$\displaystyle=-\sum_{m=1}^{N}\sum_{k=1}^{m}{\beta}^{m-k}\left[\pm{\gamma}\nabla{\mathcal{L}}({\theta}_{k})+Q_{\pm,k}\right]$

Finally, taking the exponential for $N\geq 1$, we obtain

$\displaystyle w_{\pm,N+1}$
$\displaystyle=w_{\pm,0}\exp\left(-\sum_{m=1}^{N}\sum_{k=1}^{m}{\beta}^{m-k}\left[\pm{\gamma}\nabla{\mathcal{L}}({\theta}_{k})+Q_{\pm,k}\right]\right)$
$\displaystyle=w_{\pm,0}\exp\left(\pm\sum_{m=1}^{N}\sum_{k=1}^{m}{\beta}^{m-k}Q_{\pm,k}\right)\exp\left(\mp{\gamma}\sum_{m=1}^{N}\sum_{k=1}^{m}{\beta}^{m-k}\nabla{\mathcal{L}}({\theta}_{k})\right)$
$\displaystyle=w_{\pm,0}\exp\left(-\frac{1}{1-{\beta}}\sum_{m=1}^{N}(1-{\beta}^{N+1-m})Q_{\pm,m}\right)\exp\left(\mp\frac{{\gamma}}{1-{\beta}}\sum_{m=1}^{N}(1-{\beta}^{N+1-m})\nabla{\mathcal{L}}({\theta}_{m})\right),$

where the last equality is obtained by changing the order of summation.
Following our continuous-time approach, for $N\geq 2$, we define
${\alpha}_{\pm,N+1}$ as

$\displaystyle\begin{split}{\alpha}_{\pm,N+1}&\coloneqq
w_{\pm,0}\exp\left(\pm\sum_{m=1}^{N}\sum_{k=1}^{m}{\beta}^{m-k}Q_{\pm,k}\right)\\\
&=w_{\pm,0}\exp\left(-\frac{1}{1-{\beta}}\sum_{m=1}^{N}(1-{\beta}^{N+1-m})Q_{\pm,m}\right).\end{split}$
(27)

We can now write the iterates $w_{\pm,k}$ as

$\displaystyle
w_{\pm,N+1}={\alpha}_{\pm,N+1}\exp\big{(}\mp{\gamma}\sum_{m=1}^{N}\sum_{k=1}^{m}{\beta}^{m-k}\nabla{\mathcal{L}}({\theta}_{k})\big{)}.$

Thus, the regression parameter $\theta_{N}$ becomes

$\displaystyle\theta_{N+1}$
$\displaystyle=\frac{1}{4}(w_{+,N+1}^{2}-w_{-,N+1}^{2})$
$\displaystyle=\frac{1}{4}{\alpha}_{+,N+1}^{2}\exp\big{(}-2{\gamma}\sum_{m=1}^{N}\sum_{k=1}^{m}{\beta}^{m-k}\nabla{\mathcal{L}}({\theta}_{k})\big{)}-\frac{1}{4}{\alpha}_{-,N+1}^{2}\exp\big{(}2{\gamma}\sum_{m=1}^{N}\sum_{k=1}^{m}{\beta}^{m-k}\nabla{\mathcal{L}}({\theta}_{k})\big{)}$
$\displaystyle=\frac{1}{2}\Delta_{N+1}\
\sinh\left(-2{\gamma}\sum_{m=1}^{N}\sum_{k=1}^{m}{\beta}^{m-k}\nabla{\mathcal{L}}({\theta}_{k})+\operatorname{arcsinh}\left(\frac{{\alpha}_{+,N+1}^{2}-{\alpha}_{-,N+1}^{2}}{2\Delta_{N+1}}\right)\right),$

where we recall that
${\Delta}_{N}=|w_{+,N}w_{-,N}|=|{\alpha}_{+,N}{\alpha}_{-,N}|$. Hence, similar
to the continuous case,

$\displaystyle\frac{1}{2}\operatorname{arcsinh}\left(\frac{2{\theta}_{N+1}}{{\Delta}_{N+1}}\right)-\frac{1}{2}\operatorname{arcsinh}\left(\frac{{\alpha}_{+,N+1}^{2}-{\alpha}_{-,N+1}^{2}}{2{\Delta}_{N+1}}\right)=-{\gamma}\sum_{m=1}^{N}\sum_{k=1}^{m}{\beta}^{m-k}\nabla{\mathcal{L}}({\theta}_{k}).$

For $N\geq 1$, the above identity becomes exactly

$\displaystyle\nabla\Phi_{N+1}(\theta_{N+1})=-{\gamma}\sum_{m=1}^{N}\sum_{k=1}^{m}{\beta}^{m-k}\nabla{\mathcal{L}}({\theta}_{k}),$
(28)

where the time-varying potential $\Phi_{N}:{\mathbb{R}}^{d}\to{\mathbb{R}}$ is
defined as

$\displaystyle\Phi_{N}(\theta)$
$\displaystyle=\frac{1}{4}\sum_{i=1}^{d}\left(2{\theta}_{i}\operatorname{arcsinh}\left(\frac{2{\theta}_{i}}{{\Delta}_{N,i}}\right)-\sqrt{4{\theta}_{i}^{2}+{\Delta}_{N,i}^{2}}+{\Delta}_{N,i}\right)+\langle\phi_{N},{\theta}\rangle$
$\displaystyle=\psi_{{\Delta}_{N}}({\theta})+\langle\phi_{N},{\theta}\rangle,$

where
$\phi_{N}=\frac{1}{2}\operatorname{arcsinh}\left(\frac{{\alpha}_{+,N}^{2}-{\alpha}_{-,N}^{2}}{2{\Delta}_{N}}\right)$
and $\psi_{{\Delta}_{N}}$ is the hyperbolic entropy defined in Equation 6.
Notice that with this definition we arrive at the following time-varying
momentum mirror descent for $N\geq 1$:

$\nabla\Phi_{N+1}(\theta_{N+1})=\nabla\Phi_{N}(\theta_{N})-{\gamma}\nabla{\mathcal{L}}({\theta}_{N})+{\beta}(\nabla\Phi_{N}(\theta_{N})-\nabla\Phi_{N-1}(\theta_{N-1})).$
(29)

Convergent Quantities. From Lemma 7, we have that $\alpha_{\pm,N}$ must
converge and that the limiting vectors $\alpha_{\pm,\infty}$ have non-zero
coordinates. Therefore, the series
$\sum_{m=1}^{\infty}\sum_{k=1}^{m}{\beta}^{m-k}Q_{\pm,k}$ are convergent and
their terms must hence converge to zero:
$\sum_{k=1}^{m}{\beta}^{m-k}Q_{\pm,k}\underset{m\to\infty}{\longrightarrow}0$.
Therefore,

$\displaystyle{\alpha}_{\pm,N}\to{\alpha}_{\pm,\infty}=w_{\pm,0}\exp\left(-\frac{1}{1-{\beta}}\sum_{m=1}^{\infty}Q_{\pm,m}\right).$

We now develop the formulas for $Q_{\pm,m}$ in order to arrive at the sums
$S_{\pm}$ from Lemma 2. Recall that for $m\geq 1$,
$Q_{\pm,m}=q(\delta_{\pm,m+1})+{\beta}q(-\delta_{\pm,m})$ and
$\delta_{\pm,1}=q(\delta_{\pm,1})=0$. Therefore,

$\displaystyle\sum_{m=1}^{\infty}Q_{\pm,m}$
$\displaystyle=\sum_{m=1}^{\infty}q(\delta_{\pm,m+1})+{\beta}q(-\delta_{\pm,m})$
$\displaystyle=\sum_{m=1}^{\infty}q(\delta_{\pm,m+1})+{\beta}q(-\delta_{\pm,m+1}).$

Since $\delta_{\pm,m+1}=g_{\pm,m+1}-g_{\pm,m}$, we have

$\delta_{\pm,m+1}=\begin{cases}\ln\left(\frac{w_{\pm,m+1}}{w_{\pm,m}}\right)&\text{if
}w_{\pm,m+1}\text{ and }w_{\pm,m}\text{ have the same sign},\\\
\ln\left(\left|\frac{w_{\pm,m+1}}{w_{\pm,m}}\right|\right)+\operatorname{sgn}(w_{\pm,m})i\pi&\text{if
they have different signs}.\end{cases}$

It remains to notice that since $q(z)=\exp(z)-(1+z)$, we get that

$\displaystyle q(\ln(z))=(z-1)-\ln(z)\ $ $\displaystyle\text{for
}z\in\mathbb{R}_{>0},$ $\displaystyle q(\ln(|z|)\pm i\pi)=(z-1)-(\ln(|z|)\pm
i\pi)\ $ $\displaystyle\text{for }z\in\mathbb{R}_{<0}.$

Therefore letting $r(z)=(z-1)-\ln(|z|)$ as in Lemma 2, we get

$\displaystyle q(\delta_{\pm,m+1})$
$\displaystyle=r\Big{(}\frac{w_{\pm,m+1}}{w_{\pm,m}}\Big{)}-\xi_{\pm,m}\operatorname{sgn}(w_{\pm,m})i\pi$
$\displaystyle q(-\delta_{\pm,m+1})$
$\displaystyle=r\Big{(}\frac{w_{\pm,m}}{w_{\pm,m+1}}\Big{)}+\xi_{\pm,m}\operatorname{sgn}(w_{\pm,m})i\pi,$

where $\xi_{\pm,m}=0$ if
$\operatorname{sgn}(w_{\pm,m+1})=\operatorname{sgn}(w_{\pm,m})$ and $1$
otherwise. This leads to

$\displaystyle\frac{1}{1-{\beta}}\sum_{m=1}^{\infty}Q_{\pm,m}$
$\displaystyle=\frac{1}{1-{\beta}}\sum_{m=1}^{\infty}\big{[}r\Big{(}\frac{w_{\pm,m+1}}{w_{\pm,m}}\Big{)}+{\beta}r\Big{(}\frac{w_{\pm,m}}{w_{\pm,m+1}}\Big{)}\Big{]}-\sum_{m=1}^{\infty}\xi_{\pm,m}\operatorname{sgn}(w_{\pm,m})i\pi$
$\displaystyle=S_{\pm}-\sum_{m=1}^{\infty}\xi_{\pm,m}\operatorname{sgn}(w_{\pm,m})i\pi<\infty.$

The last equality is due to the definition of $S_{\pm}$ from Lemma 2, and the
last inequality is due to the summability of $(Q_{\pm,m})_{m}$. This therefore
proves lemma Lemma 2. Now notice that

$\displaystyle{\alpha}_{\pm,\infty}^{2}=w_{\pm,0}^{2}\exp\left(-2S_{\pm}\right).$

Since $\Delta_{\infty}=|\alpha_{+,\infty}\alpha_{-,\infty}|$, we finally get
that

${\Delta}_{\infty}={\Delta}_{0}\odot\exp\Big{(}-(S_{+}+S_{-})\Big{)}.$

##### Implicit Regularisation Problem.

Notice that

$\displaystyle\nabla\Phi_{N+1}(\theta_{N+1})=-{\gamma}\sum_{m=1}^{N}\sum_{k=1}^{m}{\beta}^{m-k}\nabla{\mathcal{L}}({\theta}_{k})\in\texttt{span}(x_{1},\cdots,x_{n}).$

Let
$\Phi_{\infty}(\theta)\coloneqq\psi_{\Delta_{\infty}}(\theta)+\langle\phi_{\infty},\theta\rangle$
and consider

$\displaystyle\nabla\Phi_{\infty}(\theta^{\texttt{MGD}})=(\nabla\Phi_{\infty}(\theta^{\texttt{MGD}})-\nabla\Phi_{\infty}(\theta_{N}))+(\nabla\Phi_{\infty}(\theta_{N})-\nabla\Phi_{N}(\theta_{N}))+\nabla\Phi_{N}(\theta_{N}).$

The first two terms converge to $0$: the first due to the convergence
$\theta_{N}\to\theta^{\texttt{MGD}}$ and the second due to the uniform
convergence of $\nabla\Phi_{N}$ to $\nabla\Phi_{\infty}$ on compact sets. The
last term is in $\texttt{span}(x_{1},\cdots,x_{n})$ for all $N$. Therefore, we
get that
$\nabla\Phi_{\infty}(\theta_{\infty})\in\texttt{span}(x_{1},\cdots,x_{n})$,
and following the exact same proof as in the continuous-time framework, we
finally get that

$\displaystyle\theta^{\texttt{MGD}}=\operatorname*{argmin}_{{\theta}^{\star}\in\mathcal{S}}D_{\psi_{\Delta_{\infty}}}({\theta}^{\star},\tilde{{\theta}}_{0})$

where

$\displaystyle\tilde{{\theta}}_{0}$
$\displaystyle=\frac{{\alpha}_{+,\infty}^{2}-{\alpha}_{-,\infty}^{2}}{4}$
$\displaystyle=\frac{1}{4}\Big{(}w_{+,0}^{2}\odot\exp\left(-2S_{+}\right)-w_{-,0}^{2}\odot\exp\left(-2S_{-}\right)\Big{)}.$

∎

We recall and prove the following corollary.

See 3

###### Proof.

The corollary follows from the fact that if the iterates $w_{\pm,k}$ do not
change sign, then since $r(z)\geq 0$ for $z>0$, we get that $S_{\pm}>0$ and
$\Delta_{\infty}<\Delta_{0}$. Furthermore,
$|\tilde{\theta}_{0}|<\max(w_{+,0}^{2},w_{-,0}^{2})/4\leq\alpha^{2}$ ∎

###  D.2 Link to the Continuous-Time Result.

In this subsection we link our continuous results with the discrete when the
iterates do not cross zero. Indeed, at first sight, the discrete-time
expression for ${\Delta}_{\infty}$ might seem quite different from its
continuous-time counterpart:

$\displaystyle{\Delta}_{\infty}^{\texttt{MGD}}$
$\displaystyle=\Delta_{0}\exp\left(-\frac{1}{1-{\beta}}\sum_{k=1}^{\infty}\left[r\left(\frac{w_{+,k+1}}{w_{+,k}}\right)+r\left(\frac{w_{-,k+1}}{w_{-,k}}\right)\right]+{\beta}\left[r\left(\frac{w_{+,k}}{w_{+,k+1}}\right)+r\left(\frac{w_{-,k}}{w_{-,k+1}}\right)\right]\right)$
$\displaystyle{\Delta}_{\infty}^{\texttt{MGF}}$
$\displaystyle=\Delta_{0}\exp\left(-{\lambda}\int_{0}^{\infty}\left(\frac{\dot{w}_{+,t}}{w_{+,t}}\right)^{2}+\left(\frac{\dot{w}_{-,t}}{w_{-,t}}\right)^{2}\mathrm{d}t\right).$

However, upon closer inspection, by letting the discretisation step
${\varepsilon}=\sqrt{{\lambda}{\gamma}}=\frac{{\gamma}}{(1-{\beta})}$ from
Proposition 1 go to 0, we can recover the continuous-time result. Indeed, as
${\varepsilon}\to 0$, we expect successive iterates $w_{\pm,k}$ to be close
and hence $w_{\pm,k+1}/w_{\pm,k}\approx 1$. Now, since $r(z)\sim_{z\to
1}(z-1)^{2}/2$, we roughly have

$\displaystyle r\Big{(}\frac{w_{\pm,k+1}}{w_{\pm,k}}\Big{)}$
$\displaystyle\approx\frac{1}{2}\big{(}\frac{w_{\pm,k+1}-w_{\pm,k}}{w_{\pm,k}}\big{)}^{2}$

and

$\displaystyle r\Big{(}\frac{w_{\pm,k}}{w_{\pm,k+1}}\Big{)}$
$\displaystyle\approx\frac{1}{2}\big{(}\frac{w_{\pm,k+1}-w_{\pm,k}}{w_{\pm,k+1}}\big{)}^{2}\approx\frac{1}{2}\big{(}\frac{w_{\pm,k+1}-w_{\pm,k}}{w_{\pm,k}}\big{)}^{2}$

Putting the approximations together:

$\displaystyle\frac{1}{1-{\beta}}\sum_{k}\Big{[}r\Big{(}\frac{w_{\pm,k+1}}{w_{\pm,k}}\Big{)}+{\beta}r\Big{(}\frac{w_{\pm,k}}{w_{\pm,k+1}}\Big{)}\Big{]}$
$\displaystyle\approx\frac{1}{2}\frac{{\varepsilon}(1+{\beta})}{1-{\beta}}\sum_{k}\left(\frac{w_{\pm,k+1}-w_{\pm,k}}{\varepsilon}\right)^{2}\frac{1}{(w_{\pm,k})^{2}}\cdot{\varepsilon}$
$\displaystyle\approx\frac{1+{\beta}}{2}\frac{{\gamma}}{(1-{\beta})^{2}}\int_{0}^{\infty}\left(\frac{\dot{w}_{\pm,t}}{w_{\pm,t}}\right)^{2}\mathrm{d}t$
$\displaystyle=\frac{1+{\beta}}{2}{\lambda}\int_{0}^{\infty}\left(\frac{\dot{w}_{\pm,t}}{w_{\pm,t}}\right)^{2}\mathrm{d}t.$

Notice that in order for ${\lambda}$ to remain constant and ${\varepsilon}$ to
go to 0, we must both have ${\gamma}\to 0$ and ${\beta}\to 1$. Hence,
$(1+{\beta})/2\to 1$, and we recover the continuous-time expression for the
balancedness.

However, note that when the iterates cross zero it is unclear to the authors
how the continuous formula and its discrete counterpart compare.

##### Another Safe-Check Computation.

Recall that MGD with stepsize $\gamma$ and momentum parameter $\beta$
corresponds to the discretisation of MGF with $\lambda=\gamma/(1-\beta)^{2}$
and discretisation step $\varepsilon=\sqrt{\lambda\gamma}$. To check the
consistency between the discrete time equations and continuous time equations,
we look at the value of $\exp(-\frac{t-s}{\lambda})$ and times
’$t=m\varepsilon$’ and ’$s=k\varepsilon$’:

$\displaystyle\exp(-\frac{t-s}{\lambda})$
$\displaystyle=\exp(-\frac{(m-k)\varepsilon}{\lambda})$
$\displaystyle=\exp(-(m-k)(1-\beta))$ $\displaystyle=[\exp(\beta-1)]^{m-k}$
$\displaystyle\sim_{\beta\to 1}\beta^{m-k}.$

This small computation serves as a safe-check, affirming the correspondence
between the continuous-time analysis expression $\exp(-\frac{t-s}{\lambda})$
and its discrete-time counterpart $\beta^{m-k}$.

##  Appendix E Technical Lemmas

In this section we present various technical lemmas which allow us to prove
our main results. For ${\Delta}\in{\mathbb{R}}_{>0}^{d}$, we recall the
definition of the hyperbolic entropy function (Ghai et al., 2020)
$\psi_{{\Delta}}:{\mathbb{R}}^{d}\to{\mathbb{R}}$ at scale ${\Delta}$:

$\psi_{\Delta}({\theta})=\frac{1}{4}\sum_{i=1}^{d}\left(2{\theta}_{i}\operatorname{arcsinh}\left(\frac{2{\theta}_{i}}{{\Delta}_{i}}\right)-\sqrt{4{\theta}_{i}^{2}+{\Delta}_{i}^{2}}+{\Delta}_{i}\right).$

The following lemma shows that the potential behaves as the $\ell_{1}$-norm as
${\Delta}$ approaches $0$.

######  Lemma 4.

For ${\theta}\in{\mathbb{R}}^{d}$ the following asymptotic equivalence holds:

$\psi_{\Delta}({\theta})\underset{\scriptscriptstyle{\Delta}\to
0}{\sim}\frac{1}{4}\sum_{i=1}^{d}\ln\left(\frac{1}{{\Delta}_{i}}\right)|{\theta}_{i}|.$

###### Proof.

The lemma easily follows from the asymptotic convergence

$\operatorname{arcsinh}(x)\underset{\scriptscriptstyle|x|\to\infty}{\sim}\operatorname{sgn}(x)\ln|x|.$

∎

The following lemma is a classical result which gives a closed-form expression
to the solution of a first order ODE.

######  Lemma 5.

Let $f:{\mathbb{R}}_{\geq 0}\to{\mathbb{R}}^{d}$ be a differentiable function
and let $g:{\mathbb{R}}_{\geq 0}\to{\mathbb{R}}^{d}$ be a continuous function
such that for some ${\lambda}\neq 0$,

${\lambda}\dot{f}+f+g=0,\enspace\forall t\in{\mathbb{R}}_{\geq 0}.$

Then,

$f(t)=f(0)e^{-\frac{t}{{\lambda}}}-\frac{1}{{\lambda}}\int_{0}^{t}g(s)e^{-\frac{(t-s)}{{\lambda}}}ds.$

Moreover, we have the following formula for the integral of $f(t)$:

$\int_{0}^{T}f(t)dt={\lambda}f(0)(1-e^{-\frac{T}{{\lambda}}})-\int_{0}^{T}g(t)(1-e^{-\frac{(T-t)}{{\lambda}}})dt.$

###### Proof.

If we integrate the identity
$\frac{d}{dt}\left[f(t)e^{t/{\lambda}}\right]=-\frac{1}{{\lambda}}g(t)e^{t/{\lambda}}$,
we get that

$f(t)=f(0)e^{-\frac{t}{{\lambda}}}-\frac{1}{{\lambda}}\int_{0}^{t}g(s)e^{-\frac{(t-s)}{{\lambda}}}ds.$

As for the second part of the lemma, notice that

$\int_{0}^{T}f(t)dt=\int_{0}^{T}\left[f(0)e^{-\frac{t}{{\lambda}}}-\frac{1}{{\lambda}}\int_{0}^{t}g(s)e^{-\frac{(t-s)}{{\lambda}}}ds\right]dt.$

Hence, using Fubini, we get

$\displaystyle\int_{0}^{T}\int_{0}^{t}g(s)e^{-\frac{(t-s)}{{\lambda}}}dsdt$
$\displaystyle=\int_{0}^{T}\int_{0}^{T}g(s)\mathbf{1}_{s\leq
t}(s,t)e^{-\frac{(t-s)}{{\lambda}}}dsdt$
$\displaystyle=\int_{0}^{T}g(s)\int_{0}^{T}\mathbf{1}_{s\leq
t}(s,t)e^{-\frac{(t-s)}{{\lambda}}}dtds$
$\displaystyle=\int_{0}^{T}g(s)\int_{s}^{T}e^{-\frac{(t-s)}{{\lambda}}}dtds$
$\displaystyle=\int_{0}^{T}g(s){\lambda}(1-e^{-\frac{(T-s)}{{\lambda}}})ds,$

which concludes the proof of the lemma. ∎

The following lemma gives various properties on integrability and convergence
of the solution $f$ of the aforementioned ODE.

######  Lemma 6.

Let $f:{\mathbb{R}}_{\geq 0}\to{\mathbb{R}}^{d}$ be a differentiable function
such that $f(0)=0$ and let $g:{\mathbb{R}}_{\geq 0}\to{\mathbb{R}}^{d}$ be a
continuous function such that for some ${\lambda}\neq 0$,

${\lambda}\dot{f}+f+g=0,\enspace\forall t\in{\mathbb{R}}_{\geq 0}.$

If $g\in L^{\infty}(0,+\infty)$, then $f\in L^{\infty}(0,+\infty)$ and $\lVert
f\rVert_{\infty}\leq\lVert g\rVert_{\infty}$. Moreover, if $g\in
L^{1}(0,+\infty)$, then the following hold:

  * •

$f\in L^{1}(0,+\infty)$ and
$\int_{0}^{t}|f(s)|ds\leq\int_{0}^{t}|g(s)|ds,\enspace\forall
t\in[0,+\infty]$;

  * •

$\displaystyle\lim_{t\to\infty}f(t)=0$;

  * •

$\displaystyle\int_{0}^{\infty}f=-\int_{0}^{\infty}g$.

###### Proof.

First, assume $g\in L^{\infty}(0,\infty)$. From Lemma 5, we have that
$f(t)=-\frac{1}{{\lambda}}\int_{0}^{t}g(s)e^{-\frac{(t-s)}{{\lambda}}}ds$.
Hence,

$\displaystyle|f(t)|$ $\displaystyle\leq\frac{\lVert
g\rVert_{\infty}}{{\lambda}}\int_{0}^{t}e^{-\frac{(t-s)}{{\lambda}}}ds$
$\displaystyle=\lVert g\rVert_{\infty}(1-e^{-t/{\lambda}})\leq\lVert
g\rVert_{\infty},$

which proves the first assertion.

Second, assume $g\in L^{1}(0,\infty)$. Then,
$|f(t)|\leq\frac{1}{{\lambda}}\int_{0}^{t}|g(s)|e^{-\frac{(t-s)}{{\lambda}}}ds$.
Therefore,

$\displaystyle\int_{0}^{t}|f(s)|ds$
$\displaystyle\leq\int_{0}^{t}|g(s)|(1-e^{-\frac{(t-s)}{{\lambda}}})ds$
$\displaystyle\leq\int_{0}^{t}|g(s)|ds\leq\lVert g\rVert_{L^{1}}.$

Moving on, we will show that $\lim_{t\to\infty}f(t)=0$. Recall that
$f(t)=-\frac{1}{{\lambda}}\int_{0}^{t}g(s)e^{-\frac{(t-s)}{{\lambda}}}ds$.
Then,

$\displaystyle\left|\int_{0}^{t}g(s)e^{-\frac{(t-s)}{{\lambda}}}ds\right|$
$\displaystyle=\left|\int_{0}^{t/2}g(s)e^{-\frac{(t-s)}{{\lambda}}}ds+\int_{t/2}^{t}g(s)e^{-\frac{(t-s)}{{\lambda}}}ds\right|$
$\displaystyle\leq
e^{-\frac{t}{2{\lambda}}}\int_{0}^{t/2}|g(s)|ds+\int_{t/2}^{\infty}|g(s)|ds$
$\displaystyle\xrightarrow{t\to\infty}0.$

Finally, notice that

$\displaystyle\lim_{t\to\infty}\left[{\lambda}\int_{0}^{t}\dot{f}+\int_{0}^{t}(f+g)\right]=0\iff$
$\displaystyle{\lambda}\lim_{t\to\infty}f(t)+\int_{0}^{\infty}(f+g)=0\iff$
$\displaystyle\int_{0}^{\infty}f+\int_{0}^{\infty}g=0,$

where we used that $\lim_{t\to\infty}f(t)=0$ and the linearity of the Lebesgue
integral. ∎

With the help of Lemma 5 and Lemma 6, we can finally prove Proposition 3,
which considers ODE (4) and establishes the positivity of the balancedness for
small ${\lambda}$.

See 3

###### Proof.

We consider $\mathrm{MGF}(\lambda)$ with the diagonal-linear-network loss
$F(w)={\mathcal{L}}(u\odot v)$, where $w=(u,v)$. From the energy of the
system, defined in Equation 14 as
$E_{t}=F(w_{t})+\frac{{\lambda}}{2}\lVert\dot{w}_{t}\rVert_{2}^{2}$ with
derivative $\dot{E}_{t}=-\lVert\dot{w}_{t}\rVert_{2}^{2}$, we get that

${\mathcal{L}}({\theta}_{t})=\frac{\lVert
y\rVert_{2}^{2}}{2n}-\frac{{\lambda}}{2}\lVert\dot{w}_{t}\rVert_{2}^{2}-\int_{0}^{t}\lVert\dot{w}_{s}\rVert_{2}^{2}ds.$

Hence, since the LHS of the above equation is nonnegative, we get

$\int_{0}^{\infty}\lVert\dot{w_{t}}\rVert^{2}dt\leq\frac{\lVert
y\rVert^{2}}{2n}.$

Therefore,

$\int_{0}^{\infty}|\dot{u}_{t}^{2}-\dot{v}_{t}^{2}|dt<\frac{\lVert
y\rVert^{2}}{2n}\mathbf{1}.$

Consequently, $\dot{u}_{t}^{2}-\dot{v}_{t}^{2}\in L^{1}(0,\infty)$. Now,
notice that from ODE (4), we obtain

$\displaystyle{\lambda}(\ddot{u}_{t}u_{t}-\ddot{v}_{t}v_{t})+(\dot{u}_{t}u_{t}-\dot{v}_{t}v_{t})=0\iff$
$\displaystyle{\lambda}\frac{d}{dt}(\dot{u}_{t}u_{t}-\dot{v}_{t}v_{t})+(\dot{u}_{t}u_{t}-\dot{v}_{t}v_{t})-{\lambda}(\dot{u}_{t}^{2}-\dot{v}_{t}^{2})=0.$

Applying Lemma 5 yields

$\dot{u}_{t}u_{t}-\dot{v}_{t}v_{t}=\int_{0}^{t}(\dot{u}_{s}^{2}-\dot{v}_{s}^{2})e^{-\frac{(t-s)}{{\lambda}}}ds$

and

$u_{t}^{2}-v_{t}^{2}={\Delta}_{0}+2{\lambda}\int_{0}^{t}(\dot{u}_{s}^{2}-\dot{v}_{s}^{2})(1-e^{-\frac{(t-s)}{{\lambda}}})ds.$
(30)

Applying Lemma 6 allows us to conclude that for every $t\in[0,+\infty]$,

$\displaystyle{\Delta}_{t}$
$\displaystyle\geq{\Delta}_{0}-2{\lambda}\int_{0}^{t}|\dot{u}_{s}^{2}-\dot{v}_{s}^{2}|ds$
$\displaystyle>{\Delta}_{0}-\frac{{\lambda}\lVert
y\rVert_{2}^{2}}{n}\mathbf{1}\geq 0,$

where the last inequality is due to the inequality assumption over $\lambda$.
∎

Our final technical lemma helps with the proof of Theorem 2. The definition of
the quantities $Q_{\pm,m}$ can be found in the proof of this theorem.

######  Lemma 7.

The quantities $\alpha_{\pm,N}$ defined in eq. 27:

${\alpha}_{\pm,N+1}={\alpha}\exp\left(-\frac{1}{1-{\beta}}\sum_{m=1}^{N}(1-{\beta}^{N+1-m})Q_{\pm,m}\right),$

converge as $N\to\infty$ to vectors $\alpha_{\pm,\infty}$ with non-zero
coordinates.

###### Proof.

From 3 and 4, we have that the iterates $w_{\pm,N}$ converge towards vectors
$w_{\pm,\infty}$ such that $\Delta_{\infty}=|w_{+,\infty}\odot w_{-,\infty}|$
has non-zero coordinates. This means that there exists $N_{0}>0$ such that
$w_{\pm,N}$ do not change sign for $N\geq N_{0}$. Consequently, the imaginary
parts of $g_{\pm,N}$ are constant (equal to $0$ or $\pi$ depending on the sign
of $w_{\pm,\infty}$) for $N\geq N_{0}$, and $\delta_{\pm,N}\in{\mathbb{R}}$
for $N\geq N_{0}$. This finally means that $Q_{\pm,N}\geq 0$ for $N\geq N_{0}$
and

$\displaystyle\sum_{m=1}^{N}(1-{\beta}^{N+1-m})Q_{\pm,m}=\sum_{m=1}^{N_{0}}(1-{\beta}^{N+1-m})Q_{\pm,m}+\sum_{m=N_{0}+1}^{N}(1-{\beta}^{N+1-m})Q_{\pm,m}$

The first term converges to $\sum_{m=1}^{N_{0}}Q_{\pm,m}$ as $N\to\infty$. The
second term is increasing because $Q_{\pm,N}$ are positive for $N\geq N_{0}$
and $(1-\beta^{N+1-m})$ is increasing. Therefore, the second term also
converges to a finite value since otherwise $\alpha_{\pm,\infty}=0$, which
contradicts ${\Delta}_{\infty}=|\alpha_{+,\infty}\alpha_{-,\infty}|\neq 0$. ∎

##  Appendix F Additional Experiments

In this section of the appendix, we clarify experimental details and discuss
additional experiments.

###  F.1 MGF: A Good Continuous Surrogate

Most of our experiments deal with 2-layer diagonal networks, but before we
constrain ourselves to that tractable setting, we present a couple of
experiments on more general architectures. These experiments highlight our
observation from Section 2 that $\mathrm{MGF}(\lambda)$ serves as a good
continuous proxy for $\mathrm{MGD}({\gamma},{\beta})$ even for complicated
non-convex losses $F$ and large step sizes ${\gamma}$. We provide evidence for
that conclusion by showing that the single parameter
${\lambda}={\gamma}/(1-{\beta})^{2}$ controls the generalisation performance
of models trained with $\mathrm{MGD}({\gamma},{\beta})$.

Teacher-Student Fully Connected Network. We detail the experimental setting
which leads to Figure 2. We consider a teacher-student setup where the teacher
is a one-hidden-layer fully-connected ReLU network with $5$ hidden neurons and
the student is a one-hidden-layer fully-connected ReLU network with $20$
hidden neurons. We randomly generate $15$ inputs $x_{i}\in{\mathbb{R}}^{2}$
according to a standard multivariate normal distribution. Each $y_{i}$
corresponds to the output by the teacher network on input $x_{i}$. The student
is trained using momentum gradient descent with a square loss. Figure 2
corresponds to the test loss after the student reaches $10^{-5}$ training
error. Each grid point corresponds to the same data set and initialisation of
the student network. We observe that the quantity
$\lambda=\frac{\gamma}{(1-\beta)^{2}}$ aligns well with the level lines of the
test loss as expected from Proposition 1.

Deep Linear Network. The network used for Figure 6 contains 5 layers with
widths (30, 60, 120, 60, 1) and was trained for 1000 epochs for each pair of
momentum parameter ${\beta}$ and step size ${\gamma}$. Each network weight was
randomly initialised according to $\mathcal{N}(0,0.1^{2})$ with fixed
randomness for each $({\gamma},{\beta})$-trial. The training data was chosen
as follows:
$(x_{i})_{i=1}^{n}\stackrel{{\scriptstyle\scriptscriptstyle\text{i.i.d.}}}{{\sim}}\mathcal{N}(\mu\mathbf{1},{\sigma}^{2}I_{d})$
and $y_{i}=\langle x_{i},{\theta}_{s}^{\star}\rangle$ for $i\in[n]$ where
${\theta}_{s}^{\star}$ is $s$-sparse with nonzero entries equal to
$1/\sqrt{s}$, where $(n,d,s)=(20,30,5)$ and $(\mu,{\sigma})=(1,1)$. We show
results averaged over 5 replications.

Figure 6:  Test and train loss of a fully connected deep linear network
trained with $\mathrm{MGD}({\gamma},{\beta})$ in a noiseless sparse
overparametrised regression setting. The test loss appears considerably
correlated with the intrinsic parameter ${\lambda}={\gamma}/(1-{\beta})^{2}$,
evincing that $\mathrm{MGF}(\lambda)$ approximates
$\mathrm{MGD}({\gamma},{\beta})$ sufficiently well even on complex
architectures.

2-Layer Diagonal Linear Network. The plots from Figure 7 were obtained for a
2-layer diagonal linear network trained in the noiseless sparse
overparametrised regression setting described above. The first network layer
was initialised with the uniform initialisation ${\alpha}\mathbf{1}$, where
${\alpha}=0.01$, and the weights of the second layer were set to 0. The
momentum gradient flow evolution of the weights was simulated with the default
version of the ODE solver scipy.integrate.odeint.

Figure 7:  _Left_ : Decimal logarithm of the test loss of a 2-layer diagonal
linear network trained with $\mathrm{MGD}({\gamma},{\beta})$ for 1 million
epochs. _Right_ : Decimal logarithm of the test loss of a 2-layer diagonal
linear whose weights evolved according to $\mathrm{MGF}(\lambda)$ – where
${\lambda}={\gamma}/(1-{\beta})^{2}$ – and converged to an interpolator of the
training dataset. We observe an almost one-to-one correspondence in terms of
generalisation capacity, which demonstrates that $\mathrm{MGF}(\lambda)$
serves as a suitable continuous surrogate for $\mathrm{MGD}({\gamma},{\beta})$
in the diagonal linear setting.

###  F.2 Experiments with Diagonal Linear Networks

Having seen empirical proof that $\mathrm{MGF}(\lambda)$ approximates well the
optimisation trajectory of $\mathrm{MGD}({\gamma},{\beta})$ on complicated
models, we proceed with experiments that illustrate the conclusions of our
results for 2-layer diagonal linear networks. In particular, we provide
experimental evidence that both in the continuous and discrete-time cases, the
recovered interpolators by MGD and MGF satisfy

${\theta}^{\texttt{MGF/MGD}}=\operatorname*{argmin}_{{\theta}^{\star}\in\mathcal{S}}D_{\psi_{\Delta_{\infty}}}({\theta}^{\star},\tilde{\theta}_{0})\approx\operatorname*{argmin}_{{\theta}^{\star}\in\mathcal{S}}\psi_{\Delta_{\infty}}({\theta}^{\star}),$

as we explain underneath Theorem 1, Theorem 2, and in Section C.3.3. Indeed,
we observe that the perturbation term $\tilde{{\theta}}_{0}$ can be safely
ignored even without the assumption of strictly positive balancedness. The
asymptotic balancedness ${\Delta}_{\infty}$ then uniquely controls the
properties of the recovered solution. We now specify our experimental setting.

Experimental Details. We work in the noiseless sparse overparametrised
regression setting with uncentered data. More precisely, we let
$(x_{i})_{i=1}^{n}\stackrel{{\scriptstyle\scriptscriptstyle\text{i.i.d.}}}{{\sim}}\mathcal{N}(\mu\mathbf{1},{\sigma}^{2}I_{d})$
and $y_{i}=\langle x_{i},{\theta}_{s}^{\star}\rangle$ for $i\in[n]$ where
${\theta}_{s}^{\star}$ is $s$-sparse with nonzero entries equal to
$1/\sqrt{s}$. We train a 2-layer diagonal linear network with (M)GD and (M)GF
with the uniform initialisation $u_{0}={\alpha}\mathbf{1}$, where
${\alpha}=0.01$ and $v_{0}=0$. In order to simulate gradient flow or momentum
gradient flow on the network weights, we use the vanilla version of the ODE
solver scipy.integrate.odeint. For most of the incoming plots, we have fixed
$(n,d,s,{\sigma})=(20,30,5,1)$ and we let $\mu\in\\{0,0.5,1,1.5\\}$. In what
follows, all plots show results averaged over 5 replications.

####  F.2.1 Continuous-Time Plots

We first present a set of 3 continuous-time plots (Figure 8) for the setting
where the input data follows a Gaussian distribution
$\mathcal{N}(\mu\mathbf{1},I_{d})$ with $\mu=1$.

Experimental Setup. For a sampled dataset $(X,y)$, we train our diagonal
network with $\mathrm{MGF}(\lambda)$, ${\lambda}\in[0,1]$, and initialisation
$(u_{0},v_{0})=({\alpha}\cdot\mathbf{1},0)$ until convergence to an
interpolator 777We know that ${\theta}^{\texttt{MGF}}$ interpolates the
dataset $(X,y)$ because we also record the Train Loss
(${\theta}^{\texttt{MGF}}$), which falls under $10^{-20}$.
${\theta}^{\texttt{MGF}}$. During the training of $\mathrm{MGF}(\lambda)$, we
also take note of whether the balancedness ${\Delta}_{t}$ remains strictly
positive at all times, thereby checking the explanatory range of Section 3.3.
Having completed the MGF training, we plot the Test Loss of
${\theta}^{\texttt{MGF}}$, the $\ell_{2}$-Norm of ${\Delta}_{\infty}$, and the
$\ell_{1}$-Norm of ${\theta}^{\texttt{MGF}}$ in order to visualise the gain in
generalisation performance.

Insignificance of $\tilde{\theta}_{0}$. Now, recall from Theorem 1 that
${\theta}^{\texttt{MGF}}=\operatorname*{argmin}_{{\theta}^{\star}\in\mathcal{S}}\
D_{\psi_{\Delta_{\infty}}}({\theta}^{\star},\tilde{{\theta}}_{0})$ and that
for
$\|\tilde{\theta}_{0}\|_{\infty}\ll\lVert{\theta}^{\texttt{MGF}}\rVert_{\infty}$,
$D_{\psi_{\Delta_{\infty}}}({\theta}^{\star},\tilde{{\theta}}_{0})\approx\psi_{\Delta_{\infty}}(\theta^{\star})$.
We proved that for small values of ${\lambda}$, the balancedness remains
strictly positive at all times, which allowed us to show that
$\|\tilde{\theta}_{0}\|_{\infty}<\alpha^{2}$. We conjecture that
${\theta}^{\texttt{MGF}}\approx\operatorname*{argmin}_{{\theta}^{\star}\in\mathcal{S}}\psi_{\Delta_{\infty}}({\theta}^{\star})$
continues to hold for larger values of ${\lambda}$. We experimentally test
this claim by measuring the precise distance between ${\theta}^{\texttt{MGF}}$
and
${\theta}_{\Delta_{\infty}}^{\texttt{GF}}=\operatorname*{argmin}_{\theta^{\star}\in\mathcal{S}}\psi_{\Delta_{\infty}}(\theta^{\star})$.
Indeed, we initialise a gradient flow with initial balancedness equal to
$\Delta_{\infty}$ and such that $\theta_{0}=0$, which converges to the
predictor ${\theta}_{\Delta_{\infty}}^{\texttt{GF}}$ as discussed in Section
3.1. Hence, we can calculate the Normalised Distance between
${\theta}^{\texttt{MGF}}$ and ${\theta}_{\Delta_{\infty}}^{\texttt{GF}}$ equal
to
$\lVert{\theta}^{\texttt{MGF}}-{\theta}_{\Delta_{\infty}}^{\texttt{GF}}\rVert_{2}/\lVert{\theta}_{\Delta_{\infty}}^{\texttt{GF}}\rVert_{2}$,
and we obtain that
$\lVert{\theta}^{\texttt{MGF}}-{\theta}_{\Delta_{\infty}}^{\texttt{GF}}\rVert_{2}/\lVert{\theta}_{\Delta_{\infty}}^{\texttt{GF}}\rVert_{2}<0.01$
for ${\lambda}\in(0,1)$.

Figure 8:  Continuous-time experiments on uncentered data with mean $\mu=1$.
Here, ${\theta}^{\texttt{MGF}}$ denotes the interpolator recovered by
$\mathrm{MGF}(\lambda)$ and ${\Delta}_{\infty}$ stands for the balancedness at
infinity for $\mathrm{MGF}(\lambda)$. We observe that the test loss and
sparsity of ${\theta}^{\texttt{MGF}}$ correlate with the magnitude of
${\Delta}_{\infty}$ as predicted by Theorem 1.  Figure 9:  We observe that for
uncentered data the magnitude of the balancedness at infinity
${\Delta}_{\infty}$ correlates with the test loss of the interpolator selected
by $\mathrm{MGF}(\lambda)$. However, this relationship breaks for centered
data.

Insights from Continuous-Time Experiments. First, we observe that no matter
the mean of the data distribution888We performed the continuous-time
experiments depicted in Figure 9 for data with mean $\mu=0,0.5,1,1.5$. or the
size of ${\lambda}\in(0,1)$, the normalised distance between
${\theta}^{\texttt{MGF}}$ and ${\theta}_{\Delta_{\infty}}^{\texttt{GF}}$ is
always upper-bounded by $0.01$. Hence, we can empirically confirm our
conjecture from Theorem 1 that
${\theta}^{\texttt{MGF}}\approx{\theta}_{\Delta_{\infty}}^{\texttt{GF}}$ for
larger ${\lambda}$ when the balancedness changes sign. Second, we see that
regardless of the mean of the dataset, the balancedness at infinity (i.e., the
effective initialisation $\Delta_{\infty}$) controls the generalisation
behavior of the recovered interpolator. We can explain this observation again
through the approximate equivalence
${\theta}^{\texttt{MGF}}\approx\operatorname*{argmin}_{\theta^{\star}\in\mathcal{S}}\psi_{\Delta_{\infty}}(\theta^{\star})$.

The Effect of the Data Mean. In Figure 9, we summarise our empirical results
for data with various means. Notice that there exists a difference between the
generalisation behavior for centered and uncentered data. Indeed, for centered
data (top left), the key quantity $\lambda$ has little impact on the sparsity
of the recovered solution. This circumstance is reminiscent of the
observations from (Nacson et al., 2022) and (Even et al., 2023). However, for
uncentered data, we observe an interval
$\mathcal{I}_{\mathcal{D}_{x}}=(0,{\lambda}_{\max})$ (which depends on the
data distribution $\mathcal{D}_{x}$) for which MGF with
${\lambda}\in\mathcal{I}_{\mathcal{D}_{x}}$ outperforms GF in terms of
generalisation. Furthermore, there appears to exist a constant
${\lambda}^{\star}_{\mathcal{D}_{x}}\in\mathcal{I}_{\mathcal{D}_{x}}$ (roughly
corresponding to the minimum magnitude of ${\Delta}_{\infty}$) which brings
about the most improvement compared to gradient flow. We note that the
following tendency seems to hold empirically:

$\lim_{|\mu|\to+\infty}{\lambda}^{\star}_{\mathcal{D}_{x}}=0.$

####  F.2.2 Discrete-Time Plots

For the sake of brevity999We performed discrete-time experiments for data with
means $\mu=0,0.5,1,1.5$., we only present a single set of plots for the
discrete-time noiseless sparse recovery given in Figure 4. Our input data
follows a unit-mean Gaussian distribution $\mathcal{N}(\mathbf{1},I_{d})$.

Experimental Setup. For a sampled dataset $(X,y)$ and hyperparameter pair
(${\beta},{\gamma}$), we train our 2-layer diagonal linear network with
$\mathrm{MGD}({\gamma},{\beta})$ initialised at
$(u_{0},v_{0})=({\alpha}\mathbf{1},0)$ for 1 million epochs (which suffices
for convergence101010Again, we record the Train Loss
(${\theta}_{\gamma,\beta{\alpha}}^{\texttt{MGD}}$), which falls under
$10^{-8}$.). During the $\mathrm{MGD}({\gamma},{\beta})$ training, we also
take note of whether the iterates $w_{\pm,k}$ change sign or not thereby
checking the explanatory range of Corollary 3. Having completed the MGD
training, we plot the Test Loss of ${\theta}^{\texttt{MGD}}$, the
$\ell_{2}$-Norm of ${\Delta}_{\infty}$, and the $\ell_{1}$-Norm of
${\theta}^{\texttt{MGD}}$ in order to visualise the gain in generalisation
performance.

Insignificance of $\tilde{\theta}_{0}$. Recall from Theorem 2 that
${\theta}^{\texttt{MGD}}=\operatorname*{argmin}_{{\theta}^{\star}\in\mathcal{S}}D_{\psi_{\Delta_{\infty}}}({\theta}^{\star},\tilde{{\theta}}_{0})$.
Again, we want to characterise the recovered interpolator as
${\theta}^{\texttt{MGD}}\approx\operatorname*{argmin}_{{\theta}^{\star}\in\mathcal{S}}\psi_{\Delta_{\infty}}(\theta^{\star})$.
In order to verify empirically that the effect of the perturbation term is
negligible, we follow the same strategy as in the continuous-time case. We
initialise a gradient flow with initial balancedness equal to
$\Delta_{\infty}$ and $\theta_{0}=0$, which converges to the predictor
${\theta}_{\Delta_{\infty}}^{\texttt{GF}}$ as discussed in Section 3.1. Hence,
we can calculate the Normalised Distance between ${\theta}^{\texttt{MGD}}$ and
${\theta}_{\Delta_{\infty}}^{\texttt{GF}}$ equal to
$\lVert{\theta}_{{\gamma},{\beta},{\alpha}}^{\texttt{MGD}}-{\theta}_{\Delta_{\infty}}^{\texttt{GF}}\rVert_{2}/\lVert{\theta}_{\Delta_{\infty}}^{\texttt{GF}}\rVert_{2}$,
and we find that
$\lVert{\theta}_{{\gamma},{\beta},{\alpha}}^{\texttt{MGD}}-{\theta}_{\Delta_{\infty}}^{\texttt{GF}}\rVert_{2}/\lVert{\theta}_{\Delta_{\infty}}^{\texttt{GF}}\rVert_{2}<0.01$
for all pairs $({\gamma},\beta)$ in Figure 4. This experimentally shows that
${\theta}^{\texttt{MGD}}\approx\operatorname*{argmin}_{{\theta}^{\star}\in\mathcal{S}}\psi_{\Delta_{\infty}}(\theta^{\star})$
and that the asymptotic balancedess is the key quantity which predicts the
recovered solution.

Insights from Discrete-Time Experiments. As predicted by Theorem 2, a more
balanced solution (center plot) leads to a solution with a lower
$\ell_{1}$-norm (right plot), which in turn translates to better
generalisation (left plot). Finally, as proven in Corollary 3, the
trajectories for which the iterates do not cross zero satisfy
$\Delta_{\infty}<\Delta_{0}$, where $\Delta_{0}$ (approximately) corresponds
to the asymptotic balancedness for the pair $({\beta},{\gamma})=(0,10^{-3})$
in the bottom left corner of the center plot. Clearly, the pairs
$({\beta},{\gamma})$ for which $w_{\pm,k}$ do not change sign lead to better
generalisation than the pair $(0,10^{-3})$. Again, we note that for centered
data the story changes, and we lose the clear correspondence between small
$\lVert{\Delta}_{\infty}\rVert_{2}$ and small
$\lVert{\theta}^{\texttt{MGD}}\rVert_{1}$.


# Quantum Computation by Spin Parity Measurements with Encoded Spin Qubits

Matthew Brooks<EMAIL_ADDRESS>Charles Tahan  Laboratory for
Physical Sciences, 8050 Greenmead Dr., College Park, MD 20740, USA

###### Abstract

Joint measurements of two-Pauli observables are a powerful tool for both the
control and protection of quantum information. By following a simple recipe
for measurement choices, single- and two- qubit rotations using two-Pauli
parity and single qubit measurements are guaranteed to be unitary whilst
requiring only a single ancilla qubit. This language for measurement based
quantum computing is shown to be directly applicable to encoded double quantum
dot singlet-triplet spin qubits, by measuring spin-parity between dots from
neighboring qubits. Along with exchange interaction, a complete, leakage free,
measurement based gate set can be shown, up to a known Pauli correction. Both
theoretically exact spin-parity measurements and experimentally demonstrated
asymmetric spin-parity measurements are shown to be viable for achieving the
proposed measurement based scheme, provided some extra leakage mitigating
measurement steps. This new method of spin qubit control offers a leakage
suppressed, low resource overhead implementation of a measurement-based
control that is viable on current spin qubit processor devices.

##  I Introduction

Measurement based quantum computing (MBQC) is a proposed method of processing
quantum information that seeks to replace operations given by continuous
control fields with discrete quantum measurements[1, 2, 3, 4, 5, 6, 7, 8].
Generally, there are two routes with which this can be achieved. The first
approach is by generating some large initial entangled state containing all
the qubits of a given processor, i.e. a resource state, and performing a
series of single qubit measurements around a given measurement axis, until the
remaining unmeasured qubits of the processor are in the desired state[1, 2, 3,
4, 5, 9, 8]. This method is particularly applicable to photonic processors[9]
whereby entanglement generation is experimentally achievable, but two-qubit
gates are challenging. The second method involves the use of joint
measurements, a measurement about an axis of a shared property of two or more
qubits, to achieve non-unitary entanglement by measurement[6, 7, 10, 11, 12,
13, 8]. This method has been proposed as a method of braiding neighbouring
topological Majorana qubits via measurement of charge parity between the
qubits and a shared quantum dot[6, 7]. Additionally, it has also been shown to
be applicable by measuring joint properties of solid state spin qubits[10, 11,
12, 8]. Both approaches can be shown to provide a universal set of single and
two-qubit gates, at the cost of an at worse polynomial[14] number of ancilla
qubits.

Recent work has demonstrated the power of multiple qubit joint-Pauli parity
measurements in designing quantum error correcting (QEC) codes[15, 16].
Generally such measurements on and $N$-qubit processor can be written as

$\mathcal{P}(\nu,s)=\Gamma\left(\mathbb{I}^{\otimes
N}+(-1)^{s}\bigotimes_{i=1}^{N}\sigma_{\nu_{i}}^{i}\right)$ (1)

where $\nu$ is a vector of length $N$ with elements $\nu_{i}\in\\{0,x,y,z\\}$
that describe a particular joint measurement, $\sigma_{\mu}$ are the Pauli
matrices with $\sigma_{0}=\mathbb{I}$, $\mathbb{I}$ is the identity matrix,
$\Gamma$ is a normalising constant, and $s\in\\{0,1\\}$ is given by the either
measurement of the positive ($s=0$) or negative ($s=1$) eigenstates of the
desired observable. For example, in the case $N=2$ a joint-Pauli measurement
of $\nu=\\{z,z\\}$ is equivalent to either the
$\ket{\Psi^{+}}\bra{\Psi^{+}}=(\ket{00}\bra{00}+\ket{11}\bra{11})/2$ Bell
state projection if $s=0$ or
$\ket{\Phi^{+}}\bra{\Phi^{+}}=(\ket{01}\bra{01}+\ket{10}\bra{10})/2$ Bell
state projection if $s=1$. Note that in this notation when $\nu_{i}=0$ this
corresponds to performing no measurement on qubit $i$. A shorthand notation
for (1) will be used whereby a measurement of the type $\nu=\\{0,x,y,z\\}$ is
described as a $IXYZ$ parity check. Application of rounds of such measurements
provide robust quantum memories without the need for fixed logical qubit space
with only two-qubit joint-Pauli measurements[15], i.e. joint-measurements are
only performed between two neighbouring qubits. Whilst some work on using such
a measurement toolset for qubit control in Majorana[6, 7] and spin qubits[10,
11] systems has been done, demonstration of how to use measurements of this
type for arbitrary qubit control has not been fully characterised.
Additionally, these previous studies tend to focus on just a single entangling
measurement sequence, without consideration for universal control.

In this work we demonstrate that both single and two-qubit operations can be
achieved by rounds of two-qubit joint-Pauli measurements and a single ancilla
qubit. Then, by employing the physical observable of spin parity of spins in
two neighboring QDs, we demonstrate how to achieve such control scheme in
double quantum dot (DQD) spin qubits with only measurement and nearest-
neighbour exchange interaction. Spin qubits offer fast and precise spin-spin
operations via the exchange interaction[17, 18, 19, 20] as well as long
coherence times[21, 22]. They are however vulnerable to noise from control
voltage fluctuations, known as charge noise[23, 24, 25]. Charge noise may be
neutralised somewhat by encoding of logical qubits in two or more spins across
multiple dots[26, 27, 28, 29, 30, 20, 31]. Unfortunately, this also allows for
a new form of error where the constituent spins of a qubit couple to states
outside of the logical spin subspace, known as leakage[27, 20, 32, 33]. The
proposed joint-Pauli parity measurement control scheme offers leakage-
suppressed operation of singlet-triplet encoded qubits at their charge noise
resistant symmetric operating point (SOP)[34], without reliance on both
external magnetic fields or magnetic field gradients for control.

In this paper, first in Sec. II a general condition on how to perform MBQC
with joint-Pauli parity measurements is given. Then in Sec. III the case of a
MBQC with a physical parity observable is discussed, spin-parity measurements
between encoded DQD spin qubits. This is explored with both an ideal case of
an exact spin-parity measurement, and the lab case of an asymmetric spin-
parity measurement. A proposed entangling gate is simulated in Sec. IV,
investigating the effect of infidelity of exchange pulses and measurements.
Finally the results are summarised in Sec. V.

##  II Joint-Pauli MBQC

Projective measurements of quantum states are non-unitary processes in nature.
Therefore, when designing an MBQC scheme, a balance between the choice of
measurements and ancillæ must be found to ensure that the overall imparted
rotations on the data qubits are unitary. For consistency, we will treat the
initialisation of the ancilla qubits as another measurement step, i.e. if an
ancilla requires is initialised to the $\ket{0}$ state we will treat this as
if that qubit has been measured in the Pauli-$z$ basis with a fixed outcome.
For single and two-qubit unitaries using arbitrary joint-Pauli measurements,
the requirements of measurement choices and number of ancilla are the same:
only a single ancilla qubit is needed and each measurement step must be
selected such that the chosen observables’ multi-qubit Pauli operator does not
commute with the observable of the previous measurement step. To clarify the
requisite restrictions on the measurement choices, consider a two-qubit
system. If the second qubit is an ancilla initialised to the $\ket{+}$ state,
this is written as an $IX$ measurement. Then if a $ZX$ joint-measurement is
performed between the two qubits, any data in the data qubit will be lost and
the overall process cannot be made to be unitary since $[IX,ZX]=0$. However,
if the joint-measurement of $ZZ$ is chosen instead, then the overall process
can be made to be unitary as $[IX,ZZ]\neq 0$.

This rule of measurement selection is shown here explicitly with single qubit
unitaries. To perform a single qubit rotation with joint-Pauli measurements,
the recipe is: two-qubit system, the first in some data state $\ket{\psi}$,
the second qubit reserved as the ancilla. The ancilla is initialised, then a
joint measurement on the two qubits creates an entangled pair, the ancilla is
measured along a chosen single qubit basis to disentangle the pair such that
some rotation $U$ is performed on the data qubit. Explicitly this sequence of
measurements can be written as follows

$\begin{split}\Pi_{1-q}=\Gamma\left(\mathbb{I}^{\otimes
2}+\mathbb{I}\otimes\sigma_{\zeta}\right)\left(\mathbb{I}^{\otimes
2}+\sigma_{\nu}\otimes\sigma_{\xi}\right)&\\\ \left(\mathbb{I}^{\otimes
2}+\mathbb{I}\otimes\sigma_{\mu}\right)&\end{split}$ (2)

assuming the outcomes of all measurements are of the positive eigenstate
($s=0$). So, reading (2) from right to left the measurements are ancilla
initialisation, entanglement, and ancilla disentangling with chosen non-
identity Pauli observables $\mu$, $\nu-\xi$ and $\zeta$ respectively. To see
the effect of the measurement sequence on the data qubit, we must trace over
the ancilla qubit. Since the process should completely disentangle the ancilla
qubit, (2) can be written as

$\Pi_{1-q}=\sum_{i=\\{0,x,y,z\\}}\gamma_{i}U\otimes\sigma_{i}$ (3)

where $\gamma_{i}$ are constants. From expanding (2) and assuming the form of
(3), $U$ is given as

$U=\Gamma\left[(1+\delta_{\zeta,\mu})\mathbb{I}+i^{\epsilon_{\zeta,\xi}}(\delta_{\zeta,\xi}+i^{\epsilon_{\zeta,\xi\oplus\mu}}\delta_{\zeta,\xi\oplus\mu})\sigma_{\nu}\right]$
(4)

where $\delta_{i,j}$ is the Kronecker delta, $\varepsilon_{i,j}$ is the Levi-
Civita symbol and the notation $i\oplus j$ is given from the product of the
two Pauli matrices $\sigma_{i}\sigma_{j}=i^{\varepsilon_{i,j}}\sigma_{i\oplus
j}$. Defining $1+\delta_{\zeta,\mu}=\alpha$ and
$i^{\epsilon_{\zeta,\xi}}(\delta_{\zeta,\xi}+i^{\epsilon_{\zeta,\xi\oplus\mu}}\delta_{\zeta,\xi\oplus\mu})=\beta$,
then the unitary condition for the rotation $U$ given by

$UU^{\dagger}=\mathbb{I}=|\Gamma|^{2}\left[(|\alpha|^{2}+|\beta|^{2})\mathbb{I}+(\alpha\beta^{*}+\alpha^{*}\beta)\sigma_{\nu}\right]$
(5)

is only satisfied when

$\alpha\beta^{*}=-\alpha^{*}\beta.$ (6)

Since $\alpha$ is always real this implies that $\beta=0$ or $\pm i$. These
conditions are satisfied only when $\mu\neq\xi$ and $\xi\neq\zeta$, and
therefore the joint-Pauli MBQC scheme only returns unitary rotations if and
only if each measurement is selected such that it does not commute with the
previous measurement chosen. From (4) it is apparent that the scope of what
single qubit unitaries that can be achieved in this scheme is limited to
$\\{I,S,S^{\dagger},XH,HX,HSH,HS^{\dagger}H\\}$, up to a local Pauli
correction depending on the outcomes of each of the measurements.
Unfortunately, while this does form a Clifford set, this does not make a
universal set of single qubit gates, and so will need to be complemented by a
$T$ gate, or some tunable non-Pauli observable such as

$\mathcal{P}_{W}(\theta,s)=\frac{1}{2}\left(\mathbb{I}+(-1)^{s}e^{I\theta\sigma_{z}/2}\sigma_{x}e^{-I\theta\sigma_{z}/2}\right)$
(7)

for completeness. Equally, this can also be achieved by initialising the
ancilla in a state that is not an eigenstate of one of the Pauli matrices.

The same constraints on implementing single qubit gates apply to two-qubit
gates. For two-qubit unitaries one needs only three physical qubits, the first
a data qubit $\ket{\psi_{1}}$, the second an ancilla, and the third another
data qubit $\ket{\psi_{2}}$. After the ancilla qubit is initialised, a two-
qubit joint-measurement is done between one of the data qubits, say
$\ket{\psi_{1}}$, and the ancilla, followed by joint-measurement between the
other data qubit $\ket{\psi_{2}}$ and the ancilla, before finally the ancilla
qubit is disentangled by a chosen measurement. Again, all measurements,
including the initialisation, must obey the measurement selection rule
outlined prior, such that the resulting gate on the data qubits is unitary.
Two-qubit gates performed in this scheme can be both entangling or a product
of two single qubit gates, depending on the choice of measurements, and will
vary up to local Pauli corrections depending of the outcomes of each of the
measurement steps. An example of an implementation of an entangling gate of
this type has been previously discussed with respect to Majorana qubits[7, 6].
There, the sequence that was discussed, as limited by what is possible with
the proposed architecture, $IZI\rightarrow ZXI\rightarrow IZX\rightarrow IXI$
which evidently obeys the measurement selection rule and is equivalent to a
$CNOT$ gate on the data qubits[6].

##  III Spin-Parity MBQC

Spin qubits are a natural fit for control by joint-Pauli measurement.
Measurement of the parity of two spins, being either aligned or anti-aligned,
is a $ZZ$ joint-measurement that has previously been discussed as a means of
controlling single QD spin-1/2 qubits[10, 11]. The spin-parity measurement can
be written as follows

$\begin{split}\mathcal{P}_{\text{sp}}(s)=(1-s)\left(\ket{\uparrow\uparrow}\bra{\uparrow\uparrow}+\ket{\downarrow\downarrow}\bra{\downarrow\downarrow}\right)&\\\
+s\left(\ket{\uparrow\downarrow}\bra{\uparrow\downarrow}+\ket{\downarrow\uparrow}\bra{\downarrow\uparrow}\right)&\end{split}$
(8)

where $s\in\\{0,1\\}$ such that $s=0$ measures the spins as aligned, while
$s=1$ measures the spins as anti-aligned. Assuming access to single spin
Hadamard gates an entangling $CZ$ gate can be implemented by a $IZI\rightarrow
ZXI\rightarrow IZZ\rightarrow IXI$ measurement sequence[11]. This sequence,
similar to the Majorana sequence, follows the outlined measurement selection
rule. However, this scheme abstracts the spin parity measurement such that in
a physical implementation of the scheme, two non measurement-based $CNOT$
gates and an additional qubit is needed for each parity measurement.
Additionally, as this sequence requires single spin-1/2 qubit gates by ESR-
driven pulses, the system is susceptible to charge noise.

###  III.1 Exact Spin-Parity Measurements

Here instead we will consider DQD single-triplet qubits. The logical qubit
states are a subset of the full spin space of a DQD whereby the singlet
$\ket{S}=(\ket{\uparrow\downarrow}-\ket{\downarrow\uparrow})/\sqrt{2}=\ket{0}$
and the triplet-0
$\ket{T_{0}}=(\ket{\uparrow\downarrow}+\ket{\downarrow\uparrow})/\sqrt{2}=\ket{1}$.
These encoded qubits can be tuned to be resistant to first-order charge noise
as well as offer accurate and fast single qubit phase gates by varying the
exchange interaction between the two spins. However, universal control of such
qubits requires additional on chip engineering such as magnetic field
gradients by either variable g-factors or micro-magnets. Previous work on
parity measurement based gates in such encoded spin qubits have relied on gate
based full single qubit control[12], specifically $H$ gates, which demand such
considerations in chip design. These requirements can potentially limit future
scalability.

Suppose instead there are two of such qubits in a linear array, consisting of
four identical neighouring QDs. The first two dots encoded some state in the
singlet-triplet subspace
$\ket{\psi_{1}}=\alpha_{1}\ket{S}+\beta_{1}\ket{T_{0}}$ and the last two
encode some state $\ket{\psi_{2}}=\alpha_{2}\ket{S}+\beta_{2}\ket{T_{0}}$. If
a spin-parity measurement is done between the second and third dot, i.e. one
dot from each constituent DQD qubit, the effect of the measurement in the
qubit space, is an $XX$ joint-Pauli measurement. Assuming there was no leakage
in the preparation of the two qubit states, the spin-parity measurement will
not couple to states outside singlet-triplet subspace.

$\Qcircuit@C=0.8em@R=1em@!R{\lstick{\ket{\psi}}&\qw\gate{\frac{\pi}{2}}\multigate{1}{\mathcal{P}_{\text{sp}}(s_{1})}\qw\gate{\frac{-(-1)^{s_{1}+s_{2}}\pi}{2}}\qw\rstick{X^{s_{1}+s_{2}}H\ket{\psi}}\\\
\lstick{\ket{+}}\qw\gate{\frac{\pi}{2}}\ghost{\mathcal{P}_{\text{sp}}(s_{1})}\qw\measureD{\mathcal{M}_{ST}(s2)}}$
Figure 1: Circuit diagram of the exchange gates and measurement sequence
needed to achieve a single qubit Hadamard up to a local Pauli-$x$ correction.
All single qubit gates here are exchange phase gates of angles indicated.

$\Qcircuit@C=1em@R=1em{\lstick{(a)\ket{\psi_{1}}}&\gate{\frac{\pi}{2}}\multigate{1}{\mathcal{P}_{\text{sp}}(s_{1})}\qw\qw\gate{\frac{(-1)^{s_{2}}\pi}{2}}\qw\qw\qw\\\
\lstick{\ket{0}}\qw\ghost{\mathcal{P}_{\text{sp}}(s_{1})}\gate{\frac{(-1)^{s_{1}}\pi}{2}}\multigate{1}{\mathcal{P}_{\text{sp}}(s_{2})}\qw\measureD{\mathcal{M}_{ST}(s3)}\rstick{\ket{\Psi}}\\\
\lstick{\ket{\psi_{2}}}\qw\qw\qw\ghost{\mathcal{P}_{\text{sp}}(s_{2})}\qw\qw\qw\qw\\\
\\\ }$

@C=1em @R=1em (b)|ψ_1⟩ |  1YX |  |  |
---|---|---|---|---
|0⟩ |  YX |  1YX |  Z |  |Ψ⟩
|  @C=1em @R=1em (c) |ψ_1⟩ |  1 |  X |  H |  1 |  X^1-s_2 |  | $\ket{\Psi}$

Figure 2: (a) Circuit diagram of the phase gates and spin-parity measurement
sequence needed to achieve a two qubit entangling unitary. (b) Equivalent
circuit diagram as in (a) written with joint-Pauli parity measurements. (c)
Standard Clifford gate circuit diagram of the two qubit unitary performed by
the measurement and phase sequence given in (a) with Pauli-$x$ corrections.

Along with the native exchange based phase gates, the $XX$ parity measurement
from the spin-parity measurement of neighboring dots is sufficient to design a
complete Clifford+T gate set or single and entangling gates. Application of
phase $\pm\pi/2$-gates before and after the joint measurement gives access to
the following set of joint measurements $\\{XX,XY,YX,YY\\}$. Therefore,
measurement sequences that give unitary rotations may be implemented by
following the previously outline measurement selection rule. The remaining
ingredients needed are choice of initialisiation of ancilla qubits and the
final disentangling single qubit measurements. Initialisation of ancillæ is
flexible, as both $z$-initialisation $\ket{0}=\ket{S}$ and $x$-initialisation
$\ket{+}=(\ket{S}+\ket{T_{0}})\sqrt{2}=\ket{\uparrow\downarrow}$ are achieved
by either non-adiabatic tunneling from a $(2,0)$ to a $(1,1)$ charge regime,
or adiabatic tunneling respectively[35]. Consequently, with exchange based
phase gates $y$-initialisation
$\ket{+i}=(\ket{S}+i\ket{T_{0}})\sqrt{2}=(\ket{\uparrow\downarrow}+i\ket{\downarrow\uparrow})\sqrt{2}$
is also achievable. Equally, this technique can be used to inialise an ancilla
state for a non Clifford gate by selecting a phase that is not an eigenstate
of either Pauli-$x$ or $y$, but rather somewhere inbetween. The choice of
final disentangling measurements is limited by experiment to measurement along
the Pauli-$z$ axis, i.e. a measurement of the singlet-triplet spin state of
the qubit

$\mathcal{M}_{ST}(s)=|s-1|\ket{S}\bra{S}+s\sum_{i=0,\pm}\ket{T_{i}}\bra{T_{i}}$
(9)

where measurement outcome $s=0$ projects on the singlet state, and outcome
$s=1$ projects onto the triplet spin space. This can be achieved by either
conventional Pauli spin blockade (PSB)[36, 27] or by coupling to a
superconducting resonator[37]. As the entangling spin-parity measurements
prior to the disentangling measurement do not couple to leaked
$\ket{T_{\pm}}=\ket{\uparrow\uparrow}(\ket{\downarrow\downarrow})$ spin
states, measurement of the triplet spin space only projects onto $\ket{T_{0}}$
state making the disentanglement step an exact Pauli-$z$ measurement.

An example of how to perform a Hadamard single qubit gate by spin-parity
measurement and exchange alone is given in Fig. 1. This sequence is
effectively an $IY\rightarrow YX\rightarrow IZ$ measurement sequence. Assuming
a spin-parity measurement of the form (8), at no point in this sequence do
either of the two qubits couple to leaked spin states outside the qubit
subspace, and so this scheme can be considered leakage-free. Note however that
after the sequence, there is a measurement outcome dependent Pauli-$x$
correction. This is a common consequence of MBQC and can either be remedied by
assuming non-measurement based single qubit gates, which in this case will
require the re-introduction of a magnetic field gradient, or by considered
post-processing of the final state tomography data. An example of one such
possible entangling gate with the proposed scheme is given in Fig. 2 (a) with
the corresponding equivalent gate based circuit given in Fig. 2 (c). This
measurement sequence is effectively equivalent to an $IZI\rightarrow
YXI\rightarrow IYX\rightarrow IZI$ sequence, as shown in Fig. 2 (b), optimised
however to require a minimum number of phase gates. Although the resulting
gate is an unusual entangling primitive to operate a quantum computer with, it
is a maximally entangling gate, and generating Bell-pairs from classical
inputs in the following sequence $\ket{00}\rightarrow\ket{\Psi^{-}}$,
$\ket{01}\rightarrow\ket{\Phi^{-}}$, $\ket{10}\rightarrow\ket{\Phi^{+}}$, and
$\ket{11}\rightarrow\ket{\Psi^{+}}$. Note that this is not the only example of
an entangling two qubit unitary that can be achieved with this architecture,
as other rotations can be achieved by selecting different patterns of phase
gates.

###  III.2 Asymmetric Spin-Parity Measurements

Entangling parity measurement based gates in encoded spin qubits similar to
those given so far have previously been discussed assuming exact spin-parity
measurements and single qubit Hadamard gates[12]. Experimentally, however,
spin-parity measurements are achieved by tuning the energy levels of the two
neighboring QDs such that the $\ket{S}$ and $\ket{T_{0}}$ states quickly
decays to the (2,0)/(0,2) charge configuration singlet state $\ket{S_{L/R}}$
in the left/right dot. This therefore only leaves the spin-aligned triplet
states $\ket{T_{\pm}}$ in the (1,1) charge configuration[38, 35]. While this
does measure if the two spins are aligned or anti-aligned, this method lacks
the symmetry of an exact spin-parity measurement. This asymmetric spin-parity
measurement may be written as

$\mathcal{P}_{ST}(s)=|s-1|(\ket{T_{+}}\bra{T_{+}}+\ket{T_{-}}\bra{T_{-}})+s\ket{S}\bra{S}.$
(10)

It is however worth noting that, although decay from the $\ket{T_{0}}$ to
$\ket{S_{L/R}}$ is inherent, without some finite Zeeman energy splitting
between the $\ket{T_{0}}$ and $\ket{S}$ the decay is relatively slow, and its
rate uncontrollable. By re-introducing such a splitting, the rate of decay has
been shown to be increased by four orders of magnitude[38].

$\Qcircuit@C=1em@R=1em{\lstick{(a)\ket{\psi_{1}}}&\gate{\frac{\pi}{2}}\multigate{1}{\mathcal{P}_{ST}(s_{1})}\multigate{1}{R_{\mathcal{P}_{ST}}(s_{1})}\qw\qw\gate{\frac{(-1)^{s_{1}+s_{2}}\pi}{2}}\qw\qw\qw\\\
\lstick{\ket{0}}\qw\ghost{\mathcal{P}_{ST}(s_{1})}\ghost{R_{\mathcal{P}_{ST}}(s_{1})}\gate{\frac{(-1)^{s_{1}}\pi}{2}}\multigate{1}{\mathcal{P}_{ST}(s_{2})}\qw\measureD{\mathcal{P}_{ST}(s_{3})}\multigate{1}{R_{ST}(s_{3})}\rstick{\ket{\Psi}}\\\
\lstick{\ket{\psi_{2}}}\qw\qw\qw\qw\ghost{\mathcal{P}_{ST}(s_{2})}\qw\qw\ghost{R_{ST}(s_{3})}\qw}$

$\Qcircuit@C=1em@R=1em{\\\ \\\ \lstick{(b)\text{
}\ket{\text{Data}}}&\multigate{1}{R_{\mathcal{P}_{ST}}}\qw\raisebox{-20.00003pt}{=}\gate{\pi}\multigate{1}{\mathcal{P}_{ST}(1)}\qw\raisebox{-20.00003pt}{$\cdots$}\multigate{1}{\mathcal{P}_{ST}(0)}\qw\\\
\lstick{\ket{\text{Ancilla}}}\ghost{R_{\mathcal{P}_{ST}}}\qw\qw\ghost{\mathcal{P}_{ST}(1)}\qw\ghost{\mathcal{P}_{ST}(0)}\qw}$

$\Qcircuit@C=1em@R=1em{\\\ \\\
&\multigate{4}{R_{ST}}\qw\raisebox{-70.0001pt}{=}\qw\multimeasureD{1}{S}\\\
\lstick{\raisebox{20.00003pt}{$(c)\ket{\text{Ancilla}}$}}\ghost{R_{ST}}\qw\qswap\ghost{S}\\\
\qwx\\\ \ghost{R_{ST}}\qw\qswap\qwx\qw\qw\qw\\\
\lstick{\raisebox{20.00003pt}{$\ket{\text{Data}}$}}\ghost{R_{ST}}\qw\qw\qw\qw\qw\\\
}$

Figure 3: (a) Circuit diagram of the phase gates, asymmetric spin-parity
measurements and repeat measurement sequence needed to achieve an equivalent
two qubit entangling unitary as in Fig. 2(c). (b) Circuit diagram of the
repeat entangling measurement sequence. (b) Circuit diagram of the repeat
disentangling measurement sequence. Note that in (a) and (b) each wire denotes
a qubit, comprised of 2 QDs, while in (c) each wire denotes a single QD, as to
show the exchange $\pi$-pulse (swap gate) needed to enable repeat
measurements.

Assuming access to a measurement of this form, some considerations must be
made. Firstly, let it be assumed that for ease of experimental implementation
and consequently process efficiency, all measurement types used in this
protocol are identical, i.e. they all asymmetric spin-parity measurements as
in Eq. 10, including the final disentangling measurement. Secondly, when
entangling two neighboring qubits by measurement, leakage may occur if a spin
anti-aligned ($\ket{S}$) state is detected. To address this issue, some
additional ingredients are needed to ensure that the final rotation of a given
measurement sequence is unitary. If an entangling measurement detects a
singlet state, causing leakage, this can be remedied by employing repeat
measurements. By applying a $\pi$-phase gate to a select qubit, the same
qubits can then be measured again until they are found to be spin-aligned.
This, naturally, increases the measurement and exchange gate overhead of the
overall rotation in an uncontrolled and probabilistic manner. However, such
repeated measurements are not always needed. In the case of a two qubit gate,
as in the Fig. 3, repeat measurements are required if, the first entangling
measurement projects onto a singlet state, or if the disentangling measurement
of the ancilla projects onto triplet space. In the first case, when the first
entangling measurement projects onto the leaked state, the measurement can be
repeated after application of a $\pi$-phase gate to the $\ket{\psi_{1}}$ data
qubit until a triplet state is observed. Measurement of the triplet after a
repeat measure recovers the desired unitary, up to a local $\pi$-phase
(Pauli-$z$) correction on the $\ket{\psi_{1}}$ data qubit, which is accounted
for later in the protocol. This type of repeat measurement is show in Fig.
3(b). The second case, when the disentangling measurement projects onto a
triplet state, requires a repetition of the disentangling measurement on the
ancilla qubit to regain the desired unitary. Here, however, a $\pi$-phase gate
between one of ancilla qubits QDs and one of the $\ket{\psi_{2}}$ data qubit
QDs is employed. This is a gate acting outside of the qubit subspace, and is
equivalent to a SWAP-gate between the QDs in a spin-1/2 qubit picture. This
type of repeat measurement is shown in Fig. 3(c). Note that if both entangling
measurements project onto the spin aligned triplet states, even after repeat
measurements, then no leakage states are populated at the point of the
disentangling step and the probability of measuring the spin aligned outcome
is $0$. The necessity repeat of measurements is given in Tab. 1.

Although the outcome of quantum measurements are probabilistic, they can be
weighted for efficiency. Due to the choice of the ancilla qubits initial state
in the gate outline in Fig. 3, the probability of the first parity measurement
projecting onto the leakage inducing spin anti-aligned ($\ket{S}$) state, is
$1/3$, regardless of the input state $\ket{\psi_{1}}$. Therefore the
requirement for repeating this measurement converges as $(1/3)^{n}$ where $n$
is the number of times the measurement has already been repeated. The
disentangling measurement is also weighted regardless of input states, in such
a way that only a maximum of one repetition is ever needed. This is due to the
choice of disentangling by the asymmetric spin-parity measurement as opposed
to a singlet-triplet qubit subspace measurement. Assuming the outcomes of the
previous measurements allow for some leakage, and regardless of input state
the probability of initially projecting to the spin aligned ($\ket{T_{\pm}}$)
state is $2/3$. Whilst this is weighted such that a repeat measurement is
likely, after a single implementation of the repeat measurement scheme, the
probability of projecting to the spin anti-aligned ($\ket{S}$) state, and thus
completing the two qubit unitary, is always $1$. Algorithm 1 details how both
of the repeat measurement cycles are to be used.

Algorithm 1 Two-qubit gate with Repeat Measurements

$(D_{1},D_{2})\rightarrow Q_{1}$, $(D_{3},D_{4})\rightarrow A$,
$(D_{5},D_{6})\rightarrow Q_{2}$

Data qubits $Q_{1}\rightarrow\ket{\psi_{1}}$ and
$Q_{2}\rightarrow\ket{\psi_{2}}$

Ancilla $A$ $\rightarrow\ket{0}$

Exchange Pulse: $\frac{\pi}{2}\rightarrow Q_{1}$

Parity Measurement: $Q_{1}\leftrightarrow A$ ($D_{2},D_{3}$), output $s_{1}$

if $s_{1}=1$ then

$s_{\text{Repeats}}=1$

while $s_{\text{Repeats}}>0$ do

Exchange Pulse: $\pi\rightarrow Q_{1}$

Parity Measurement: $A\leftrightarrow Q_{1}$ ($D_{1},D_{2}$), update output
$s_{\text{Repeats}}$

end while

end if

Exchange Pulse: $\frac{(-1)^{s_{1}}\pi}{2}\rightarrow A$

Parity Measurement: $A\leftrightarrow Q_{2}$ ($D_{4},D_{5}$), output $s_{2}$

Exchange Pulse: $\frac{(-1)^{s_{1}+s_{2}}\pi}{2}\rightarrow Q_{1}$

Disentangling Parity Measurement: $A$ ($D_{3},D_{4}$), output $s_{3}$

if $s_{3}=0$ then

Exchange Pulse: $\pi\rightarrow(D_{4},D_{5})$

Disentangling Parity Measurement: $A$ ($D_{3},D_{4}$)

end if

$Q_{1}\otimes Q_{2}=\ket{\Psi}$

Some further improvement to the gate protocol in Fig. 3 may be achieved by
replacing the second entangling parity measurement and final disentangling
parity measurements with initialising to the singlet state. This can be done
by tuning the dots of that given measurement such that within a reasonable
time ($\mathcal{O}(1\,\mathrm{\mu s})$) the two electrons will decay into a
singlet state with $(0,2)/(2,0)$ charge configuration. Locking these
measurement outcomes by this re-initialisation step improves the gate in two
ways. Firstly, this eliminates the need for Pauli corrections to the final
qubit state after the protocol as the gate in Fig. 2(c) is locked such that
$s_{2}=s_{3}=1$. Secondly, the need to ever repeat the final disentangling
measurement is eliminated as decay to the singlet state will correct for any
leakage accumulated. This will on average decrease the average exchange gate
and measurement overhead and therefore the average gate time.

In recent work[35], this asymmetric spin-parity measurement technique has been
combined with adiabatic re-initialisation of the (1,1) from the (2,0)/(0,2)
charge configuration $\ket{S_{L/R}}\rightarrow\ket{\uparrow\downarrow}$. While
this allows for perfect spin-parity measurements within the qubit space of a
DQD encoded qubit, when employing an asymmetric measurement to entangle two of
such qubits together, the adiabatic initialisation step can effectively be
treated as a second measurement in spin space that leads to loss of encoded
information. This is a direct consequence of the violation of the commutation
rule outlined prior. As such, the protocols described require that the
$\ket{S}$ in the (1,1) charge configuration is initailased after a
$\ket{S_{L/R}}$ state is measured.

Measurement Outcome ($s_{1},s_{2},s_{3}$) | $R_{\mathcal{P}_{ST}}$ | $R_{ST}$
---|---|---
(0,0,0) | No | -
(0,0,1) | No | No
(0,1,0) | No | No
(0,1,1) | No | Yes
(1,0,0) | Yes | -
(1,0,1) | Yes | Yes
(1,1,0) | Yes | No
(1,1,1) | Yes | Yes
Table 1: Table of the 8 possible measurement outcomes ($s_{1},s_{2},s_{3}$) of
the MBQC sequence proposed in Fig. 3(a) with the corresponding required repeat
measurement types, $R_{\mathcal{P}_{ST}}$ the repeat entangling measurement
and $R_{ST}$ the repeat disentangling measurement, to achieve a leakage free
entangling two-qubit unitary. Note that entries of ‘-’ have measurement
probability of $0$, and are therefore not considered.

##  IV Entangling Gate Fidelity and Leakage

Figure 4: Histogram data of simulations of the proposed two qubit parity
measurement based entangling gate with exchange pulse errors. Each data set is
derived from $10^{4}$ iterations of the simulation and a random $1\%$ error in
the exchange pulses ($\varepsilon^{\text{max}}_{J}=0.01$). (a) Percentage
distribution of number of measurements needed to complete the entangling gate
for the input state $\ket{\psi_{1}\psi_{2}}=\ket{SS}$ for when all measurement
steps are employed as measurements (red), and when the final measurement step
is replaced by the discussed initialisation by decay (blue). (b) The
infidelity distribution of the entangling gate for the input state
$\ket{\psi_{1}\psi_{2}}=\ket{SS}$ compared to the infidelity distribution of a
single exchange pulse with the same simulated errors. (c) Percentage
distribution of number of measurements needed to complete the entangling gate
for random the input states. (d) The infidelity distribution of the entangling
gate for random the input states

Figure 5: Histogram data of simulations of the proposed two qubit parity
measurement based entangling gate with measurement fidelity errors, treating
all measurement steps are measurements. Each data set is derived from $10^{4}$
iterations of the simulation and given values of maximum measurement fidelity
$\varepsilon^{\text{max}}_{M}$ and exchange pulse
$\varepsilon^{\text{max}}_{J}$ errors. (a) Order of magnitude distribution
infidelity (red) and leakage (blue) of simulated errors
$\varepsilon^{\text{max}}_{M}=1\%$ and $\varepsilon^{\text{max}}_{J}=0\%$. (b)
Order of magnitude distribution of infidelity and leakage of simulated errors
$\varepsilon^{\text{max}}_{M}=2.5\%$ and $\varepsilon^{\text{max}}_{J}=0\%$.
(c) Order of magnitude distribution of infidelity and leakage of simulated
errors $\varepsilon^{\text{max}}_{M}=5\%$ and
$\varepsilon^{\text{max}}_{J}=0\%$. (d) Order of magnitude distribution of
infidelity and leakage of simulated errors $\varepsilon^{\text{max}}_{M}=1\%$
and $\varepsilon^{\text{max}}_{J}=1\%$, including the infidelity of the
simulation with $\varepsilon^{\text{max}}_{J}=0\%$ (green) for comparison.

The proposed two qubit gate given in Fig. 3(a) can been simulated to show
expected average number of measurements, overall gate fidelity and possible
leakage, accounting for infidelity in exchange gate and spin-parity
measurements. Measurements in the simulation are projective, and treated as
having random outcomes with the correct weighting. Initially, only exchange
gate errors were considered, the results of these simulations are given in
Fig. 4. The simulation is run with a number of variants to the protocol and
input states. Firstly, a simulation of the simplest experimental
implementation is given in Fig. 4(a), starting from an initial input state of
$\ket{\psi_{1}\psi_{2}}=\ket{SS}$ and ending in the state
$\ket{\Psi^{-}}=(\ket{SS}-\ket{T_{0}T_{0}})/\sqrt{2}$ after the implemented
unitary. This possible experiment is discussed in greated detail in App. A.
Here the distribution of the number of measurements needed when the final step
to disentangle the ancilla is implemented as a measurement, is compared to
when the disentangling is done by initialisation, so discussed in Sec. III.2.
As expected, this shifts the distribution favorably toward fewer measurements
and therefore a shorter overall gate time. Then the overall gate fidelity is
probed by assuming some artificial small deflection of each exchange gate by a
randomly selected
$\varepsilon_{J}\in[-\varepsilon^{\text{max}}_{J},\varepsilon^{\text{max}}_{J}]$.
In Fig. 4(b) the fidelity of the parity based gate with the final measurement
and final re-initialisation are compared, along with a comparative fidelity
distribution of a single phase gate of a random rotation angle acting on a
random input state under an identically selected random error. Here,
interestingly, the measurement based two qubit gates employing three or more
error-prone exchange gates demonstrate a favorable fidelity distribution than
a single phase gate. This is due to the gates reliance on interleaved
projective measurement steps mitigating the accumulation of errors from the
noisy exchange pulses. Lastly, the measurement based protocol is simulated
with random input states in Fig. 4(c) and (d) for a fuller characterisation of
the average number of measurements and fidelity of the gate protocol.

From Fig. 4(a) and (c), it is shown that the average number of measurements
and exchange gates is equivalently between $3-5$ per execution of the proposed
gate. Assuming parity readout times between $1-10\,\mathrm{\mu s}$[38, 35],
entangling two qubit gate times of $5-50\,\mathrm{\mu s}$ can be achieved.
While this is long compared to exchange gate times, $\sim
10s-100s\,\mathrm{ns}$ for two qubit gates[39], this is still shorter than the
$T_{1}\gg 100\,\mathrm{\mu s}$[35] for single dot spin-$1/2$ qubits. While
this may not seem like obvious point of comparison, as the qubits manipulated
here are encoded here $S-T_{0}$ qubits, the interstitial measurements of the
dots serve to effectively reset the spins in the processor such that a
straightforward comparison of the gate time to the decoherence times of
$S-T_{0}$ qubits is not a complete measure of quality. Instead, as the two
outer dots of the $6$ dot device needed to perform such a gate are never
measured, it is the decoherence times of these single spin-$1/2$ dots that
limit the lifetime of a processor employing this control scheme. Thus the
comparison of the proposed measurement based gate time with the $T_{1}\gg
100\,\mathrm{\mu s}$ for single spin-$1/2$ qubits is apt.

There is no leakage induced in the proposed gate from simulated errors in the
exchange pulses. To investigate possible leakage errors in the process,
imperfect readout fidelities are included. These are modeled the same as
errors in exchange pulses with some random small error
$\varepsilon_{M}\in[-\varepsilon^{\text{max}}_{M},\varepsilon^{\text{max}}_{M}]$
applied to each measurement, projecting onto the opposite outcome as is read
out. The results of these simulations are given in Fig. 5, assuming all
measurement steps are treated as measurements (no singlet re-initialisation).
In all the simulated values of $\varepsilon^{\text{max}}_{M}$, there are two
clear peaks in the distribution of leakage. The peaks lower in the
distribution of leakage ($10^{-8}-10^{-10}$) probability correspond on average
to iterations of the simulated experiment where no repeat measurements are
needed, whilst the second higher peak ($10^{-3}-10^{-6}$) correspond to
iterations of the simulation where measurements repetitions are needed.
Decreasing the readout fidelity (increasing $\varepsilon^{\text{max}}_{M}$)
pushes these two peaks closer together. In Fig. 5(d) errors in both
measurements and the exchange pulses are compared to when perfect exchange
gates are assumed with imperfect measurements. Here it is clear that the
overall infidelity of the final state is perturbed most by the addition of the
exchange pulse error from the large jump in the peak of the distribution of
the infidelity. Overall the effect of leakage in the measurement based unitary
is small, and is totally dependent on the fidelity of measurements used.
Provided reported parity measurement fidelity of $99.98\%$[35], the effect of
leakage is effectively negligible.

##  V Discussion

In this work, measurements of joint-Pauli observables were discussed as a
method of MBQC, demonstrating a straightforward recipe for designing single-
and two- qubit gates. Combined with only one ancilla qubit, initialised to an
eigenstate of a chosen Pauli observable, and a disentangling single qubit
measurement, a unitary rotation on the data qubits are guaranteed by
considered choice of measurements. Specifically, so long as each measurement
steps’ observable does not commute with that of the previous step, including
the ancilla initialisation, the rotation will be unitary. Then, a physical
parity observable in the form of spin-parity between two adjacent QDs was
considered as a means to control DQD $\ket{S}-\ket{T_{0}}$ encoded qubits.
Provided single-qubit phase gates by the natural spin-spin exchange
interaction, a spin-parity measurement between spins from two neighbouring
qubits is equivalent to an $XX$ parity measurement that can be tuned to a $YY$
parity measurement. The protocol is completely leakage-free with low overheads
in terms of physical qubits, measurements and exchange gates. Then, a form of
asymmetric spin-parity measurement shown in recent experiments was considered.
Here, due to the projection of anti-symmetric spin states onto the maximally
entangled $\ket{S_{L/R}}$ state, probabilistic coupling to leaked spin state
is to be expected, but can be accounted for by the use of repeated
measurements. Lastly, a two qubit parity-MBQC entangling gate is simulated
with errors in the applied exchange pulses and measurements. The simulations
show that the overall gate fidelity is robust against exchange infidelity, due
to the projective measurements, and that leakage from measurement infidelity
is effectively negligible.

Parity measurements of the form discussed also form a complete tool-set for
quantum error correcting (QEC) codes. These include proposals for robust
quantum memories via measurement-based QEC colour codes, as well as
dynamically generated codes without a fixed logical basis. Our proposal for
parity-MBQC with encoded spin qubits is compatible with such methods of QEC,
and so is arguably scalable. Finally, it is believed that such gates are
viable on current experimental spin qubit processors which offer spin-parity
measurements and high fidelity control of the exchange interaction[35].

##  VI Acknowledgments

We acknowledge helpful discussions with U. Güngördü and S. Hoffman.

##  Appendix A Experimental Implementation

Here, details on how to perform a two qubit entangling gate by parity
measurements on a linear six quantum dot device like the one in Ref.[35] is
given. Ideally, in a device tailored as a small MBQC processor, sensing dots
acting as quantum point contacts (QPCs) to read out changes in occupation of
adjacent qubit dots would be positioned such that all the measurements needed
to perform the desired gate set would be done so directly. In the case of a
linear six dot device, encoding three DQD qubits whereby the outer two are
data qubits and the middle is the ancilla, a minimum of four sensing dots is
needed to directly access all necessary measurements. These include two at
each end of the array to read out the occupation of the first and last dot, as
well as two in parallel to the array above/below each one of the middle two
dots to measure the occupation of both of the ancilla qubit dots. The outer
sensing dots are used only to initialise and perform the final state readout
on the two data qubits. This is done by Pauli spin-blockade style readout[36],
tilting the detuning of the potentials toward the outer two dots to detect
possible occupation change (a spin singlet state). The middle two sensing dots
are assumed to be used for both initialisation and readout of the ancilla DQD
qubit state as well as parity measurements between each dot of the ancilla
qubit, and their neighboring data dot. Choice of measuring either the qubit
basis (singlet-triplet) or spin-parity, in this case the perviously
experimentally achieved asymmetric spin-parity measurement, depends on the
choice of wait time at the measurement detuning[38]. Such an ideal device is
shown in Fig. 6.

Figure 6: Example of a six dot device with which the proposed MBQC experiment
could ideally be performed. The data dots in red, each charged with an
electron in blue, are labeled $D_{i}$, and are grouped in pairs by the blue
boxes labeling if they DQD pairs encode one of the two data qubits $Q_{i}$ or
the ancilla $A$. The purple dots represent sensing (measurement) dots labeled
$M_{i}$. In the case of a non ideal device that could still perform the
proposed experiment (Ref.[35]), $M_{3}$ and $M_{4}$ and do not exist, and so
layers of additional exchange $SWAP$ gates are needed to achieve the same
protocol with just $M_{1}$ and $M_{2}$

The experiment described here is arguably the simplest proof-of-principal
experiment of the two-qubit measurement based unitary described in Fig. 3.
This version of the protocol takes the input states
$\ket{\psi_{1}\psi_{2}}=\ket{SS}=\ket{00}$ and performs a unitary rotation
such that the output state is a maximally entangled Bell state
$\ket{\Psi^{-}}=(\ket{00}-\ket{11})/\sqrt{2}$. Such an experiment simplifies
the initialisation step of the protocol, and allows for some of the exchange
pulses from the full arbitrary input implementation of the gate given in Fig.
3(a) to be omitted. However, if instead the two-qubit entangling gate is to be
performed on arbitrary input states, these may be initialised in such a device
by a combination of either starting with an initialised $\ket{S}$ or $\ket{+}$
depending on the how the $(0,2)\rightarrow(1,1)$ tunneling is handled[35],
exchange pulses, and applications of the measurement based single qubit gates
shown in Fig. 1.

The proof of concept experiment is as follows. Assuming a linear six dot array
$D_{1},D_{2},\dots D_{6}$ with an ideal measurement setup of four sensing dots
$M_{1},M_{2},M_{3}$ and $M_{4}$, an example two qubit entangling gate by
measurement would be achieved as follows. Firstly, the three DQD encoded qubit
states are prepared, in the case of the two data qubits,
$Q_{1}\rightarrow\\{D_{1},D_{2}\\}$ and $Q_{2}\rightarrow\\{D_{5},D_{6}\\}$,
into two identical spin singlet states by tunneling from a
$(0,2)\rightarrow(1,1)$ charge configuration, as detected by the two sensing
dots $M_{1}$ and $M_{2}$ coupled to qubit dots $D_{1}$ and $D_{6}$
respectively. This initialises a $\ket{\psi_{\text{init}}}=\ket{00}$ in the
two qubit logical space. Equally, the ancilla DQD qubit
$A\rightarrow\\{D_{3},D_{4}\\}$ is prepared into a spin singlet state by
monitoring the charge transition from either the intermediary sensing dots
$M_{3}$ or $M_{4}$ coupled to $D_{3}$ or $D_{4}$ respectively. Once all the
qubits are prepared then a $\pi/2$ pulse is applied between $D_{1}$ and
$D_{2}$ and the first entangling measurement can be performed. In this case,
the spin-parity between $Q_{1}$ and $A$ is checked. This is done by tilting
the detuning potentials between $D_{2}$ and $D_{3}$ for the appropriate period
of time such the sensing dot $M_{3}$ coupled to $D_{3}$ detects the desired
parity. If the parity is detected to be anti-aligned ($\ket{S}$) then the
measurement must be repeated after a $\pi$ exchange pulse is applied to
$Q_{1}$. This is repeated until the spin aligned ($\ket{T_{\pm}}$) state is
detected. Next, if the initial measurement of the parity of $Q_{1}$ and $A$
was spin aligned ($\ket{T_{\pm}}$) that a $\pi/2$ pulse is applied to $A$,
else if any repeat measurements were needed, a $-\pi/2$ or $3\pi/2$ pulse is
applied instead. Then the parity of dots $D_{4}$ and $D_{5}$ must be measured
in the same manner as that of $D_{2}$ and $D_{3}$ with the sensing dot $M_{4}$
coupled to $D_{4}$. If the outcome of this second parity measurement matches
that of the initial outcome of the first parity measurement, then a $\pi/2$
exchange pulse must be applied to $Q_{1}$, else if the outcomes are not
equivalent a $-\pi/2$ or $3\pi/2$ exchange pulse is needed. Finally, the
disentangling measurement can be done with either the sensor dots $M_{3}$ or
$M_{4}$ coupled to $D_{3}$ of $D_{4}$ respectively, and must also probe the
parity of the neighboring dots. Here as well, the application of up to one
repeat measurements may be needed (see Tab. 1). If the anti-aligned
($\ket{S}$) state is detected, the sequence is complete and no repetition is
needed. Else if the outcome of the disentangling measurement is the spin anti-
aligned ($\ket{T_{\pm}}$) state, then the measurement must be repeated by
applying a $\pi$ exchange pulse between dots $D_{4}$ and $D_{5}$, as shown in
Fig. 3(c), then measuring again. This should result in measuring the spin
anti-aligned ($\ket{S}$) state, after which the sequence is complete. This
measurement based gate sequence will leave the two qubits $Q_{1}$ and $Q_{2}$
in the maximally entangled Bell state
$\ket{\Psi^{-}}=(\ket{SS}-\ket{T_{0}T_{0}})/\sqrt{2}$, up to a known Pauli
correction (see Fig. 2(c)). The outcome can be verified by state
tomography[40], which can account for the Pauli correction. As previously
discussed in Sec.III.2, if the final measurement of the ancilla qubit is
replaced with a initialisation by decay to the charge $(0,2)/(2,0)$ state,
then the need for repetitions of the final measurement is eliminated,
improving the average gate time of the protocol. Here however, nothing is
changed by initialising instead of measuring the second parity measurement
(between $D_{4}$ and $D_{5}$), as the final state is equivalent up to the
Pauli corrections determined by this outcome.

Finally, if, as in the case of Ref.[35], the six dot device is purely linear,
with no parallel sensing dots $M_{3}$ or $M_{4}$ coupled to $D_{3}$ and
$D_{4}$ respectively, only those coupled to dots $D_{1}$ and $D_{6}$, then the
same experiment can be achieved, only with sequences of $SWAP$ gates such the
correct information is measured by the sensing dots. These swap gates are
given by $\pi$ exchange pulses between neighboring dots. In the case of the
first entangling measurement between qubit $Q_{1}$ and the ancilla $A$, a
$SWAP$ gate sequence exchanging $D_{1}\longleftrightarrow D_{3}$ is needed
such that the parity can be measured with the sensing dot coupled to $D_{1}$.
Equally the second entangling measurement between qubit $Q_{2}$ and ancilla
$A$ must employ a $SWAP$ sequence $D_{4}\longleftrightarrow D_{6}$ such that
the sensing dot coupled to $D_{6}$ can probe the necessary parity. Then
finally, a $SWAP$ sequence $D_{1}\longleftrightarrow D_{3}$,
$D_{3}\longleftrightarrow D_{5}$ and $D_{3}\longleftrightarrow D_{4}$ is
needed to effectively $SWAP$ qubit $Q_{2}$ with the ancilla $A$. Now the final
disentangling measurement of the ancilla can be achieved with the sensing dot
coupled to $D_{6}$.

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