# Noisy Qudit vs Multiple Qubits : Conditions on Gate Efficiency

Denis Janković<EMAIL_ADDRESS>Université de Strasbourg, CNRS,
Institut de Physique et Chimie des Matériaux de Strasbourg (IPCMS), UMR 7504,
23, Rue du Loess, 67000 Strasbourg, France  Institute of Nanotechnology (INT),
Karlsruhe Institute of Technology, P.O. Box 3640, 76021 Karlsruhe, Germany
Jean-Gabriel Hartmann<EMAIL_ADDRESS>Université de
Strasbourg, CNRS, Institut de Physique et Chimie des Matériaux de Strasbourg
(IPCMS), UMR 7504, 23, Rue du Loess, 67000 Strasbourg, France  Mario Ruben
<EMAIL_ADDRESS>Institute of Nanotechnology (INT), Karlsruhe Institute of
Technology, P.O. Box 3640, 76021 Karlsruhe, Germany  Institute for Quantum
Materials and Technologies (IQMT), Karlsruhe Institute of Technology, P.O. Box
3640, 76021 Karlsruhe, Germany  Centre Européen de Sciences Quantiques (CESQ),
Institut de Science et d’Ingénierie Supramoléculaires (ISIS), Université de
Strasbourg, 8, Allée Gaspard Monge, 67000 Strasbourg, France  Paul-Antoine
Hervieux<EMAIL_ADDRESS>Université de Strasbourg, CNRS, Institut de
Physique et Chimie des Matériaux de Strasbourg (IPCMS), UMR 7504, 23, Rue du
Loess, 67000 Strasbourg, France

(September 3, 2024)

###### Abstract

As qubit-based platforms face near-term technical challenges in terms of
scalability, qudits, $d$-level quantum bases of information, are being
implemented in multiple platforms as an alternative for Quantum Information
Processing (QIP). It is, therefore, crucial to study their efficiencies for
QIP compared to more traditional qubit platforms, specifically since each
additional quantum level represents an additional source of environmental
coupling. We present a comparative study of the infidelity scalings of a qudit
and $n$-qubit systems, both with identical Hilbert space dimensions and noisy
environments. The first-order response of the Average Gate Infidelity (AGI) to
the noise in the Lindblad formalism, which was found to be gate-independent,
was calculated analytically in the two systems being compared. This yielded a
critical curve $(d^{2}-1)/3\log_{2}(d)$ of the ratio of their respective
figure of merits, defined as the gate time in units of decoherence time. This
quantity indicates how time-efficient operations on these systems are relative
to decoherence timescales, and the critical curve is especially useful for
precisely benchmarking qudit platforms with smaller values of $d$. The curve
delineates regions where each system has a higher rate of increase of the AGI
than the other. This condition on gate efficiency was applied to different
existing platforms. Specific qudit platforms were found to possess gate
efficiencies competitive with state-of-the-art qubit platforms. Numerical
simulations complemented this work and allowed for discussion of the
applicability and limits of the linear response formalism.

##  I  Introduction

The paradigmatic bases of information in Quantum Information Processing (QIP)
are qubits: two-level individually addressable quantum systems. However,
several QIP platforms have recently been proposed that instead make use of
$d$-level systems, referred to as qudits [1, 2, 3, 4]. In its infancy,
classical computing did experiment with ternary, quaternary, or higher-
dimensional, bases of information, before eventually settling on the simplest
(bits), when near-zero error rates and easy scalability were attained [5].
Analogously, it could be argued that quantum computing is likely to follow a
similar trend in the long-term; as fault-tolerant platforms emerge and
technologies mature, the industry could indeed fully settle on multiqubit
systems. However, QIP research is currently not in the noise-free regime but
near it, and in order to reach significant quantum supremacy [6], increasing
the total Hilbert space dimension of the physical platform is a primordial
requirement. As such, there is a current race to increase the number $n$ of
coupled qubits ($d=2$) with superconducting platforms leading the way with
$n=51$ [7] or $n=433$ [8]. While in general the Hilbert space dimension
increases exponentially in the number of sites, the relatively slow $2^{n}$
scaling of qubits, compared to $d^{n}$ for qudits, is proving challenging,
necessitating ever-more robust systems and complex control mechanisms.

Given these current technical challenges, most qudit-based platforms that have
been physically implemented argue for near-term advantages over equivalent
multiqubit implementations. Thus, the principal motivations of qudit platforms
over qubits include: (i) the underlying physical systems having lower
decoherence rates [9], (ii) using the redundancy in additional levels for
quantum error correction [10, 11], (iii) the higher density of information per
physical system (site) [12] or (iv) more robust flying quantum memories [13,
14]. Furthermore, qudits present fundamental theoretical advantages, enabling
novel QIP capabilities offered by $\bigotimes SU(d)$ vs. $\bigotimes SU(2)$ of
qubits [15], and therefore a fault-tolerant qudit quantum computer indeed
remains conceivable. Hence, qudits provide an alternative scaling solution by
linearly increasing $d$ \- instead of scaling up $n$, the number of sites, as
well as increasing efficiency through single qudit gates operating on larger
computational subspaces [7, 3, 16, 17]. However, one of the disadvantages
raised for qudits is the larger number of error channels compared to multiple
qubits [18]. In this context, a study of the near-term viability of qudits is
needed to investigate the interplay between computational efficiency and noise
error rates in higher dimensions.

In this work, we consider one single qudit versus multiqubit systems, in the
context of near noise-free implementations. We undertake an inquiry to
determine under what conditions on the applied gates a single qudit system
does not lose more computational information than an equivalent multiqubit
system, even when the qudit system initially presents more potential error
channels. For this purpose, a standard measure to quantify the loss of
computational information, that we study, is the Average Gate Infidelities
(AGI), as defined by Nielsen [19] where the average is over the Haar measure.
The choice of the AGI ensures the calculated fidelity is not dependent on the
input state and therefore remains relevant even if the gate is applied in
later stages of a quantum algorithm. We conduct an in-depth analysis in which
we compare the computational fidelity of a single qudit and $n$-qubit systems,
both with identical dimensions of Hilbert space, undergoing arbitrary unitary
transformations, and evolving under the influence of comparable noisy
conditions. Our benchmark for successful analysis is defined by a lower first-
order response of the AGI to the environmental noise, providing a measure of
computational fidelity independent of initial states. We investigate, for
increasing values of $d$, the respective growth rates of the AGIs with respect
to error rates $\gamma$ and dimensionless gate time $\gamma t$. The latter
quantifies the gate efficiency by indicating how time-efficient operations on
these systems are relative to decoherence timescales, therefore this paper
presents a study of the first-order connection between the AGI and this time-
efficiency $\gamma t$.

In other words, this study aims to investigate how the AGI scales
proportionally with both the error rate and the speed at which gate operations
are performed, as well as the dimension $d$ of the qudit. Additionally, this
study aims to provide a benchmarking tool for if a qudit platform, for a given
($\gamma$,$t$,$d$) specification, can compensate for its greater number of
potential error channels by leveraging advantageous decoherence times and gate
speeds. Both of those quantities depend intrinsically on the physical platform
implementing the single qudit or multiqubit system, in particular their
coupling to the environment, the mathematical form of the control pulses’
Hamiltonian, and the addressing speed. In particular, given a single qudit
platform and a multiqubit platform with equivalent Hilbert space
dimensionality, and specifying a fixed pair of parameters ($\gamma$,$t$), one
could conduct a comparative analysis to determine if the qudit platform
exhibits sufficiently low decoherence and sufficiently rapid gate time to
achieve computational fidelities that are competitive with the multiqubit
platform. Or, similarly, since increasing $d$ on a single site in a given
qudit platform is a prevailing goal for some platforms [12], assuming $\gamma
t$ remains of the same order of magnitude, this study would also allow setting
theoretical upper limits on the value of $d$ in order to remain advantageous.

In sec.II a gate-independent formula is presented for the first-order response
in $\gamma t$ of the AGI to Markovian noise in the Lindblad formalism. The
first-order formalism corresponds to the quasi-errorless regime of near-term
QIP systems. Expressions for the linear dependency of the AGI in $\gamma t$
for a single qudit, multiqubits and also multiqudits are derived for an
arbitrary collapse operator. A comparison is then made between the rate of
increase of the AGI of a single qudit vs. equivalent multiple qubits.

In sec.III numerical simulations, performed with the Python package QuTiP
[20], complement and illustrate the analytical results. Discussions of the
applicability and limits of the linear response formalism for AGI are given
and the following aspects are studied: (i) the applicable range of $\gamma t$
and its dependency on the dimension of the qudit; (ii) the extent of the gate-
independence of the result; (iii) the applicability to noise models other than
pure dephasing; and finally (iv) the conditions on gate times for which either
qudits or multiple qubits are advantageous. This latter aspect is then
examined in more detail with respect to existing platforms by taking into
account their respective decoherence rate and gate operation time.

##  II  Infidelity induced by a perturbative non-unitary noise

###  II.1  Fluctuation-dissipation relation

Consider a qudit, a $d$-level quantum system whose dynamics are governed by
the Lindblad master equation [21]:

$\frac{\text{d}\rho}{\text{d}t}=-i\left[H,\rho\right]+\sum_{k=1}^{K}\gamma_{k}\left(L_{k}\rho
L_{k}^{\dagger}-\frac{1}{2}\left\\{L_{k}^{\dagger}L_{k},\rho\right\\}\right),$
(1)

where $\rho(t)$ is the density matrix of the system at time $t$, $H$ the
Hamiltonian of the system, $L_{k}$ the so-called collapse operators
characterizing the Markovian noise, and $\gamma_{k}$ the decay parameters for
each of the $K$ noise processes.

The aim is to study the effect of a single collapse operator
$\sqrt{\gamma_{1}}L_{1}=\sqrt{\gamma}L$ on short timescales and under small-
amplitude noise, i.e., $\gamma t\ll 1$. Under these assumptions, one can
consider an ansatz of the form 111We define here that $g(\gamma,t)=o(\gamma
t)\Leftrightarrow g(\gamma,t)/(\gamma t)\rightarrow 0\text{ as }\gamma\text{
and }t\rightarrow 0$. It comprises here terms of the form
$\left(\gamma^{l}t^{k}\right)_{l+k\geq 3}$, a more in-depth discussion is
available in Appendix B.

$\rho(t)=\rho^{*}-\gamma tM+o(\gamma t),$ (2)

with $\rho^{*}$ the noiseless target state, which is the solution of
$\dot{\rho}=-i\left[H,\rho\right]$ after time $t$, and $M$ the perturbation
matrix resulting from the presence of a small-amplitude noise.

One can easily see that the use of (2) in (1) leads to

$\displaystyle
M=\frac{1}{2}\left\\{L^{\dagger}L,\rho^{*}\right\\}-L\rho^{*}L^{\dagger}.$ (3)

Consider now a quantum operation bringing the initial state $\rho_{0}$ to a
final state $\rho(t)$ at time $t$. One define the fidelity $\mathcal{F}$ of
this final state relative to some target state $\rho^{*}$ [23] as

$\mathcal{F}(\rho(t),\rho^{*})\equiv\left[\Tr\left(\sqrt{\sqrt{\rho(t)}\rho^{*}\sqrt{\rho(t)}}\right)\right]^{2}.$
(4)

Subsequently, the infidelity, is then defined as

$\mathscr{E}\equiv 1-\mathcal{F}.$ (5)

Since $\rho^{*}$ is a pure state
($\rho^{*}=\ket{\varphi^{*}}\bra{\varphi^{*}}$) Eq.(4) simplifies to [23]

$\mathcal{F}(\rho(t),\rho^{*})=\Tr\left(\rho(t)\rho^{*}\right).$ (6)

Finally, substituting Eq.(2) into Eq.(6) leads to (see A.1)

$\mathscr{E}(\rho^{*})=\gamma t\Delta_{*}L+o(\gamma t).$ (7)

where $\Delta_{*}L=\langle L^{\dagger}L\rangle_{*}-\langle
L^{\dagger}\rangle_{*}\expectationvalue{L}_{*}$ with $\langle
L^{\dagger}L\rangle_{*}\equiv\Tr\left(\rho^{*}L^{\dagger}L\right)$.

###  II.2  Average Gate Fidelity

Only $\mathscr{E}(\rho^{*})$ for a specific $\rho^{*}$ was obtained
beforehand. However, is there a state-independent approach to obtaining the
infidelity of a quantum gate under small-amplitude noise? One defines the
quantum gate $U$ applied during a time duration $t$, whose resulting operation
brings all initial states $\rho_{0}$ to all corresponding
$\rho^{*}=U\rho_{0}U^{\dagger}$. There is then a definition of the average
gate fidelity of a quantum channel $\mathcal{E}$ attempting to implement the
unitary operation $U$ with a noisy environment which reads as follows [19]

$\displaystyle\bar{\mathcal{F}}(\mathcal{E},U)$ $\displaystyle=\int
d\rho_{0}\,\mathcal{F}({\rho(t)},\rho^{*})$ (8) $\displaystyle=\int
d\rho_{0}\,\left\langle
U^{\dagger}\mathcal{E}[\rho_{0}]U\right\rangle_{0}=\int
d\rho_{0}\,\left\langle\left(\mathcal{U^{\dagger}\circ
E}\right)[\rho_{0}]\right\rangle_{0},$

where the normalized integral is over the Fubini-Study measure on pure states
(sometimes called the Haar measure) [24], $\mathcal{U}^{\dagger}[\rho]\equiv
U^{\dagger}\rho U$ and $\mathcal{E}[\rho_{0}]=\rho(t)$.

Introducting $\tilde{E}_{k}=E_{k}U$ the Kraus operators such that

$\rho(t)=\left(\mathcal{U^{\dagger}\circ
E}\right)[\rho_{0}]=\sum_{k}\tilde{E}_{k}\rho_{0}\tilde{E}_{k}^{\dagger}=\sum_{k}E_{k}\rho^{*}E_{k}^{\dagger},$
(9)

the Average Gate Fidelity $\bar{\mathcal{F}}$ given in (8) can be rewritten as
[25, 26]

$\bar{\mathcal{F}}(\mathcal{E},U)=\frac{d+\sum_{k}\left|\Tr\left(\tilde{E}_{k}U^{\dagger}\right)\right|^{2}}{d(d+1)}=\frac{d+\sum_{k}\left|\Tr\left(E_{k}\right)\right|^{2}}{d(d+1)}\;.$
(10)

Using Eq.(3), one seeks the sets of Kraus operators
$\left\\{\tilde{E}_{k}\right\\}$ or $\left\\{E_{k}\right\\}$ such that

$\displaystyle\forall$
$\displaystyle\rho_{0},\qquad\sum_{k}\tilde{E}_{k}\rho_{0}\tilde{E}_{k}^{\dagger}=U\rho_{0}U^{\dagger}-\gamma
t\frac{1}{2}\left\\{L^{\dagger}L,U\rho_{0}U^{\dagger}\right\\}+\gamma
tLU\rho_{0}U^{\dagger}L^{\dagger}+o(\gamma t)$ (11a) $\displaystyle\forall$
$\displaystyle\rho^{*},\qquad\sum_{k}E_{k}\rho^{*}E_{k}^{\dagger}=\rho^{*}-\gamma
t\frac{1}{2}\left\\{L^{\dagger}L,\rho^{*}\right\\}+\gamma
tL\rho^{*}L^{\dagger}+o(\gamma t).$ (11b)

One can see that the following two sets would work up to the first order in
$\gamma t$

$\displaystyle\tilde{E}_{0}$ $\displaystyle=$
$\displaystyle\left(\mathbb{1}_{d}-\frac{\gamma
t}{2}L^{\dagger}L\right)U,\;\;\tilde{E}_{1}=\sqrt{\gamma t}LU,$ (12a)
$\displaystyle E_{0}$ $\displaystyle=$
$\displaystyle\mathbb{1}_{d}-\frac{\gamma
t}{2}L^{\dagger}L,\;\;\;\;\;\;\;\;\;E_{1}=\sqrt{\gamma t}L.$ (12b)

In order to use Eq.(12) in Eq.(8), it is necessary to calculate the traces of
the operators. Let us consider a pure dephasing channel of a qudit coupled to
a thermal environment through the operator $J_{z}$ (In general the coupling of
a qudit, or qubit, to a thermal environment can be represented by a linear
combination, or mixture, of collapse operators, though a pure dephasing
channel will typically be present and can be represented by the operator
$J_{z}$. As a toy model, let us consider a coupling term dominated by a pure
dephasing channel.) i.e. $\mathcal{E}_{z}$ with $L=J_{z}$ [27] . One obtains a
gate\- (and Hamiltonian-) independent result for the Average Gate Fidelity
which reads as (see A.2)

$\overline{\mathcal{F}}(\mathcal{E}_{z})=1-\frac{\gamma t}{12}d(d-1)+o(\gamma
t).$ (13)

In other words the Average Gate Infidelity is given by

$\overline{\mathscr{E}}(\mathcal{E}_{z})=\frac{\gamma t}{12}d(d-1)+o(\gamma
t),$ (14)

or more generally for an arbitrary quantum channel $\mathcal{X}$ with collapse
operator $L$

$\overline{\mathscr{E}}(\mathcal{X})=\frac{\gamma
t}{d+1}\left(\Tr(L^{\dagger}L)-\frac{1}{d}\left|\Tr(L)\right|^{2}\right)+o(\gamma
t).$ (15)

Note that it is always possible to find a traceless collapse operator $L$
emulating $\mathcal{X}$ [21], so the previous expression can be, in this case,
simplified as follows

$\overline{\mathscr{E}}(\mathcal{X})=\frac{\gamma
t}{d+1}\Tr(L^{\dagger}L)+o(\gamma t).$ (16)

It follows from (16) that increasing the dimension $d$ of the Hilbert space
would also increase the robustness of qudit gates to a dimension-independent
quantum channel (if $L$ is independent of $d$).

###  II.3  Qudits vs. qubits

Now let us apply the same technique as described above to another system: an
ensemble of $n$ identical dephasing qubits (Hilbert space of dimension
$d=2^{n}$). In order to compare it with the qudit analysis in the previous
subsection, each individual qubit decoheres with the same rate (has the same
type, and strength, of environmental coupling) through its spin operator
$S_{z}$ in the same way as the individual qudit (with $d=2$). Considering any
additional coupling mechanism to the environment arising from inter-qubit
interactions would only further disadvantage the multi-qubit implementation.
Our considerations then provide a best-case scenario for comparable qubits.
This yields the master equation

$\frac{\text{d}\rho}{\text{d}t}=-i\left[H,\rho\right]+\sum_{k=1}^{n}L_{k}\rho
L_{k}^{\dagger}-\frac{1}{2}\sum_{k=1}^{n}\left\\{L_{k}^{\dagger}L_{k},\rho\right\\}\;,$
(17)

with

$L_{k}=\underbrace{\mathbb{1}_{2}^{(1)}\otimes\dots\otimes\mathbb{1}_{2}^{(k-1)}}_{k-1}\otimes
S_{z}^{(k)}\otimes\underbrace{\mathbb{1}^{(k+1)}_{2}\otimes\dots\otimes\mathbb{1}^{(n)}_{2}}_{n-k}$
(18)

for $k\in\llbracket 1,n\rrbracket$.

Using the same reasoning as for dephasing qudits one obtains $n+1$ Kraus
operators to first order in $\gamma t$

$\displaystyle E_{0}=\mathbb{1}_{2^{n}}-\sum_{k=1}^{n}\frac{\gamma
t}{2}L_{k}^{\dagger}L_{k},\;\;\;\;\;E_{k}=\sqrt{\gamma t}L_{k}.$ (19)

In this case (see A.3),

$\overline{\mathscr{E}}(\mathcal{E}_{z})=\frac{\gamma
t}{4}\frac{n2^{n}}{2^{n}+1}+o(\gamma t)=\frac{\gamma
t}{4}\frac{\log_{2}(d)d}{d+1}+o(\gamma t)\;.$ (20)

Let us stress that (20) yields the same result as Abad  _et al._ [28] in the
case of identically dephasing qubits with no energy relaxation.

The analytical expressions (16) and (20) are one of the main results of this
work.

Following those last two results, two expressions for the AGI have been found:
for a single qudit, one finds an infidelity that scales as $d^{2}$:
$\overline{\mathscr{E}_{d}}(\mathcal{E}_{z})=c_{d}\gamma t$ in (14) and for an
ensemble of $n$ qubits one finds an infidelity that scales as $\log_{2}(d)$:
$\overline{\mathscr{E}_{b,n}}(\mathcal{E}_{z})=c_{b,n}\gamma t$ in (20).
Moreover, in the case of pure dephasing, one can define the $T_{2,d}$
dephasing time between two energy-adjacent levels for a qudit. It then shares
the same expression (in terms of $\gamma$) as the typical $T_{2,b}$ dephasing
time of a single qubit, namely $\dfrac{1}{T_{2}}=\dfrac{\gamma}{2}$.

The ratio between two average gate infidelities, of duration $t_{d}$ and
$t_{b,n}$ for qudit and $n$ qubits, respectively, becomes

$\frac{\overline{\mathscr{E}_{d}}(\mathcal{E}_{z})}{\overline{\mathscr{E}_{b,n}}(\mathcal{E}_{z})}=\frac{c_{d}t_{d}/T_{2,d}}{c_{b,n}t_{b,n}/T_{2,b}}.$
(21)

Therefore, in order for a single qudit ($d=2^{n}$) to outperform an ensemble
of $n$ qubits in noise-robustness, i.e., to have a smaller AGI, the following
inequality must hold true

$\frac{t_{b,n}/T_{2,b}}{t_{d}/T_{2,d}}>\frac{c_{d}}{c_{b,n}}=\frac{d^{2}-1}{3\log_{2}(d)}=\frac{4^{n}-1}{3n}.$
(22)

This expression quantifies the requirements on the figure of merit which is
the gate time in units of decoherence time $\tau_{d}=t_{d}/T_{2,d}$ relative
to $\tau_{b,n}=t_{b,n}/T_{2,b}$ in order for the qudit to yield higher-
fidelity gates. Moreover, it confirms that the infidelity of an ensemble of
$n$ identical qubits and the infidelity of a single qudit will generally not
have the same linear behaviour in $\gamma t$ even if they have the same
$T_{2}$, thus simply having $\tau_{d}<\tau_{b,n}$ is not sufficient to
guarantee a more noise-resilient qudit. Moreover, Eq.22 provides a more
precise condition on the ratio of figure of merits than a simple qualitative
result such as $\frac{d^{2}}{\log_{2}(d)}$, while maintaining the expected
$O(d^{2}/\log_{2}(d))$ behaviour as $d\rightarrow\infty$. In particular, see
A.2 for the full analytical calculations, including the derivation of the non-
trivial factor $\frac{1}{3}$.

On a side note, the previous calculations could also be applied to an ensemble
of $N$ qudits under identical pure dephasing, in which case we have

$\overline{\mathscr{E}_{d,N}}(\mathcal{E}_{z})=\frac{\gamma
t}{12}\frac{Nd^{N}}{d^{N}+1}(d^{2}-1)+o(\gamma t),$ (23)

and for $2^{n}=d^{N}$ one obtains

$\frac{t_{b,n}/T_{2,b}}{t_{d,N}/T_{2,d}}>\frac{c_{d,N}}{c_{b,n}}=\frac{d^{2}-1}{3\log_{2}(d)}.$
(24)

Let us also note that for $L$ arbitrary (23) yields

$\overline{\mathscr{E}_{d,N}}(\mathcal{X})=\gamma
tN\frac{d^{N-1}}{d^{N}+1}\left(\Tr(L^{\dagger}L)-\frac{1}{d}\left|\Tr(L)\right|^{2}\right)+o(\gamma
t).$ (25)

And if each qudit $k$ has a different set of noise parameters ($\gamma_{k}$,
$L_{k}$), an even more general formula arises :

$\overline{\mathscr{E}_{d,N}}(\mathcal{X})=\frac{d^{N-1}}{d^{N}+1}\sum_{k=1}^{N}\gamma_{k}t\left(\Tr(L_{k}^{\dagger}L_{k})-\frac{1}{d}\left|\Tr(L_{k})\right|^{2}\right)+o(\gamma_{k}t).$
(26)

This formula through its general form, can be applied to any qudits whose
physical implementation implies different collapse operators from the ones
considered in this paper.

###  II.4  Process fidelity and average of the fluctuation-dissipation
relation

One may link the fluctuation-dissipation relation obtained in (7) with the
results regarding average gate infidelities from Eq. (15),

$\overline{\mathscr{E}}(\mathcal{X})={\gamma t}\int
d\rho^{*}\mathscr{E}(\rho^{*})={\gamma t}\int d\rho^{*}\Delta_{*}L+o(\gamma
t).$ (27)

This integral over the Fubini-Study measure can formally be computed using
Weingarten calculus methods [29] (see A.4) which can be expressed as

$\int d\rho\Delta
L=\frac{1}{d+1}\Tr(L^{\dagger}L)-\frac{1}{d(d+1)}\left|\Tr(L)\right|^{2},$
(28)

leading to (15).

In contrast to this formal approach, a more physically-informed approach to
obtain the same result was proposed in the previous subsections.

Furthermore, it is possible to express all the computed average gate
infidelities as process/entanglement infidelities $\mathscr{E}^{\text{(p)}}$
making use of the relation
$D\mathscr{E}^{\text{(p)}}=(D+1)\overline{\mathscr{E}}$, with $D=d$, $2^{n}$
or $d^{{N}}$ the dimension of the Hilbert space [30]. This yields the
expression

$\mathscr{E}^{\text{(p)}}_{d,n}(\mathcal{X})=\gamma
t\frac{n}{d}\left(\Tr(L^{\dagger}L)-\frac{1}{d}\left|\Tr(L)\right|^{2}\right)+o(\gamma
t),$ (29)

which is linear in the number of subsystems $n$. Likewise we have

$\displaystyle\mathscr{E}^{\text{(p)}}_{d}(\mathcal{E}_{z})$ $\displaystyle=$
$\displaystyle\frac{\gamma t}{12}(d^{2}-1)+o(\gamma t),$ (30a)
$\displaystyle\mathscr{E}^{\text{(p)}}_{b,n}(\mathcal{E}_{z})$
$\displaystyle=$ $\displaystyle\frac{\gamma t}{4}n+o(\gamma t).$ (30b)

Note that (30b) has been verified experimentally, for example by Ozaeta and
McMahon [31].

##  III  Simulations and Discussion

All simulations were done using the Python package QuTiP [20] version 4.7,
SciPy version 1.7.3, and NumPy version 1.21.5. For a given dimension, gate,
noise parameter and collapse operator, we calculated at different times $t\,$:
(i) the propagator superoperators and (ii) the corresponding AGIs.
Subsequently, a linear fit for the function $\overline{\mathscr{E}}=1-c\gamma
t$ is performed on the AGIs to obtain the gradient $c$. In order to obtain a
statistically relevant number of gates, many random realizations were
generated using a high-performance cluster and with the Bristol[32] package in
Python and implemented by computing the corresponding hamiltonians through
gradient-ascent methods using the GRAPE algorithm [33] implementation in the
pulse optimization module of the QuTiP package.

###  III.1  Fit and deviation from the linear behaviour

This procedure was then applied to single qudits of dimension $d$ under pure
dephasing, with $H=\mathbb{0}_{d}$ and small $\gamma t\in[0,10^{-4}]$
($\gamma\sim 10^{-4}$ in some molecular magnets such as in the experiments of
Godfrin  _et al._ [16]). For example, The simulations were performed for even
dimensions $d\in\llbracket 2,22\rrbracket$. Fitting the AGIs
$\overline{\mathscr{E}_{d}}(\mathcal{E}_{z})=c_{d}\gamma t$ as a function of
$\gamma t$ yielded the slopes $c_{d}$ that are shown in Fig.1, along with
their analytical expression as a function of $d$ predicted in (14).

Figure 1: Rate of increase of
$\overline{\mathscr{E}_{d}}(\mathcal{E}_{z})=c_{d}(J_{z})\gamma t$ as a
function of qudit dimension for $H=\mathbb{0}_{d}$ and $\gamma
t\in[0,10^{-4}]$. The circled dots show the numerical results. The solid curve
presents the expected analytical result given by (14).

The same simulations were repeated for larger values of $\gamma t\in[5\times
10^{-4},1\times 10^{-2}]$ and $H=\mathbb{0}_{d}$. The AGIs were then computed
and are shown in Fig.2 alongside the linear infidelity predicted in (14).

Figure 2: Average gate infidelities as a function of $\gamma t$. The data
points show the computed values. The solid lines represent the linear
theoretical behaviour. Each colour/marker pair corresponds to a different
value of $d$.

For more insight, Fig.3 shows the relative deviation of the computed
infidelities from the expected first-order linear behaviour for a broader
range of $\gamma t$ up to $5\times 10^{-2}$.

Figure 3: Relative deviation
$1-\frac{\overline{\mathscr{E}_{d}}^{\text{sim}}}{\overline{\mathscr{E}_{d}}^{\text{th}}}$
as a function of $\gamma t$ for $H=\mathbb{0}_{d}$.
$\overline{\mathscr{E}_{d}}^{\text{sim}}$ and
$\overline{\mathscr{E}_{d}}^{\text{th}}$ were obtained from numerical
computations and (14) respectively. Each marker corresponds to a different
value of $d$.

Average gate infidelities linear in $\gamma t$ with gradients
$\frac{d(d-1)}{12}$ were expected in the case of a single qudit under pure
dephasing, according to (14). Fig.1 supports this for small values of $\gamma
t$: a least-squares fit of the computed gradients yields the expected
relationship with $1-R^{2}<10^{-5}$. Simulations for larger values of $\gamma
t$ (Fig. 2 ) highlight deviations from this linear behaviour. These originate
from $o(\gamma t)$ terms of the form $(\gamma t)^{k>1}$ (see Eq. (51)).
Moreover, for fixed values of $\gamma t$, as $d$ increases, the amplitude of
this deviation is observed to increase (Fig.3). This implies that the range of
$\gamma t$ values for which the AGI can be treated linearly diminishes with
increasing qudit dimension. Assuming a prefactor of the order $d^{4}$ for the
$(\gamma t)^{2}$ term in the AGI series expansion (as 47 hints), this provides
an estimate of the range for which the deviation from linearity is negligible:
$\gamma t\ll 1$ and $\dfrac{1}{d^{2}}$.

###  III.2  Gate dependence

While the linearity of the AGI does not scale well with $d$, Eq. (14) has
another important characteristic that deserves to be studied: the gate
independence of the AGI. This was investigated over a large number of random
gates for a given dimension $d$. Random unitary quantum gates in $U(d)$ were
sampled from the circular unitary ensemble, which represents a uniform
distribution over the unitary square matrices of dimension $d$, also known as
the Haar measure on the unitary group $U(d)$, and implemented on a qudit
through a Hamiltonian obtained by gradient-ascent methods. We decided to model
qudits as ladder systems, with one pulse per transition between adjacent
levels as considered for example in the experiments of Godfrin  _et al._ [16],
for a single-molecule magnet ($\text{TbPc}_{2}$, qudit with $d=4$), the $d-1$
pulses are then each represented by a control Hamiltonian in the interaction
picture. Other physical implementations and reference frames for the pulses
can be considered, and the deviation from linearity they cause needs to be
studied in more detail. The AGIs were then computed for $\gamma
t\in[10^{-5},10^{-3}]$ , which lie in the typical ranges observed in current
platforms as seen in Table 2 in subsection III.4,  and their rate of increase
as a function of $\gamma t$ was fitted. Fig.4 shows the statistical
distributions of the relative deviations from the linear behaviour of the
obtained rates.

Figure 4: Statistical distributions of the relative deviation from the linear
behaviour in (14) of the numerically obtained infidelity gradients $c_{d}$ for
$N_{g}=5000$ gates for $\gamma t\in[10^{-5},10^{-3}]$, as a function of the
dimension $d\in\llbracket 3,8\rrbracket$. The candlestick bar chart should be
interpreted as indicated in the upper right. The lower right inset shows the
same results for $d\in\llbracket 2,4\rrbracket$.

Considering Fig.4, the relative deviation from the linear behaviour for
different random gates seems to exceed 1‰ rarely and was not observed outside
the $<$1% range. Moreover, the range of deviations decreases as the dimension
$d$ increases. The inset highlights a noticeable irregularity for $d=2$, a
single qubit, where the relative deviation is of the order of 1%. Note that
for the $H=\mathbb{0}_{d}$ case simulated in Fig.1, this shift remained $<$1‱,
coinciding with the dashed line in Fig.4, including the case $d=2$. Beginning
at $d=2$, the gradient distributions appear broad and off-centred from the
$H=\mathbb{0}_{d}$ case. As $d$ increases further, the distributions become
progressively concentrated around $0$. The gate-dependence also arises from
$o(\gamma t)$ terms, with $\gamma t^{2}$ being dominant in the $\gamma
t^{2}||H||\gtrsim 1$ regime (see Eq. (50)). Therefore, the AGI can only be
considered gate-independent when $\gamma t\ll 1$ and $\gamma
t\ll\dfrac{1}{||H||t}$. An informative figure showing the deviation from
linearity and gate-dependence at higher values of $\gamma t$ is available in
Appendix B.1.

###  III.3  Other cases than pure dephasing

Fig.5 shows AGI rates of increase for channels different from pure dephasing
namely: $\overline{\mathscr{E}_{d}}(\mathcal{E}_{x})$,
$\overline{\mathscr{E}_{d}}(\mathcal{E}_{+})$, and
$\overline{\mathscr{E}_{d}}(\mathcal{E}_{x,y,z})$ corresponding to bit-flip,
amplitude damping and depolarizing channels respectively. The simulations were
performed again with $H=\mathbb{0}_{d}$, small $\gamma t\in[0,10^{-6}]$ and
for even dimensions $d\in\llbracket 2,22\rrbracket$.

Figure 5: Rate of increase of
$\overline{\mathscr{E}_{d}}(\mathcal{X})=c_{d}(J)\gamma t$ as a function of
the qudit dimension with $H=\mathbb{0}_{d}$ and $\gamma t\in[0,10^{-4}]$ . The
markers show the numerical results. The solid curves represent the expected
linear responses. Each marker/colour pair corresponds to a different error
channel $\mathcal{X}$, with collapse operators $J$ specified in the legend.

Consider ${\cal R}$, the unitary transformation representing a change of
basis, such as a $3D$ real-space rotation. The average gate fidelity defined
in (8) is invariant under the transformation $\rho\rightarrow{\cal
R}^{\dagger}\rho{\cal R}$. This is supported by a comparison of the results
for $L=J_{z}$ and $L=J_{x}$ in Fig.1 and Fig.5, respectively, since the two
gradients appear to share the same dependency in $d$. Moreover, let
$\\{l_{k}\\}$ be an ensemble of traceless collapse operators with
corresponding error channels $\\{e_{k}\\}$, then define $L=\sum_{k}l_{k}$ and
associated error channel $\mathcal{E}$. From (10) and (12b), as long as
$\Tr\left(l_{k}^{\dagger}l_{j}\right)=0,\,\forall j\neq k$ then
$\overline{\mathscr{E}}(\mathcal{E})=\sum_{k}\overline{\mathscr{E}}(e_{k})$.
Fig. 5 again supports such behaviour since simulations with the collapse
operator $L=J_{+}{\equiv}J_{x}+iJ_{y}$ yield gradients twice as large as, and
$L=J_{x}+J_{y}+J_{z}$ yield gradients three times as large as, the $L=J_{z}$
case.

###  III.4  A single qudit vs an ensemble of qubits

An ensemble of $n$ qubits were simulated under identical pure dephasing, with
$H=\mathbb{0}_{2^{n}}$ and small $\gamma t\in[0,10^{-4}]$. The simulations
were performed for $n\in\llbracket 1,7\rrbracket$. Fitting the AGI
$\overline{{\mathscr{E}_{b,n}}(\mathcal{E}_{z})}=c_{b,n}\gamma t$ as a
function of $\gamma t$ yielded the slopes $c_{b,n}$ that are shown in Fig.6,
along with their analytical expression as a function of $d$ given in (20).

Figure 6: Rate of increase of
$\overline{\mathscr{E}_{b,n}}(\mathcal{E}_{z})=c_{b,n}(J_{z})\gamma t$ as a
function of qudit dimension $d=2^{n}$ with $H=\mathbb{0}_{2^{n}}$ and $\gamma
t\in[0,10^{-4}]$. The circled dots show the numerical results. The solid curve
presents the expected theoretical result. The dashed line shows
$\mathscr{E}^{\text{(p)}}_{b,n}(\mathcal{E}_{z})$ given in (30b) which is
linear in $n={\mathrm{log}}_{2}(d)$.

The same simulations were performed on a single qudit with dimension $d=2^{n}$
and Fig.7 shows the ratios $\frac{c_{d}}{c_{b,n}}$ for $n\in\llbracket
1,6\rrbracket$ as well as the theoretical curve provided by Eq. (22) on which
the points should be falling. According to the same Eq. (22), this curve also
highlights the critical values of $\tau_{b}/\tau_{d}$, denoting the figure of
merit $\tau_{k}=t_{k}/T_{2,k}=\gamma_{k}t_{k}/2$, with respect to qudit/qubits
advantage in terms of the rate of increase of the AGI.

Figure 7: Potential range for $\tau_{b}/\tau_{d}$. The rounded circles show
the numerical values obtained for $c_{d}/c_{b,n}$. The solid curve comes from
(22) and highlights the theoretical critical values of ${T_{2,d}}/{T_{2,b}}$.

The AGI gradients obtained for an ensemble of $n$ qubits under identical pure
dephasing were expected to follow a $\frac{d\log_{2}(d)}{4(d+1)}$ relationship
as a function of $d$ (20). Fig.6 justifies this for small values of $\gamma t$
with the least-square fit now yielding $1-R^{2}<10^{-7}$. Finally, Fig. 7
provides quantitative data for the ratio of the decoherence times of a single
qudit vs an ensemble of qubits. Some values of interest are summarised in
Table 1. For example, in order for a qu-$8$-it (qudit with $d=8$)  to present
a computational fidelity advantage over $3$ qubits for a fixed gate time, the
qudit platform needs a coherence time at least $7$ times longer than the
multiqubit platform. Note that an intuitive scaling such as
$\frac{d^{2}}{\log_{2}(d)}$ would indicate a much more demanding constraint of
$21.5$.

Number $n$ of qubits | $1$ | $2$ | $3$ | $6$
---|---|---|---|---
Dimension $d$ of the qudit | $2$ | $4$ | $8$ | $64$
Critical $\tau_{b}/\tau_{d}$ | $1$ | $2.5$ | $7$ | $227.5$
Table 1: Ratios of gate times in units of decoherence times between qubits and
qudits for specific values of $n$ and $d$. $\tau_{b}/\tau_{d}$ needs to be
larger than the critical values in order for a single qudit to be advantageous
vs an equivalent ensemble of $n$ qubits.

| $\;\,d$ | $\;\,n$ | $T_{2}$ | $t_{n}$ | $\tau_{n}$ | ref.
---|---|---|---|---|---|---
| $\;2$ | $\;2$ |  $\sim$ 10 µs | 60 ns | $\sim 10^{-2}$ |  [34] (2020a)
| $\;2$ | $\;2$ |  $\sim 2$ µs |  $51$ ns | $\sim 10^{-2}$ |  [35] (2020)
| $\;2$ | $\;17$ |  $\sim$ 30 µs |  $\sim$ 100 ns | $\sim 10^{-3}$ |  [36] (2022)
qubits | $\;2$ | $\;24$ |  $\sim 100$ ms |  $\sim 200$ µs | $\sim 10^{-3}$ |  [37] (2021)
| $\;4$ | $\;1$ | 0.32 ms |  $\sim$ 100 ns | $\sim 10^{-4}$ |  [1] (2018)
| $\;4$ | $\;1$ |  $\sim 100$ µs |  $\sim 150$ ns | $\sim 10^{-3}$ |  [38, 4] (’20,’23)
| $\;3$ | $\;2$ |  $\sim 100$ ms |  $\sim 100$ µs | $\sim 10^{-3}$ |  [3] (2022)
| $\;4$ | $\;2$ |  $\infty$222considered unlimited by the source authors | – | 0 |  [2] (2022)
qudits | $\;52$ | $\;1$ | – |  $\sim 100$ ns333no universal gates for the moment, only specific quantum operations implemented |  – |  [39] (2020)

superconducting qubits, Rydberg atoms, trapped ions
molecular nuclear spins, photonic qudits

Table 2: Decoherence times ($T_{2}$) and gate times ($t_{n}$) of different
qubit/qudit platforms. $d$ and $n$ are the maximum dimension and number of
qudits an operation was applied to, while $\tau_{n}=t_{n}/T_{2}$ is the figure
of merit.

From Table 2, state-of-the-art single qudit platforms, such as trapped ions
[3], present coherence times of the order of $100$ms for a single qu-$7$-it,
orders of magnitude longer than superconducting qubits [31, 40, 41, 42].
Trapped ions present $\gamma t\approx 10^{-3}$, while $\gamma t\approx
10^{-2}$ for superconducting qubits; this ratio of 10 would allow qudits with
$d\lesssim 10$ to still be advantageous i.e., according to (22), the single
qu-$7$-it would still maintain a higher average gate fidelity over one gate
acting on the whole Hilbert Space than the multiqubit platform. Another
comparison with superconducting qubits could be molecular nuclear qudits,
where some proposals put $\gamma t\approx 10^{-4}$ [1], and whose coherence
times are $\sim 6-7$ times larger than the superconducting qubit case. With
figure of merits $\tau$ $\sim 100$ larger than superconduction platforms,
single molecular nuclear qudits with $d\lesssim 40$ are still advantageous
over equivalent superconducting qubits, i.e. $n\sim 5$. Such high-$d$ qudit
platforms can still be conceivable, given that some specific quantum
operations on $d=52$ have already successfully been implemented on, for
example, Rydberg atoms [39]. However, it remains to be seen if universal
quantum gate generation will become easily achievable in practice with such
high $d$.

Finally, one can compare (22) and (24) to discuss conditions on $N$ qudits
outperforming $N\log_{2}(d)$ qubits. From this, if a single qudit outperforms
$\log_{2}(d)$ qubits, the advantage remains conserved as long as the
multiqudit gate time scales slower from $1$ qudit to $N$ qudits than the
multiqubit from $\log_{2}(d)$ to $N\log_{2}(d)$ qubits.

##  IV  Conclusion

Given the rapid development of quantum computing platforms with very different
physical properties, such as decoherence time or Hilbert space dimension (see
Table II), there is a growing need for detailed elaboration of the tradeoffs
between their information density and noise error rates. By combining
analytical results and numerical simulations, we have performed a comparative
study of gate efficiency for systems composed of sets of qubits or qudits. A
fluctuation-dissipation-like relation for the gate infidelity of an operation
on a pure state was derived. We then put forward a physically-informed method
to obtain the first-order effect of Markovian noise on the average gate
infidelity (AGI). A connection was made between the latter and the first gate-
independent result. The rate of increase of the AGI of a single qudit vs
equivalent multiple qubits under pure dephasing was compared. This yielded a
critical curve of the ratio of their respective gate times in units of
decoherence time, a quantity indicating how time-efficient operations on a
particular system are. Values on either side of the curve specify which of the
two systems had a higher rate of increase of the AGI. To compete in terms of
gate fidelity, as the dimension increases, the efficiency of qudit gates must
not simply always be larger than the multiqubit one by a factor
$O(d^{2}/\log_{2}(d))$, but precisely by a factor
$\frac{d^{2}-1}{3\log_{2}(d)}$, which makes a significant difference for lower
values of $d$ for which it provides less demanding constraints. Additionally,
analytical expressions of linear response for arbitrary collapse operators and
a general multiqudit system were presented (see (16) and (25)). They may be
useful to those working in the field of quantum computing e.g. to, as
mentioned in Sec.I, benchmark qudit platforms either in terms of maximal
practical $d$ or in terms of conditions on the figure of merit to compensate
for the greater noise scaling, in comparison with current state-of-the-art
multiqubit platforms.

Numerical simulations contributed to the discussion on the validity and limits
of the linear response assumption. This further restricted the ranges of
possible $\gamma t\ll 1$ accounting for qudit dimension, gate, and noise type.
For example, the larger the dimension, the lower the relative gate-dependent
response. Finally, after simulations supported the analytical critical curve,
different current platforms were studied with respect to this condition on
gate time efficiency. Given equivalent Hilbert space dimensions, viable qudit
platforms 444 leveraging advantageous decoherence times and gate speeds to
compensate for the higher rate of increase in AGI. capable of outperforming
equivalent state-of-the-art multiqubit ones in gate fidelity have been found
for pure dephasing. Moreover, this performance could be extended to qudits
with $d$ as large as $\sim 40$ in the case of nuclear molecular magnets, for
example. Some multiqubit platforms still outperform any existing qudit
platform regarding scalability in the number of subsystems. However, it is
conceivable that some scalable qudit platforms continue to outperform
equivalent multiqubit systems in terms of attainable fidelity. Further study
of how multiqudit and multiqubit gate times scale with the number of
subsystems is needed. Moreover, this study was limited to first-order noise
responses. However, using the notation ”qu-$j$-it” for a qudit of dimension
$d=j$, through carefully chosen quantum error correction schemes, it was
recently demonstrated it is possible to entirely remove the first-order
response of logical qu-$k$-its embedded in physical qu-$d$-its ($k<d$) through
carefully chosen encodings [10, 11]. An additional study of higher-order
responses of logical qudits vs. physical qubits would therefore also be
required to assess the viability of these logical error-resilient qudits. This
could elucidate if the presence of single qudit advantage, as quantified by
this paper, is robust to system scaling and if qudits will remain useful
beyond the NISQ era.

###### Acknowledgements.

This work was funded by the French National Research Agency (ANR) through the
Programme d’Investissement d’Avenir under contract ANR-11-LABX-0058_NIE and
ANR-17-EURE-0024 within the Investissement d’Avenir program
ANR-10-IDEX-0002-02. J-G.H. also acknowledges QUSTEC funding from the European
Union’s Horizon 2020 research and innovation program under the Marie
Skłodowska-Curie Grant Agreement No. 847471. The authors would like to
acknowledge the High Performance Computing Center of the University of
Strasbourg for supporting this work by providing scientific support and access
to computing resources. Part of the computing resources were funded by the
Equipex Equip@Meso project (Programme Investissements d’Avenir) and the CPER
Alsacalcul/Big Data.

##  Appendix A Complementary Derivations

###  A.1 Fluctuation-dissipation relation

Substituting Eq.(3) and Eq.(2) into Eq.(6) and Eq.(5) leads to

$\mathscr{E}(\rho^{*})=\gamma
t\left(\frac{1}{2}\Tr\left(\rho^{*}\left\\{L^{\dagger}L,\rho^{*}\right\\}\right)-\Tr\left(\rho^{*}L\rho^{*}L^{\dagger}\right)\right)+o(\gamma
t).$ (31)

The trace being invariant by cyclic permutations and ${\rho^{*}}^{2}=\rho^{*}$
leads to the simplication

$\frac{1}{2}\Tr\left(\rho^{*}\left\\{L^{\dagger}L,\rho^{*}\right\\}\right)=\Tr\left(\rho^{*}L^{\dagger}L\right)\equiv\langle
L^{\dagger}L\rangle_{*}\;.$ (32)

Moreover

$\displaystyle\Tr\left(\rho^{*}L\rho^{*}L^{\dagger}\right)$
$\displaystyle=\Tr\left(\ket{\varphi^{*}}\expectationvalue{L}{\varphi^{*}}\bra{\varphi^{*}}L^{\dagger}\right)$
(33)
$\displaystyle=\expectationvalue{L}_{*}\Tr\left(\rho^{*}L^{\dagger}\right)$
(34) $\displaystyle=\langle L^{\dagger}\rangle_{*}\expectationvalue{L}_{*}\;.$
(35)

Accounting for the above results, one finally obtains

$\mathscr{E}(\rho^{*})=\gamma t\left(\langle L^{\dagger}L\rangle_{*}-\langle
L^{\dagger}\rangle_{*}\expectationvalue{L}_{*}\right)+o(\gamma t),$ (36)

which can be rewritten as Eq. (7).

###  A.2 Average Gate Infidelity for the Pure Dephasing Channel of one qudit

In a trivial way we have $\Tr(E_{1})\propto\Tr(J_{z})=0$, and

$\displaystyle\Tr(E_{0})=d-\frac{\gamma t}{2}\Tr(J_{z}^{2})$
$\displaystyle=d-\frac{\gamma
t}{2}\sum_{k=0}^{d-1}\left(\frac{d-1-2k}{2}\right)^{2}$
$\displaystyle=d-\frac{\gamma
t}{8}\left[d(d-1)^{2}-4(d-1)\sum_{k=0}^{d-1}k+4\sum_{k=0}^{d-1}k^{2}\right]$
$\displaystyle=d-\frac{\gamma
t}{8}\left[d(d-1)^{2}-2d(d-1)^{2}+4\frac{d(d-1)(2d-1)}{6}\right]$
$\displaystyle=d-\frac{\gamma t}{24}d(d^{2}-1),$

which results in

$\left|\Tr(E_{0})\right|^{2}=d^{2}-\frac{\gamma t}{12}d^{2}(d^{2}-1)+o(\gamma
t)\;.$ (37)

Therefore

$\overline{\mathcal{F}}(\mathcal{E}_{z})=\frac{d+d^{2}-\frac{\gamma
t}{12}d^{2}(d^{2}-1)}{d(d+1)}+o(\gamma t)\;,$ (38)

which leads to the simplified expression (13).

###  A.3 Average Gate Infidelity for the pure dephasing channel of $n$ qubits

Since $\Tr(E_{k})=0\;\forall k\neq 0$, only $\Tr(E_{0})$ is left and is given
by

$\displaystyle\Tr(E_{0})$ $\displaystyle=2^{n}-\frac{\gamma
t}{2}\sum_{k=1}^{n}\left[\Tr(\left.S^{2}_{z}\right.^{(k)})\prod_{j\neq
k}\Tr(\left.\mathbb{1}_{2}^{2}\right.^{(j)})\right]$ (39)
$\displaystyle=2^{n}-n\frac{\gamma t}{8}2^{n},$

leading to

$\left|\Tr(E_{0})\right|^{2}=2^{2n}-\frac{\gamma t}{4}n2^{2n}+o(\gamma t)\;,$
(40)

which allows to obtain (20) using (10).

###  A.4 Average over the Fubini-Study measure of the uncertainty of $L$

First, rewriting $\int\text{d}\rho\Tr(\rho M^{\dagger}M)$ yields

$\int dU\,\Tr(U\rho U^{\dagger}M^{\dagger}M)=\Tr\left[\left(\int dU\,U\rho
U^{\dagger}\right)M^{\dagger}M\right]$

, where the integration is performed over the uniform Haar measure in the
space of unitaries. Using the identity

$\int dU\,UXU^{\dagger}=\frac{\operatorname{Tr}(X)I}{d}$

valid for any linear operator $X$, for the special case of $\rho$ pure one
obtains

$\int d\rho\,\,\text{Tr}(\rho
M^{\dagger}M)=\frac{1}{d}\text{Tr}(M^{\dagger}M).$ (41)

Now, rewriting $\int\text{d}\rho\Tr(\rho M^{\dagger}M)$ yields

$\displaystyle\int$ $\displaystyle dU\Tr(U\rho U^{\dagger}M)^{2}=$
$\displaystyle\sum_{\begin{subarray}{c}i,j,k,l\\\ m,n,p,q\end{subarray}}$
$\displaystyle\int$ $\displaystyle
dU\,U_{ij}U_{k\ell}\bar{U}_{mn}\bar{U}_{pq}\rho_{jn}\rho_{\ell
q}M_{mi}M_{pk}.$

Collins  _et al._ [29] provides formulae to integrate polynomials of unitary
matrices

$\displaystyle\int_{U_{d}}dUU_{ij}U_{k\ell}\bar{U}_{mn}\bar{U}_{pq}=$
$\displaystyle\frac{1}{d^{2}-1}\left[(\delta_{im}\delta_{jn}\delta_{kp}\delta_{\ell
q}+\delta_{ip}\delta_{jq}\delta_{km}\delta_{\ell
n}){\color[rgb]{1,1,1}\frac{1}{d}}\right.$
$\displaystyle{\color[rgb]{1,1,1}\frac{1}{d}}-\left.\frac{1}{d}(\delta_{im}\delta_{jq}\delta_{kp}\delta_{\ell
n}+\delta_{ip}\delta_{jn}\delta_{km}\delta_{\ell q})\right]\;,$ (42)

which contracting the indices gives

$\int d\rho\,\,\left|\text{Tr}(\rho
M)\right|^{2}=\frac{1}{d(d+1)}\left(\text{Tr}(M^{\dagger}M)+|\text{Tr}(M)|^{2}\right).$
(43)

Finally subtracting (A.4) from (41) leads to (28).

##  Appendix B Higher-order effects of the collapse operators

###  B.1 Complementary figure : deviation from linearity and gate dependence

Figure 8: Simulated AGIs of $N_{g}=4400$ gates for $d=4,8$ in solid lines. The
dashed lines correspond to the expected linear behaviour at small $\gamma t$.

Fig.8 allows for observation of (i) the deviation from linear behaviour as
$\gamma t$ increases, (ii) for higher $d$, this deviation becomes noticeable
for smaller values of $\gamma t$ and, (iii) the infidelity becomes
increasingly gate-dependent as $\gamma t$ increases.

###  B.2 Full expansion of the density matrix

The density matrix $\rho(t)$ can be decomposed as

$\rho(t)=\rho^{*}+{\sum_{l=1}\sum_{k=1}}\rho_{lk}\gamma^{l}t^{k}\;.$ (44)

Substituting Eq. (44) in (1) yields the following results:

  * •

$\rho_{11}=\mathcal{D}[\rho^{*}]:=\sum_{k}L_{k}\rho^{*}L_{k}^{\dagger}-\frac{1}{2}\\{L_{k}^{\dagger}L_{k},\rho^{*}\\}$
.

  * •

for $l\geq 2$, $k=1$, $\rho_{l,1}=0$ .

  * •

for $l=1$, $k\geq 2$,

$k\rho_{1k}=-i[H,\rho_{1(k-1)}]-\dot{\rho}_{1(k-1)}\;.$ (45)

  * •

$\forall l,k\geq 2$,

$k\rho_{lk}=-i[H,\rho_{l(k-1)}]-\dot{\rho}_{l(k-1)}+\mathcal{D}[\rho_{(l-1)(k-1)}]\;.$
(46)

It can be linked to (3) by noticing that $M=-\rho_{11}$.

Moreover, for $k=l=2$ we obtain

$\rho_{22}=\mathcal{D}[\rho_{11}]=\mathcal{D}\left[\mathcal{D}[\rho^{*}]\right]\;.$
(47)

Finally we have

$\rho_{12}=\frac{i}{2}\left(\mathcal{D}\left[[H,\mathcal{\rho^{*}}]\right]-[H,\mathcal{\rho^{*}}]\right),$
(48)

and

$\rho_{13}=-\frac{i}{3}[H,\rho_{12}]-\frac{\dot{\rho}_{12}}{3}.$ (49)

This gives us the following expansion

$\rho(t)=\rho^{*}+\gamma t\rho_{11}+\gamma t^{2}\rho_{12}+\gamma
t^{3}\rho_{13}+(\gamma t)^{2}\rho_{22}+\epsilon,$ (50)

with $\epsilon=O\left(\gamma^{l}t^{k}\right)_{l+k\geq 5}$.

Interestingly, it can be proven by induction that if $H=\mathbb{0}_{d}$ then,

$\rho(t)=\rho^{*}+\sum_{k}(\gamma t)^{k}\rho_{kk},$ (51)

with $\rho_{kk}=\frac{1}{k!}\mathcal{D}^{(k)}[\rho^{*}]$.

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  * Note [2] Leveraging advantageous decoherence times and gate speeds to compensate for the higher rate of increase in AGI.