + 2\varepsilon_v^{(2)}(t_{k,r}, d) \right) \quad (C.138)
$$

Where in line eq. (C.123) we have used the definitions of $I_{\text{MW}}^{(k)}$ and $\bar{H}_{\text{MWC}_2}^{(k)}$ together with eq. (C.87) and recalled the Hamiltonian eq. (C.68):

$$
\bar{H}_{\text{MC}_2}^{[k,1]} = H_{\text{C}_2} + \sum_{\substack{l=1 \\ l \neq k}}^{N_g} I_{\text{M}}^{(l)} \otimes I_{\text{C}_2}^{(l)}. \qquad (C.139)
$$

In line eq. (C.124), we have defined $t_y = yt_1 = yT_0/N_g$, $y \in \mathbb{R}$; used $|t_{k-1,r}\rangle_{\text{MC}_2} = |t_{k-1,r}\rangle_{\text{M}\setminus\{M_{\#k-1}, M_{\#k}\}C_2}$
$|t_{k-1,r}\rangle_{M_{\#k-1}} |t_{k-1,r}\rangle_{M_{\#k}}$, and recalled that $\bar{H}_{\text{MC}_2}^{(k)}$ acts trivially on $M_{\#k}$ and $I_{\text{M}}^{(k)}$ acts trivially on $\bar{M}_{\#k}C_2$. In
line eq. (C.125) we have then calculated $e^{-iy\bar{H}_{\text{MC}_2}^{(k)}} |t_{k-1,r}\rangle_{\bar{M}_{\#k}C_2}$ following the same steps as in lines eq. (C.91)
to eq. (C.95) but for a time $y$ rather than a time $t_1$. We have also used the fact that the spectrum of $I_{\text{M}}^{(l)}$ is bounded
(eq. (C.48)). In line eq. (C.126), we have explicitly calculated the corresponding 2-norm in the basis $\{|\Omega_{il}\rangle_{M_{\#k,l}}\}_{i,l}$
on $\text{M}$ and $\{|\theta_k\rangle_{C_2}\}_k$ on $C_2$. In line eq. (C.130), we have used the fact that $q \in S_d(t_{k-1+y,r}d/T_0)$ is equivalent to
$q \in \{[-d/2+[k-1+y]d/N_g+rd], [-d/2+[k-1+y]d/N_g+rd]+1, [-d/2+[k-1+y]d/N_g+rd]+2, \dots, [-d/2+[k-1+y]d/N_g+rd]+d-1\}$. Since $[-d/2+[k-1+y]d/N_g+rd]+d-1 = [-d/2+d-1+[k-1+y]d/N_g+rd] \le d/2+[k-1+y]d/N_g+rd$, we have that $q$ takes on $d$ values in the interval $q \in [-d/2+[k-1+y]d/N_g+rd, d/2+[k-1+y]d/N_g+rd]$.
In line eq. (C.131) we have used the definition in Lemma C.3. In line eq. (C.132), we have performed the change
of variable $q$ to $q' = q/d - (k-1+y)/N_g - r$ and substituted for the definitions of $\bar{V}_0(\cdot)$, $x_0^{(k)}$, $\psi_{\text{nor}}^{(2)}(\cdot, \cdot)$. In
line eq. (C.134), we have spilt the interval into subintervals and used the bound $V_B(\cdot) \le (\pi(\cdot))^{2N_2}$. For the 1st term
in line eq. (C.136) we have used the conditions $N_g-2, N_2-2 \in \mathbb{N}_{>0}$ (stated in the Lemma) and observed that
$q' + [1/2+y-1]/N_g + 1/2 \in [1/8, 7/8]$. For the second term, we have used the bound $V_B(x) \le 1$ for all $x \in \mathbb{R}$ in
the case of $p=0, -1$ and $V_B(x) \le (\pi x)^{-2N_2}$ for or all $x \in \mathbb{R}$ in the cases $p \ne 0$. We have then used the fact that
$N_g-2, N_2-2 \in \mathbb{N}_{>0}$ to generate the final bound.
$\blacksquare$

**Theorem 2** (Optimal quantum frequential computers only require a classical internal bus). For all gate sets $\mathcal{U}_G$, initial memory states $|0\rangle_M \in \mathcal{C}_M$ and initial logical states $|0\rangle_S \in \mathcal{P}(\mathcal{H}_S)$, there exists $|0\rangle_C$, $|0\rangle_{C_2}$, $\{|t_{j,l}\rangle_C, |t_{j,l}\rangle_{MC_2}\}_{j=1,2,...,N_g; l=0,1,...,L}$, $N_g$, $H_{M_0SC}$ parametrised by the power $P > 0$ and a dimensionless parameter $\bar{\epsilon}$ (where elements $|t_{j,l}\rangle_C, |t_{j,l}\rangle_{MC_2}$ satisfy eqs. (IV.1) and (IV.8) respectively), such that for all $j = 1, 2, 3, ..., N_g; l = 0, 1, 2, ..., L$ and fixed $\bar{\epsilon} > 0$, the large-$P$ scaling is as follows

$$
T \left( e^{-it_{j,l} H_{\text{MWSCC}_2}} |0\rangle_M |0\rangle_{C_2} |0\rangle_W |0\rangle_S |0\rangle_C, \right. \quad (\text{IV.12})
$$

$$
\le \left( \sum_{r=0}^{l} \sum_{k=1}^{j} \tilde{d}(\mathfrak{m}_{r,k}) \right) h(\bar{\epsilon}) \text{ poly}((L+1)P') ((L+1)P')^{-1/\sqrt{\bar{\epsilon}}}, \quad (\text{IV.13})
$$
where $|0\rangle\langle0|_{C_2}$, tr$_M[|t_{j,l}\rangle\langle t_{j,l}|_{MC_2}] \in C_{C_2}$ and

$$f = \frac{1}{T_0} (T_0^2(L+1)P')^{1-\bar{\epsilon}} + \delta f'', \quad |\delta f''| \le \frac{1}{T_0} + \mathcal{O}\left(\text{poly}((L+1)P')((L+1)P')^{-1/\sqrt{\bar{\epsilon}}}\right) \text{ as } P' \to \infty. \quad (\text{IV.14})$$

*Proof.* Recall eq. (B.122): $N_g = [d^{1-\epsilon_g}]$. Let $\epsilon_g = \bar{\epsilon}$ so that we are in the regime where lemma C.2 holds, and $N_g \le d^{1-\bar{\epsilon}}$. Choosing

$$n_2 = d^{\bar{\epsilon}/4} \qquad (\text{C.140})$$

we find

$$\frac{d}{N_g n_2^2} \ge d^{\bar{\epsilon}/2}. \qquad (\text{C.141})$$

Therefore, so long as $N_2$ is d-independent, from lemma C.3 it follows that $\epsilon_v^{(2)}(t_1, d)$ decays faster than any polynomial in $d$ for all fixed $\bar{\epsilon} \in (0, 1/6)$. (As a side comment, note that we would not have been able to calculate its decay rate if $N_2$ were d-dependent. This is because the function $C_0(N_2)$ is unknown. This will limit the rate at which $n_2^{-2N_2}$ can decay as we will now see.) We now choose the parameter $N_2$,

$$N_2 = [2/\bar{\epsilon}^{1+1/2}] \ge 2/\bar{\epsilon}^{1+1/2}. \qquad (\text{C.142})$$

Therefore

$$n_2^{-2N_2} \le d^{-1/\sqrt{\bar{\epsilon}}} \qquad (\text{C.143})$$

for all $\bar{\epsilon} \in (0, 1/6)$.

<figure><img src="image_11.png" /></figure>

Therefore, using the above upper bounds for $\epsilon_v^{(2)}(t_1, d)$ and $n_2^{-2N_2}$, we can plug the bounds of lemmas C.2, C.4 and C.5 into the r.h.s. of lemma C.1 to achieve

$$\|e^{-it_{j,l}H_{\text{MWSCC}_2}} |0\rangle_M |0\rangle_W |0\rangle_S |\Psi(0)\rangle_C |\Psi(0)\rangle_{C_2} - |t_{j,l}\rangle_{MC_2} |t_{j,l}\rangle_W |t_{j,l}\rangle_S |\Psi(t_{j,l}d/T_0)\rangle_C\|_2 \qquad (\text{C.144})$$

$$\le \left( \sum_{r=0}^{l} \sum_{k=1}^{j} \tilde{d}(\mathfrak{m}_{r,k}) \right) h(\bar{\epsilon}) \text{poly}(d) d^{-3/\sqrt{\bar{\epsilon}}} + \text{poly}'(d) d^{-1/\sqrt{\bar{\epsilon}}}, \qquad (\text{C.145})$$

where poly(d), poly'(d) are $\bar{\epsilon}$-independent polynomials and $h(\bar{\epsilon}) \ge 0$ is d-independent. All poly(d), poly'(d), and $h(\bar{\epsilon})$ are independent from the elements of $\{\tilde{d}(\mathfrak{m}_{r,k})\}_{l,j}$. Therefore,

$$\|e^{-it_{j,l}H_{\text{MWSCC}_2}} |0\rangle_M |0\rangle_W |0\rangle_S |\Psi(0)\rangle_C |\Psi(0)\rangle_{C_2} - |t_{j,l}\rangle_{MC_2} |t_{j,l}\rangle_W |t_{j,l}\rangle_S |\Psi(t_{j,l}d/T_0)\rangle_C\|_2 \qquad (\text{C.146})$$

$$\le \left( \sum_{r=0}^{l} \sum_{k=1}^{j} \tilde{d}(\mathfrak{m}_{r,k}) \right) h(\bar{\epsilon}) \text{poly}''(d) d^{-1/\sqrt{\bar{\epsilon}}}, \qquad (\text{C.147})$$

where poly''(d) is independent from the elements of $\{\tilde{d}(\mathfrak{m}_{l,j})\}_{l,j}$ and $\bar{\epsilon}$.

We will now express $d$ in terms of the power $P'$. For this, we start by calculating the mean energy of the initial state. Defining $\rho_{\text{MWSCC}_2}^0 := |0\rangle_M |0\rangle_W |0\rangle_S |\Psi(0)\rangle_C |\Psi(0)\rangle_{C_2} (\langle M|0|W\rangle_S \langle 0|C\rangle_S \langle \Psi(0)|_{C_2}\langle \Psi(0)|)$, we find by direct calculation,

$$E'_0 := \text{tr}[\rho_{\text{MWSCC}_2}^0 H_{\text{MWSCC}_2}] = \langle C|0|H_C|0\rangle_C + \langle W|0|C\rangle_C \left( \sum_{l=1}^{N_g} I_{W_l}^{(l)} \otimes I_C^{(l)} \right) |0\rangle_W |0\rangle_C + \langle C_2|0|H_{C_2}|0\rangle_{C_2} \qquad (\text{C.148})$$

$$= \frac{2\pi}{T_0} \tilde{n}_0(d+1) + \delta E' + \langle C_2|0|H_{C_2}|0\rangle_{C_2}, \qquad (\text{C.149})$$

$$= \frac{2\pi}{T_0} (\tilde{n}_0 + \tilde{n}_{0,2})(d+1) + \delta E'', \qquad (\text{C.150})$$

where in line eq. (C.149) we have used eqs. (B.142) and (B.144) and where $|\delta E''|$ satisfies the same bound as $\delta E'$, namely eq. (B.145). To achieve the last line, we have used that $\langle 0|H_C|0\rangle_{C_2}$ is the same as $\langle 0|H_C|0\rangle_C$ after mapping the pair $\{\tilde{n}_0, \sigma\}$ to $\{\tilde{n}_{0,2}, \sigma_2\}$, as is readily verifiable from their definitions.
Using the relation

$$P' := \frac{E'_0}{T_0(L+1)}, \qquad (C.151)$$

from the main text and eq. (C.150),

$$d = \frac{T_0^2 P'(L+1)}{2\pi(\tilde{n}_0 + \tilde{n}_{0,2})} + \delta d', \qquad \delta d' := 1 - \frac{T_0(L+1)}{2\pi(\tilde{n}_0 + \tilde{n}_{0,2})} \delta E''. \qquad (C.152)$$

therefore, up to an additive vanishing term, $\delta d' - 1$, we have that $d$ scales linearly with $P'$, thus using lemma B.5 to lower bound the l.h.s. of eq. (C.147) in term of trace distance, we find eq. (IV.12).

To achieve the scaling of the gate frequency $f$ with power $P'$, i.e. eq. (IV.14), we can proceed analogously to eq. (B.149) for $\varepsilon_g = \bar{\varepsilon}$. This leads to

$$f = \frac{1}{T_0} \left( \frac{T_0^2(L+1)}{2\pi(\tilde{n}_0 + \tilde{n}_{0,2})} P' \right)^{1-\bar{\varepsilon}} + \delta f'', \qquad |\delta f''| \le \frac{1}{T_0} + \mathcal{O}\left(\text{poly}\left((L+1)P'\right)\left((L+1)P'\right)^{-1/\sqrt{\bar{\varepsilon}}}\right) \text{ as } P' \to \infty. \qquad (C.153)$$

Hence eq. (IV.14) is achieved by choosing $\tilde{n}_0 = \tilde{n}_{0,2} = 1/\pi$ [which is permitted since $\tilde{n}_0, \tilde{n}_{0,2} \in (0, 1]$].

To finalise the proof, using the definition of the set of non-squeezed states $C_{C_2}$ from appendix A 2, we need to show that $|0\rangle\langle 0|_{C_2}, \text{tr}_M[|t_{j,l}\rangle\langle t_{j,l}|_{MC_2}] \in C_{C_2}$. We show this property now. For $|0\rangle\langle 0|_{C_2}$ this follows immediately from the fact that $\sigma_2 = \sqrt{d}$ [from eq. (C.58)], the definitions from appendix A 2 and the proof that $|t_j\rangle_C \in C_C$ (which can be found in the paragraph after eq. (B.150)). From the definition eq. (C.49) a direct calculation of $\Delta t_{C_2}(\text{tr}_M[|t_{j,l}\rangle\langle t_{j,l}|_{MC_2}])$ yields

$$\Delta t_{C_2} \left( \text{tr}_M \left[ |t_{j,l}\rangle\langle t_{j,l}|_{MC_2} \right] \right) = \sum_{k \in S_d(t_{j,l}d/T_0)} C_2 \langle \theta_k | t_{C_2}^2 | \psi_{\text{nor}}^{(2)}(t_{j,l}d/T_0, k) |^2 | \theta_k \rangle_{C_2} - \left( C_2 \langle \theta_k | t_{C_2} | \psi_{\text{nor}}^{(2)}(t_{j,l}d/T_0, k) |^2 | \theta_k \rangle_{C_2} \right)^2 \qquad (C.154)$$

$$= C_2 \langle \Psi(t_{j,l}d/T_0) | t_{C_2}^2 | \Psi(t_{j,l}d/T_0) \rangle_{C_2} - \left( C_2 \langle \Psi(t_{j,l}d/T_0) | t_{C_2} | \Psi(t_{j,l}d/T_0) \rangle_{C_2} \right)^2, \qquad (C.155)$$

for $j = 1, 2, 3, \dots, N_g, l = 0, 1, 2, \dots, L$ and where we have defined $|\Psi(t_{j,l}d/T_0)\rangle_{C_2}$ analogously to $|\Psi(t_{j,l}d/T_0)\rangle_C$ but on $C_2$ rather than $C$, i.e.

$$|\Psi(t_{j,l}d/T_0)\rangle_{C_2} := \sum_{k \in S_d(t_{j,l}d/T_0)} \psi_{\text{nor}}^{(2)}(t_{j,l}d/T_0, k) |\theta_k\rangle_{C_2}, \qquad (C.156)$$

where $\psi_{\text{nor}}^{(2)}$ is defined in eq. (C.65). Similarly

$$\Delta H_{C_2} \left( \text{tr}_M \left[ |t_{j,l}\rangle\langle t_{j,l}|_{MC_2} \right] \right) = \sum_{k \in S_d(t_{j,l}d/T_0)} C_2 \langle \theta_k | H_{C_2}^2 | \psi_{\text{nor}}^{(2)}(t_{j,l}d/T_0, k) |^2 | \theta_k \rangle_{C_2} - \left( C_2 \langle \theta_k | H_{C_2} | \psi_{\text{nor}}^{(2)}(t_{j,l}d/T_0, k) |^2 | \theta_k \rangle_{C_2} \right)^2 \qquad (C.157)$$

$$= C_2 \langle \Psi(t_{j,l}d/T_0) | H_{C_2}^2 | \Psi(t_{j,l}d/T_0) \rangle_{C_2} - \left( C_2 \langle \Psi(t_{j,l}d/T_0) | H_{C_2} | \Psi(t_{j,l}d/T_0) \rangle_{C_2} \right)^2. \qquad (C.158)$$

for $j = 1, 2, 3, \dots, N_g, l = 0, 1, 2, \dots, L$. Therefore, from the definition of $C_{C_2}$ in appendix A 2 it follows that $\text{tr}_M[|t_{j,l}\rangle\langle t_{j,l}|_{MC_2}] \in C_{C_2}$ for $j = 1, 2, 3, \dots, N_g, l = 0, 1, 2, \dots, L$.

# Appendix D: Nonequilibrium steady-state dynamics

## 1. Setup of the dynamical semigroup

Here we define the dynamical semigroup. Some of the parameter choices will be left to the proofs in the subsequent section. A generic generator for the dynamics of a dynamical semigroup can be written in the from

$$\mathcal{L}(\cdot) = -i[H, (\cdot)] + \mathcal{D}(\cdot), \quad \mathcal{D}(\cdot) = \sum_j J^j(\cdot)J^j - \frac{1}{2}\{J^j\dagger J^j, (\cdot)\}, \qquad (D.1)$$
where $H$ is self-adjoint and the dissipative terms are formed by a set $\{J^j\}_j$ of arbitrary linear operators [37, 38]. The evolution operator for a time $\tau \ge 0$ is then $e^{\tau \mathcal{L}}(\cdot)$.

In our case, we choose $H = H'_{M_0SWC}$ and

$$J_C^j = \sqrt{2v_j} \mathcal{C}|0\rangle\langle\theta_j|_{\mathcal{C}}, \quad v_j > 0 \qquad (D.2)$$

$j = 0, 1, \dots, d-1$ where $|0\rangle_{\mathcal{C}}$ is the unperturbed initial state of the oscillator and $\{|\theta_j\rangle_{\mathcal{C}}\}_{j=0}^{d-1}$ is the discrete Fourier transform basis of the energy eigenbasis of $H_C$ defined in eq. (B.25).

Before discussing the dissipative pert, let us fix the form of the Hamiltonian to be

$$H'_{M_0SWC} = H_C + \sum_{l=1}^{N_g-1} I_{M_0SW}^{(l)} \otimes I_C^{(l)}, \qquad (D.3)$$

where the terms are as in appendix C. E.g. for $I_{M_0SW}^{(l)}$ see definition in eq. (C.4).

This form of the dissipater leads to the renewal of the oscillator state. The renewal itself is modelled by the stochastic jump, occurring with some probability $P(t)$ during an infinitesimal time state $[\tau, \tau + \delta\tau]$ taking the state from $\rho_{M_0SWC}(\tau)$ to $\mathcal{D}_C^{\text{re}}(\rho_{M_0SWC}(\tau))$ where

$$\mathcal{D}_C^{\text{re}}(\cdot) := \sum_{j=0}^{d-1} J_C^j(\cdot) J_C^{j\dagger}. \qquad (D.4)$$

Note that for our choice of $\{J_j\}_j$ this is many-to-one. If a renewal operation does not occur in said infinitesimal time interval, then the state is mapped to $\rho_{M_0SWC}(\tau)$ to $\mathcal{L}^{\text{no re}}(\rho_{M_0SWC}(\tau))$ where

$$\mathcal{L}_{M_0SWC}^{\text{no re}}(\cdot) := -i[H_{M_0SWC}(\cdot)] + \mathcal{D}_C^{\text{no re}}(\cdot), \quad \mathcal{D}_C^{\text{no re}}(\cdot) = -\frac{1}{2} \sum_{j=0}^{d-1} \{J_C^{j\dagger} J_C^j(\cdot)\}. \qquad (D.5)$$

Note that the renewal process therefore maps all input states to the state of the oscillator to its initial state: $\mathcal{D}_C^{\text{re}}(\rho_C) = |0\rangle\langle 0|_{\mathcal{C}}$ for all $\rho_C \in \mathcal{S}(\mathcal{H}_C)$. This is exactly the state after each cycle when it evolves according to its free dynamics without any perturbations due to it controlling the implementation of gates, in other words, according to $e^{jT_0H_C}|0\rangle_{\mathcal{C}} = |0\rangle_{\mathcal{C}}$, $j \in \mathbb{N}_{\ge 0}$. Therefore, if the renewal process occurs periodically with period $T_0$ it will correct for the small perturbative errors incurred by the oscillator due to its implementation of the logical gates required for the computation.

The form of eq. (D.5) allows for significant simplification. Namely

$$e^{\tau \mathcal{L}_{M_0SWC}^{\text{no re}}(\cdot)} = e^{-i\tau G_{M_0SWC}(\cdot)} e^{i\tau G_{M_0SWC}^{\dagger}(\cdot)} \qquad (D.6)$$

where

$$G_{M_0SWC} := H_{M_0SWC} - iV_C, \quad V_C := \sum_{j=0}^{d-1} v_j |\theta_j\rangle\langle\theta_j|_{\mathcal{C}} > 0. \qquad (D.7)$$

We choose

$$V_C = \gamma_0 I_C^{(N_g)} + \frac{\epsilon_b}{2T_0} \mathbb{1}_C, \qquad (D.8)$$

where $I_C^{(N_g)}$ is the positive semidefinite interaction term used in the proofs of Theorems 1 and 2 defined in eq. (B.26), $\epsilon_b > 0$ is a small constant (to be specified later), $\gamma_0 > 0$ is a scale factor (also to be specified later) and $\mathbb{1}_C$ the identity operator

Since $\mathcal{L}_{M_0SWC}(\cdot) = \mathcal{L}_{M_0SWC}^{\text{no re}}(\cdot) + \mathcal{D}_C^{\text{re}}(\cdot)$ we see that these are the only two processes which can occur in any infinitesimal time step.

By writing the total transcribed time as $\tau = \tau_l + t$, where $\tau_l$ denotes a total number of renewals $l$ in time interval $[0, \tau_l]$, we can write the state of the quantum frequential computer at time $\tau$ as a decomposition of the ensemble estate $\rho_{M_0SWC}(\tau)$ into partitions $\{\rho(t|\tau_l)\}_{l=0}^{\infty}$, where each state $\rho(t|\tau_l)$ has been renewed a total of $l$ times in the interval $[0, \tau_l]$ and zero times in the interval $(\tau_l, \tau_l + t]$. In particular:

$$\rho_{M_0SWC}(\tau) = \sum_{l=0}^{\infty} P(t|\tau_l) \rho_{M_0SWC}(t|\tau_l), \qquad (D.9)$$
where {$P(t|\tau_l)$}$_l$ are the probabilities of said events occurring.

The states {$P(0|\tau_l)$}$_l$ have just undergone a renewal process and are thus the output of channel eq. (D.4). As a consequence, they are of the form

$$
\rho_{M_0 \text{SWC}}(0|\tau_l) = \text{tr}_C[\rho_{M_0 \text{SWC}}(0|\tau_l)] \otimes |0\rangle\langle 0|_C. \qquad (D.10)
$$

Furthermore, it is a product state with memory cell M₀,₁ due to eq. (V.3). It is convenient for the following proofs to work with pure states, we will therefore purify the state on M₀SW via ancillary systems leading to a state

$$
|\rho(0|\tau_l)\rangle_{M_0\text{SWCA}} := |m_{l,1}\rangle_{M_{0,1}A_1} |(0|\tau_l)\rangle_{\bar{M}_{0,1}WS\bar{A}_1} |0\rangle_C \quad (D.11)
$$

which satisfies tr<sub>A</sub>[|ρ(0|τ<sub>l</sub>)⟩⟨ρ(0|τ<sub>l</sub>)|<sub>M<sub>0</sub>SWCA</sub>] = ρ<sub>M<sub>0</sub>SWC</sub>(0|τ<sub>l</sub>). In this appendix we used the convention M̄<sub>0,k</sub> = M<sub>0</sub>\M<sub>0,k</sub><fnref n="15" /> and each memory cell M<sub>0,k</sub> we associate with its own ancilla A<sub>k</sub> and additional ancillae for WS. Here Ā<sub>1</sub> denotes the total ancilla system after the removal of A<sub>1</sub>. We will use the notation |m⟩<sub>M<sub>0,k</sub>A<sub>k</sub></sub> = |m⟩<sub>M<sub>0,k</sub></sub> |m⟩<sub>A<sub>k</sub></sub> for all m ∈ G ∪ {0} since said states on M<sub>0,k</sub> are already pure.

The state $\rho_{M_0\text{SWC}}(t|\tau_l)$ can be calculated from the state $|\rho(0|\tau_l)\rangle_{M_0\text{SWC}}$ by evolving it according to the dynamical semigroup while conditioning on a renewal event not occurring. Had this been the only process involved, the dynamics would have been given by $\rho_{M_0\text{SWC}}(t|\tau_l) = e^{t\mathcal{L}_{M_0\text{SWC}}^{\text{no re}}}(\rho_{M_0\text{SWC}}(0|\tau_l))/P(t|\tau_l)$ where $P(t|\tau_l) = \text{tr}[e^{t\mathcal{L}_{M_0\text{SWC}}^{\text{no re}}}(\rho_{M_0\text{SWC}}(0|\tau_l))]$. However, recall that the bus is updating the memory cells $M_0$ in a similar fashion to the autonomous and explicit formulation in Theorem 2, although here modelled implicitly for convenience. In particular, we have assumed that eq. (V.3) holds and complete ignorance on the other memory cells contained in $M_0$. It is convenient to write this assumption in terms of a unitary transformation denoted $U_{M_{0,k}A_k M_{0,k-1}A_{k-1}}$ on the states in $M_0$ which have been purified. Therefore, noting that the channel eq. (D.6) preserves purity, the states $\{\rho_{M_0\text{SWC}}(t|\tau_l)\}_{l=0}^{\infty}$ for $t \in [t_{k-1}, t_k)$ and $k \in \mathbb{N}_{>0}$ are given by<fnref n="16" />

$$
\rho_{M_0 \text{SWC}}(t|\tau_l) P(t|\tau_l) = \rho_{M_0 \text{SWC}}(t|\tau_l), \qquad (D.12)
$$

$$
\rho_{M_0 \text{SWC}}(t|\tau_l) = \text{tr}_A [ |\rho(t|\tau_l)\rangle \langle \rho(t|\tau_l)|_{M_0 \text{SWCA}} ], \qquad (D.13)
$$

$$
|\rho(t|\tau_l)\rangle_{M_0\text{SWCA}} := e^{-i(t-t_{k-1})G_{M_0\text{SWC}}} U_{M_{0,k}A_k M_{0,k-1}A_{k-1}}^{(l,k)} |(k,l)\rangle_{M_0\text{SWCA}}, \quad (D.14)
$$

where<fnref n="17" />

$$
|(k, l)\rangle_{M_0 \text{SWCA}} := e^{-it_1 G_{M_0 \text{SWC}}} U_{M_{0,k-1}A_{k-1}M_{0,k-2}A_{k-2}}^{(l,k-1)} |(k-1, l)\rangle_{M_0 \text{SWCA}}, \quad k-1 \in \mathbb{N}_{>0} \quad (D.15)
$$

$$
|(1, l)\rangle_{M_0 \text{SWCA}} := |(0|\tau_l)\rangle_{M_0 \text{SWCA}}, \quad U_{M_{0,1}A_k M_{0,0}A_0}^{(l,1)} = \mathbb{1}_{M_{0,1}A_1 M_{0,0}A_0}, \quad 17 \qquad (D.16)
$$

and the conditional probability $P(t|\tau_l)$ is defined by taking the trace on both sides of eq. (D.12) and noting that
$\text{tr}[\rho_{M_0\text{SWCA}}(t|\tau_l)] = 1$.

The set $\{ U_{M_{0,k}A_k M_{0,k-1}A_{k-1}}^{(l,k)} \}_{k=1}^{N_g-1}$ uniquely determine the full set $\{ U_{M_{0,k}A_k M_{0,k-1}A_{k-1}}^{(l,k)} \}_{k=1}^{\infty}$ since

$$
U_{M_{0,k}A_k M_{0,k-1}A_{k-1}}^{(l,k)} = U_{M_{0,k+qN_g}A_{k+qN_g} M_{0,k-1+qN_g}A_{k-1+qN_g}}^{(l,k+qN_g)} \quad (D.17)
$$

for $q \in \mathbb{N}_{>0}$.

Finally, we now introduce the idealised states for which we wish to understand how well the actual dynamics approximates. We denote them by $\{|t_k|\tau_l\rangle\}_{M_0\text{SWA}}^{N_g-1}$ and define them as follows

$$
|[t_k|\tau_l]\rangle_{M_0\text{SWA}} = U_{M_{0,k+1}A_{k+1}M_{0,k}A_k}^{(l,k+1)} U_{(m_{l,k})} |[t_{k-1}|\tau_l]\rangle_{M_0\text{SWA}}, \quad (D.18)
$$

$$
|[0|\tau_l]\rangle_{M_0\text{SWA}} = |\rho(0|\tau_l)\rangle_{M_0\text{SWCA}} = |m_{l,1}\rangle_{M_{0,1}A_1} |(0|\tau_l)\rangle_{\bar{M}_{0,1}WS\bar{A}_1} |0\rangle_C \quad (D.19)
$$

$$
|[t_k|\tau_l]_C = \delta(k) e^{-\frac{\epsilon_b t_k}{2T_0}} |t_k]_C, \quad \delta(k) = e^{-\gamma_0 \beta(k) N_g}, \quad \gamma_0 = \tilde{\gamma}_0 d^{\tilde{\epsilon}^2}, \qquad (D.20)
$$

<footnote n="15">¹⁵This is the convention used in all of appendix D. Note that it differs from the convention used in appendix C where $\bar{M}_{0,k} = M \setminus M_{0,k}$.</footnote>

<footnote n="16">¹⁶Kets and bras containing left and right bold font brackets ( and ) (or [ and ]) indicate that the states are not necessarily normalised and to distinguish themselves from their normalised counterparts which will use normal brackets ( and ) (or [ and ]).</footnote>

<footnote n="17">¹⁷In the following we have denoted by $U_{M_{0,k}A_k M_{0,k-1}A_{k-1}}^{(l,k)}$ the unitary denoted by $U_{M_0A}^{(l,k)}$ in the main text. This is to avoid notational clutter in the main text while permitting more expression in the appendix. Furthermore, note that memory cell and ancilla $M_{0,0}A_0$ have not been defined. Such systems are not required, and only appear here. Since this operator is proportional to the identity, such systems are only introduced for notational convenience.</footnote>
$$ \|[t_k|\tau_l]\rangle_C = \frac{\|[t_k|\tau_l]\rangle_C}{\|[t_k|\tau_l]\rangle_C\|_2} = |t_k\rangle_C = |\Psi(t_k d/T_0)\rangle_C, \qquad (D.21) $$

$k = 1, 2, 3...$ where $\tilde{\gamma}_0 > 0$ is any positive $d$-independent parameter, $\beta(k)$ is the largest non-negative integer such that $k = \beta(k) N_g + r$, where $0 \le r < N_g$. Noticed that the choice $|t_k\rangle_C = |\Psi(t_k d/T_0)\rangle_C$ is identical to that of the proofs to Theorems 1 and 2 (namely eq. (B.34)). Recall that $|\Psi(t_k d/T_0)\rangle_C$ is periodic: $|\Psi(t_k d/T_0)\rangle_C = |\Psi(t_{k+N_g} d/T_0)\rangle_C$, c.f. eq. (C.34). And where $U(\mathfrak{m}_{l,N_g}) := 1$ (since there is no gate being applied in this time step), and we assume periodic boundary conditions $U(\mathfrak{m}_{l,k+qN_g}) = U(\mathfrak{m}_{l,k})$, $k = 1, 2, ..., N_g$, $q \in \mathbb{N}_{>0}$. This condition is only relevant in the event that the renewal process occurs late, which we will show in the proof of Theorem 3 is extremely unlikely.<fnref n="18" /> The only constraint on the unitaries $\{U_{\mathfrak{M}_{0,k}A_k\mathfrak{M}_{0,k-1}A_{k-1}}^{(l,k)}\}_{k=2}^{N_g-1}$ is that they are such that eq. (V.3) is satisfied. This amounts to property

$$ \|[t_k|\tau_l]\rangle_{\mathfrak{M}_{0,SWA}} = |\mathfrak{M}_{l,k+1}\rangle_{\mathfrak{M}_{0,k+1}A_{k+1}} \|[t_k|\tau_l]\rangle_{\bar{\mathfrak{M}}_{0,k+1}SW\bar{A}_{k+1}}, \qquad (D.22) $$

for $k = 1, 2, ..., N_g - 2$ (for $k = 0$, the state $\|[t_k|\tau_l]\rangle_{\mathfrak{M}_{0,SWA}}$ already satisfies condition eq. (D.22) by virtue of eq. (D.19)). Notice that due to the periodicity of the control states, $\{|\Psi(t_k d/T_0)\rangle_C\}_k$, where is an implied periodic boundary conditions on the states $\{|\mathfrak{M}_{l,k+1}\rangle_{\mathfrak{M}_{0,k+1}A_{k+1}}\}_k$ in eq. (D.22). Physically, the nature of the imposed boundary conditions is not very relevant since the probability of the dynamics not being renewed after each cycle of the control, is very small as 1) in theorem 3 shows.

We now make a definition needed for the following lemma, for $j \in \mathbb{N}_{>0}$, let $\beta'(j)$ be the largest number in $\mathbb{N}_{\ge 0}$ s.t.

$$ j = \beta'(j)N_g + r, \qquad (D.23) $$

for some $r \in \{1, 2, ..., N_g - 1\}$

## 2. Proof of Theorem 3

Before proving theorem 3, we start by proving a lemma which we will use multiple times in the proof of theorem 3.

**Lemma D.1.** For all $l, j \in \mathbb{N}_{\ge 0}$, $\bar{\varepsilon} > 0$

$$ \| |\rho(t_j|\tau_l)\rangle_{\mathfrak{M}_{0,SWCA}} - \|[t_j|\tau_l]\rangle_{\mathfrak{M}_{0,SWA}} \|[t_j|\tau_l]\rangle_C \|_2 \le e^{-\frac{\varepsilon_b}{2T_0}t_j} \left( \sum_{q=1}^{N_g} \tilde{d}(\mathfrak{m}_{l,q}) \right) \frac{1 - e^{-\gamma_0 N_g (1+\beta'(j))}}{1 - e^{-\gamma_0 N_g}} g(\bar{\varepsilon}) \text{poly}(P) P^{-1/\sqrt{\bar{\varepsilon}}}, \qquad (D.24) $$

where the r.h.s. is $\varepsilon_b$ independent. As in the main text, $g(\bar{\varepsilon}) > 0$ is $P$ independent while $\text{poly}(P)$ is an $\bar{\varepsilon}$-independent polynomial in $P$. The product $g(\bar{\varepsilon}) \text{poly}(P)$ is $l, j$ and $\varepsilon_b$ independent.

Recall that $P$ here is defined as in eq. (III.6). In the present context, the Hamiltonian is $H'_{\mathfrak{M}_{0,SWC}}$ rather than Hamiltonian $H_{\mathfrak{M}_{0,SWC}}$ used in the definition of $P$ in eq. (III.6). However, since these two Hamiltonians only differ in their interaction terms, as the initial state is close to being orthogonal on both of them (this is easily verifiably as per the calculation in eqs. (B.128) and (B.129)), and thus $\tilde{P} := \text{tr}[H'_{\mathfrak{M}_{0,SWC}}\rho^0_{\mathfrak{M}_{0,SWC}}]/T_0$ (where $\rho^0_{\mathfrak{M}_{0,SWC}}$ denotes the initial state as described in section V) satisfies $\tilde{P} = P + \delta P$, where $\delta P \to 0$ as $P \to \infty$. Thus there exists $P_0 > 0$, $x_0 > 0$ such that for all $P \ge P_0$ we have $x_0 \tilde{P} \ge P$ and hence $g(\bar{\varepsilon}) \text{poly}(\tilde{P}) P^{-1/\sqrt{\bar{\varepsilon}}} \le g(\bar{\varepsilon}) x_0^{-1/\sqrt{\bar{\varepsilon}}} \text{poly}(P) \tilde{P}^{-1/\sqrt{\bar{\varepsilon}}}$. Thus by redefining $g(\bar{\varepsilon})$ as $g(\bar{\varepsilon}) x_0^{-1/\sqrt{\bar{\varepsilon}}}$ we have that in the r.h.s. of eq. (D.24), $P$ can be replaced with $\tilde{P}$ if desired. Ultimately, in Theorem 3 we wish to describe our results as a function of $P''$ which is the more physical definition of power. As we will show in the proof of Theorem 3 below, $P''$ is also proportional to $P$ and hence a similar argument to that stated here will also us to exchange the variables.

**Proof.** The proof will follow similar steps to that of Theorem 1, but with important differences. We will only briefly cover the steps which follow analogously to that of Theorem 1 for brevity.

<footnote n="18">¹⁸We could have alternatively imposed a hard “cut-off” condition where we define the unitary to be the identity for $k \ge N_g$ and a similar theorem would follow.</footnote>
To start with, note that we can write $|\rho(t_j|\tau_l)\rangle_{M_0SWCA}$ in the form

$$|\rho(t_j|\tau_l)\rangle_{M_0SWCA} = \Delta_j \Delta_{j-1} \dots \Delta_1 |\rho(0|\tau_l)\rangle_{M_0SWCA}, \quad \Delta_j = U_{M_{0,j+1}A_{j+1}M_{0,j}A_j}^{(l,j+1)} e^{-it_1 G_{M_0SWC}} \quad (D.25)$$

for $j = 0, 1, 2, \dots$. Therefore, by making the association $\{|\Phi_j\rangle = \|[t_j|\tau_l]\rangle_{M_0SWA} \|[t_j|\tau_l]\rangle_C\}_{j=0}^{\infty}$, applying lemma B.4 we find for $j = 0, 1, 2, \dots$.

$$e^{\frac{\epsilon_b}{2T_0} t_j} \left\| |\rho(t_j|\tau_l)\rangle_{M_0SWCA} - \|[t_j|\tau_l]\rangle_{M_0SWA} \|[t_j|\tau_l]\rangle_C \right\|_2 \quad (D.26)$$

$$\le e^{\frac{\epsilon_b}{2T_0} t_j} \sum_{q=1}^{j} \left\| \|[t_q|\tau_l]\rangle_{M_0SWA} \|[t_q|\tau_l]\rangle_C - U_{M_{0,q+1}A_{q+1}M_{0,q}A_q}^{(l,q+1)} e^{-it_1 G_{M_0SWC}} \|[t_{q-1}|\tau_l]\rangle_{M_0SWA} \|[t_{q-1}|\tau_l]\rangle_C \right\|_2 \quad (D.27)$$

$$\le \sum_{q=1}^{j} \left\| \|[t_q|\tau_l]\rangle_{M_0SWA} \delta(q) |t_q\rangle_C - U_{M_{0,q+1}A_{q+1}M_{0,q}A_q}^{(l,q+1)} e^{-it_1 G'_{M_0SWC}} \|[t_{q-1}|\tau_l]\rangle_{M_0SWA} \delta(q-1) |t_{q-1}\rangle_C \right\|_2 \quad (D.28)$$

$$= \sum_{q=1}^{j} \left\| U(\mathfrak{m}_{l,q}) \|[t_{q-1}|\tau_l]\rangle_{M_0SWA} \delta(q) |t_q\rangle_C - e^{-it_1 G'_{M_0SWC}} \|[t_{q-1}|\tau_l]\rangle_{M_0SWA} \delta(q-1) |t_{q-1}\rangle_C \right\|_2, \quad (D.29)$$

where in the third line we have defined $G'_{M_0SWC} := G_{M_0SWC} - i \frac{\epsilon_b}{2T_0}$ while in the last line we used the unitary invariance of the two-norm.

Define

$$G'_{M_0SWC} := \begin{cases} H_C + I_{M_0SW}^{(k)} \otimes I_C^{(k)} & \text{if } k = 1, 2, \dots, N_g - 1 \\ H_C - i\gamma_0 I_C^{(N_g)} & \text{if } k = N_g \end{cases} \quad (D.30)$$

Thus adding and subtracting an appropriate term in eq. (D.29) we find

$$e^{\frac{\epsilon_b}{2T_0} t_j} \left\| |\rho(t_j|\tau_l)\rangle_{M_0SWCA} - \|[t_j|\tau_l]\rangle_{M_0SWA} \|[t_j|\tau_l]\rangle_C \right\|_2 \quad (D.31)$$

$$\le \sum_{q=1}^{j} \left\| U(\mathfrak{m}_{l,q}) \|[t_{q-1}|\tau_l]\rangle_{M_0SWA} \delta(q) |t_q\rangle_C - e^{-it_1 G'_{M_0SWC}} \|[t_{q-1}|\tau_l]\rangle_{M_0SWA} \delta(q-1) |t_{q-1}\rangle_C \right\|_2 \quad (D.32)$$

$$+ \delta(q-1) \left\| e^{-it_1 G'_{M_0SWC}} \|[t_{q-1}|\tau_l]\rangle_{M_0SWA} |t_{q-1}\rangle_C - e^{-it_1 G'_{M_0SWC}} \|[t_{q-1}|\tau_l]\rangle_{M_0SWA} |t_{q-1}\rangle_C \right\|_2. \quad (D.33)$$

To proceed, for line eq. (D.32) we can replace $G'_{M_0SWC}$ with

$$G'_{SWC} := \begin{cases} H_C + I_S^{(q, \mathfrak{m}_{l,q})} \otimes I_C^{(q)} & \text{if } q = 1, 2, \dots, N_g - 1 \text{ and } \mathfrak{m}_{l,q} \in \mathcal{G} \\ H_C + I_W^{(q, \mathfrak{m}_{l,q})} \otimes I_C^{(q)} & \text{if } q = 1, 2, \dots, N_g - 1 \text{ and } \mathfrak{m}_{l,q} = \mathbf{0} \\ H_C - i \gamma_0 I_C^{(N_g)} & \text{if } q = N_g \end{cases} \quad (D.34)$$

since $\|[t_{q-1}|\tau_l]\rangle_{M_0SWA}$ is proportional to $|\mathfrak{m}_{l,q}\rangle_{M_{0,q}A_q}$ due to eq. (D.22).

For line eq. (D.33), we can proceed analogously to lines eq. (B.10) to the end of the proof. Crucially, we note that the final results still follow even when the generator is not trace-preserving nor self-adjoint (such as is the case for $G'_{M_0SWC}$). Thus we find

$$e^{\frac{\epsilon_b}{2T_0} t_j} \left\| |\rho(t_j|\tau_l)\rangle_{M_0SWCA} - \|[t_j|\tau_l]\rangle_{M_0SWA} \|[t_j|\tau_l]\rangle_C \right\|_2 \quad (D.35)$$

$$\le \sum_{q=1}^{j} \left( \left\| \|[t_{q-1}|\tau_l]\rangle_{M_0SWA} \delta(q) |t_q\rangle_C - U^{\dagger}(\mathfrak{m}_{l,q}) e^{-it_1 G'_{SWC}} \|[t_{q-1}|\tau_l]\rangle_{M_0SWA} \delta(q-1) |t_{q-1}\rangle_C \right\|_2 \right) \quad (D.36)$$

$$+ \delta(q-1) t_1 \max_{x \in [0, t_1]} \left\| (G'_{M_0SWC} - G'_{M_0SWC}) e^{-ix G'_{M_0SWC}} \|[t_{q-1}|\tau_l]\rangle_{M_0SWA} \delta(q-1) |t_{q-1}\rangle_C \right\|_2 \quad (D.37)$$

$$= \sum_{q=1}^{j} \left( \left\| \|[t_{q-1}|\tau_l]\rangle_{M_0SWA} \delta(q) |t_q\rangle_C - U^{\dagger}(\mathfrak{m}_{l,q}) e^{-it_1 G'_{SWC}} \|[t_{q-1}|\tau_l]\rangle_{M_0SWA} \delta(q-1) |t_{q-1}\rangle_C \right\|_2 \right) \quad (D.38)$$
$$ + \delta(q-1) t_1 \max_{x \in [0, t_1]} \left\| \left( G_{\text{SWC}}^{(q, m_{l,q})} - G'_{\text{M}_0 \text{SWC}} \right) e^{-ix} G_{\text{SWC}}^{(q, m_{l,q})} \left| [t_{q-1}|\tau_l] \right\rangle_{\text{M}_0 \text{SWA}} |t_{q-1}\rangle_C \right\|_2, \quad (\text{D.39}) $$

where in the last equality we have noted that we can exchange $G_{M_0SWC}^{(q)'}$ with $G_{SWC}^{(q,m_{l,q})}$ for the same reasons as this exchange was possible before. Recall

$$ U(m_{l,k}) = \begin{cases} e^{iI_S^{(k,m_{l,k})}} & \text{if } m_{l,k} \in \mathcal{G} \text{ and } k \neq N_g \\ e^{iI_W^{(k,m_{l,k})}} & \text{if } m_{l,k} = \mathbf{0} \text{ and } k \neq N_g \\ \mathbf{1} & \text{if } k = N_g. \end{cases} \quad (\text{D.40}) $$

Therefore, by expanding $||[t_{q-1}|\tau_l]||_{M_0SWA}$ in the eigenbasis of $I_S^{(q,m_{l,q})} \otimes I_W^{(q,m_{l,q})}$ (and any basis for $M_0A$) we can bound line eq. (D.39) analogously to as in the proof of lemma B.2. Crucially, for this to work in the case $j = N_g$ it is important to note that Theorem IX.1 (Moving the clock through finite time with a potential) from [26] which is used in the proof to lemma B.2, applies when the potential function is from $\mathbb{R}$ to $\mathbb{R} \cup \mathbb{H}^-$, where $\mathbb{H}^- := \{a_0 + ib_0 | a_0 \in \mathbb{R}, b_0 < 0\}$, i.e. not only to potential functions of the form $\mathbb{R}$ to $\mathbb{R}$ as in the cases we have considered thus far. For the $j = N_g$ case the potential function is $\mathbb{R}$ to $i\mathbb{R}_{\le 0}$. This small change modifies the $\varepsilon_v(t_1, d)$ function slightly, for this $j = N_g$ case. As such we can upper bound line eq. (D.39) by lines eqs. (B.38) and (B.39) up to a modification in $\varepsilon_v(t_1, d)$ [defined in line eq. (B.40)] when $q = N_g$ which we will detail below starting in the paragraph above eq. (D.59).

As for upper bounding line eq. (D.38), for $q = 1, 2, \dots, q \neq mN_g$, $m \in \mathbb{N}_{>0}$, we have that $\delta(q) = \delta(q-1)$ and can be taken outside of the two-norm as a common multiplicative factor. Then, by expanding $||[t_{q-1}|\tau_l]||_{M_0SWA}$ in the eigenbasis of $I_S^{(q,m_{l,q})} \otimes I_W^{(q,m_{l,q})}$ (and any basis for $M_0A$) it follows identically to the proof of lemma B.3 and as such, line eq. (D.38) is upper bounded by line eq. (B.77). For $q = mN_g$, $m \in \mathbb{N}_{>0}$ there are some modifications due to the difference in the potential function mentioned above. As such, we calculate it here for completeness. Noting $\delta(q) = e^{-\gamma_0}\delta(q-1)$, for $q = mN_g$, the square of line eq. (D.38) reduces to

$$ \begin{aligned} & \delta^2(q-1) \left\| [t_{N_g-1}|\tau_l] \right\|_{M_0SWA} e^{-\gamma_0} \left\| (t_{N_g} d/T_0) \right\|_C - e^{-it_1(H_C-i\gamma_0 I_C^{(N_g)})} \left\| [t_{N_g-1}|\tau_l] \right\|_{M_0SWA} \left\| (t_{N_g-1} d/T_0) \right\|_C \right\|_2^2 && (\text{D.41}) \\ & \le \delta^2(q-1) \left( e^{-2\gamma_0} + \left\| e^{-it_1(H_C-i\gamma_0 I_C^{(N_g)})} \left\| (t_{N_g-1} d/T_0) \right\|_C \right\|_2^2 - 2e^{-\gamma_0} \Re \left[ C \left\langle (t_{N_g} d/T_0) \left\| e^{-it_1(H_C-i\gamma_0 I_C^{(N_g)})} \left\| (t_{N_g-1} d/T_0) \right\|_C \right\rangle \right] \right) && (\text{D.42}) \end{aligned} $$

$$ \le \delta^2(q-1) \left( e^{-2\gamma_0} + 3 \left\| \varepsilon(t_1, d) \right\|_C \|2 + \left\| \varepsilon(t_1, d) \right\|_C \|2\|_2^2 + \sum_{k \in S_d(d)} e^{-2\gamma_0} \int_{k-d/N_g}^{k-d+d/(2N_g)} dy I_{C,d}^{(N_g)}(y) |\psi_{\text{nor}}(d; k)|^2 \right) \quad (\text{D.43}) $$

$$ \le \delta^2(q-1) \left( e^{-2\gamma_0} + 3 \left\| \varepsilon(t_1, d) \right\|_C \|2\|_2 + \left\| \varepsilon(t_1, d) \right\|_C \|2\|_2^2 \right. \\ \left. + d \max_{k' \in [-1/2, 1/2]} e^{-2\gamma_0 \frac{2\pi}{d} \int_{k'-d/(2N_g)}^{k'+d/(2N_g)} dy' \bar{V}_0 \left( \frac{2\pi}{d} y' + x_0^{(N_g)} \right)} |\psi_{\text{nor}}(d; dk' + d)|^2 \right) \quad (\text{D.45}) $$

$$ \le \delta^2(q-1) \left( e^{-2\gamma_0} + 3 \left\| \varepsilon(t_1, d) \right\|_C \|2\|_2 + \left\| \varepsilon(t_1, d) \right\|_C \|2\|_2^2 \right. \\ \left. + A_{\text{nor}}^2 \left( 2e^{-\frac{\pi}{8}\frac{d^2}{\sigma^2 N_g^2}} + d \max_{q \in [-1,1]} e^{-2\gamma_0 \frac{2\pi}{d} \int_{-1/2+q/4}^{1/2-q/4} dy'' \bar{V}_0 \left( \frac{2\pi}{N_g} y'' + x_0^{(N_g)} \right)} \right) \right) \quad (\text{D.47}) $$

$$ \le \delta^2(q-1) \left( e^{-2\gamma_0} + 3 \left\| \varepsilon(t_1, d) \right\|_C \|2\|_2 + \left\| \varepsilon(t_1, d) \right\|_C \|2\|_2^2 \right. \\ \left. + A_{\text{nor}}^2 \left( 2e^{-\frac{\pi}{8}\frac{d^2}{\sigma^2 N_g^2}} + d e^{-2\gamma_0 - 2\gamma_0(4\pi n A_0) \left( \left( \frac{2N_g}{\pi^2 n} \right)^{2N} + \frac{\pi^2}{3} \left( \frac{1}{2\pi^2 n} \right)^{2N} \right)} \right) \right) \quad (\text{D.49}) $$

$$ \le \delta^2(q-1) \left( e^{-2\gamma_0} + 3 \left\| \varepsilon(t_1, d) \right\|_C \|2\|_2 + \left\| \varepsilon(t_1, d) \right\|_C \|2\|_2^2 + A_{\text{nor}}^2 \left( 2e^{-\frac{\pi}{8}\frac{d^2}{\sigma^2 N_g^2}} + d e^{-2\gamma_0} \right) \right) \quad (\text{D.50}) $$

Where in line eq. (D.43) we have employed Theorem IX.1 (Moving the clock through finite time with a potential) from [26] now with the pure imaginary potential function. In line eq. (D.45) we have noted that $k \in S_d(d)$ is the same as $k \in \{-d/2+d\}, [-d/2+d]+1, \dots, [-d/2+d]+d-1\}$. Therefore, since $[-d/2+d]+d-1 = [-d/2+d+d-1] \le$
$d/2 + d$ we can generate an upper bound by replacing $\sum_{k \in S_d(d)} (\cdot)$ with $d \max_{k \in [-d/2+d, d/2+d]} (\cdot)$. We have then performed a change of variable for $k$ and $y$ in the integral, followed by substituting for the definition of $I_{C,d}^{(N_g)}(\cdot)$. In line eq. (D.47), we have split the interval $[-1/2, 1/2]$ over which we are maximising into three partitions, $[-1/2, 1/2] = [-1/2, -1/(4N_g)] \cup (-1/(4N_g), 1/(4N_g)) \cup [1/(4N_g), 1/2]$ and maximised over them individually. Over the 1st and last interval, we have upper bounded the exponentiated integral by one, while we have bounded the modulus-squared wave function by one in the middle interval. Finally, we have performed a change of variable for $k' \in (-1/(4N_g), 1/(4N_g))$ to $q \in (-1, 1) \subset [-1, 1]$ followed by a change of integration variable. In to achieve line eq. (D.49), we have written the last exponential in line eq. (D.47) in the form $2\gamma_0(-1 + [1 - \beta_{\text{Int}}])$ with $\beta_{\text{Int}} := \frac{2\pi}{d} \int_{-1/2+q/4}^{1/2-q/4} dy'' \bar{V}_0(\frac{2\pi}{N_g}y'' + x_0^{(N_g)})$ followed by noting that we have already upper bounded $[1 - \beta_{\text{Int}}]^2$ between lines eq. (B.91) and eq. (B.98). Since it follows from the normalization of $\bar{V}_0(\cdot)$ that $\beta_{\text{Int}} \le 1$, $[1 - \beta_{\text{Int}}] = \sqrt{[1 - \beta_{\text{Int}}]^2}$ and hence we have used said bound to generate line eq. (D.49).

We now calculate $\| |\epsilon(t_1, d)\rangle_C \|_2$ noting that it takes on a slightly different expression to that of previous proofs due to the relevant potential function having the $\gamma_0$ prefactor. Our potential function $\bar{V}_0(\cdot)$ satisfies

$$ \sup_{k \in \mathbb{N}_{>0}} \left( 2 \max_{x \in [0, 2\pi]} \left| \frac{d^{(k-1)}}{dx^{(k-1)}} \bar{V}_0(x) \right| \right)^{1/k} \le 2nC_0, \qquad (\text{D.51}) $$

and $b$ was any upper bound to the l.h.s. We thus set $b = 2nC_0$ [See Eqs. (F217) and (F218) pg 63 in and text between them in [27] and recall that we have set $\delta = 1$ in this manuscript]. Furthermore, we have the relations

$$ d^{\epsilon_5} = \frac{d}{\bar{v}\sigma}, \qquad \epsilon_5 > 0, \qquad (\text{D.52}) $$

$$ \bar{v} = \frac{\pi\alpha_0\kappa}{\ln(\pi\alpha_0\sigma^2)} b, \qquad (\text{D.53}) $$

$$ \sigma = d^{\epsilon_6}, \qquad (\text{D.54}) $$

$$ \epsilon_5 < \epsilon_6, \qquad (\text{D.55}) $$

$$ \epsilon_{\nu}(t, d) = |t| \frac{d}{T_0} \left( \mathcal{O} \left( \frac{\sigma^3}{\bar{v}\sigma^2/d + 1} \right)^{1/2} + \mathcal{O} \left( \frac{d^2}{\sigma^2} \right) \right) \exp \left( -\frac{\pi}{4} \frac{\alpha_0^2}{\left( \frac{d}{\sigma} + \bar{v} \right)^2} \left( \frac{d}{\sigma} \right)^2 \right) + \mathcal{O} \left( |t| \frac{d^2}{\sigma^2} + 1 \right) e^{-\frac{\pi}{4} \frac{d^2}{\sigma^2}} + \mathcal{O} \left( e^{-\frac{\pi}{2} \sigma^2} \right), \qquad (\text{D.56}) $$

where in our notation $\| |\epsilon(t, d)\rangle_C \|_2 = \epsilon_\nu(t, d)$ (see eqs. (F219), (F33), (F214), (F213) and (F38) respectively in [27]). These give rise to eq. (B.40).

We now have that the potential function of interest is $\gamma_0 \bar{V}_0(\cdot)$, and hence the value for $\epsilon_\nu(t, d)$ will be modified. In particular, we have

$$ \sup_{k \in \mathbb{N}_{>0}} \left( 2 \max_{x \in [0, 2\pi]} \left| \frac{d^{(k-1)}}{dx^{(k-1)}} \gamma_0 \bar{V}_0(x) \right| \right)^{1/k} \le 2n\gamma_0 C_0, \qquad (\text{D.57}) $$

leading to $b = 2n\gamma_0 C_0$. We call that $\gamma$ was chosen with the parametrization

$$ \gamma_0 = \bar{\gamma}_0 d^{\epsilon_p}, \qquad (\text{D.58}) $$

with $\epsilon_p = \bar{\epsilon}^2$, and where we assume $\bar{\gamma}_0$ to be $d$ independent. let us now justify this parametrization. From above we find that

$$ \epsilon_{\nu}(t, d) = |t| \text{poly}(d) \exp \left( -\frac{\pi}{4} \frac{\alpha_0^2}{(\bar{\gamma}_0 + d^{\epsilon_5}/\sigma)^2} d^{2(\epsilon_5 - \epsilon_p)} \right). \qquad (\text{D.59}) $$

This reduces to eq. (B.40) in the limit $\bar{\gamma}_0 \to 1$, $\epsilon_p \to 0$ as expected. From eq. (D.59) we observe that we need to choose $\epsilon_p$ such that $\epsilon_5 - \epsilon_p > 0$ and parametrize $\epsilon$ in terms of $\bar{\epsilon}$. To do so, recall $\epsilon_5 = \eta \bar{\epsilon}$ (eq. (B.119)) and that in the case under consideration, where the optimal logical frequency $f$ is asymptotically achievable, we have using eq. (B.151)

$$ \epsilon_5 = \eta \bar{\epsilon} = \frac{2}{2\bar{\epsilon} + 2\sqrt{\bar{\epsilon} + 1}} \bar{\epsilon}^2 \qquad (\text{D.60}) $$

where we set $\epsilon_p = \bar{\epsilon}^2$ such that

$$ \epsilon_5 - \epsilon_p = \left( \frac{2}{2\bar{\epsilon} + 2\sqrt{\bar{\epsilon} + 1}} - 1 \right) \bar{\epsilon}^2 \qquad (\text{D.61}) $$
for $\bar{\epsilon} \in (1 - \sqrt{3}/2, 0)$ so that the r.h.s. is positive. Therefore, in this parameter range the quantities $\| |\epsilon(t_1, d)\rangle_C \|_2$ in eq. (D.50) decay exponentially in $d^{2(\epsilon_5 - \epsilon_p)}$ for $q = N_g$ (for $q = 1, 2, \dots, N_g - 1$ they decay exponentially in $d^{2\epsilon_5}$ since they are given by eq. (B.40)). Furthermore, from eq. (D.58) and eq. (B.126) we have that the other terms also decay exponentially in $d^x$ for some $x > 0$ solely determined by $\bar{\epsilon}$.

We thus have that $\| |\rho(t_j|\tau_l)\rangle_{M_0SWCA} - |[t_j|\tau_l]\rangle_{M_0SWCA} \|_2$ is upper bounded by the r.h.s. of eq. (III.7) up to the replacement of $\tilde{d}(\mathfrak{m}_k)$ with $\tilde{d}(\mathfrak{m}_{l,k})$ since the gate being applied is distinct in the case at hand. Putting everything together we thus obtain for $j \in \mathbb{N}_{\ge 0}$

$$ \| |\rho(t_j|\tau_l)\rangle_{M_0SWCA} - \| [t_j|\tau_l]\rangle_{M_0SWCA} \|_2 \le e^{-\frac{\epsilon_b}{2T_0}t_j} \left( \sum_{k=1}^{j} \tilde{d}(\mathfrak{m}_{l,k}) \delta(k-1) \right) g(\bar{\epsilon}) \text{poly}(P) P^{-1/\sqrt{\bar{\epsilon}}}, \quad (D.62) $$

$$ \le e^{-\frac{\epsilon_b}{2T_0}t_j} \left( \sum_{q=1}^{N_g} \sum_{k=0}^{\beta'(j)} \tilde{d}(\mathfrak{m}_{l,k}N_g+q) \delta(kN_g+q-1) \right) g(\bar{\epsilon}) \text{poly}(P) P^{-1/\sqrt{\bar{\epsilon}}}, \quad (D.63) $$

$$ \le e^{-\frac{\epsilon_b}{2T_0}t_j} \left( \sum_{q=1}^{N_g} \tilde{d}(\mathfrak{m}_{l,q}) \left( \sum_{k=0}^{\beta'(j)} e^{-\gamma_0 N_g k} \right) g(\bar{\epsilon}) \text{poly}(P) P^{-1/\sqrt{\bar{\epsilon}}}, \quad (D.64) $$

where in line eq. (D.64), we have taken into account the definition of $\delta(k)$ in eq. (D.20).

<figure><img src="image_12.png" /></figure>

We can now prove theorem 3.

**Theorem 3** (Nonequilibrium steady-state optimal quantum frequential computers exist). For all gate sets $\mathcal{U}_G$, initial gate sequences $(\mathfrak{m}_{l,k})_{l,k}$ with elements in $\mathcal{G}$, and initial logical states $|0\rangle_S \in \mathcal{P}(\mathcal{H}_S)$, there exists $|0\rangle_C$, $\{|t_j|\tau_l\rangle_C\}_{j=1,2,\dots,N_g}$, $N_g$, $\mathcal{L}_{M_0SWC}$ parametrised by the power $P'' > 0$ and a dimensionless parameter $\bar{\epsilon}$ (where elements $|[t_j|\tau_l]\rangle_C$, satisfy eq. (V.5)), such that for all $j = 1, 2, 3, \dots, N_g$; $l \in \mathbb{N}_{\ge 0}$ and fixed $\bar{\epsilon} > 0$, the following large-$P''$ scaling hold simultaneously

1) Given that $l \in \mathbb{N}_{\ge 0}$ renewals occurred in the time interval $[0, \tau_l]$, the probability that the next renewal occurs in the interval $[\tau_l + T_0 - t_1, \tau_l + T_0]$ is:

$$ \int_{\tau_l + T_0 - t_1}^{\tau_l + T_0} dt P(t, +1|\tau_l) = 1 - \epsilon_r, \quad 0 < \epsilon_r \le \left( \sum_{k=1}^{N_g} \tilde{d}(\mathfrak{m}_{l,k}) \right) g(\bar{\epsilon}) \text{poly}(P'') P''^{-1/(2\sqrt{\bar{\epsilon}})}, \quad (V.9) $$

2) The deviations in the state between renewals are small: For $j = 1, 2, \dots, N_g$,

$$ T\left(\rho_{M_0SC}(t_j|\tau_l), \left\| [t_j|\tau_l] \right\|_{M_0SWCA} \left\| [t_j|\tau_l] \right\|_C\right) \le \left( \sum_{k=1}^{j} \tilde{d}(\mathfrak{m}_{l,k}) \right) g(\bar{\epsilon}) \text{poly}(P'') P''^{-1/\sqrt{\bar{\epsilon}}}, \quad (V.10) $$

3) The gate frequency has the asymptotically optimal scaling in terms of power:

$$ f = \frac{1}{T_0} (T_0^2 P'')^{1-\bar{\epsilon}} + \delta f', \quad |\delta f'| \le \frac{1}{T_0} + \mathcal{O}\left(\text{poly}(P'') P''^{-1/(2\sqrt{\bar{\epsilon}})}\right) \text{ as } P'' \to \infty. \quad (V.11) $$

*Proof.* We start with the proof of 1). This requires the calculation of the probability that the $(l+1)$th renewal occurs in the interval $(\tau_l + t_{N_g}, \tau_l + t_{N_g-1})$, namely

$$ \epsilon_r = 1 - \lim_{\epsilon \to 0} \int_{\tau_l + t_{N_g-1} + \epsilon}^{\tau_l + t_{N_g} - \epsilon} dt P(t, +1|\tau_l) \quad (D.65) $$

The probability, $P(t, +1|\tau_l)$, can be evaluated by conditioning on not renewing in the interval $t \in (0, t)$ given that the $l$th renewal occurred at time $\tau_l$, followed by the $(l+1)$th renewal occurring at time $t$. This can be calculated by applying the renewal generator $\mathcal{D}_C^{\text{re}}(\cdot)$ to $|\rho(t|\tau_l)\rangle_{M_0SWCA}$ followed by taking the trace. Thus from eqs. (D.4) and (D.14) we find that $P(t, +1|\tau_l) = \text{tr}[2V_C |\rho(t|\tau_l)\rangle\langle\rho(t|\tau_l)|_{M_0SWCA}]$. For $\epsilon > 0$, in the interval $t \in (\tau_l + t_{N_g-1}, \tau_l + t_{N_g-1} + \epsilon)$, $|\rho(t|\tau_l)\rangle_{M_0SWCA} = e^{-i(t-t_{N_g-1})G_{M_0SWC}} |\rho(t_{N_g-1}|\tau_l)\rangle_{M_0SWCA}$ and in the interval $t \in (\tau_l + t_{N_g} - \epsilon, \tau_l + t_{N_g})$, $|\rho(t|\tau_l)\rangle_{M_0SWCA} = e^{-i(t-t_{N_g-1})G_{M_0SWC}} |\rho(t_{N_g-1}|\tau_l)\rangle_{M_0SWCA}$. Therefore, we can solve the integral analytically to find

$$ \epsilon_r = 1 - \lim_{\epsilon \to 0^+} \int_{\tau_l + t_{N_g-1} + \epsilon}^{\tau_l + t_{N_g} - \epsilon} dt P(t, +1|\tau_l) \quad (D.66) $$
$$ = 1 - \lim_{\epsilon \to 0^+} \left( \mathrm{tr} \left[ |\rho(t_{N_g-1} + \epsilon|\tau_l)\rangle\langle\rho(t_{N_g-1} + \epsilon|\tau_l)|_{\mathrm{M}_0\mathrm{SWCA}} \right] - \mathrm{tr} \left[ |\rho(t_{N_g} - \epsilon|\tau_l)\rangle\langle\rho(t_{N_g} - \epsilon|\tau_l)|_{\mathrm{M}_0\mathrm{SWCA}} \right] \right) \quad (\mathrm{D}.67) $$

$$ = 1 + \| |\rho(t_{N_g}|\tau_l)\rangle_{\mathrm{M}_0\mathrm{SWCA}} \|_2^2 - \| |\rho(t_{N_g-1}|\tau_l)\rangle_{\mathrm{M}_0\mathrm{SWCA}} \|_2^2. \quad (\mathrm{D}.68) $$

We can now use lemma D.1 to exchange $|\rho(t_{N_g}|\tau_l)\rangle_{\mathrm{M}_0\mathrm{SWCA}}$ and $|\rho(t_{N_g-1}|\tau_l)\rangle_{\mathrm{M}_0\mathrm{SWCA}}$ for $|[t_{N_g}|\tau_l]\rangle_{\mathrm{M}_0\mathrm{SWA}}$ $|[t_{N_g}|\tau_l]\rangle_{\mathrm{C}}$ and $|[t_{N_g-1}|\tau_l]\rangle_{\mathrm{M}_0\mathrm{SWA}}$ $|[t_{N_g-1}|\tau_l]\rangle_{\mathrm{C}}$ respectively up to the small errors dictated by the r.h.s. of eq. (D.24). Recall that $\| \|[t_{N_g}|\tau_l]\rangle_{\mathrm{M}_0\mathrm{SWA}} \|[t_{N_g}|\tau_l]\rangle_{\mathrm{C}} \|_2$ is exponentially small in $d^{\bar{\epsilon}^2}$, while $\| \|[t_{N_g-1}|\tau_l]\rangle_{\mathrm{M}_0\mathrm{SWA}} \|[t_{N_g-1}|\tau_l]\rangle_{\mathrm{C}} \|_2^2 = e^{-\epsilon_b} = 1 - \epsilon_b + \mathcal{O}(\epsilon_b)^2$. This gives us

$$ \int_{\tau_l + T_0 - t_1}^{\tau_l + T_0} dt P(t, +1|\tau_l) = 1 - \epsilon_r, \quad 0 < \epsilon_r \le \epsilon_b + \mathcal{O}(\epsilon_b)^2 + \left( \sum_{k=1}^{N_g} \tilde{d}(\mathfrak{m}_{l,k}) \right) g(\bar{\epsilon}) \mathrm{poly}(P) P^{-1/\sqrt{\bar{\epsilon}}}. \quad (\mathrm{D}.69) $$

We now set the free parameter $\epsilon_b$. We choose

$$ \epsilon_b = \left( \sum_{q=1}^{N_g} \tilde{d}(\mathfrak{m}_{l,q}) \right) P^{-1/(2\sqrt{\bar{\epsilon}})}. \quad (\mathrm{D}.70) $$

[The reason for this choice will become apparent later when bounding $|\delta E_1^{\mathrm{before re}}|$. See line eq. (D.126)]. Thus plugging in to eq. (D.69) we find

$$ \int_{\tau_l + T_0 - t_1}^{\tau_l + T_0} dt P(t, +1|\tau_l) = 1 - \epsilon_r, \quad 0 < \epsilon_r \le \left( \sum_{k=1}^{N_g} \tilde{d}(\mathfrak{m}_{l,k}) \right) g(\bar{\epsilon}) \mathrm{poly}(P) P^{-1/(2\sqrt{\bar{\epsilon}})}. \quad (\mathrm{D}.71) $$

We now calculate $|\langle E^{\mathrm{re}} \rangle|$ as a function of $P$. Since from eq. (V.8) we have $P'' \ge |\langle E^{\mathrm{re}} \rangle|/T_0$, this in conjunction with eq. (D.71) will allow us to prove item 1) in Theorem 3.

$$ -\langle E^{\mathrm{re}} \rangle = \int_0^\infty ds \mathrm{tr}[H'_{\mathrm{M}_0\mathrm{SWC}} \mathcal{D}_{\mathrm{M}_0\mathrm{SWC}}^{\mathrm{re}}(\rho_{\mathrm{M}_0\mathrm{SWC}}(s|\tau_l))] \quad (\mathrm{D}.72) $$

$$ = 2 \int_0^\infty ds \mathrm{tr}[H_C |0\rangle\langle 0|_C \otimes \sum_{j=1}^{d-1} v_{jC} \langle \theta_j | \rho_{\mathrm{M}_0\mathrm{SWC}}(s|\tau_l) | \theta_j \rangle_C] \quad (\mathrm{D}.73) $$

$$ + 2 \sum_{l=1}^{N_g-1} \int_0^\infty ds \mathrm{tr} \left[ \left( I_{\mathrm{M}_0\mathrm{SW}}^{(l)} \otimes I_C^{(l)} \right) \left( |0\rangle\langle 0|_C \otimes \sum_{j=1}^{d-1} v_{jC} \langle \theta_j | \rho_{\mathrm{M}_0\mathrm{SWC}}(s|\tau_l) | \theta_j \rangle_C \right) \right] \quad (\mathrm{D}.74) $$

$$ = c \langle 0 | H_C | 0 \rangle_C \int_0^\infty ds P(t, +1|\tau_l) \quad (\mathrm{D}.75) $$

$$ + 2 \sum_{l=1}^{N_g-1} \mathrm{tr}[I_C^{(l)} |0\rangle\langle 0|_C] \int_0^\infty ds \mathrm{tr}[I_{\mathrm{M}_0\mathrm{SW}}^{(l)} V_C \rho_{\mathrm{M}_0\mathrm{SWC}}(s|\tau_l)] \quad (\mathrm{D}.76) $$

$$ = c \langle 0 | H_C | 0 \rangle_C \quad (\mathrm{D}.77) $$

$$ - \sum_{l=1}^{N_g-1} \mathrm{tr}[I_C^{(l)} |0\rangle\langle 0|_C] \int_0^\infty ds \mathrm{tr}[I_{\mathrm{M}_0\mathrm{SW}}^{(l)} \frac{d}{ds} \rho_{\mathrm{M}_0\mathrm{SWC}}(s|\tau_l)] \quad (\mathrm{D}.78) $$

$$ = c \langle 0 | H_C | 0 \rangle_C + \sum_{l=1}^{N_g-1} \mathrm{tr}[I_C^{(l)} |0\rangle\langle 0|_C] \mathrm{tr}[I_{\mathrm{M}_0\mathrm{SW}}^{(l)} \rho_{\mathrm{M}_0\mathrm{SWC}}(0|\tau_l)] \quad (\mathrm{D}.79) $$

$$ = \mathrm{tr}[H'_{\mathrm{M}_0\mathrm{SWC}} \rho_{\mathrm{M}_0\mathrm{SWC}}(0|\tau_l)] \quad (\mathrm{D}.80) $$

$$ = P T_0 + \delta \langle E^{\mathrm{re}} \rangle_1 \quad (\mathrm{D}.81) $$

Where in eq. (D.80), we have used eq. (D.10) and where

$$ \delta E_1^{\mathrm{re}} := \mathrm{tr}[H'_{\mathrm{M}_0\mathrm{SWC}} \rho_{\mathrm{M}_0\mathrm{SWC}}(t|\tau_l)] - \mathrm{tr}[\rho_{\mathrm{M}_0\mathrm{SC}}^0 \rho_{\mathrm{M}_0\mathrm{SC}}(t|\tau_l)] \quad (\mathrm{D}.82) $$
$$ = \sum_{q=1}^{N_g-1} \mathrm{tr}[I_{M_0 \mathrm{SWC}}^{(q)} |\mathfrak{m}_{l,1}\rangle\langle\mathfrak{m}_{l,1}|_{M_{0,1}A_1} |(0|\tau_l)\rangle\langle(0|\tau_l)|_{\bar{M}_{0,1}WS\bar{A}_1}] \mathrm{tr}[I_C^{(q)} |\Psi(0)\rangle\langle\Psi(0)|_C] \quad (\mathrm{D.83}) $$

$$ - \sum_{q=1}^{N_g} \mathrm{tr}[I_{M_0 \mathrm{SC}}^{(q)} |\mathfrak{m}_{l,1}\rangle\langle\mathfrak{m}_{l,1}|_{M_{0,1}} \rho_{M_0 \mathrm{S}}^0] \mathrm{tr}[I_C^{(q)} |\Psi(0)\rangle\langle\Psi(0)|_C], \quad (\mathrm{D.84}) $$

$$ |\delta E_1^{\mathrm{re}}| \le g(\bar{\epsilon}) \mathrm{poly}(P) P^{-1/\sqrt{\bar{\epsilon}}}. \quad (\mathrm{D.85}) $$

In inequalities eqs. (D.84) and (D.85), we have recalled definition eq. (III.6) and noted that the interaction terms have a vanishing contribution to the energy as proven in the proof of Theorem 2 from eq. (B.141) onwards.

Thus

$$ P'' \ge |\langle E^{\mathrm{re}} \rangle|/T_0 = P + \delta E_1^{\mathrm{re}}/T_0. \quad (\mathrm{D.86}) $$

Therefore, there exists $x_0(\bar{\epsilon}) > 0$, $P_0(\bar{\epsilon}) > 0$ such that $P'' \ge x_0(\bar{\epsilon})P$ for all $P \ge P_0(\bar{\epsilon})$, $\bar{\epsilon} > 0$. This concludes the proof of item 1) in Theorem 3.

For item 2) in Theorem 3, start by observing that the two-norm distance between two normalised states is an upper bound to the trance distance between said states (recall lemma B.5). Then note that lemma D.1 can easily be repeated for the normalised version of the states in eq. (D.24). This yields item 2) in 2-norm distance, up to a replacement of $P''$ with $P$. Using eq. (D.81), we can then convert from $P$ to $P''$.

We now prove item 3) in Theorem 3. Since we are using the same parametrizations as in Theorem 1, for case b), we have that eq. (B.152) holds, namely

$$ f = \frac{1}{T_0} \left( \frac{T_0^2}{2\pi\bar{n}_0} P \right)^{1-\bar{\epsilon}} + \delta f', \quad |\delta f'| \le \frac{1}{T_0} + \mathcal{O}\left(\mathrm{poly}(P) P^{-1/\sqrt{\bar{\epsilon}}}\right) \text{ as } P \to \infty. \quad (\mathrm{D.87}) $$

In order to convert from $P$ to $P''$, we will need an upper bound for $P''$ in terms of $P$ (we already have a lower bound via eq. (D.86).) This is

Observe that

$$ \frac{d}{dt} \mathrm{tr}[H'_{M_0 \mathrm{SWC}} \rho_{M_0 \mathrm{SWC}}(t|\tau_l)] = \mathrm{tr}[H'_{M_0 \mathrm{SWC}} \mathcal{D}_C^{\mathrm{no re}} (\rho_{M_0 \mathrm{SWC}}(t|\tau_l))], \quad (\mathrm{D.88}) $$

in the intervals where $\mathrm{tr}[H'_{M_0 \mathrm{SWC}} \rho_{M_0 \mathrm{SWC}}(t|\tau_l)]$ is differentiable. Therefore, defining $N(t)$ as the largest integer $N$ s.t. $t_{N+1} < t$ Recalling eq. (V.7) we have

$$ \langle E^{\mathrm{after re}} \rangle = - \int_0^\infty dt P(t, +1|\tau_l) \int_0^t \mathrm{tr}[H'_{M_0 \mathrm{SWC}} \mathcal{D}_C^{\mathrm{no re}} (\rho_{M_0 \mathrm{SWC}}(s|\tau_l))] \mathrm{d}s. \quad (\mathrm{D.89}) $$

$$ = - \int_0^\infty dt P(t, +1|\tau_l) \times \quad (\mathrm{D.90}) $$

$$ \left( \sum_{j=0}^{N(t)} \lim_{\epsilon \to 0^+} \int_{t_j+\epsilon}^{t_{j+1}-\epsilon} \mathrm{tr}[H'_{M_0 \mathrm{SWC}} \mathcal{D}_{M_0 \mathrm{SWC}}^{\mathrm{no re}} (\rho_{M_0 \mathrm{SWC}}(s|\tau_l))] + \lim_{\epsilon \to 0^+} \int_{t_{N+1}+\epsilon}^{t} \mathrm{tr}[H'_{M_0 \mathrm{SWC}} \mathcal{D}_{M_0 \mathrm{SWC}}^{\mathrm{no re}} (\rho_{M_0 \mathrm{SWC}}(s|\tau_l))] \right) \quad (\mathrm{D.91}) $$

$$ = - \int_0^\infty dt P(t, +1|\tau_l) \quad (\mathrm{D.92}) $$

$$ \left( \sum_{j=0}^{N(t)} \lim_{\epsilon \to 0^+} \left[ \mathrm{tr}[H'_{M_0 \mathrm{SWC}} \rho_{M_0 \mathrm{SWC}}(t_{j+1} - \epsilon|\tau_l)] - \mathrm{tr}[H'_{M_0 \mathrm{SWC}} \rho_{M_0 \mathrm{SWC}}(t_j + \epsilon|\tau_l)] \right] \right) \quad (\mathrm{D.93}) $$

$$ + \lim_{\epsilon \to 0^+} \left( \mathrm{tr}[H'_{M_0 \mathrm{SWC}} \rho_{M_0 \mathrm{SWC}}(t|\tau_l)] - \mathrm{tr}[H'_{M_0 \mathrm{SWC}} \rho_{M_0 \mathrm{SWC}}(t_{N+1} + \epsilon|\tau_l)] \right) \quad (\mathrm{D.94}) $$

$$ = \int_0^\infty dt P(t, +1|\tau_l) \mathrm{tr}[H'_{M_0 \mathrm{SWC}} \rho_{M_0 \mathrm{SWC}}(0|\tau_l)] + \delta E_1^{\mathrm{after re}} \quad (\mathrm{D.95}) $$

$$ - \int_0^\infty dt P(t, +1|\tau_l) \mathrm{tr}[H'_{M_0 \mathrm{SWC}} \rho_{M_0 \mathrm{SWC}}(t|\tau_l)] \quad (\mathrm{D.96}) $$

$$ = P T_0 + \delta E_1^{\mathrm{re}} + \delta E_1^{\mathrm{after re}} - \int_0^\infty dt P(t, +1|\tau_l) \mathrm{tr}[H'_{M_0 \mathrm{SWC}} \rho_{M_0 \mathrm{SWC}}(t|\tau_l)], \quad (\mathrm{D.97}) $$
$$ \leq P T_0 + \delta E_1^{\text{re}} + \delta E_1^{\text{after re}}. \qquad (D.98) $$

In line eq. (D.98) we have lower bounded the integral by zero since $\text{tr}[H'_{M_0SWC} \rho_{M_0SWC}(t|\tau_l)] \ge 0$ because $H'_{M_0SWC} \ge 0$ by definition and thus the integral is non-negative.

$$ |\delta E_1^{\text{after re}}| \le \int_0^\infty dt P(t, +1|\tau_l) \sum_{j=0}^\infty \left| \lim_{\epsilon \to 0^+} \left[ \text{tr}[H'_{M_0SWC} \rho_{M_0SWC}(t_{j+1} - \epsilon|\tau_l)] - \text{tr}[H'_{M_0SWC} \rho_{M_0SWC}(t_{j+1} + \epsilon|\tau_l)] \right] \right| \qquad (D.99) $$

$$ - \lim_{\epsilon \to 0^+} \text{tr}[H'_{M_0SWC} \rho_{M_0SWC}(t_{N+1} + \epsilon|\tau_l)] \qquad (D.100) $$

$$ + \sum_{j=0}^{\infty} \left| \text{tr} \left[ H'_{M_0SWC} \left( |(t_{j+1}|\tau_l)\rangle\langle(t_{j+1}|\tau_l)|_{M_0SWCA} \right. \right. \right. \qquad (D.101) $$

$$ \left. \left. - U_{M_{0,j+2}A_{j+2}M_{0,j+1}A_{j+1}}^{(l,j+2)} |(t_{j+1}|\tau_l)\rangle\langle(t_{j+1}|\tau_l)|_{M_0SWCA} U_{M_{0,j+2}A_{j+2}M_{0,j+1}A_{j+1}}^{(l,j+2)\dagger} \right] \right| \qquad (D.102) $$

$$ \le \sum_{j=0}^{\infty} \left| \text{tr} \left[ \left( I_{M_0SW}^{(j+1)} \otimes I_C^{(j+1)} \bar{\delta}_{j+1,N_g} + I_{M_0SW}^{(j+2)} \otimes I_C^{(j+2)} \bar{\delta}_{j+2,N_g} \right) \left( |(t_{j+1}|\tau_l)\rangle\langle(t_{j+1}|\tau_l)|_{M_0SWCA} \right) \right. \right. \qquad (D.103) $$

$$ \left. \left. - U_{M_{0,j+2}A_{j+2}M_{0,j+1}A_{j+1}}^{(l,j+2)} |(t_{j+1}|\tau_l)\rangle\langle(t_{j+1}|\tau_l)|_{M_0SWCA} U_{M_{0,j+2}A_{j+2}M_{0,j+1}A_{j+1}}^{(l,j+2)\dagger} \right] \right| \qquad (D.104) $$

$$ \le \sum_{j=0}^{\infty} \left\| \left( I_{M_0SW}^{(j+1)} \otimes I_C^{(j+1)} \bar{\delta}_{j+1,N_g} \right)^{1/2} |(t_{j+1}|\tau_l)\rangle_{M_0SWCA} \right\|_2 + \left\| \left( I_{M_0SW}^{(j+2)} \otimes I_C^{(j+2)} \bar{\delta}_{j+2,N_g} \right)^{1/2} |(t_{j+1}|\tau_l)\rangle_{M_0SWCA} \right\|_2 \qquad (D.105) $$

$$ + \left\| \left( I_{M_0SW}^{(j+1)} \otimes I_C^{(j+1)} \bar{\delta}_{j+1,N_g} \right)^{1/2} U_{M_{0,j+2}A_{j+2}M_{0,j+1}A_{j+1}}^{(l,j+2)} |(t_{j+1}|\tau_l)\rangle_{M_0SWCA} \right\|_2 \qquad (D.106) $$

$$ + \left\| \left( I_{M_0SW}^{(j+2)} \otimes I_C^{(j+2)} \bar{\delta}_{j+1,N_g} \right)^{1/2} U_{M_{0,j+2}A_{j+2}M_{0,j+1}A_{j+1}}^{(l,j+2)} |(t_{j+1}|\tau_l)\rangle_{M_0SWCA} \right\|_2 \qquad (D.107) $$

$$ \le \sum_{j=0}^{\infty} \bar{\delta}_{j+1,N_g} \left\| \left( I_{M_0SW}^{(j+1)} \otimes I_C^{(j+1)} \right)^{1/2} |(t_{j+1}|\tau_l)\rangle_{M_0SWCA} \left\| |(t_{j+1}|\tau_l)\rangle_C \right\|_2 \qquad (D.108) $$

$$ + \bar{\delta}_{j+2,N_g} \left\| \left( I_{M_0SW}^{(j+2)} \otimes I_C^{(j+2)} \right)^{1/2} |(t_{j+1}|\tau_l)\rangle_{M_0SWCA} \left\| |(t_{j+1}|\tau_l)\rangle_C \right\|_2 \qquad (D.109) $$

$$ + \bar{\delta}_{j+1,N_g} \left\| \left( I_{M_0SW}^{(j+1)} \otimes I_C^{(j+1)} \right)^{1/2} U_{M_{0,j+2}A_{j+2}M_{0,j+1}A_{j+1}}^{(l,j+2)} |(t_{j+1}|\tau_l)\rangle_{M_0SWCA} \left\| |(t_{j+1}|\tau_l)\rangle_C \right\|_2 \qquad (D.110) $$

$$ + \bar{\delta}_{j+2,N_g} \left\| \left( I_{M_0SW}^{(j+2)} \otimes I_C^{(j+2)} \right)^{1/2} U_{M_{0,j+2}A_{j+2}M_{0,j+1}A_{j+1}}^{(l,j+2)} |(t_{j+1}|\tau_l)\rangle_{M_0SWCA} \left\| |(t_{j+1}|\tau_l)\rangle_C \right\|_2 \qquad (D.111) $$

$$ + \bar{\delta}_{j+1,N_g} \left\| \left( I_{M_0SW}^{(j+1)} \otimes I_C^{(j+1)} \right)^{1/2} |\epsilon(j+1)\rangle \right\|_2 + \bar{\delta}_{j+2,N_g} \left\| \left( I_{M_0SW}^{(j+2)} \otimes I_C^{(j+2)} \right)^{1/2} |\epsilon(j+1)\rangle \right\|_2 \qquad (D.112) $$

$$ + \bar{\delta}_{j+1,N_g} \left\| \left( I_{M_0SW}^{(j+1)} \otimes I_C^{(j+1)} \right)^{1/2} U_{M_{0,j+2}A_{j+2}M_{0,j+1}A_{j+1}}^{(l,j+2)} |\epsilon(j+1)\rangle \right\|_2 \qquad (D.113) $$

$$ + \bar{\delta}_{j+2,N_g} \left\| \left( I_{M_0SW}^{(j+2)} \otimes I_C^{(j+2)} \right)^{1/2} U_{M_{0,j+2}A_{j+2}M_{0,j+1}A_{j+1}}^{(l,j+2)} |\epsilon(j+1)\rangle_{M_0SWCA} \right\|_2 \qquad (D.114) $$

$$ \le \sum_{j=0}^{\infty} 2\sqrt{4\pi} e^{-\epsilon_b t_{j+1}/(2T_0)} \delta_{(j+1)} \left( \bar{\delta}_{j+1,N_g} \left\| \left( I_C^{(j+1)} \right)^{1/2} |t_{j+1}\rangle_C \right\|_2 + \bar{\delta}_{j+2,N_g} \left\| \left( I_C^{(j+2)} \right)^{1/2} |t_{j+1}\rangle_C \right\|_2 \right) \qquad (D.115) $$
$$ + 2\sqrt{4\pi} \left( \bar{\delta}_{j+1, N_g} \left\| \left( I_C^{(j+1)} \right)^{1/2} \right\|_2 + \bar{\delta}_{j+2, N_g} \left\| \left( I_C^{(j+2)} \right)^{1/2} \right\|_2 \right) \| |\epsilon(j+1)\rangle \|_2 \quad (D.116) $$

$$ \le g(\bar{\epsilon})\mathrm{poly}(P)e^{-1/\sqrt{\bar{\epsilon}}} \left( \sum_{j=0}^{\infty} e^{-\epsilon_b t_{j+1}/(2T_0)} \delta_{(j+1)} \right) \quad (D.117) $$

$$ + \mathrm{poly}(P) \left( \sum_{j=0}^{\infty} \| |\epsilon(j+1)\rangle_C \|_2 \right) \quad (D.118) $$

$$ \le g(\bar{\epsilon})\mathrm{poly}(P)e^{-1/\sqrt{\bar{\epsilon}}} \left( N_g \sum_{k=0}^{\infty} e^{-\epsilon_b k/2} e^{-\gamma_0 N_g k} \right) \quad (D.119) $$

$$ + g(\bar{\epsilon})\mathrm{poly}(P)P^{-1/\sqrt{\bar{\epsilon}}} \left( \sum_{q=1}^{N_g} \tilde{d}(\mathfrak{m}_{l,q}) \right) \left( \sum_{j=1}^{\infty} e^{-\frac{\epsilon_b}{2T_0} t_j} \frac{1 - e^{-\gamma_0 N_g (1+\beta'(j))}}{1 - e^{-\gamma_0 N_g}} \right) \quad (D.120) $$

$$ \le g(\bar{\epsilon})\mathrm{poly}(P) e^{-1/\sqrt{\bar{\epsilon}}} \frac{N_g}{1 - e^{-(\gamma_0 N_g + \epsilon_b/2)}} \quad (D.121) $$

$$ + g(\bar{\epsilon})\mathrm{poly}(P)P^{-1/\sqrt{\bar{\epsilon}}} \left( \sum_{q=1}^{N_g} \tilde{d}(\mathfrak{m}_{l,q}) \right) \left( (N_g + 1) \sum_{j=0}^{\infty} e^{-\frac{\epsilon_b}{2} \frac{1}{1 - e^{-\gamma_0 N_g (1+j)}}} \right) \quad (D.122) $$

$$ \le g(\bar{\epsilon})\mathrm{poly}(P) e^{-1/\sqrt{\bar{\epsilon}}} \quad (D.123) $$

$$ + g(\bar{\epsilon})\mathrm{poly}(P)P^{-1/\sqrt{\bar{\epsilon}}} \left( \sum_{q=1}^{N_g} \tilde{d}(\mathfrak{m}_{l,q}) \right) \left( \frac{1}{1 - e^{-\epsilon_b/2}} \right) \quad (D.124) $$

$$ = g(\bar{\epsilon})\mathrm{poly}(P)P^{-1/\sqrt{\bar{\epsilon}}} \left( \sum_{q=1}^{N_g} \tilde{d}(\mathfrak{m}_{l,q}) \right) \left( \frac{2}{\epsilon_b + \mathcal{O}(\epsilon_b)^2} \right) \quad (D.125) $$

$$ \le g(\bar{\epsilon})\mathrm{poly}(P)P^{-1/(2\sqrt{\bar{\epsilon}})} \quad (D.126) $$

In lines eqs. (D.103) and (D.104) we have used the fact that $I_{M_0SW}^{(j)}$ has support on $M_{0,j}$SW only, and the unitary invariance of the trace. We have also defined $\bar{\delta}_{j+1,N_g}$ as

$$ \bar{\delta}_{q,r} = \begin{cases} 0 & \text{if } q = r \\ 1 & \text{otherwise.} \end{cases} \quad (D.127) $$

In line eq. (D.112) we have defined $|\epsilon(j+1)\rangle = |(t_{j+1}|\tau_l)\rangle_{M_0SWCA} - |[t_{j+1}|\tau_l]\rangle_{M_0SWA} |[t_{j+1}|\tau_l]\rangle_C$. In line eq. (D.115) we have used definitions eqs. (B.100), (C.4) and (D.20) together with the definition of the two norm.

In line eq. (D.117) we have used $(\bar{\delta}_{j+1,N_g} \left\| \left( I_C^{(j+1)} \right)^{1/2} |t_{j+1}\rangle_C \right\|_2 + \bar{\delta}_{j+2,N_g} \left\| \left( I_C^{(j+2)} \right)^{1/2} |t_{j+1}\rangle_C \right\|_2)$
$\le g(\bar{\epsilon})\mathrm{poly}(P)e^{-1/\sqrt{\bar{\epsilon}}}$ (where the r.h.s. is j-independent) which follows from the proof on lemma B.2 and the parametrization of P in terms of d from the proof of Theorem 1. Similarly, in line eq. (D.118) we have used that $(\bar{\delta}_{j+1,N_g} \left\| \left( I_C^{(j+1)} \right)^{1/2} \right\|_2 + \bar{\delta}_{j+2,N_g} \left\| \left( I_C^{(j+2)} \right)^{1/2} \right\|_2) \le \mathrm{poly}(P)$ (where the r.h.s. is j-independent) which follows from eq. (B.26) and the parametrization of P in terms of d in the proof of Theorem 1. In line eq. (D.120) we have applied lemma D.1. In lines eqs. (D.123) and (D.124) we have used that $N_g = \mathrm{poly}(P)$. In line eq. (D.126) we used definition eq. (D.70).

Thus using the definition of $P''$, eq. (V.8), and eq. (D.98), we conclude an upper bound on $P''$ of

$$ P'' = \frac{\langle E^{\text{after re}} \rangle}{T_0} + \frac{|\langle E^{\text{re}} \rangle|}{T_0} \le 2P + \frac{2\delta E_1^{\text{re}} + \delta E_1^{\text{after re}}}{T_0}. \quad (D.128) $$

Therefore, taking into account eq. (D.86), we conclude that there exists $\kappa_1 \in [1, 2]$ such that

$$ P = \frac{P''}{\kappa_1} + \delta P'', \quad |\delta P''| \le 2|\delta E_1^{\text{re}}| + |\delta E_1^{\text{after re}}|. \quad (D.129) $$
Recall that $\tilde{n}_0$ is a free parameter of the model in the interval $(0, 1)$ [see text below eq. (B.126)]. We can therefore choose $\tilde{n}_0 = 1/(2\pi\kappa_1)$. Finally, to remove the $\delta P''$ from the bound on $f$, we Tailor expand about the point $\delta P'' = 0$ analogously to the expansion in $\delta d$ in eq. (B.149). This concludes the derivation of item 3) in Theorem 3. ■

### 3. Generation of a classical signal when each renewal process occurs

Consider the mapping $\{J_C^j\}_j \to \{\tilde{J}_{CR}^j\}_j$, where $\tilde{J}_C^j := J_C^j \otimes J_R$, $J_R := |1\rangle\langle 0|_R + |2\rangle\langle 1|_R + \dots + |N_T\rangle\langle N_T - 1|_R + |0\rangle\langle N_T|_R$ and $\{J_C^j\}_j$ are defined in appendix D 1. It can readily be seen from the dynamical semigroup that if the register on R is initiated to $|0\rangle_R$—a product state with the rest of the system—then the dynamics leads to a probabilistic mixture over product states with the register and the rest of the system with the register in one of the states $C_R := \{|l\rangle_R\}_l$. Moreover, the state of the register keeps track of the partitioning of the ensemble into the number of renewals which have occurred at time $t$. To see this, note that if the $l$th renewal occurs in the infinitesimal interval $[t, t + \delta t]$, then the register transitions from the state $|l-1\rangle_R$ to $|l\rangle_R$ within said infinitesimal interval. This follows inductively by using the Markovian property of the dynamical semigroup and expand the dynamics to leading order in $\delta t$ at time $t$. When the register runs out of memory (i.e. after recording $N_T \in N_{>0}$ renewals) it resets to $|0\rangle_R$ and starts again. This is to say, the solution to the new dynamical semigroup at time $\tau = \tau_l + t$ can be written in the form

$$ \rho_{M_0 \text{SWCR}}(\tau) = \sum_{l=0}^{\infty} P(t|\tau_l) \rho_{M_0 \text{SWC}}(t|\tau_l) \otimes |l \text{ mod } N_T\rangle\langle l \text{ mod } N_T|_R, \quad (D.130) $$

where $\text{tr}_R[\rho_{M_0 \text{SWCR}}(\tau)]$ is the solution to the dynamical semigroup described in appendix D 1 and to which Theorem 3 applies—c.f eq. (D.9). Here the modulo arithmetic is required due to the resetting of the counter when it runs out of memory. In practice one could just choose $N_T$ large enough so that it does not run out over the relevant timescales over which the quantum frequential computer is running.

Therefore, as discussed in section VI, this classical register serves as a counter which changes approximately periodically once every $1/f_{\text{bus}} = T_0$. It can thus be classically monitored in order to keep the oscillator on $C_2$ synced.

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