# Collective preparation of large quantum registers with high fidelity

Lorenzo Buffoni  Department of Physics and Astronomy, University of Florence,
50019 Sesto Fiorentino, Italy  Michele Campisi  NEST, Istituto Nanoscienze-CNR
and Scuola Normale Superiore, I-56127 Pisa, Italy

###### Abstract

We report on the preparation of a large quantum register of 5612 qubits, with
the unprecedented high global fidelity of $F\simeq 0.9956$. This was achieved
by applying an improved cooperative quantum information erasure (CQIE)
protocol [Buffoni, L. and Campisi, M., Quantum 7, 961 (2023)] to a
programmable network of superconducting qubits featuring a high connectivity.
At variance with the standard method based on the individual reset of each
qubit in parallel, here the quantum register is treated as a whole, thus
avoiding the well-known orthogonality catastrophe wehereby even an extremely
high individual reset fidelity $f$ results in vanishing global fidelities
$F=f^{N}$ with growing number $N$ of qubits.

Quantum computers are inherently susceptible to external disturbances that
greatly hinder the possibility of achieving quantum advantage. For this
reason, quantum computing research has focused on error correction [1, 2, 3]
and mitigation [4], fault tolerance [5] and other techniques aimed at
achieving higher fidelities in the output state of quantum computations.

Fidelity may be degraded because of various sources acting at different stages
in the course of a quantum computation. In this regard, it is useful to
distinguish between the sources of error incurred during the execution of
quantum circuits, (i.e. gate noise) and SPAM (state preparation and
measurement) errors. Here we focus on the latter, and in particular we
concentrate on the problem of preparing a quantum register composed of many
qubits in a high fidelity state.

Efforts in that direction have been aimed so far exclusively at increasing
individual qubit preparation fidelity. However, that is not a scalable
strategy; as thoroughly discussed in [6], if each qubit is individually
prepared with a fidelity $f=(1-\varepsilon)$, then the total register fidelity
$F$ would dramatically diminish with the number of qubits, $N$:

$\displaystyle F=f^{N}=(1-\varepsilon)^{N}\,,$ (1)

a phenomenon analogous to Anderson “orthogonality catastrophe” [7]. For
example, with the current state of the art SPAM fidelity of $f=0.997$ for
superconducting qubits [8], the global fidelity would drop to $F\simeq 0.45$
with $N=262$, and $F\simeq 4.7\times 10^{-8}$ for $N=5612$.

In order to overcome this major block, here we abandon the paradigm of $N$
individual qubit resets occurring in parallel and embrace instead a new
paradigm based on collective effects, to achieve a genuinely global
preparation of the whole quantum register. The quest for collective advantages
is currently in the limelight of intense research in stochastic and quantum
thermodynamics [9], with applications that span from metrology, [10], to
quantum batteries [11, 12, 13, 14, 15, 16, 17, 18, 19], (quantum) heat
engines, [20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 23, 31, 32, 33, 34, 35,
36, 37, 38, 39] and quantum transport [40, 41, 42]. While this activity is
almost exclusively of theoretical character (Ref. [19] is an exception), the
present study is one of the first to practically demonstrate a collaborative
advantage on a physical quantum hardware, and the first to demonstrate its
application to high fidelity quantum register preparation.

In a previous work [43] we have demonstrated a multi-qubit reset mechanism
where an ensemble of $N=256$ qubits starting in a completely mixed state was
brought close to a pure state featuring all qubits being in the computational
state $|0\rangle$. The method, dubbed “cooperative quantum information
erasure” (CQIE), uses both cooperative and quantum effects to achieve the
record _erasure action_ (i.e., the product of energy cost per qubit and reset
time) of $10^{-22}$ erg s/bit occurring (within our experimental error) at
Landauer cost of $kT\ln 2$ per bit.

Here we demonstrate the employment of CQIE to prepare large quantum registers
with unprecedented global fidelity. The CQIE method employed here is a variant
of that originally employed in [43], featuring now a more effective choice of
protocol parameters and their time-dependence. We implemented it on a
programmable network of superconducting qubits, specifically, the D-Wave
Advantage 6.4 quantum annealing processor [44] and achieved exceptionally high
values of the measured global fidelity. For example, we demonstrate a global
SPAM fidelity of $F\simeq 0.9996$ on a register of $262$ qubits, and $F\simeq
0.9956$ with as many as $5612$ qubits. To the best of our knowledge such high
values of global SPAM fidelity were never reported in the literature for such
large qubit registers.

We compared the results obtained for various network sizes (up to $N=5612$)
with those obtained from a variant of CQIE featuring non-interacting qubits,
that implements a non-cooperative, parallel reset of each qubit. While the
global fidelity dramatically decreased in this case, as predicted by Eq. 1, it
remained almost constant with increasing $N$ in the cooperative case, thus
demonstrating the genuinely cooperative nature of the method. This was
corroborated by the observation that CQIE global resets on networks with
intermediate levels of connectivity results in intermediate values of global
fidelity.

##  I CQIE working principle

CQIE is a method to reset a network of qubits to a state that is as close as
possible to the pure state featuring each and all qubits being in their
computational state $|0\rangle$, regardless of its initial state [43]. CQIE
leverages cooperative, quantum and dissipative phenomena to achieve that. The
idea is that the quantum register is a spin network featuring spontaneous
symmetry breaking, subject to an external magnetic field, i.e. a many-body
quantum system with Hamiltonian

$\displaystyle
H=-\mathcal{B}_{x}\sum_{i=1}^{N}\sigma_{i}^{x}-\mathcal{B}_{z}\sum_{i=1}^{N}\sigma_{i}^{z}-\mathcal{J}\sum_{i,j}\alpha_{i,j}\sigma_{i}^{z}\sigma_{j}^{z}\,.$
(2)

The coefficients $\alpha_{i,j}$ define the adjacency matrix of the network’s
graph, and the register is immersed in an environment at temperature $T$. The
method consists in changing the three parameters
$\mathcal{B}_{x},\mathcal{B}_{z},\mathcal{J}$ in time in a cyclical fashion,
according to the following general scheme [43]. The protocol starts with no
magnetic fields $\mathcal{B}_{x},\mathcal{B}_{z}$ at a value of $\mathcal{J}$
above the critical value $\mathcal{J}_{C}$ separating the ferromagnetic phase,
$\mathcal{J}>\mathcal{J}_{C}$, from the paramagnetic phase,
$\mathcal{J}<\mathcal{J}_{C}$ (note that $\mathcal{J}_{C}$ generally depends
on the environment’s temperature $T$). One then decreases $\mathcal{J}$ so as
to decrease the height of the Landau free energy barrier thus allowing the
system to get demagnetised. This de-magnetisation is highly enhanced by
turning on a transverse field $B_{x}$, that allows tunnelling through the
barrier. Once in the paramagnetic phase, one goes back to the initial point,
but now adding some longitudinal field $B_{z}$, to drive the network to a
state of high net magnetisation along the direction of $B_{z}$, regardless of
its initial state.

As demonstrated in [43] the method works excellently. On a 2D Ising network of
$N=256$ superconducting qubits with nearest neighbour interaction, we
demonstrated that each qubit in the network was reset to the $|0\rangle$ state
with an individual fidelity of $f=0.999$, while the global network fidelity
$F$ which is the focus of the present work was not addressed.

The cooperative nature of the CQIE method comes from the physics of
spontaneous symmetry breaking and is enabled by qubit-qubit interactions. The
quantum nature comes from quantum tunnelling and is enabled by the transverse
field $\mathcal{B}_{x}$. Dissipation also plays a crucial role as it is
impossible to increase the purity of the quantum register if it evolves
unitarily, i.e., without interacting with an environment: the latter provides
in fact the bin where to dump the register mixedness (i.e. its “impurity”).

##  II Improving the CQIE cycle

Figure 1 shows the cyclic time-dependent protocol,
$\mathcal{B}(t),\mathcal{B}_{z}(t),\mathcal{J}(t)$, that was employed in Ref.
[43] to implement CQIE on a programmable network of superconducting qubits
(i.e., a quantum annealer), see the red curves. As stressed in [43] that cycle
is only one of the many cycles that one can employ to implement CQIE. There is
in fact plenty of room for optimising the cycle in order to maximise the
effectiveness of the CQIE protocol. While a systematic optimisation is beyond
the scope of the present study, we nonetheless proceeded to a heuristic
improvement of the path, based on trial and error runs

Figure 1: (Red) Original (unoptimized) CQIE protocol [43]. (Blue) Optimized
CQIE protocol.

These runs were implemented on a D-Wave Advantage 6.3 processor that
implements a connectivity graph with up to $15$ couplings per qubit called
Pegasus topology [45]. The variation in time of the control fields can be
tuned with a resolution up to $0.01\mu s$ on these processors. The complete
results of the optimization runs are reported in the appendix. They provided
us with three critical insights. The first is that the higher the longitudinal
($\mathcal{B}_{z}$) field, the higher the final magnetization. Second, the
later and the faster that field is turned off the more effective the protocol
is. The third is that the transverse field ($\mathcal{B}_{x}$) is so effective
at de-correlating the qubits that it is in fact not necessary that the
critical point $\mathcal{J}_{C}$ be crossed.

These three key insights led us to define a protocol of duration $\tau=30\mu
s$ with stronger interactions than the original CQIE, a stronger transverse
field and a higher longitudinal field. This improved protocol, depicted in
blue in Fig.1, was employed to prepare large quantum registers with
unprecedented high fidelity, as discussed below.

Note that the colour coding used in Fig.1 is used in all figures of the
manuscript: red curves refer to the un-optimised protocol, blue curves refer
to optimised protocol.

##  III Results

Figure 2: Measured global fidelity as a function of the number of qubits in
the register. Red and blue curves refer, respectively to un-optimised and
optimised CQIE protocols depicted in Fig. 4. Dotted line refers to individual
reset ($\alpha_{i,j}=0$). Dashed line refers to the 2D square lattice with
nearest neighbors interaction. Solid lines refer to maximal connectivity
subgraphs available on the hardware.

Figure 2 reports the fidelity of preparation of the state $|00\dots 0\rangle$
as a function of the number $N$ of qubits in the quantum register, as measured
for various protocols implemented onto a programmable network of
superconducting qubits, for $N$ up to $N=324$, and various adjacency matrices
$\alpha_{i,j}$. Specifically, we implemented them on the D-Wave Advantage 6.4
processor with up to 5612 active qubits connected with the Pegasus topology
[45]. The initial state of the network featured each qubit being randomly
either in the up $|1\rangle$ or down state $|0\rangle$ of the Pauli operator
$\sigma_{z}$ with equal probability. Each point is obtained by running the
same protocol on the same network (as defined by its adjacency matrix
$\alpha_{i,j}$), followed by the measurement of the Pauli operator
$\sigma_{z}^{i}$ of each qubit, for a total of $\mathcal{N}=10^{5}$ times. The
global fidelity was computed as the relative frequency of observing the target
state $|0,0,\dots 0\rangle$.

Both in Fig. 2 and Fig. 3, dotted curves refer to individual reset, i.e.
$\alpha_{i,j}=0$ for all $i,j$’s; Dashed curves refer to 2D nearest neighbour
square lattice Ising connectivity; Solid curves refer to maximal connectivity
subgraphs available on the hardware. Since the hardware connectivity is
obtained by tiling together in the 2D plane of the chip patches of densely
connected qubits, we used D-Wave’s own graph generation tools [44] to sample
patches of increasing size from the Pegasus topology [45] and turned on all
the available couplers of the chip inside these patches to achieve the maximal
connectivity sub-networks of a given size. Such a network features larger
connectivity (up to 15 neighbours per qubit)

Fig. 2 evidences that the fidelity increases as the connectivity increases, as
expected. With reference to to the three red curves (relative to the un-
optimised protocol) note how fidelity is drastically increased when moving
from the individual reset to the 2D nearest neighbour lattice. It further
increases when moving to maximal connectivity. Finally, cycle optimisation
further raises global fidelity which remains above the exceptional value of
$0.9995$ for $N$ up to $324$ (blue solid curve).

Figure 3: Measured global fidelity as a function of the number of qubits in
the register. The solid blue curve is for the optimised cycle (see Fig. 4) run
on maximal connectivity sub-graphs available on the hardware. The dotted blue
curve is for the optimised individual reset cycle (i.e. featuring no
interactions among the qubits $\alpha_{i,j}=0$. The inset shows a close-up of
the solid curve.

Figure 3 shows the global fidelity as obtained by applying the optimised
protocol onto maximal connectivity graphs of yet larger sizes. This resulted
in exceptionally high global fidelities, with $F$ as large as $0.9956$ for
$N=5612$, see blue solid line. For completeness, the network sizes were:
$N=$40, 262, 678, 1284, 2078, 3061, 4241, 5612, and the according average
connectivities were $c=$4.1, 6.02672, 6.56932, 6.81075, 6.93744, 7.01993,
7.08866, 7.14326, respectively. For comparison, Fig. 3 reports as well the
results obtained when applying the non-cooperative version of the optimised
protocol (dotted blue line). Note how the fidelity dramatically drops down to
about $0.35$ for the largest size $N$. We have checked that the data in fact
obeys the expected Eq. (1), with $f$ being calculated from the data as the
relative frequency over all runs and all qubits, to find any single qubit in
the $|0\rangle$ state, which amounted to $f=1-(2.11\pm 0.08)\times 10^{-4}$.

As a side remark, it is interesting to note that our optimisation not only
improved the collective reset protocol but also resulted in an improved
individual reset protocol. To see that compare the red dotted line in Fig.2
with the blue dotted line and Fig.3.

##  IV Conclusion and outlook

By abandoning the customary method of preparing a quantum register via
individual initialization of each qubit and embracing instead a global
preparation approach, we have demonstrated here the achievement of
exceptionally high values of global fidelities of the prepared state, that
largely surpass the highest fidelities that can be currently obtained with
individual preparation strategies. This was practically demonstrated on a
programmable network of superconducting qubits, specifically a D-Wave
Advantage 6.4 processor.

While this is an exceptional result _per se_ , its application to gate-based
quantum computers is not immediate. While the presented method crucially rests
on the existence of interactions with a thermal environment, gate-based
quantum computers should be as isolated as possible from the environment. The
present work then suggests that the next generations of quantum computers
should be designed in such a way as to have tunable environment interactions
so that they can efficiently perform both logically irreversible gates, e.g.,
reset to a specific state, which according to Landauer principle require
dissipation [46, 47], and logically reversible quantum gates, which require
complete absence of environmental influences. Meanwhile, since we are still in
the so-called NISQ (noisy intermediate scale quantum) era [48] one can
leverage on noise and attempt to implement the CQIE scheme on a current gate-
based quantum computer, e.g., by means of Trotterisation techniques. Research
in this direction is currently ongoing.

##  V Acknowledgments

Access to D-Wave services was provided by CINECA through the competitive
ISCRA-C project “COOPERA”. L.B. was funded by PNRR MUR Project No.
SOE0000098-ThermoQT financed by the European Union–Next Generation EU.

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##  Appendix A The protocol

In order to perform the collective erasure of [43] we need an Ising
Hamiltonian that we can control with time-dependent fields across its critical
point (see [43] for the details of the protocol). We implemented the protocol
using the controllable spin Hamiltonian of the D-Wave Advantage quantum
annealer that is:

$\displaystyle\frac{H(t)}{h}=-\frac{A(s)}{2}\sum_{i}\sigma^{x}_{i}+\frac{B(s)}{2}\left[g(t)\cdot\bar{h}\sum_{i}\sigma^{z}_{i}+J\sum_{\langle
i,j\rangle}\sigma^{z}_{i}\sigma^{z}_{j}\right],$ (3) $\displaystyle
g(t)\;\text{and}\;s\equiv s(t)\in[0,1].$ (4)

where $s(t)$ and $g(t)$ are both piecewise linear functions of time and $A(s)$
and $B(s)$ are energies expressed in GHz units as given by the constructor
(for reference see [44, 43]). In particular to implement the desired time
evolution we used the following protocol:

$s(t)=\begin{cases}1-\frac{(1-\bar{s})t}{10}&0\mu s\leq t\leq 10\mu s\\\
\bar{s}&10\mu s\leq t\leq 20\mu s\\\ 1-\frac{(1-\bar{s})(30-t)}{10}&20\mu
s\leq t\leq 30\mu s\end{cases}$ (5)

and

$g(t)=\begin{cases}0&0\mu s\leq t\leq 10\mu s\\\ \frac{t-10}{10}&10\mu s\leq
t\leq 20\mu s\\\ \frac{30-t}{10}&20\mu s\leq t\leq 30\mu s\end{cases}.$ (6)

That can be also seen more clearly pictured in Fig.4. In this description, it
is worth noting that the transverse field component $A(s)$ becomes greater
than zero, and thus the protocol becomes quantum, only for values for
$\bar{s}\leq 0.5$. For the characterization below, we will take only two
values $\bar{s}=0.6$ (classical protocol) and $\bar{s}=0.4$ (quantum
protocol). The other parameters that we will vary to characterize the protocol
in the following are $\bar{h}$, a dimensionless value that determines the
maximum local field applied to the qubits, and $J$, the dimensionless
parameter determining the coupling between qubits. The connectivity $\langle
i,j\rangle$ can be any subgraph of the full connectivity graph of the D-Wave
Advantage 6.3 called Pegasus graph [44] for which each qubit is connected on
average to $15$ neighbors. To begin, we will start by considering square
lattice graphs in order to leverage the well-known statistical properties of
the 2D Ising model, but we will relax this assumption later in the paper and
eventually take advantage of the full connectivity of the device.

Figure 4: (Top panel) plot of the driving fields $s(t)$ and $g(t)$ during the
protocol. (Bottom panel) the respective plot of the energies $A(s)/2$ and and
$B(s)/2$ as a function of time. The quantum case is reported with the only
difference, with respect to the classical case, that the term $A(s)/2$ is
different from zero.

In the main text, we used a modified version of $g(t)$ that is a quenched
version of the one employed above.

$g(t)=\displaystyle{\begin{cases}0&0\mu s\leq t\leq 10\mu s\\\
\frac{t-10}{10}&10\mu s\leq t\leq 20\mu s\\\ 1&20\mu s\leq t\leq 30\mu s\\\
\frac{30.01-t}{0.01}&30\mu s\leq t\leq 30.01\mu s\,.\end{cases}}$ (7)

##  Appendix B Protocol characterization

To begin our characterization, we found the critical point of the 2D Ising
model by sampling the equilibrium magnetization of the system at $\bar{h}=0$
and varying $J$. In this way, we located the second-order phase transition on
the magnetization at $J=0.08\pm 0.01$. This corresponds to an estimated
temperature $T=33\pm 9$ mK using the well-known relation between critical
temperature and coupling [49]. An independent temperature estimate using the
pseudo-likelihood method described in [50] gave us a consistent estimate of
$T=31\pm 4$ mK.

Figure 5: Panel (a): average magnetization $\langle m\rangle$ and
corresponding standard deviation as a function of the local field strength
$\bar{h}$ for the classical erasure protocol. A non-cooperative protocol (i.e.
lattice size $1$) is compared against $5\times 5$ and $10\times 10$
cooperative protocols on the Ising lattice. Panel (b): average magnetization
and corresponding standard deviation as a function of the local field strength
$\bar{h}$ for the quantum erasure protocol for various lattice sizes as
explained above. Panel (c): average magnetization and corresponding standard
deviationas a function of the number of qubits $N$ for a quantum cooperative
protocol in an Ising lattice of increasing size (up to $18\times 18$). The
dashed region marks the lattice size above which one has to resort to minor
embedding techniques to fit the lattice into the fixed connectivity of the
D-Wave Advantage 6.3 chip.

We then proceeded to a detailed investigation of the role that the various
components play in the realization of this protocol. First, we fixed $J=0.12$
and varied the maximum longitudinal field $\bar{h}$ for both the classical and
the quantum case and for lattices of various sizes spanning from the uncoupled
(individual) reset of size $1$ up to a $10\times 10$ lattice. As a figure of
merit, we used the average normalized magnetization of the system defined as
$\langle m\rangle=\sum_{i}\sigma^{z}_{i}/N$. As can be seen in panels (a) and
(b) of Fig.5 it is clear that the higher the longitudinal field employed the
better the results. However, note that a larger field means more energy
expenditure. Regarding the cooperative effects, in the classical case, we do
not see a big contribution from the cooperative effects, indeed the average
magnetizations of the cooperative protocols are lower than the individual ones
except for protocols in the high $\bar{h}$ regime. For the quantum case, we
can nevertheless observe a clear effect on the quality of the result (i.e.
higher average magnetization) given by cooperative effects as the size of the
system increases. Note that the quantum protocol also magnetizes the system
faster and with lower variance than the classical one. This is most probably
because introducing the transverse field is very effective in decorrelating
the spins and bringing them into a purely paramagnetic state, which makes it
easy to align them in the same direction using the local field $\bar{h}$. On
the contrary, the classical protocol cannot bring the coupling all the way to
zero due to limitations on the control of $B(s)$. Due to the finite size of
the system and hysteresis effects, the system will still have some magnetic
domains and not be purely decorrelated and thus harder to align with the
external field.

To clarify this behaviour, we fixed the size of the system to a $10\times 10$
lattice and $\bar{h}=0.25$ and varied the values of $J$ such that, during the
protocol, the system was spending most of the time closer to the ferromagnetic
phase or closer to the paramagnetic phase. The results of this experiment
showed that the quantum protocol performs slightly better by increasing $J$
(i.e. being more ferromagnetic) while the classical protocol performs slightly
better by decreasing $J$ (i.e. easier to decorrelate). This observation is
coherent with the explanation given above, even if it is not a definitive
proof for which we would need a device capable of implementing a classical
protocol that can turn $B(s)$ to zero without introducing unwanted transverse
fields.

To conclude the characterization of the protocol, we also investigated how,
fixing all the other parameters, the quantum cooperative erasure scales with
the size of the system. This observation is crucial for the following as if we
want to use this protocol to reset multi-qubit registers the cooperative
effects need to increase as the number of qubits increases. As one can see
from panel (c) of Fig.5 a bigger system results in a higher final average
magnetization after the protocol. Interestingly, we used this analysis to
start relaxing the constraints on the 2D topology of the system. Indeed the
last points of the graph, in the region contained by the dashed line, are not
the result of a so-called minor embedding [44] of the 2D Ising model on the
physical connectivity of the chip. Indeed, it happens that above a certain
size, one can no longer fit the 2D lattice onto the fixed Pegasus topology of
the chip. The usual workaround for these situations is to perform a minor
embedding by creating chains of qubits that are strongly ferromagnetically
coupled and thus act as one unique logical qubit allowing one to circumvent
the topological constraints of the chip. However, for our protocol, adding
this minor embedding step means no longer working with an exact 2D Ising
Hamiltonian with all its theoretical guarantees. Confirming that the protocol
works just as well in the presence of minor embedding is crucial to move
forward into the next section where we will then be able to move beyond the
constraints of the 2D topology.

##  Appendix C High fidelity registers

Having characterized all these properties of our protocol the main question we
want to pose is whether this cooperative reset can be exploited to prepare
high fidelity multi-qubit registers in an efficient way. We will then focus
exclusively on the quantum protocol and taking a relatively high value of
$\bar{h}=1$. The reset fidelity that we are interested in, is the overlap
between the target reset state $\sigma=\ket{0}^{\otimes N}$ and the (unknown)
state $\rho$ at the end of the protocol. Computing the fidelity then amounts
to measuring the state of the system at the end of the protocol and counting
the fraction of occurrences of the state $\ket{0}^{\otimes N}$.

First, we established a baseline by computing the reset fidelity of the
individual reset protocol, i.e. the protocol with $J=0$, for up to $N=324$
qubits. The results are shown as the dotted line in Fig.2, where we can
observe that the fidelity decreases quickly dropping below $0.5$ for a
register of around $200$ qubits. Since we know that Eq.(1) can be interpreted
as an effective temperature model as described in [6] it is interesting to use
the experimental fidelities to extract the effective temperature from the
theoretical scaling:

$F=\left(1+e^{-\beta\Delta E}\right)^{-N}.$ (8)

Indeed, we can fit the parameter beta above and obtain for the individual
erasure an effective temperature of $T=32.56\pm 0.06mK$. This is consistent
with our previous temperature estimates of the environment. We then
implemented the full cooperative erasure protocol with $J=0.12$ for the Ising
model and also with the full Pegasus topology to understand if having more
connectivity would benefit the performance of the protocol with respect to the
Ising case. The results can be seen as dashed and solid lines respectively in
Fig.2, where we see that in both cases we are able to prepare registers of
dimensions up to $300$ qubits at fidelities $>90\%$ by exploiting the
cooperative effects, in a regime where the individual erasure would have
failed miserably. We then fitted the effective temperature of Eq.(2) on both
cases. In contrast to the individual reset, the estimated temperature of the
cooperative protocols on Ising and Pegasus are $T=23.1\pm 0.1mK$ and
$T=20.78\pm 0.05mK$ respectively, which are significantly lower than the
estimated initial temperature. In this regard, we can also see our protocol as
an algorithmic cooling protocol whereby we have lowered the effective
temperature of the ensemble of qubits. In order to push the protocol even
further we tried a variant of the protocol where the $g(t)$ is kept constant
in the final stroke up until the last possible second and it gets quenched
down to zero in $0.01\mu s$ at the end, which is reported in Eq.7. The aim is
to make the system more robust against fluctuations. The result is the solid
blue line of Fig.2 where we were able to obtain a $99.95\%$ fidelity for the
collective preparation of $324$ qubits corresponding to an effective
temperature of $14.0\pm 0.2mK$ which is significantly below the measured
temperature of ca 33 mK and even lower than D-Wave fridge nominal temperature
that is reported to be $16.0\pm 0.1mK$ in its own datasheet [44]. It is
interesting to notice that, while this protocol was originally implemented on
a 2D Ising model to take advantage of the known phase transition, it works
well also for the Pegasus topology with ferromagnetic interaction. It thus
seems that we do not necessarily need to cross a phase transition for this
cooperative effect to take place as the transverse field is good at
decorrelating spins in itself without the need of entering a paramagnetic
phase. We could then assume that, generally, the denser the topology, the
better the outcome of the cooperative erasure.

##  Appendix D Scaling estimate

Figure 6: Average single-qubit fidelity computed measured for the qubits in
the cooperative protocol (orange dots). The functions $(1-\alpha/N)$ (blue
dashed line) and $(1-\hat{\alpha}/N)$ (red dashed line) are reported for
visual reference regarding the scaling of $f(N)$ in the cooperative case.
Figure 7: Measured global reset fidelity as a function of the number of qubits
for the cooperative protocol (orange dots). The two possible models proposed
above are reported as dashed lines $(1-\hat{\alpha}/N)^{N}$ (red) and
$(1-\alpha)^{N/N_{0}}$ (orange).

Knowing that for the individual (non-cooperative) case, the fidelity should
scale as $F=f^{N}=(1-\alpha)^{N}$ we can fit the value of $\alpha$ from the
blue data in Fig.3 obtaining a value of $\alpha=2.11\pm 0.08\times 10^{-4}$.
For the cooperative data, we can follow two approaches. The first comes by
observing that in the cooperative case, we have that $f$ is not a constant but
a function of the number of qubits (as can be observed from the orange points
in Fig.6). For example, we can assume that $f(N)=(1-\alpha/N)$, by using the
same value of alpha found above we see that we should expect slightly higher
single-qubit fidelities (blue dashed line in Fig.6). But if we try to find the
best fit of this functional form we get $f(N)=(1-\hat{\alpha}/N)$ with
$\hat{\alpha}=5.0\pm 0.2\times 10^{-4}$ (red dashed line in Fig.6). However,
what seems to be a good fit for the single qubit scaling turns out to be
somehow unsatisfactory when dealing with the global fidelity. Indeed, as we
can see in Fig.7 if we look at the measured values of the global fidelity
(orange dots) the curve corresponding to the scaling
$F=f(N)^{N}=(1-\hat{\alpha}/N)^{N}$ (red dashed line) doesn’t look that good.
We can then turn to the second approach, that is to consider
$F=f^{N/N_{0}}=(1-\alpha)^{N/N_{0}}$ again we can fit the value of $N_{0}$
from the experimental data obtaining a value of $N_{0}=329\pm 38$
(corresponding to the orange dashed line in Fig.7) which looks like a better
fit to the global fidelity.