# Shortcuts to Adiabaticity for Fast Qubit Readout in Circuit Quantum
Electrodynamics

F. A. Cárdenas-López  International Center of Quantum Artificial Intelligence
for Science and Technology (QuArtist) and
Physics Department, Shanghai University, 200444 Shanghai, China  Xi Chen
<EMAIL_ADDRESS>Department of Physical Chemistry, University of the
Basque Country UPV/EHU, Bilbao, Spain  EHU Quantum Center, University of the
Basque Country UPV/EHU, 48940 Leioa, Spain

###### Abstract

We propose how to engineer the longitudinal coupling to accelerate the
measurement of a qubit longitudinally coupled to a cavity, motivated by the
concept of shortcuts to adiabaticity. Different modulations are inversely
designed from inverse engineering, counter-diabatic driving and genetic
algorithm, for achieving optimally large values of the signal-to-noise ratio
(SNR) at nanosecond scale. By comparison, we demonstrate that our protocols
outperform the usual periodic modulations on the pointer state separation and
SNR. Finally, we show a possible implementation considering state-of-the-art
circuit quantum electrodynamics architecture, estimating the minimal time
allowed for the measurement process.

##  I Introduction

Retrieving information from a quantum system is at the heart of quantum
information processing applications, in which an accurate and reliable quantum
measurement is requisite. The most common measurement strategy is dispersive
readout consisting in coupling an auxiliary system whose observables depend on
the system state. Specifically, in superconducting quantum circuit (SC) or
circuit quantum electrodynamics (cQED) [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12,
13, 14, 15], the qubit measurement is carried out through an auxiliary
oscillator whose frequency relies on the qubit state [16, 17, 18, 19, 20].
However, dispersive readout has the drawback that its performance is limited
by the detuning between the qubit and the oscillator which bounds the number
of thermal photons and induces losses via Purcell effect as well [21, 22, 23].
A way to circumvent this limitation relies on engineer the longitudinal
interaction between the quantum systems [24, 25, 26, 27] yielding the non-
demolition quantum measurement that is faster and more robust than the
previous dispersive approach. Moreover, the parametric modulation of the
external magnetic flux on a cavity-qubit system leads to rapid and
unconditional reset mechanism [28].

In the past decade, shortcuts to adiabaticity (STA) [29] have experienced a
huge development, with the various applications of quantum computing and more
generally quantum technologies [30]. The methods of STA provide efficient
control of quantum systems, by accelerating the slow adiabatic processes, and
overcoming obstacles from systematic errors or environmental noise, see review
[30]. Recently, STA have been generalized to complex open systems [31, 32],
which offer the opportunity for designing a counter-diabatic pulse, to
accelerate a dissipative process in cQED [33].

In this article, we propose STA method for elaborating the modulation of
longitudinal coupling between a two-level system with a cavity mode to
accelerate the qubit measurement. By using inverse engineering, we obtain
large values for the cavity pointer state separation and the signal-to-noise
ratio (SNR) on a short timescale. Besides, the SNR is exponentially enhanced
when the cavity is prepared in a squeezed state. For completeness, we further
discuss the measurement process speed-up by counter-diabatic driving, and
their possible implementations. Moreover, we use genetic algorithm, a search-
based optimization technique, to find optimal or near-optimal modulation.
Finally, a feasible experimental model regarding state-of-the-art cQED
architecture is considered and the minimal time for the measurement process is
estimated with the bound of coupling strength.

##  II Longitudinal cavity-qubit interaction

We consider an LC oscillator of frequency $\omega_{r}$ longitudinally coupled
to a two-level system of frequency $\omega_{q}$ with time-dependent coupling
strength $g_{z}(t)$ described through the Hamiltonian [26] ($\hbar=1$)

$\displaystyle\mathcal{H}$ $\displaystyle=$
$\displaystyle\frac{\omega_{q}}{2}\sigma^{z}+\omega_{r}\hat{a}^{{\dagger}}\hat{a}+g_{z}(t)\sigma^{z}(\hat{a}^{{\dagger}}+\hat{a}).$
(1)

Here $\sigma^{z}$ is the $z-$component Pauli matrix describing the two-level
system, and $a^{{\dagger}}$ ($a$) is the creation (annihilation) operator of
the LC oscillator. This Hamiltonian corresponds to a state-dependent displaced
oscillator. When the cavity acts as a pointer state, we can perform high-
fidelity quantum measurement on the qubit state since the cavity state
displaces upwards or downwards on its phase state according to the qubit
state. Furthermore, as the longitudinal interaction commutes with the free
terms of the Hamiltonian such process is a quantum nondemolition measurement
(QND) [26].

We aim to engineer $g_{z}(t)$ inversely to accelerate the QND measurement
process. For doing so, we propose a solution for the Schödinger equation of
$\mathcal{H}$ as
$|\Psi(x,t)\rangle=e^{-iE_{\rm{LC}}t/\hbar}\mathcal{V}(t)|\varphi(x,t)\rangle|\xi\rangle$,
where $E_{\rm{LC}}=\omega_{r}(n+1/2)$ and $|\varphi(x,t)\rangle$ correspond to
the eigenenergies and eigenfunctions of the LC oscillator
$\mathcal{H}_{\rm{LC}}=\omega_{r}\hat{a}^{{\dagger}}\hat{a}$. Besides,
$|\xi\rangle$ describes the qubit state. As a consequence,
$\mathcal{V}(t)=e^{i\theta(t)}e^{-i\dot{g}_{c}(t)\sigma^{z}(a^{{\dagger}}+a)/\omega_{r}^{2}}e^{-g_{c}(t)\sigma^{z}(a^{{\dagger}}-a)/\omega_{r}}$
is a unitary transformation that eliminates the longitudinal coupling [34],
where $\theta(t)=-\int_{0}^{t}\mathcal{L}_{g}(t^{\prime})dt^{\prime}$
corresponds to a phase relating the coupling strength $g_{z}(t)$ with the
auxiliary variable $g_{c}(t)$ through the Lagrangian

$\displaystyle\mathcal{L}_{g}(t)$ $\displaystyle=$
$\displaystyle\frac{\dot{g}^{2}_{c}(t)}{\omega_{r}^{3}}-\frac{g^{2}_{c}(t)}{\omega_{r}}+\frac{2g_{c}(t)g_{z}(t)}{\omega_{r}}.$
(2)

To guarantee that $|\Psi(x,t)\rangle$ corresponds to the exact solution of the
time-dependent Schödinger equation, the classical variables must obey the
equation of motion, (see Appendix A for the detailed calculation)

$\displaystyle\ddot{g}_{c}(t)+\omega_{r}^{2}[g_{c}(t)-g_{z}(t)]=0,$ (3)

which is nothing but the Euler-Lagrange equation from Eq. (2). In what follow,
we use Eq. (3) to engineer inversely the modulation of suitable coupling
strength $g_{z}(t)$, in order to accelerate measurement process.

Figure 1: Coupling strength modulation $g_{c}(t)$ and $g_{z}(t)$ calculated
with the reserve engineering approach. Blue and orange solid line stand for
$g_{c}(t)$ for the polynomial and trigonometric ansatzes, respectively,
whereas blue dots and orange squares corresponds to $g_{z}(t)$. We have
performed the simulation with the parameters $\kappa/2\pi=1~{}{\rm{MHz}}$,
$g_{z0}/2\pi=21~{}{\rm{MHz}}$, $\omega_{r}/2\pi=6.6~{}{\rm{GHz}}$ for a
designed time $t_{f}=\pi/(100\kappa)$ [26].

Sharing the concept of STA [29], we require that $g_{c}(t)$ fulfill the
following boundary conditions [35]:

$\displaystyle g_{c}(0)=0;~{}g_{c}(t_{f})=0,~{}~{}~{}~{}~{}~{}$ (4a)
$\displaystyle\dot{g}_{c}(0)=\ddot{g}_{c}(0)=\dot{g}_{c}(t_{f})=\ddot{g}_{c}(t_{f})=0.$
(4b)

The flexibility left in the inverse-engineering approach permits us to add a
constrain over the final cavity displacement i.e.,

$\displaystyle\int_{0}^{t_{f}}g_{c}(s)ds=g_{z0}\pi/(2\kappa).$ (5)

There exist a vast number of function fulfilling the above criteria, we
suggest a polynomial ansätze of the form
$g_{c}(t)=\sum_{\ell=0}^{6}b_{\ell}t^{\ell}$ that leads to

$\displaystyle g_{c}(t)$ $\displaystyle=$ $\displaystyle-\frac{70\pi
g_{z0}t^{3}(t-t_{f})^{3}}{\kappa t_{f}^{7}}.$ (6)

However, this is not the unique solution for Eq. (3) with the initial and
final boundary conditions, see Eqs. (4a), (4b) and (5). For the generality, we
assume an alternative trigonometric ansätze of the form
$g_{c}(t)=\sum^{6}_{m}\mathcal{A}_{m}(t)\sin(m\pi t/t_{f})$, resulting in

$\displaystyle g_{c}(t)=\frac{3g_{z0}\pi^{2}}{2\kappa
t_{f}}\sin\left(\frac{\pi t}{2t_{f}}\right)\cos^{5}\left(\frac{\pi
t}{2t_{f}}\right).$ (7)

Consequently, we obtain $g_{z}(t)$ by substituting $g_{c}(t)$ into Eq. (3), as
illustrated in Fig. 1, where the experimental parameters in an realistic cQED
[26] yield $g_{z}(t)\simeq g_{c}(t)$ for large frequency $\omega_{r}$ of LC
oscillator, and the measuring time $t_{f}$ can be shortened if the larger
coupling strength is allowed.

Figure 2: (a) Relative pointer state separation for the output cavity field
$\hat{a}_{{\rm{out}}}$ as function of $t/t_{f}$ for the inverse engineering
with polynomial (orange dotted) and trigonometric (red dashed) ansatzes for
longitudinal coupling strength. We compare with the case with the conventional
sinusoidal modulation (blue solid). Function $F(\kappa,t_{f})$ as a function
of the decay rate and the final time $t_{f}$ for both polynomial (b) and
trigonometric (c) modulations. All parameters are the same as those in Fig. 2.

##  III Qubit readout

Now, we consider the evolution of the pointer state with the modulation of
$g_{c}(t)$ given in Eqs. (6) and (7). The dynamical equation of the cavity
field regarding losses [36] reads
$\dot{\hat{a}}=ig_{c}(t)\sigma^{z}-\kappa\hat{a}/2-\sqrt{\kappa}\hat{a}_{\rm{in}}$,
where $\kappa$ is the decay rate of the LC oscillator, and $\hat{a}_{\rm{in}}$
is the input cavity operator taking into account the effect of an additional
subsystem (measurement apparatus). By assuming both $\hat{a}_{\rm{in}}$ and
$\hat{a}$ are in their vacuum state, the solution of this equation yields

$\displaystyle\langle\hat{a}(t)\rangle=-i\langle\sigma^{z}\rangle e^{-\kappa
t/2}\int_{0}^{t}g_{c}(s)e^{\kappa s/2}ds.$ (8)

Depending on the value of $\langle\sigma^{z}\rangle$, the field will be
displaced upward or downward on the phase space. To quantify this separation,
we further define the relative separation
$d=|\langle\hat{a}_{{\rm{out}},e}(t)\rangle-\langle\hat{a}_{{\rm{out}},g}(t)\rangle|$,
where
$\langle\hat{a}_{{\rm{out}},\ell}(t)\rangle=\sqrt{\kappa}\langle\hat{a}(t)\rangle$
is the output field averaged with respect to the qubit states
$|\ell\rangle=\\{|e\rangle,|g\rangle\\}$.

Figure 2 shows the dependence of relative separation $d$ on $t/t_{f}$. Fig. 2
(a) demonstrates that the modulations designed from STA produce the pointer
state separation that is ten times larger than the normal sinusoidal
modulation in Ref. [26] at timescale $t_{f}=\pi/(100~{}\kappa)\approx
30~{}\rm{ns}$, thus yielding a fast qubit readout. For completeness, we
further compare in Fig. 2 (b) the performance of our designed modulations with
polynomial and trigonometric ansätzes. Though at shorter time trigonometric
modulation performs better than polynomial one, at final time $t_{f}$ both
modulations reach the same values, due to the fixed boundary condition (5). To
compare their performance for different $t_{f}$ we plot the quantity
$F(\kappa,t_{f})\equiv e^{-\kappa t_{f}/2}\int_{0}^{t_{f}}g_{c}(s)e^{\kappa
s/2}ds$ for different $\kappa$ and $t_{f}$. Notice that $F(\kappa,t_{f})$
gives us the maximal cavity displacement at fixed $\kappa$ and $t_{f}$,
respectively. From Fig. 2 (b) and Fig. 2 (c), we observe both modulations
start to behave similarly, as long as $t$ approaches $t_{f}$. Actually, we can
also check the fluctuation on the cavity in the final readout. Based on it, we
cannot attribute the enhanced performance to different engineered pulses,
which only depends on the final boundary conditions.

Figure 3:  Signal-to-noise ratio (SNR) as function of the dimensionless
measuring time $\tau/t_{f}$ (a) regarding both sinusoidal modulation (blue
solid) and the inverse-engineered protocol with polynomial (orange dotted) and
trigonometric (red dashed) ansatzes. In (b) we also plot the SNR but now
considering the cavity with a single-mode squeezed state with squeezing
parameter $\theta=\pi/4$ and $r=20~{}(\rm{dB})\equiv e^{2r}=100$ and homodyne
angle $\phi=\pi/4$ for a time $t_{f}=30~{}\rm{ns}$. The other system
parameters for the calculation are the same as those in Fig. 2.

Moreover, distant pointer state permits us to perform high-fidelity qubit
measurement quantified through the signal-to-noise ratio (SNR) corresponding
to the ratio between the homodyne signal with its fluctuations. We define the
signal as
$|\langle\hat{\mathcal{M}}_{e}\rangle-\langle\hat{\mathcal{M}}_{g}\rangle|$
where $\langle\hat{\mathcal{M}}_{k}\rangle$ is the average of the homodyne
operator
$\hat{\mathcal{M}}(\tau)=\sqrt{\kappa}\int_{0}^{\tau}ds(a^{{\dagger}}_{{\rm{out}}}(t)\exp(i\phi)+a_{{\rm{out}}}(t)\exp(-i\phi))$
with respect to the qubit state, and we refer to fluctuations by
$\sqrt{\langle\hat{\mathcal{M}}_{{\rm{N}}e}^{2}\rangle+\langle\hat{\mathcal{M}}_{{\rm{N}}g}^{2}\rangle}$
with
$\hat{\mathcal{M}}_{{\rm{N}}\ell}=\hat{\mathcal{M}}_{\ell}-\langle\hat{\mathcal{M}}_{\ell}\rangle$
as the noise homodyne operator. The SNR is then defined as

$\displaystyle{\rm{SNR}}(\kappa\tau)=\frac{|\langle\hat{\mathcal{M}}_{e}\rangle-\langle\hat{\mathcal{M}}_{g}\rangle|}{\sqrt{\langle\hat{\mathcal{M}}_{{\rm{N}}e}^{2}\rangle+\langle\hat{\mathcal{M}}_{{\rm{N}}g}^{2}\rangle}}.$
(9)

In Fig. 3 (a), we plot the SNR as a function of the dimensionless integration
time $\tau/t_{f}$ using the polynomial (red dashed) and trigonometric (orange
dotted) modulations, see Eqs. (6) and (7), designed from inverse engineering
approach. We see an enhancement in the SNR, achieving the values approximately
ten times larger than the sinusoidal modulation proposed at the time scale
$t_{f}$. Furthermore, at short measuring time $\kappa\tau\ll 1$ we obtain the
asymptotic scaling of SNR as
${\rm{SNR}}(\kappa\tau)\approx(\kappa\tau)^{9/4}$, which approves the
enhancement in time on quantum nondemolition measurement. Our result agrees
with the statement in Ref. [26] that the readout performance can be improved
by quantum control methods.

Moreover, we obtain the further improvement of SNR by considering the LC
oscillator prepared in a single-mode squeezed orthogonal to the field
displacement. The squeezed state only modifies the noise homodyne operator as
$\hat{\mathcal{M}}_{{\rm{N}}\ell}=\kappa\tau(\cosh(2r)+\sinh(2r)\cosh(2(\phi-\theta)))$
[26], where $r$ and $\theta$ are the squeeze parameters and $\phi$ is the
homodyne angle. By choosing $\phi-\theta=\pi/2\mod\pi$, we finally achieve
$\hat{\mathcal{M}}_{{\rm{N}}\ell}=\kappa\tau\exp(-2r)$, that leads to the
exponential improvement on the SNR illustrated in Fig. 3 (b).

##  IV Comparision to Counter-diabatic driving

An alternative way to accelerate the qubit readout relies upon the counter-
diabatic driving [30]. Similar to STA for Rabi model [37], the counter-
diabatic term for the Hamiltonian in Eq. (1) is calculated as

$\displaystyle\mathcal{H}_{\rm{CD}}=-i\frac{\dot{g}_{z}(t)}{\omega_{r}}\sigma^{z}(a^{{\dagger}}-a).$
(10)

This interaction could be implemented in cQED architecture with a tunable
capacitive interaction [38], which has not been easily implemented yet. To
circumvent this problem, we utilize the multiple Schrödinger/interaction
pictures [39] and express the Hamiltonian $\mathcal{H}+\mathcal{H}_{\rm{CD}}$
in a rotating frame, which has different structure, but the same underlying
physics. By using
$\mathcal{U}(t)=\exp(-i\dot{g}_{z}(t)\sigma^{z}(a^{{\dagger}}+a)/\omega_{r}^{2})$,
we arrive at the effective Hamiltonian
$\bar{\mathcal{H}}=\omega_{q}\sigma^{z}/2+\omega_{r}\hat{a}^{{\dagger}}\hat{a}+\tilde{g}_{z}(t)\sigma^{z}(\hat{a}^{{\dagger}}+\hat{a})$,
where $\tilde{g}_{z}(t)=g_{z}(t)+\ddot{g}_{z}(t)/\omega_{r}^{2}$ is the
effective longitudinal coupling. In this new frame, it only requires a new
modulation on the coupling strength $\tilde{g}_{z}(t)$ rather than complicated
implementation, for instance, in a current experiment on open cQED [33].

Moreover, such approximate counter-diabatic driving can be implemented by
using Floquet engineering (FE) [40, 41, 42, 43]. Here, we add a high-frequency
driving with a complex time dependency to emulate the counter-diabatic term
only using the operators available on the system Hamiltonian. In what follows
we will calculate the FE Hamiltonian such that its dynamics corresponds to the
same as $\mathcal{H}_{\rm{CD}}$. Essentially, to obtain the state preparation,
up to the phase factor, it is not necessary to include the original
Hamiltonian $\mathcal{H}$. For doing so, we define the Floquet engineering
Hamiltonian as follows:

$\displaystyle\mathcal{H}_{\rm{FE}}(t)=\Omega\nu\sin(\nu
t)(\sigma^{z}+a^{{\dagger}}a)+\lambda(t)\sigma^{z}(a^{{\dagger}}+a),~{}~{}$
(11)

where $\nu\ll\omega_{r}$ is an arbitrary frequency, and $\Omega$ is a free
parameter. Next, we express $\mathcal{H}_{\rm{FE}}$ in the rotating frame
described by the following unitary transformation

$\displaystyle\hat{U}(t)=\exp\bigg{[}i\Omega\cos\nu
t(\sigma^{z}+a^{{\dagger}}a)\bigg{]}.$ (12)

In the rotating frame, the effective Hamiltonian
$\tilde{\mathcal{H}}_{\rm{FE}}(t)=\hat{U}^{{\dagger}}(t)\mathcal{H}_{\rm{FE}}\hat{U}(t)-i\hat{U}^{{\dagger}}(t)\dot{\hat{U}}(t)$
reads

$\displaystyle\tilde{\mathcal{H}}_{\rm{FE}}=\lambda(t)\hat{U}^{{\dagger}}(t)\sigma^{z}(a^{{\dagger}}+a)\hat{U}(t).$
(13)

Using the Bakker-Campbell-Hausdorff formula, we express the transformed
Hamiltonian as

$\displaystyle\tilde{\mathcal{H}}_{\rm{FE}}(t)=\lambda(t)\sigma^{z}(a^{{\dagger}}e^{-i\Omega\cos\nu
t}+ae^{i\Omega\cos\nu t}).$ (14)

We proceed by calculating the average of the Hamiltonian over the period
$T=2\pi/\nu$ to get the first term of the Magnus expansion

$\displaystyle\tilde{\mathcal{H}}^{(0)}_{\rm{FE}}(t)=\frac{1}{T}\int_{0}^{T}\lambda(t)\sigma^{z}(a^{{\dagger}}e^{-i\Omega\cos\nu
t}+ae^{i\Omega\cos\nu t})dt.$ (15)

To obtain that $\tilde{\mathcal{H}}^{(0)}_{\rm{FE}}=\mathcal{H}_{\rm{CD}}$ we
require that

$\displaystyle\frac{1}{T}\int_{0}^{T}\lambda(t)e^{-i\Omega\cos\nu t}dt$
$\displaystyle=$ $\displaystyle-i\frac{\dot{g}_{z}(t)}{\omega_{r}},$ (16)
$\displaystyle\frac{1}{T}\int_{0}^{T}\lambda(t)e^{i\Omega\cos\nu t}dt$
$\displaystyle=$ $\displaystyle i\frac{\dot{g}_{z}(t)}{\omega_{r}}.$ (17)

We proceed by assuming that $\lambda(t)=\sum_{n}C_{n}\cos(n\nu t)$, and we
define $\alpha=\nu t$. In this case, the integrals above defined read

$\displaystyle\sum_{n}\frac{C_{n}}{\pi}\int_{0}^{\pi}\cos(n\alpha)e^{-i\Omega\cos\alpha}d\alpha$
$\displaystyle=$ $\displaystyle-i\frac{\dot{g}_{z}(t)}{\omega_{r}}.$ (18)

Notice that this integral looks similar to the definition of the Bessel
function

$\displaystyle
J_{n}(z)=\frac{i^{-n}}{\pi}\int_{0}^{\pi}e^{iz\cos\theta}\cos(n\theta)d\theta.$
(19)

With this definition Eq. (17) and Eq. (17) read

$\displaystyle\sum_{n}i^{n}C_{n}J_{n}(\Omega)$ $\displaystyle=$
$\displaystyle-i\frac{\dot{g}_{z}(t)}{\omega_{r}},$ (20)
$\displaystyle\sum_{n}i^{n}C_{n}J_{n}(-\Omega)$ $\displaystyle=$
$\displaystyle i\frac{\dot{g}_{z}(t)}{\omega_{r}}.$ (21)

These conditions are meet when $n=2m+1$, $\forall m\in\mathbb{Z}$. Thus, for
$n=1$ we have $\lambda(t)=C_{1}\cos(\nu t)$ leading to

$\displaystyle C_{1}$ $\displaystyle=$
$\displaystyle\frac{\dot{g}_{z}(t)}{\omega_{r}J_{1}(\Omega)}.$ (22)

Finally, the Floquet engineered Hamiltonian takes the following form:

$\mathcal{H}_{\rm{FE}}(t)=\Omega\nu\sin(\nu
t)(\sigma^{z}+a^{{\dagger}}a)+\frac{\dot{g}_{z}(t)}{\omega_{r}J_{1}(\Omega)}\cos(\nu
t)\sigma^{z}(a^{{\dagger}}+a),$ (23)

where $\nu\ll\omega_{r}$ is an arbitrary frequency, $\Omega$ is a free
parameter, and $J_{1}$ is the Bessel function of the first kind. By
implementing the only Floquet Hamiltonian (not including the original
Hamiltonian $\mathcal{H}$), we show the relative distance between the pointer
states and the SNR can be significantly enhanced for different values of
$\nu$.

Figure 4: (a) Relative pointer state separation for the output cavity field
$\hat{a}_{out}$ as function of $t/t_{f}$ for different values of the frequency
$\nu$. (b) Signal-to-noise ratio (SNR) as function of the dimensionless
measuring time $\tau/t_{f}$. We have performed the simulation with the
parameters $\kappa/2\pi=1$ MHz, $g_{z0}/2\pi=21$ MHz, $\omega_{r}/2\pi=6.6$
GHz for $t_{f}=\pi/(100\kappa)$, and $\Omega=1$.

In Fig. 4, we show the relative distance between the pointer states and the
signal-to-noise ratio (SNR) for the Floquet Hamiltonian in Eq. (23) for
different values of $\nu$. As expected by increasing the frequency on the
modulation, larger pointer state separation we achieve leading to larger
values of the SNR.

##  V Genetic algorithm for optimization

Now, we turn to the genetic algorithm [44], an optimization subroutine based
on the fundamentals of natural selection, in order to complement our inverse-
engineering method. We formulate the optimization problem by assuming
$g_{c}(t)=\sum_{m}c_{m}\cos(m\pi t/t_{f})+d_{m}\sin(m\pi t/t_{f})$, where the
coefficients $\\{c_{m},d_{m}\\}$ are to be optimized according the constrains
provided by Eqs. (4a), (4b), and (5), respectively. Fig. 5 illustrates the
design for $g_{z}$ and corresponding SNR by fixing the number of coefficients.
Surprisingly, the increase of the numbers of coefficients does not always lead
to better SNR, as depicted in the inline of Fig. 5 (b), where the performance
of the modulation with $8$ coefficients surpasses that with 20. In this sense,
the genetic algorithms provide a simpler but efficient modulation for
achieving the same SNR at shorter time $t_{f}/2$. Of course, other
optimization techniques, i.e., machine learning [45] and pulse shaping [16,
46] can be incorporated as well, since there exists the freedom left in
inverse-engineering method mentioned above.

Figure 5: (a) The modulation of coupling strength $g_{z}(t)$ optimized by the
genetic algorithm containing 8 (blue), 12 (orange), and 20 (green)
coefficients, respectively. (b) The corresponding SNR obtained with the
genetic algorithm. We have performed the simulation with the same parameters
as Fig. 2, except for the measuring time being $t_{f}/2$.

##  VI Physical implementation

###  VI.1 Circuit Hamiltonian

We shall shed light on the experimental implementation for a two-level system
coupled to an oscillator via longitudinal interaction. The circuit consist in
an LC resonator of capacitance $C_{r}$ and inductance $L_{r}$ coupled to a
transmon qubit [47] through a SQUID. The transmon qubit consists of a
capacitor $C_{B}$ parallel-connected Josephson junction of capacitance $C_{J}$
and tunable Josephson energy $E_{J}(\phi_{x})$. Moreover, we bias the circuit
with an external gate voltage $V_{g}$ connected to the transmon with the gate
capacitance $C_{g}$. On the other hand, the SQUID is modeled as a tunable
Josephson junction with effective capacitance $C_{JS}$, and Josephson energies
$E_{JS}(\varphi_{x})$. We write the Lagrangian of the circuit in terms of the
flux nodes of each device $\\{\psi_{J},\psi_{{\rm{r}}}\\}$ related with the
voltage drop across their respective brach
$\psi_{\ell}=\int_{-\infty}^{t}V_{\ell}(x,t^{\prime})dt^{\prime}$ leading to

$\displaystyle\mathcal{L}_{c}$ $\displaystyle=$
$\displaystyle\frac{C_{g}}{2}(V_{g}-\dot{\psi}_{J})^{2}+\frac{C_{T}}{2}\dot{\psi}_{J}^{2}+\frac{C_{r}}{2}\dot{\psi}_{r}^{2}$
(24) $\displaystyle+$
$\displaystyle\frac{C_{S}}{2}(\dot{\psi}_{J}-\dot{\psi}_{r})^{2}-\frac{\psi_{r}^{2}}{2L_{r}}+E_{J}(\phi_{x})\cos\bigg{(}\frac{\psi_{J}}{\varphi_{0}}\bigg{)}$
$\displaystyle+$ $\displaystyle
E_{JS}(\varphi_{x})\cos\bigg{(}\frac{\psi_{J}-\psi_{r}}{\varphi_{0}}\bigg{)},$

where $\varphi_{0}=\hbar/(2e)$ is the quantum magnetic flux and $e$ is the
electron charge. Besides, $C_{T}=C_{B}+C_{J}$ is the effective transmon
capacitance. We calculate the canonical conjugate momenta
$P_{\ell}=\partial\mathcal{L}_{c}/\partial[\dot{\psi}_{\ell}]$

$\displaystyle P_{J}$ $\displaystyle=$
$\displaystyle(C_{T}+C_{S}+C_{g})\dot{\psi}_{J}-C_{S}\dot{\psi}_{r}-C_{g}V_{g},$
(25) $\displaystyle P_{r}$ $\displaystyle=$
$\displaystyle(C_{r}+C_{S})\dot{\psi}_{r}-C_{S}\dot{\psi}_{J}.$ (26)

Figure 6: Schematic illustration of the experimental proposal: a transmon
qubit formed by a capacitor $C_{B}$ parallel-connected to a tunable inductor
is biased by a gate voltage $V_{g}$ through a gate capacitor $C_{g}$, the two-
level system coupled to an LC resonator of capacitance $C_{r}$ and inductance
$L_{r}$ via an asymmetric SQUID threaded by an external magnetic flux
$\varphi_{x}$. Furthermore, we describe the circuit in terms of their fluxes
node $\psi_{J}$ and $\psi_{r}$.

Here we have dropped the terms proportional to $V_{g}^{2}$. In matrix form we
have $\vec{P}=\hat{\mathcal{C}}~{}\vec{\Psi}-\vec{Q}_{g}$, where
$\vec{P}^{T}=(P_{J},P_{r})$ and $\vec{\Psi}^{T}=(\psi_{J},\psi_{r})$ is the
charge and flux vectors, respectively. Furthermore,
$\vec{Q}_{g}^{T}=(C_{g}V_{g},0)$ is the gate charge vector, and
$\hat{\mathcal{C}}$ is the capacitance matrix

$\displaystyle\hat{\mathcal{C}}=\begin{pmatrix}C_{\Sigma}&-C_{S}\\\
-C_{S}&C_{\gamma}\end{pmatrix},$ (27)

where $C_{\Sigma}=C_{T}+C_{S}+C_{g}$ and $C_{\gamma}=C_{r}+C_{S}$ are
effective transmon and resonator capacitances, respectively. We obtain the
circuit Hamiltonian thought the Legendre transformation
$\mathcal{H}=\vec{P}^{T}\vec{\Psi}-\mathcal{L}_{c}$ Where
$\vec{\Psi}=\hat{\mathcal{C}}^{-1}(\vec{P}+\vec{Q}_{g})$ with
$\hat{\mathcal{C}}^{-1}$ beings the inverse of the capacitance matrix. The
circuit Hamiltonian reads

$\displaystyle\mathcal{H}$ $\displaystyle=$
$\displaystyle\frac{C_{\gamma}P_{J}^{2}}{2(C_{\Sigma}C_{\gamma}-C_{S}^{2})}+\frac{C_{\Sigma}P_{r}^{2}}{2(C_{\Sigma}C_{\gamma}-C_{S}^{2})}+\frac{C_{S}P_{J}P_{r}}{(C_{\Sigma}C_{\gamma}-C_{S}^{2})}$
(28) $\displaystyle+$
$\displaystyle\frac{C_{\gamma}P_{J}Q_{g}}{(C_{\Sigma}C_{\gamma}-C_{S}^{2})}+\frac{C_{S}P_{r}Q_{g}}{(C_{\Sigma}C_{\gamma}-C_{S}^{2})}+\frac{\psi_{r}^{2}}{2L_{r}}$
$\displaystyle-$ $\displaystyle
E_{J}(\phi_{x})\cos\bigg{(}\frac{\psi_{J}}{\varphi_{0}}\bigg{)}-E_{JS}(\varphi_{x})\cos\bigg{(}\frac{\psi_{J}-\psi_{r}}{\varphi_{0}}\bigg{)}.$

To proceed, we assume that the SQUID works on a parameter regime where the
capacitive interaction is smaller than the Josephson energy, regarding only
inductive interaction between the subsystems [24, 25, 26, 27]. Thus, for small
$C_{S}$ we neglect the capacitive interaction and we rewrite the potential
energy obtaining

$\displaystyle\mathcal{H}$ $\displaystyle=$
$\displaystyle\frac{C_{\gamma}P_{J}^{2}}{2(C_{\Sigma}C_{\gamma}-C_{S}^{2})}+\frac{C_{\Sigma}P_{r}^{2}}{2(C_{\Sigma}C_{\gamma}-C_{S}^{2})}+\frac{C_{\gamma}P_{J}Q_{g}}{(C_{\Sigma}C_{\gamma}-C_{S}^{2})}$
(29) $\displaystyle+$
$\displaystyle\frac{\psi_{r}^{2}}{2L_{r}}-E_{J}(\phi_{x})\cos\bigg{(}\frac{\psi_{J}}{\varphi_{0}}\bigg{)}$
$\displaystyle-$ $\displaystyle
E_{JS}(\varphi_{x})\cos\bigg{(}\frac{\psi_{J}-\psi_{r}}{\varphi_{0}}\bigg{)}.$

We now assume that the SQUID works in the linear regime [57] meaning that the
mostly of the current flows through the transmon. Hence, the resonator phase
is well locate allowing to expand the potential energy up to its leading order
in $\psi_{r}/\varphi_{0}$ [58]

$\displaystyle\mathcal{H}$
$\displaystyle=\frac{C_{\gamma}P_{J}^{2}}{2(C_{\Sigma}C_{\gamma}-C_{S}^{2})}+\frac{C_{\Sigma}P_{r}^{2}}{2(C_{\Sigma}C_{\gamma}-C_{S}^{2})}+\frac{C_{\gamma}P_{J}Q_{g}}{(C_{\Sigma}C_{\gamma}-C_{S}^{2})}$
(30) $\displaystyle-$ $\displaystyle
E_{J}(\phi_{x})\cos\bigg{(}\frac{\psi_{J}}{\varphi_{0}}\bigg{)}-E_{JS}(\varphi_{x})\cos\bigg{(}\frac{\psi_{J}}{\varphi_{0}}\bigg{)}+\frac{\psi_{r}^{2}}{2L_{r}}$
$\displaystyle-$ $\displaystyle
E_{JS}(\varphi_{x})\cos\bigg{(}\frac{\psi_{J}}{\varphi_{0}}\bigg{)}\psi_{r}.$

Figure 7: (a)-(c) Pauli matrix coefficient
$\alpha_{k}={\rm{Tr}}[\sigma^{k}\cos(\hat{\theta}_{J})]$ for the coupling
operator as a function of the external magnetic fluxes $\phi_{x}$ and
$\varphi_{x}$. (d) Low-lying energy spectrum of the transmon Hamiltonian
$\mathcal{H}_{T}$ as a function of the external magnetic fluxes $\phi_{x}$ and
$\varphi_{x}$. We have performed the simulation choosing the parameters
$E_{J}/\hbar=2\pi\times 20~{}{\rm{GHz}}$, $E_{C}=E_{J}/67$,
$E_{\Sigma}/\hbar=2\pi\times 30~{}{\rm{GHz}}\equiv 1.5~{}E_{J}$, yielding
$\omega_{q}=2\pi\times 3.28~{}{\rm{GHz}}$ and $n_{g}=0.5$.

We proceed to quantize the circuit Hamiltonian by promoting the classical
variables to quantum operators. For the transmon qubit the charge of the
circuit is proportional to the number of Cooper-pair
$P_{J}\rightarrow-2e\hat{n}_{J}$ and its conjugate variable corresponds to the
phase drop $\hat{\theta}_{J}=\psi_{J}/\varphi_{0}$ satisfying commutation
relation $[\hat{P}_{J},e^{i\hat{\theta}_{J}}]=i$. For the LC resonator, the
operators satisfies $[\hat{\psi}_{J},\hat{P}_{J}]=i\hbar$. The quantum circuit
Hamiltonian reads

$\displaystyle\mathcal{H}$ $\displaystyle=$ $\displaystyle
E_{C}(\hat{n}_{J}-n_{g})^{2}-E_{\tilde{J}}(\phi_{x},\varphi_{x})\cos\big{(}\hat{\theta}_{J}\big{)}$
(31) $\displaystyle+$
$\displaystyle\hbar\omega_{r}\hat{a}^{{\dagger}}\hat{a}-\frac{E_{JS}(\varphi_{x})}{\varphi_{0}}\sqrt{\frac{\hbar\omega_{r}L_{r}}{2}}(\hat{a}^{{\dagger}}+\hat{a})\cos\big{(}\hat{\theta}_{J}\big{)}.~{}~{}~{}~{}$

Here, $E_{C}=2e^{2}C_{\gamma}/(C_{\Sigma}C_{\gamma}-C_{S}^{2})$ and
$E_{\tilde{J}}(\phi_{x},\varphi_{x})=E_{J}(\phi_{x})+E_{JS}(\varphi_{x})$
correspond to the charge energy and the effective Josephson energy of the
transmon, respectively, $n_{g}=Q_{g}/2e^{2}$ stands for the dimensionless gate
charge. Besides,
$\omega_{r}=\sqrt{C_{\Sigma}/((C_{\Sigma}C_{\gamma}-C_{S}^{2})L_{r})}$ is the
oscillator frequency. It is convenient to divide the circuit Hamiltonian in
three parts

$\displaystyle\mathcal{H}_{T}$ $\displaystyle=$ $\displaystyle
E_{C}(\hat{n}_{J}-n_{g})^{2}-E_{\tilde{J}}(\phi_{x},\varphi_{x})\cos\big{(}\hat{\theta}_{J}\big{)},$
(32) $\displaystyle\mathcal{H}_{r}$ $\displaystyle=$
$\displaystyle\hbar\omega_{r}\hat{a}^{{\dagger}}\hat{a},$ (33)
$\displaystyle\mathcal{H}_{I}$ $\displaystyle=$
$\displaystyle-\frac{E_{JS}(\varphi_{x})}{\varphi_{0}}\sqrt{\frac{\hbar\omega_{r}L_{r}}{2}}(\hat{a}^{{\dagger}}+\hat{a})\cos\big{(}\hat{\theta}_{J}\big{)},$
(34)

corresponding to the transmon, resonator and interaction Hamiltonian
respectively.

###  VI.2 Two-level approximation

Next, we turn to illustrate that in the two-level approximation of the
transmon qubit, the Hamiltonian $\mathcal{H}_{I}$ leads to a longitudinal
oscillator qubit interaction. We express this Hamiltonian in the charge basis
$|n_{J}\rangle$ choosing $|g\rangle\equiv|0\rangle$ and
$|e\rangle\equiv|1\rangle$ the Hamiltonian reads (up to terms proportional to
$n_{g}^{2}$

$\displaystyle\mathcal{H}_{T}$ $\displaystyle=$ $\displaystyle
E_{C}(1-2n_{g})|e\rangle\langle
e|-\frac{E_{\tilde{J}}(\phi_{x},\varphi_{x})}{2}\sigma^{x},$ (35)
$\displaystyle\mathcal{H}_{I}$ $\displaystyle=$
$\displaystyle-\frac{E_{JS}(\varphi_{x})}{2\varphi_{0}}\sqrt{\frac{\hbar\omega_{r}L_{r}}{2}}(\hat{a}^{{\dagger}}+\hat{a})\sigma^{x},$
(36)

where $\sigma^{x}=|1\rangle\langle 0|+|0\rangle\langle 1|$ For a external gate
charge $n_{g}=0.5$ we obtain that the first term of $\mathcal{H}_{T}$
vanishes. Then it is possible to write as follows

$\displaystyle\mathcal{H}_{T}$ $\displaystyle=$
$\displaystyle\frac{\hbar\omega_{q}(\phi_{x},\varphi_{x})}{2}\sigma^{z},$ (37)

where $\omega_{q}=\sqrt{E_{\tilde{J}}^{2}(\phi_{x},\varphi_{x})}/\hbar$ is the
transition frequency of the qubit. Similar to the coupling operator. To prove
that, In Fig. 7, we have calculated the coefficients
$\alpha_{k}={\rm{Tr}}[\sigma^{k}\cos(\hat{\theta}_{J})]$ writing the operator
$\cos\big{(}\hat{\theta}_{J}\big{)}$ in the diagonal basis of
$\mathcal{H}_{T}$ observing that there is no contributions of either
$\sigma^{x}$ or $\sigma^{y}$. Therefore, the interaction Hamiltonian can be
expressed as follows

$\displaystyle\mathcal{H}_{I}$ $\displaystyle=$
$\displaystyle-\frac{E_{JS}(\varphi_{x})}{2\varphi_{0}}\sqrt{\frac{\hbar\omega_{r}L_{r}}{2}}(\hat{a}^{{\dagger}}+\hat{a})\sigma^{z}.$
(38)

Finally, the circuit Hamiltonian reads

$\displaystyle\mathcal{H}$ $\displaystyle=$
$\displaystyle\hbar\omega_{r}\hat{a}^{{\dagger}}\hat{a}+\frac{\hbar\omega_{q}}{2}\sigma^{z}+\hbar
g_{z}(t)(\hat{a}^{{\dagger}}+\hat{a})\sigma^{z},$ (39) $\displaystyle
g_{z}(t)$ $\displaystyle=$
$\displaystyle\frac{\omega_{q}}{2\varphi_{0}}\sqrt{\frac{\hbar\omega_{r}L_{r}}{2}}.$
(40)

In Fig. 7 we have also plot the energy spectrum of the transmon Hamiltonian as
function of both external magnetic fluxes $\phi_{x}$ and $\varphi_{x}$. We see
that the energy spectrum of the two-level system does not exhibit abrupt
changes along $\varphi_{x}$ corresponding to the tunable coupling strength.
Consequently, it is possible to switch the coupling strength without modifying
the energy spectrum of the qubit.

###  VI.3 Estimation of the coupling strength and minimal time

From the last subsection, we have obtained that under suitable conditions the
circuit corresponds to a qubit longitudinally coupled to an oscillator with
coupling strength given by

$\displaystyle
g_{z}(t)=\frac{\omega_{q}}{2\varphi_{0}}\sqrt{\frac{\hbar\omega_{r}L_{r}}{2}}.$
(41)

From this expression we observe that the coupling strength depends mainly on
four parameters, the qubit frequency, the sum of the Josephson energy of the
SQUID, and the resonator capacitance and inductance. Thus considering
consistent cQED values it is possible to estimate the maximal value of
$g_{z}(t)$. To achieve larger values of the coupling strength that yields to
faster measuring time we require large impedance [59]. In this direction,
technological progress has made possible to engineer inductances in the $\mu$H
regime using arrays of Josephon junctions or taking into account kinetic
inductors [49, 50, 51, 52]. To estimate the value of the coupling strength we
regard $\phi_{x}/\varphi_{0}=\varphi_{x}/\varphi_{0}=\pi/4$, hence the qubit
frequency turns into $\omega_{q}=\sqrt{E_{C}^{2}+dE_{\Sigma}^{2}}/\hbar$. For
realistic cQED parameters $E_{J}/\hbar=2\pi\times 20~{}{\rm{GHz}}$,
$E_{C}=E_{J}/67$ [26], $E_{\Sigma}/\hbar=2\pi\times 30~{}{\rm{GHz}}\equiv
1.5~{}E_{J}$ [53] and $d=0.02$ we obtain $\omega_{q}=2\pi\times
3.28~{}{\rm{GHz}}$. Moreover, for an LC oscillator (or transmission line
resonator) having values $\omega_{r}L_{r}=200~{}{{\rm{k}}}\Omega$ [49] we
achieve $g_{z}(t)=2\pi\times 2.57~{}\rm{GHz}$ corresponding to
$\max(g_{z})/\omega_{r}\equiv 0.5793$.

With this maximal value, it is possible to estimate the minimal time required
to measure the qubit $t_{{\rm{min}}}=\pi/(2\omega_{r})$, on a subnanosecond
time scale, from optimal control theory, see the detailed discussion in
Appendix B.

##  VII Conclusion

In summary, the methods of STA, including inverse engineering and counter-
diabaticity, have been worked out for designing the longitudinal qubit-cavity
coupling to accelerate the qubit measurement. Remarkably, by engineering the
modulations, the pointer state separation is significantly enhanced,
accompanied by a large SNR. We also see an exponential enhancement when the
cavity in a single-mode is prepared in squeezed state. In addition, genetic
algorithm are discussed also for the optimization. In the cQED platform,
tunable capacitive interaction is required to implement counter-diabatic
driving, which makes the inverse-engineering approach more feasible to speed
up the measurement process, with the realistic cQED architecture. We estimate
an upper bound for the coupling strength that set the low bound for measuring
time. Last but not least, we hope our result can be experimentally verified
with circuit design implementing longitudinal coupling of superconducting
qubits [26, 54], and applicable to electronic spin readout as well [55, 56].

## ACKNOWLEDGMENTS

This work is partially supported from NSFC (12075145), STCSM
(2019SHZDZX01-ZX04), QMiCS (820505) and OpenSuperQ (820363) of the EU Flagship
on Quantum Technologies, EU FET Open Grant Quromorphic (828826) and EPIQUS
(899368), QUANTEK project (KK-2021/00070), and the Basque Government through
Grant No. IT1470-22. X.C. acknowledges the Ramón y Cajal program
(RYC-2017-22482).

##  Appendix A Elimination of the longitudinal coupling

Let us consider a two-level system longitudinally coupled to a oscillator
described by the Hamiltonian Eq. (1), where $\omega_{q}$ is the transition
frequency of the qubit, $\omega_{r}$ is the oscillator frequency. Furthermore,
$\sigma^{z}$ is the $z-$component Pauli matrix describing the two-level
system, and $a^{{\dagger}}$ ($a$) is the creation (annihilation) operator of
the oscillator. We will show that the unitary transformation

$\displaystyle\mathcal{V}(t)=e^{i\theta(t)}e^{-i\frac{\dot{g}_{c}(t)\sigma^{z}(a^{{\dagger}}+a)}{\omega_{r}^{2}}}e^{-\frac{g_{c}(t)}{\omega_{r}}\sigma^{z}(a^{{\dagger}}-a)},$

presented in the manuscript eliminates the longitudinal coupling strength
leading to the effective Hamiltonian
$\mathcal{H}_{{\rm{eff}}}=\mathcal{V}^{{\dagger}}\mathcal{H}\mathcal{V}-i\dot{\mathcal{V}}\mathcal{V}^{{\dagger}}\equiv\omega_{r}a^{{\dagger}}a+\omega_{q}\sigma^{z}/2$.
Here, $\theta(t)$ corresponds to a phase defined as

$\displaystyle\theta(t)=-\int_{0}^{t}\mathcal{L}_{g}(t^{\prime})dt,$ (42)

where $\mathcal{L}_{g}$ is a Lagrangian that relates the quantities $g_{z}(t)$
and $g_{c}(t)$ through the following relation:

$\displaystyle\mathcal{L}_{g}(t)$ $\displaystyle=$
$\displaystyle\frac{\dot{g}^{2}_{c}(t)}{\omega_{r}^{3}}-\frac{g^{2}_{c}(t)}{\omega_{r}}+\frac{2g_{c}(t)g_{z}(t)}{\omega_{r}}.$
(43)

For this derivation it is convenient to divide in three the effective
Hamiltonian in three terms regarding the free terms
($\mathcal{H}_{{\rm{eff}},1}$), the transformed longitudinal interaction
($\mathcal{H}_{{\rm{eff}},2}$) and the terms appearing due the transformation
$\mathcal{H}_{{\rm{eff}},3}=-i\dot{\mathcal{V}}\mathcal{V}^{{\dagger}}$. For
the free terms we obtain

$\displaystyle\mathcal{H}_{{\rm{eff}},1}$ $\displaystyle=$
$\displaystyle\mathcal{V}^{{\dagger}}\bigg{(}\frac{\omega_{q}}{2}\sigma^{z}+\omega_{r}\hat{a}^{{\dagger}}\hat{a}\bigg{)}\mathcal{V}$
(44) $\displaystyle=$
$\displaystyle\frac{\omega_{q}}{2}\sigma^{z}+\omega_{r}\hat{a}^{{\dagger}}\hat{a}+\frac{i\dot{g}_{c}(t)}{\omega_{r}}\sigma^{z}[\hat{a}^{{\dagger}}+\hat{a},\hat{a}^{{\dagger}}\hat{a}]$
$\displaystyle+$ $\displaystyle
g_{c}(t)\sigma^{z}[\hat{a}^{{\dagger}}-\hat{a},\hat{a}^{{\dagger}}\hat{a}]-\frac{\dot{g}_{c}^{2}(t)}{2\omega_{r}^{3}}[\hat{a}^{{\dagger}}+\hat{a},[\hat{a}^{{\dagger}}+\hat{a},\hat{a}^{{\dagger}}\hat{a}]]$
$\displaystyle+$
$\displaystyle\frac{i\dot{g}_{c}(t)g_{c}(t)}{2\omega_{r}^{2}}[\hat{a}^{{\dagger}}+\hat{a},[\hat{a}^{{\dagger}}-\hat{a},\hat{a}^{{\dagger}}\hat{a}]]$
$\displaystyle+$
$\displaystyle\frac{i\dot{g}_{c}(t)g_{c}(t)}{2\omega_{r}^{2}}[\hat{a}^{{\dagger}}-\hat{a},[\hat{a}^{{\dagger}}+\hat{a},\hat{a}^{{\dagger}}\hat{a}]]$
$\displaystyle+$
$\displaystyle\frac{g_{c}^{2}(t)}{2\omega_{r}}[\hat{a}^{{\dagger}}-\hat{a},[\hat{a}^{{\dagger}}-\hat{a},\hat{a}^{{\dagger}}\hat{a}]].$

In this derivation, we have used the Baker-Campbell-Hausdorff formulae keeping
terms up to second order in the coupling strength $g_{c}$ and $\dot{g}_{c}$,
respectivelly. After solving the commutators we arrive at

$\displaystyle\mathcal{H}_{{\rm{eff}},1}$ $\displaystyle=$
$\displaystyle\frac{\omega_{q}}{2}\sigma^{z}+\omega_{r}\hat{a}^{{\dagger}}\hat{a}$
(45) $\displaystyle-$
$\displaystyle\frac{i\dot{g}_{c}(t)}{\omega_{r}}\sigma^{z}(\hat{a}^{{\dagger}}-\hat{a})-g_{c}(t)\sigma^{z}(\hat{a}^{{\dagger}}+\hat{a})$
$\displaystyle+$
$\displaystyle\frac{\dot{g}_{c}^{2}(t)}{\omega_{r}^{3}}+\frac{g_{c}^{2}(t)}{\omega_{r}}.$

With the same procedure we obtain the transformed longitudinal coupling
$\mathcal{H}_{{\rm{eff}},2}$ as follows

$\displaystyle\mathcal{H}_{{\rm{eff}},2}=g_{z}(t)\sigma^{z}(\hat{a}^{{\dagger}}+\hat{a})-\frac{2g_{c}(t)g_{z}(t)}{\omega_{r}}.$
(46)

Similar for $\mathcal{H}_{{\rm{eff}},3}$ we obtain

$\displaystyle\mathcal{H}_{{\rm{eff}},3}$ $\displaystyle=$
$\displaystyle-\dot{\theta}(t)+\frac{\ddot{g}_{c}(t)}{\omega_{r}^{2}}\sigma^{z}(\hat{a}^{{\dagger}}+\hat{a})$
(47) $\displaystyle+$
$\displaystyle\frac{i\dot{g}_{c}(t)}{\omega_{r}}\sigma^{z}(\hat{a}^{{\dagger}}-\hat{a})-\frac{2\dot{g}^{2}_{c}(t)}{\omega_{r}^{3}}.$

Finally, we arrive at the effective Hamiltonian

$\displaystyle\mathcal{H}_{{\rm{eff}}}$ $\displaystyle=$
$\displaystyle\frac{\omega_{q}}{2}\sigma^{z}+\omega_{r}\hat{a}^{{\dagger}}\hat{a}-\dot{\theta}(t)$
(48) $\displaystyle+$
$\displaystyle\bigg{[}\frac{\ddot{g}_{c}(t)}{\omega_{r}^{2}}-g_{c}(t)+g_{z}(t)\bigg{]}\sigma^{z}(\hat{a}^{{\dagger}}+\hat{a})$
$\displaystyle-$
$\displaystyle\bigg{[}\frac{\dot{g}_{c}^{2}(t)}{\omega_{r}^{3}}-\frac{g_{c}^{2}(t)}{\omega_{r}}+\frac{2g_{c}(t)g_{z}(t)}{\omega_{r}}\bigg{]}.$

From the effective Hamiltonian in Eq. (48) we see that for achieve
$\mathcal{H}_{{\rm{eff}}}=\mathcal{V}^{{\dagger}}\mathcal{H}\mathcal{V}-i\dot{\mathcal{V}}\mathcal{V}^{{\dagger}}\equiv\omega_{r}\hat{a}^{{\dagger}}\hat{a}+\omega_{q}\sigma^{z}/2$,
we require satisfy two conditions;

$\displaystyle\dot{\theta}(t)=\frac{\dot{g}_{c}^{2}(t)}{\omega_{r}^{3}}-\frac{g_{c}^{2}(t)}{\omega_{r}}+\frac{2g_{c}(t)g_{z}(t)}{\omega_{r}},$
(50) $\displaystyle\ddot{g}_{c}(t)+\omega_{r}^{2}[g_{c}(t)-g_{z}(t)]=0.$

Notice that Eq. (50) is exactly the same as the Lagrangian in Eq. (43), and
the condition Eq. (50) is nothing but the Euler-Lagrange equation for the
Lagrangian $\mathcal{L}_{g}$. Thus, we concluded that the unitary
transformation $\mathcal{V}(t)$ permit to us to express the system Hamiltonian
$\mathcal{H}$, see Eq. (1), in a frame without longitudinal coupling strength.

##  Appendix B Time-optimal control

Given the freedom left in reverse engineering based on Eq. (3), we combine it
with optimal control theory (OCT) to find the minimal time for measurement
according to the maximal coupling strength obtained above. In order to account
for boundary conditions, we first enlarge the control system as
$g_{d}=\dot{g}_{c}$. The state of the system $X=(g_{c},g_{d})^{\intercal}$
satisfies the following differential system:

$\dot{X}=AX+uB,$ (51)

with the control $u=g_{z}$ and the matrices $A$ and $B$ defined as follows:

$\displaystyle A=\begin{pmatrix}0&1\\\
-\omega_{r}^{2}&0\end{pmatrix}~{}\mbox{and}~{}B=\begin{pmatrix}0\\\
\omega_{r}^{2}\end{pmatrix}.$ (52)

We reformulate Eq. (3) into time-optimal control problem, by defining
dynamical equations

$\displaystyle\dot{g}_{c}$ $\displaystyle=$ $\displaystyle g_{d},$ (53)
$\displaystyle\dot{g}_{d}$ $\displaystyle=$
$\displaystyle-\omega^{2}_{r}(g_{c}-u).$ (54)

To minimize the time $J=\int_{0}^{t_{f}}dt$, we apply the Pontryagin maximum
principle to find the control $\\{u,X(t)\\}$ under the constraint $0\leq u\leq
u_{m}$ ($u_{m}=\max(g_{z})$), consistent with the boundary conditions. The
optimal control Hamiltonian is
$H_{c}=p_{0}+p_{g}g_{d}+p_{d}\omega^{2}_{r}(u-g_{c})$, with $p_{0}$ being the
constant and $(p_{g},p_{d})$ being the multipliers. The Pontryagin maximum
principle tells [60] us that the adjoint state is the solution of the
following differential equations,

$\displaystyle\dot{p}_{g}$ $\displaystyle=$ $\displaystyle\omega^{r}p_{d},$
(55) $\displaystyle\dot{p}_{d}$ $\displaystyle=$ $\displaystyle-p_{g},$ (56)

from which the adjoint equations can be obtained as

$\displaystyle p_{d}(t)=A_{d}\cos(\omega_{r}t)+B_{d}\sin(\omega_{r}t),$ (57)

where $A_{d}$ and $B_{d}$ can be fixed by initial values $p_{d}(0)$. We can
also introduce the switching function $\Phi=p_{d}$, such that

$\displaystyle u=\left\\{\begin{array}[]{rcl}\max(g_{z}),&&{\Phi>0}\\\
0,&&{\Phi<0}\end{array}\right.,$ (60)

In the singular case we have $p_{d}=p_{g}=0$ on a non-zero time interval, and
this extremal cannot be reached since ${p_{d},p_{g}}$ are continuous. When
$u=0$, we have $\ddot{g}_{c}+\omega^{2}_{r}g_{c}=0$, yielding

$\displaystyle g_{c}(t)$ $\displaystyle=$ $\displaystyle
A_{0}\cos(\omega_{r}t)+B_{0}\sin(\omega_{r}t),$ (61) $\displaystyle g_{d}(t)$
$\displaystyle=$ $\displaystyle-
A_{0}\omega_{r}\sin(\omega_{r}t)+B_{0}\omega_{r}\cos(\omega_{r}t),$ (62)

where $A_{0}$ and $B_{0}$ are constants determined later by boundary
conditions. From this, we deduce that $g_{d}$ and $g_{c}$ cannot be
simultaneously equal to zero at initial or final times, so $u=0$ does not
correspond to the first or last bang. However, we can still estimate the
minimal time when $u=u_{m}$ on $[0,t_{f}]$. Therefore, from the initial
boundary condition, we obtain:

$\displaystyle g_{c}(t)$ $\displaystyle=$ $\displaystyle
u_{m}(1-\cos(\omega_{r}t)),$ (63) $\displaystyle g_{d}(t)$ $\displaystyle=$
$\displaystyle\omega_{r}u_{m}\sin(\omega_{r}t).$ (64)

We can notice that $\omega_{r}t_{f}=2k\pi$ ($k=1,2,3...$), which gives the
minimal measurement time $t_{\min}=\pi/(2\omega_{r})$ ($k=1$), which is on the
subnanosecond time scale with the system parameters used here. Of course, one
may consider other optimal problems with various constraints as well.

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${\bm{q}}={\bm{q}}_{U}\equiv(\mu_{\star}^{2}-\lambda\psi_{1},\mu_{1}^{2},\psi_{2},0,0)$.
Using the formula for block matrix inversion, we have

${\bm{A}}(\mu_{\star}^{2}-\lambda\psi_{1},\mu_{1}^{2},\psi_{2},0,0)^{-1}=\begin{bmatrix}(\mu_{\star}^{2}-\lambda\psi_{1}){\mathbf{I}}_{N}+\mu_{1}^{2}{\bm{Q}}&{\bm{Z}}^{\mathsf{T}}\\\
{\bm{Z}}&\psi_{2}{\mathbf{I}}_{n}\end{bmatrix}^{-1}=\begin{bmatrix}({\bm{U}}_{c}-\psi_{2}^{-1}{\bm{Z}}^{\mathsf{T}}{\bm{Z}}-\psi_{1}\lambda{\mathbf{I}}_{N})^{-1}&\cdots\\\
\cdots&\cdots\end{bmatrix}.$

Then we have

$\partial_{s_{2}}G_{d}(0,{\bm{q}}_{U})=\frac{1}{d}{\rm
Tr}\left(\begin{bmatrix}\overline{{\bm{M}}}^{-1}&\cdots\\\
\cdots&\cdots\end{bmatrix}\begin{bmatrix}{\bm{Q}}&{\bm{0}}\\\
{\bm{0}}&{\bm{0}}\end{bmatrix}\right)={\rm
Tr}(\overline{{\bm{M}}}^{-1}{\bm{Q}})/d.$

Applying similar argument to compute other derivatives, we get

  1. 1.

${\rm
Tr}(\overline{{\bm{M}}}^{-1}{\bm{\Theta}}{\bm{\Theta}}^{\mathsf{T}})/d^{2}={\rm
Tr}(\overline{{\bm{M}}}^{-1}{\bm{Q}})/d=\partial_{s_{2}}G_{d}(0,{\bm{q}}_{U})$.

  2. 2.

$\mu_{1}\cdot{\rm
Tr}({\bm{Z}}\overline{{\bm{M}}}^{-1}{\bm{\Theta}}{\bm{X}}^{\mathsf{T}})/d^{2}={\rm
Tr}(\overline{{\bm{M}}}^{-1}{\bm{Z}}_{1}^{\mathsf{T}}{\bm{Z}})/d=-\psi_{2}\partial_{p}G_{d}(0,{\bm{q}}_{U})/2$.

  3. 3.

${\rm
Tr}({\bm{Z}}\overline{{\bm{M}}}^{-1}{\bm{Z}}^{\mathsf{T}}{\bm{X}}{\bm{X}}^{\mathsf{T}})/d^{2}={\rm
Tr}({\bm{Z}}\overline{{\bm{M}}}^{-1}{\bm{Z}}^{\mathsf{T}}{\bm{H}})/d=\psi_{2}^{2}\partial_{t_{2}}G_{d}(0,{\bm{q}}_{U})-\psi_{2}^{2}$.

  4. 4.

${\rm
Tr}({\bm{Z}}\overline{{\bm{M}}}^{-1}{\bm{Z}}^{\mathsf{T}})/d=\psi_{2}^{2}\partial_{t_{1}}G_{d}(0,{\bm{q}}_{U})-\psi_{2}^{2}$
.

  5. 5.

${\rm
Tr}(\overline{{\bm{M}}}^{-2}{\bm{Q}})/d=-\partial_{s_{1}}\partial_{s_{2}}G_{d}(0,{\bm{q}}_{U})$.

  6. 6.

$(2/d\psi_{2})\cdot{\rm
Tr}({\bm{Z}}_{1}^{{\mathsf{T}}}{\bm{Z}}\overline{{\bm{M}}}^{-2})=\partial_{s_{1}}\partial_{p}G_{d}(0,{\bm{q}}_{U})$.

  7. 7.

${\rm
Tr}(\overline{{\bm{M}}}^{-2}{\bm{Z}}^{\mathsf{T}}{\bm{H}}{\bm{Z}})/(d\psi_{2}^{2})=-\partial_{s_{1}}\partial_{t_{2}}G_{d}(0,{\bm{q}}_{U})$.

  8. 8.

${\rm
Tr}(\overline{{\bm{M}}}^{-2}{\bm{Z}}^{\mathsf{T}}{\bm{Z}})/(d\psi_{2}^{2})=-\partial_{s_{1}}\partial_{t_{1}}G_{d}(0,{\bm{q}}_{U})$.

Combining these equations concludes the proof. ∎

###### Proof of Lemma 2.

We prove this lemma by assuming that ${\bm{\beta}}$ follows a different
distribution:
${\bm{\beta}}\sim\mathcal{N}({\bm{0}},(\|{F}_{1}\|_{2}^{2}/d){\mathbf{I}}_{d})$.
The case when ${\bm{\beta}}\sim{\rm Unif}(\mathbb{S}^{d-1}({F}_{1}))$ can be
treated similarly.

By directly calculating the variance, we can show that, there exists scalers
$(c_{ik}^{(d)})_{k\in[K_{i}]}$ with $c_{ik}^{(d)}=\Theta_{d}(1)$, and matrices
$({\bm{A}}_{ik},{\bm{B}}_{ik})_{k\in[K_{i}]}\subseteq\\{{\mathbf{I}}_{N},{\bm{Q}},{\bm{Z}}^{\mathsf{T}}{\bm{H}}{\bm{Z}},{\bm{Z}}^{\mathsf{T}}{\bm{Z}}\\}$,
such that the variance of $\Psi_{i}$’s can be expressed in form

$\text{Var}_{{\bm{\varepsilon}},{\bm{\beta}}}(\Psi_{i})=\frac{1}{d}\sum_{k=1}^{K_{i}}c_{ik}^{(d)}{\rm
Tr}({\overline{\bm{M}}}^{-1}{\bm{A}}_{ik}{\overline{\bm{M}}}^{-1}{\bm{B}}_{ik})/d.$

For example, by Lemma 8, we have

$\displaystyle\text{Var}_{{\bm{\beta}}\sim\mathcal{N}({\bm{0}},({F}_{1}^{2}/d){\mathbf{I}}_{d})}(\Psi_{1})=$
$\displaystyle~{}\lambda_{d,1}(\sigma)^{4}\text{Var}_{{\bm{\beta}}\sim\mathcal{N}({\bm{0}},({F}_{1}^{2}/d){\mathbf{I}}_{d}))}({\bm{\beta}}^{\mathsf{T}}{\bm{\Theta}}^{\mathsf{T}}{\overline{\bm{M}}}^{-1}{\bm{\Theta}}{\bm{\beta}})=2\lambda_{d,1}(\sigma)^{4}{F}_{1}^{4}\|{\bm{\Theta}}^{\mathsf{T}}{\overline{\bm{M}}}^{-1}{\bm{\Theta}}\|_{F}^{2}/d^{2}$
$\displaystyle=$ $\displaystyle~{}c_{1}^{(d)}{\rm
Tr}({\overline{\bm{M}}}^{-1}{\bm{Q}}{\overline{\bm{M}}}^{-1}{\bm{Q}})/d^{2},$

where $c_{1}^{(d)}=2d^{2}\lambda_{d,1}(\sigma)^{4}{F}_{1}^{4}=O_{d}(1)$. The
variance of $\Psi_{2}$ and $\Psi_{3}$ can be calculated similarly.

Note that each ${\rm
Tr}({\overline{\bm{M}}}^{-1}{\bm{A}}_{ik}{\overline{\bm{M}}}^{-1}{\bm{B}}_{ik})/d$
can be expressed as an entry of $\nabla_{\bm{q}}^{2}G_{d}(0;{\bm{q}})$ (c.f.
Eq. (47)), and by Proposition 4, they are of order $O_{d,\mathbb{P}}(1)$. This
gives

$\text{Var}_{{\bm{\varepsilon}},{\bm{\beta}}}(\Psi_{i})=o_{d,\mathbb{P}}(1).$

Similarly, for the same set of scalers $(c_{ik}^{(d)})_{k\in[K_{i}]}$ and
matrices $({\bm{A}}_{ik},{\bm{B}}_{ik})_{k\in[K_{i}]}$, we have

$\text{Var}_{{\bm{\varepsilon}},{\bm{\beta}}}(\Phi_{i})=\frac{1}{d}\sum_{k=1}^{K_{i}}c_{ik}{\rm
Tr}({\overline{\bm{M}}}^{-2}{\bm{A}}_{ik}{\overline{\bm{M}}}^{-2}{\bm{B}}_{ik})/d.$

Note that for two semidefinite matrices ${\bm{A}},{\bm{B}}$, we have ${\rm
Tr}({\bm{A}}{\bm{B}})\leq\|{\bm{A}}\|_{{\rm op}}{\rm Tr}({\bm{B}})$. Moreover,
note we have $\|{\overline{\bm{M}}}\|_{{\rm op}}=O_{d,\mathbb{P}}(1)$ (by
Assumption 4). This gives

$\text{Var}_{{\bm{\varepsilon}},{\bm{\beta}}}(\Phi_{i})=o_{d,\mathbb{P}}(1).$

This concludes the proof. ∎

###  D.6 Auxiliary Lemmas

The following lemma (Lemma 5) is a reformulation of Proposition 3 in (Ghorbani
et al., 2019). We present it in a stronger form, but it can be easily derived
from the proof of Proposition 3 in (Ghorbani et al., 2019). This lemma was
first proved in (El Karoui, 2010) in the Gaussian case. (Notice that the
second estimate —on $Q_{k}({\bm{\Theta}}{\bm{X}}^{\mathsf{T}})$— follows by
applying the first one whereby ${\bm{\Theta}}$ is replaced by
${\bm{W}}=[{\bm{\Theta}}^{{\mathsf{T}}}|{\bm{X}}^{{\mathsf{T}}}]^{{\mathsf{T}}}$

######  Lemma 5.

Let
${\bm{\Theta}}=({\bm{\theta}}_{1},\ldots,{\bm{\theta}}_{N})^{\mathsf{T}}\in\mathbb{R}^{N\times
d}$ with $({\bm{\theta}}_{a})_{a\in[N]}\sim_{iid}{\rm
Unif}(\mathbb{S}^{d-1}(\sqrt{d}))$ and
${\bm{X}}=({\bm{x}}_{1},\ldots,{\bm{x}}_{n})^{\mathsf{T}}\in\mathbb{R}^{n\times
d}$ with $({\bm{x}}_{i})_{i\in[n]}\sim_{iid}{\rm
Unif}(\mathbb{S}^{d-1}(\sqrt{d}))$. Assume $1/c\leq n/d,N/d\leq c$ for some
constant $c\in(0,\infty)$. Then

$\displaystyle\mathbb{E}\Big{[}\sup_{k\geq
2}\|Q_{k}({\bm{\Theta}}{\bm{\Theta}}^{\mathsf{T}})-{\mathbf{I}}_{N}\|_{{\rm
op}}^{2}\Big{]}$ $\displaystyle=o_{d}(1)\,,$ (48)
$\displaystyle\mathbb{E}\Big{[}\sup_{k\geq
2}\|Q_{k}({\bm{\Theta}}{\bm{X}}^{\mathsf{T}})\|_{{\rm op}}^{2}\Big{]}$
$\displaystyle=o_{d}(1).$ (49)

Notice that the second estimate —on
$Q_{k}({\bm{\Theta}}{\bm{X}}^{\mathsf{T}})$— follows by applying the first one
—Eq. (48)— whereby ${\bm{\Theta}}$ is replaced by
${\bm{W}}=[{\bm{\Theta}}^{{\mathsf{T}}}|{\bm{X}}^{{\mathsf{T}}}]^{{\mathsf{T}}}$,
and we use $\|Q_{k}({\bm{\Theta}}{\bm{X}}^{\mathsf{T}})\|_{{\rm
op}}\leq\|Q_{k}({\bm{W}}{\bm{W}}^{\mathsf{T}})-{\mathbf{I}}_{N+n}\|_{{\rm
op}}$.

The following lemma (Lemma 6) can be easily derived from Lemma 5. Again, this
lemma was first proved in (El Karoui, 2010) in the Gaussian case.

######  Lemma 6.

Let
${\bm{\Theta}}=({\bm{\theta}}_{1},\ldots,{\bm{\theta}}_{N})^{\mathsf{T}}\in\mathbb{R}^{N\times
d}$ with $({\bm{\theta}}_{a})_{a\in[N]}\sim_{iid}{\rm
Unif}(\mathbb{S}^{d-1}(\sqrt{d}))$. Let activation function $\sigma$ satisfies
Assumption 2. Assume $1/c\leq N/d\leq c$ for some constant $c\in(0,\infty)$.
Denote

${\bm{U}}=\Big{(}\mathbb{E}_{{\bm{x}}\sim{\rm
Unif}(\mathbb{S}^{d-1}(\sqrt{d}))}[\sigma(\langle{\bm{\theta}}_{a},{\bm{x}}\rangle/\sqrt{d})\sigma(\langle{\bm{\theta}}_{b},{\bm{x}}\rangle/\sqrt{d})]\Big{)}_{a,b\in[N]}\in\mathbb{R}^{N\times
N}.$

Then we can rewrite the matrix ${\bm{U}}$ to be

${\bm{U}}=\lambda_{d,0}(\sigma)^{2}\bm{1}_{N}\bm{1}_{N}^{\mathsf{T}}+{\mu}_{1}^{2}{\bm{Q}}+{\mu}_{\star}^{2}({\mathbf{I}}_{N}+{\bm{\Delta}}),$

with ${\bm{Q}}={\bm{\Theta}}{\bm{\Theta}}^{\mathsf{T}}/d$ and
$\mathbb{E}[\|{\bm{\Delta}}\|_{{\rm op}}^{2}]=o_{d}(1)$.

In the following, we show that, under sufficient regularity condition of
$\sigma$, we have $\lambda_{d,0}(\sigma)=O(1/d)$.

######  Lemma 7.

Let $\sigma\in C^{2}(\mathbb{R})$ with
$|\sigma^{\prime}(x)|,|\sigma^{\prime\prime}(x)|<c_{0}e^{c_{1}|x|}$ for some
$c_{0},c_{1}\in\mathbb{R}$. Assume that
$\mathbb{E}_{G\sim\mathcal{N}(0,1)}[\sigma(G)]=0$. Then we have

$\lambda_{d,0}(\sigma)\equiv\mathbb{E}_{{\bm{x}}\sim{\rm
Unif}(\mathbb{S}^{d-1}(\sqrt{d}))}[\sigma(x_{1})]=O(1/d).$

###### Proof of Lemma 7.

Let ${\bm{x}}\sim{\rm Unif}(\mathbb{S}^{d-1}(\sqrt{d}))$ and
$\gamma\sim\chi(d)/\sqrt{d}$ independently. Then we have
$\gamma{\bm{x}}\sim\mathcal{N}({\bm{0}},{\mathbf{I}}_{d})$, so that by the
assumption, we have $\mathbb{E}[\sigma(\gamma x_{1})]=0$.

As a consequence, by the second order Taylor expansion, and by the
independence of $\gamma$ and ${\bm{x}}$, we have (for
$\xi(x_{1})\in[\gamma,1]$)

$\displaystyle|\lambda_{d,0}(\sigma)|=$
$\displaystyle~{}|\mathbb{E}[\sigma(x_{1})]|\leq|\mathbb{E}[\sigma(x_{1})]-\mathbb{E}[\sigma(\gamma
x_{1})]|\leq\Big{|}\mathbb{E}[\sigma^{\prime}(x_{1})x_{1}]\mathbb{E}[\gamma-1]\Big{|}+\Big{|}(1/2)\mathbb{E}[\sigma^{\prime\prime}(\xi(x_{1})x_{1})(\gamma-1)^{2}]\Big{|}$
$\displaystyle\leq$
$\displaystyle~{}\Big{|}\mathbb{E}[\sigma^{\prime}(x_{1})x_{1}]\Big{|}\cdot\Big{|}\mathbb{E}[\gamma-1]\Big{|}+(1/2)\mathbb{E}\Big{[}\sup_{u\in[\gamma,1]}\sigma^{\prime\prime}(ux_{1})^{2}\Big{]}^{1/2}\mathbb{E}[(\gamma-1)^{4}]^{1/2}.$

By the assumption that
$|\sigma^{\prime}(x)|,|\sigma^{\prime\prime}(x)|<c_{0}e^{c_{1}|x|}$ for some
$c_{0},c_{1}\in\mathbb{R}$, there exists constant $K$ that only depends on
$c_{0}$ and $c_{1}$ such that

$\sup_{d}\Big{|}\mathbb{E}[\sigma^{\prime}(x_{1})x_{1}]\Big{|}\leq
K,~{}~{}~{}~{}~{}\sup_{d}\Big{|}(1/2)\mathbb{E}\Big{[}\sup_{u\in[\gamma,1]}\sigma^{\prime\prime}(ux_{1})^{2}\Big{]}^{1/2}\Big{|}\leq
K.$

Moreover, by property of the $\chi$ distribution, we have

$|\mathbb{E}[\gamma-1]|=O(d^{-1}),~{}~{}~{}~{}~{}~{}\mathbb{E}[(\gamma-1)^{4}]^{1/2}=O(d^{-1}).$

This concludes the proof. ∎

The following lemma is a simple variance calculation and can be found as Lemma
C.5 in (Mei & Montanari, 2019). We restate here for completeness.

######  Lemma 8.

Let ${\bm{A}}\in\mathbb{R}^{n\times N}$ and ${\bm{B}}\in\mathbb{R}^{n\times
n}$. Let ${\bm{g}}=(g_{1},\ldots,g_{n})^{\mathsf{T}}$ with
$g_{i}\sim_{iid}\mathbb{P}_{g}$, $\mathbb{E}_{g}[g]=0$, and
$\mathbb{E}_{g}[g^{2}]=1$. Let ${\bm{h}}=(h_{1},\ldots,h_{N})^{\mathsf{T}}$
with $h_{i}\sim_{iid}\mathbb{P}_{h}$, $\mathbb{E}_{h}[h]=0$, and
$\mathbb{E}_{h}[h^{2}]=1$. Further we assume that ${\bm{h}}$ is independent of
${\bm{g}}$. Then we have

$\displaystyle\text{Var}({\bm{g}}^{\mathsf{T}}{\bm{A}}{\bm{h}})=$
$\displaystyle~{}\|{\bm{A}}\|_{F}^{2},$
$\displaystyle\text{Var}({\bm{g}}^{\mathsf{T}}{\bm{B}}{\bm{g}})=$
$\displaystyle~{}\sum_{i=1}^{n}B_{ii}^{2}(\mathbb{E}[g^{4}]-3)+\|{\bm{B}}\|_{F}^{2}+{\rm
Tr}({\bm{B}}^{2}).$

##  Appendix E Proof of Theorem 1

Here we give the whole proof for $U$. The proof for $T$ is the same.

For fixed
$A^{2}\in\Gamma_{U}\equiv\\{{\mathcal{A}}_{U}(\lambda,\psi_{1},\psi_{2}):\lambda\in{\Lambda_{U}}\\}$,
we denote

$\lambda_{\star}(A^{2})=\inf_{\lambda}\Big{\\{}\lambda:{\mathcal{A}}_{U}(\lambda,\psi_{1},\psi_{2})=A^{2}\Big{\\}}.$

By the definition of $\Gamma_{U}$, the set
$\\{\lambda:{\mathcal{A}}_{U}(\lambda,\psi_{1},\psi_{2})=A^{2}\\}$ is non-
empty and lower bounded, so that $\lambda_{\star}(A^{2})$ can be well-defined.
Moreover, we have $\lambda_{\star}(A^{2})\in{\Lambda_{U}}$. It is also easy to
see that we have

$\lambda_{\star}(A^{2})\in\operatorname*{arg\,min}_{\lambda\geq
0}\Big{[}\overline{\mathcal{U}}(\lambda,\psi_{1},\psi_{2})+\lambda
A^{2}\Big{]}.$ (50)

###  E.1 Upper bound

Note we have

$\displaystyle U(A,N,n,d)=$
$\displaystyle~{}\sup_{(N/d)\|{\bm{a}}\|_{2}^{2}\leq
A^{2}}\Big{(}R({\bm{a}})-\widehat{R}_{n}({\bm{a}})\Big{)}$ $\displaystyle\leq$
$\displaystyle~{}\inf_{\lambda}\sup_{(N/d)\|{\bm{a}}\|_{2}^{2}\leq
A^{2}}\Big{(}R({\bm{a}})-\widehat{R}_{n}({\bm{a}})-\psi_{1}\lambda(\|{\bm{a}}\|_{2}^{2}-\psi_{1}^{-1}A^{2})\Big{)}$
$\displaystyle\leq$
$\displaystyle~{}\inf_{\lambda}\Big{[}\overline{U}(\lambda,N,n,d)+\lambda
A^{2}\Big{]}$ $\displaystyle\leq$
$\displaystyle~{}\overline{U}(\lambda_{\star}(A^{2}),N,n,d)+\lambda_{\star}(A^{2})A^{2}.$

Note that $\lambda_{\star}(A^{2})\in{\Lambda_{U}}$, so by Lemma 5, in the
limit of Assumption 3, we have

$U(A,N,n,d)\leq\overline{\mathcal{U}}(\lambda_{\star}(A^{2}),\psi_{1},\psi_{2})+\lambda_{\star}(A^{2})A^{2}+o_{d,\mathbb{P}}(1)={\mathcal{U}}(A,\psi_{1},\psi_{2})+o_{d,\mathbb{P}}(1),$

where the last equality is by Eq. (50). This proves the upper bound.

###  E.2 Lower bound

For any $A^{2}>0$, we define a random variable $\hat{\lambda}(A^{2})$ (which
depend on ${\bm{X}}$, ${\bm{\Theta}}$, ${\bm{\beta}}$, ${\bm{\varepsilon}}$)
by

$\hat{\lambda}(A^{2})=\inf\Big{\\{}\lambda:\lambda\in\operatorname*{arg\,min}_{\lambda\geq
0}\Big{[}\overline{U}(\lambda,N,n,d)+\lambda A^{2}\Big{]}\Big{\\}}.$

By Proposition 1, the set is should always be non-empty, so that
$\hat{\lambda}(A^{2})$ can always be well-defined.

Moreover, since $\lambda_{\star}(A^{2})\in{\Lambda_{U}}$, by Assumption 4, as
we have shown in the proof in Proposition 2, we can uniquely define
$\overline{\bm{a}}_{U}(\lambda_{\star}(A^{2}))$ with high probability, where

$\overline{\bm{a}}_{U}(\lambda_{\star}(A^{2}))=\operatorname*{arg\,max}_{{\bm{a}}}\Big{[}R({\bm{a}})-\widehat{R}_{n}({\bm{a}})-\psi_{1}\lambda_{\star}(A^{2})\|{\bm{a}}\|_{2}^{2}\Big{]}.$

As a consequence, for a small $\varepsilon>0$, the following event
${\mathcal{E}}_{\varepsilon,d}$ can be well-defined with high probability

$\displaystyle{\mathcal{E}}_{\varepsilon,d}=$
$\displaystyle~{}\Big{\\{}\psi_{1}\|\overline{\bm{a}}_{U}(\lambda_{\star}(A^{2}))\|_{2}^{2}\geq
A^{2}-\varepsilon\Big{\\}}\cap\Big{\\{}\hat{\lambda}(A^{2}+\varepsilon)\leq\lambda_{\star}(A^{2})\Big{\\}}$
$\displaystyle=$
$\displaystyle~{}\Big{\\{}A^{2}-\varepsilon\leq\psi_{1}\|\overline{\bm{a}}_{U}(\lambda_{\star}(A^{2}))\|_{2}^{2}\leq
A^{2}+\varepsilon\Big{\\}}.$

Now, by Proposition 2, in the limit of Assumption 3, we have

$\lim_{d\to\infty}\mathbb{P}_{{\bm{X}},{\bm{\Theta}},{\bm{\beta}},{\bm{\varepsilon}}}({\mathcal{E}}_{\varepsilon,d})=1,$
(51)

and we have

$\displaystyle{\mathcal{U}}(\lambda_{\star}(A^{2}),\psi_{1},\psi_{2})=\overline{\mathcal{U}}(\lambda_{\star}(A^{2}),\psi_{1},\psi_{2})+o_{d,\mathbb{P}}(1).$
(52)

By the strong duality as in Proposition 1, for any $A^{2}\in\Gamma_{U}$, we
have

$\displaystyle U(A,N,n,d)=$
$\displaystyle~{}\overline{U}(\hat{\lambda}(A^{2}),N,n,d)+\hat{\lambda}(A^{2})A^{2}.$

Consequently, for small $\varepsilon>0$, when the event
${\mathcal{E}}_{\varepsilon,d}$ happens, we have

$\displaystyle~{}U((A^{2}+\varepsilon)^{1/2},N,n,d)$ $\displaystyle=$
$\displaystyle~{}\sup_{{\bm{a}}}\Big{(}R({\bm{a}})-\widehat{R}_{n}({\bm{a}})-\psi_{1}\hat{\lambda}(A^{2}+\varepsilon)\cdot\big{(}\|{\bm{a}}\|_{2}^{2}-\psi_{1}^{-1}(A^{2}+\varepsilon)\big{)}\Big{)}$
$\displaystyle\geq$
$\displaystyle~{}R(\overline{\bm{a}}_{U}(\lambda_{\star}(A^{2})))-\widehat{R}_{n}(\overline{\bm{a}}_{U}(\lambda_{\star}(A^{2})))-\psi_{1}\hat{\lambda}(A^{2}+\varepsilon)\cdot\big{(}\|\overline{\bm{a}}_{U}(\lambda_{\star}(A^{2}))\|_{2}^{2}-\psi_{1}^{-1}(A^{2}+\varepsilon)\big{)}$
$\displaystyle\geq$
$\displaystyle~{}R(\overline{\bm{a}}_{U}(\lambda_{\star}(A^{2})))-\widehat{R}_{n}(\overline{\bm{a}}_{U}(\lambda_{\star}(A^{2})))-\psi_{1}\hat{\lambda}(A^{2}+\varepsilon)\cdot\big{(}\|\overline{\bm{a}}_{U}(\lambda_{\star}(A^{2}))\|_{2}^{2}-\psi_{1}^{-1}(A^{2}-\varepsilon)\big{)}$
$\displaystyle\geq$
$\displaystyle~{}R(\overline{\bm{a}}_{U}(\lambda_{\star}(A^{2})))-\widehat{R}_{n}(\overline{\bm{a}}_{U}(\lambda_{\star}(A^{2})))-\psi_{1}\lambda_{\star}(A^{2})\cdot\big{(}\|\overline{\bm{a}}_{U}(\lambda_{\star}(A^{2}))\|_{2}^{2}-\psi_{1}^{-1}(A^{2}-\varepsilon)\big{)}$
$\displaystyle=$
$\displaystyle~{}\overline{U}(\lambda_{\star}(A^{2}),N,n,d)+\lambda_{\star}(A^{2})\cdot(A^{2}-\varepsilon).$

As a consequence, by Eq. (51) and (52), we have

$U((A^{2}+\varepsilon)^{1/2},N,n,d)\geq\overline{\mathcal{U}}(\lambda_{\star}(A^{2}),\psi_{1},\psi_{2})+\lambda_{\star}(A^{2})\cdot(A^{2}-\varepsilon)-o_{d,\mathbb{P}}(1)={\mathcal{U}}(A,\psi_{1},\psi_{2})-\varepsilon\lambda_{\star}(A^{2})-o_{d,\mathbb{P}}(1).$

where the last equality is by the definition of ${\mathcal{U}}$ as in
Definition 2, and by the fact that
$\lambda_{\star}(A^{2})\in\operatorname*{arg\,min}_{\lambda\geq
0}[\overline{\mathcal{U}}(\lambda,\psi_{1},\psi_{2})+\lambda A^{2}]$. Taking
$\varepsilon$ sufficiently small proves the lower bound. This concludes the
proof of Theorem 1.

##  Appendix F Technical background

In this section we introduce additional technical background useful for the
proofs. In particular, we will use decompositions in (hyper-)spherical
harmonics on the $\mathbb{S}^{d-1}(\sqrt{d})$ and in Hermite polynomials on
the real line. We refer the readers to (Efthimiou & Frye, 2014; Szego, Gabor,
1939; Chihara, 2011; Ghorbani et al., 2019; Mei & Montanari, 2019) for further
information on these topics.

###  F.1 Functional spaces over the sphere

For $d\geq 1$, we let
$\mathbb{S}^{d-1}(r)=\\{{\bm{x}}\in\mathbb{R}^{d}:\|{\bm{x}}\|_{2}=r\\}$
denote the sphere with radius $r$ in ${\mathbb{R}}^{d}$. We will mostly work
with the sphere of radius $\sqrt{d}$, $\mathbb{S}^{d-1}(\sqrt{d})$ and will
denote by $\gamma_{d}$ the uniform probability measure on
$\mathbb{S}^{d-1}(\sqrt{d})$. All functions in the following are assumed to be
elements of $L^{2}(\mathbb{S}^{d-1}(\sqrt{d}),\gamma_{d})$, with scalar
product and norm denoted as $\langle\,\cdot\,,\,\cdot\,\rangle_{L^{2}}$ and
$\|\,\cdot\,\|_{L^{2}}$:

$\displaystyle\langle
f,g\rangle_{L^{2}}\equiv\int_{\mathbb{S}^{d-1}(\sqrt{d})}f({\bm{x}})\,g({\bm{x}})\,\gamma_{d}({\rm
d}{\bm{x}})\,.$ (53)

For $\ell\in{\mathbb{Z}}_{\geq 0}$, let $\tilde{V}_{d,\ell}$ be the space of
homogeneous harmonic polynomials of degree $\ell$ on ${\mathbb{R}}^{d}$ (i.e.
homogeneous polynomials $q({\bm{x}})$ satisfying $\Delta q({\bm{x}})=0$), and
denote by $V_{d,\ell}$ the linear space of functions obtained by restricting
the polynomials in $\tilde{V}_{d,\ell}$ to $\mathbb{S}^{d-1}(\sqrt{d})$. With
these definitions, we have the following orthogonal decomposition

$\displaystyle
L^{2}(\mathbb{S}^{d-1}(\sqrt{d}),\gamma_{d})=\bigoplus_{\ell=0}^{\infty}V_{d,\ell}\,.$
(54)

The dimension of each subspace is given by

$\displaystyle\dim(V_{d,\ell})=B(d,\ell)=\frac{2\ell+d-2}{\ell}{\ell+d-3\choose\ell-1}\,.$
(55)

For each $\ell\in{\mathbb{Z}}_{\geq 0}$, the spherical harmonics $\\{Y_{\ell
j}^{(d)}\\}_{1\leq j\leq B(d,\ell)}$ form an orthonormal basis of
$V_{d,\ell}$:

$\langle Y^{(d)}_{ki},Y^{(d)}_{sj}\rangle_{L^{2}}=\delta_{ij}\delta_{ks}.$

Note that our convention is different from the more standard one, that defines
the spherical harmonics as functions on $\mathbb{S}^{d-1}(1)$. It is immediate
to pass from one convention to the other by a simple scaling. We will drop the
superscript $d$ and write $Y_{\ell,j}=Y_{\ell,j}^{(d)}$ whenever clear from
the context.

We denote by ${\mathsf{P}}_{k}$ the orthogonal projections to $V_{d,k}$ in
$L^{2}(\mathbb{S}^{d-1}(\sqrt{d}),\gamma_{d})$. This can be written in terms
of spherical harmonics as

$\displaystyle{\mathsf{P}}_{k}f({\bm{x}})\equiv$
$\displaystyle~{}\sum_{l=1}^{B(d,k)}\langle
f,Y_{kl}\rangle_{L^{2}}Y_{kl}({\bm{x}}).$ (56)

Then for a function $f\in L^{2}(\mathbb{S}^{d-1}(\sqrt{d}))$, we have

$f({\bm{x}})=\sum_{k=0}^{\infty}{\mathsf{P}}_{k}f({\bm{x}})=\sum_{k=0}^{\infty}\sum_{l=1}^{B(d,k)}\langle
f,Y_{kl}\rangle_{L^{2}}Y_{kl}({\bm{x}}).$

###  F.2 Gegenbauer polynomials

The $\ell$-th Gegenbauer polynomial $Q_{\ell}^{(d)}$ is a polynomial of degree
$\ell$. Consistently with our convention for spherical harmonics, we view
$Q_{\ell}^{(d)}$ as a function $Q_{\ell}^{(d)}:[-d,d]\to{\mathbb{R}}$. The set
$\\{Q_{\ell}^{(d)}\\}_{\ell\geq 0}$ forms an orthogonal basis on
$L^{2}([-d,d],\tilde{\tau}_{d})$ (where $\tilde{\tau}_{d}$ is the distribution
of $\langle{\bm{x}}_{1},{\bm{x}}_{2}\rangle$ when
${\bm{x}}_{1},{\bm{x}}_{2}\sim_{i.i.d.}{\rm
Unif}(\mathbb{S}^{d-1}(\sqrt{d}))$), satisfying the normalization condition:

$\displaystyle\langle
Q^{(d)}_{k},Q^{(d)}_{j}\rangle_{L^{2}(\tilde{\tau}_{d})}=\frac{1}{B(d,k)}\,\delta_{jk}\,.$
(57)

In particular, these polynomials are normalized so that $Q_{\ell}^{(d)}(d)=1$.
As above, we will omit the superscript $d$ when clear from the context (write
it as $Q_{\ell}$ for notation simplicity).

Gegenbauer polynomials are directly related to spherical harmonics as follows.
Fix ${\bm{v}}\in\mathbb{S}^{d-1}(\sqrt{d})$ and consider the subspace of
$V_{\ell}$ formed by all functions that are invariant under rotations in
${\mathbb{R}}^{d}$ that keep ${\bm{v}}$ unchanged. It is not hard to see that
this subspace has dimension one, and coincides with the span of the function
$Q_{\ell}^{(d)}(\langle{\bm{v}},\,\cdot\,\rangle)$.

We will use the following properties of Gegenbauer polynomials

  1. 1.

For ${\bm{x}},{\bm{y}}\in\mathbb{S}^{d-1}(\sqrt{d})$

$\displaystyle\langle
Q_{j}^{(d)}(\langle{\bm{x}},\cdot\rangle),Q_{k}^{(d)}(\langle{\bm{y}},\cdot\rangle)\rangle_{L^{2}(\mathbb{S}^{d-1}(\sqrt{d}),\gamma_{d})}=\frac{1}{B(d,k)}\delta_{jk}Q_{k}^{(d)}(\langle{\bm{x}},{\bm{y}}\rangle).$
(58)

  2. 2.

For ${\bm{x}},{\bm{y}}\in\mathbb{S}^{d-1}(\sqrt{d})$

$\displaystyle
Q_{k}^{(d)}(\langle{\bm{x}},{\bm{y}}\rangle)=\frac{1}{B(d,k)}\sum_{i=1}^{B(d,k)}Y_{ki}^{(d)}({\bm{x}})Y_{ki}^{(d)}({\bm{y}}).$
(59)

Note in particular that property 2 implies that –up to a constant–
$Q_{k}^{(d)}(\langle{\bm{x}},{\bm{y}}\rangle)$ is a representation of the
projector onto the subspace of degree-$k$ spherical harmonics

$\displaystyle({\mathsf{P}}_{k}f)({\bm{x}})=B(d,k)\int_{\mathbb{S}^{d-1}(\sqrt{d})}\,Q_{k}^{(d)}(\langle{\bm{x}},{\bm{y}}\rangle)\,f({\bm{y}})\,\gamma_{d}({\rm
d}{\bm{y}})\,.$ (60)

For a function $\sigma\in L^{2}([-\sqrt{d},\sqrt{d}],\tau_{d})$ (where
$\tau_{d}$ is the distribution of
$\langle{\bm{x}}_{1},{\bm{x}}_{2}\rangle/\sqrt{d}$ when
${\bm{x}}_{1},{\bm{x}}_{2}\sim_{iid}{\rm Unif}(\mathbb{S}^{d-1}(\sqrt{d}))$),
denoting its spherical harmonics coefficients $\lambda_{d,k}(\sigma)$ to be

$\displaystyle\lambda_{d,k}(\sigma)=\int_{[-\sqrt{d},\sqrt{d}]}\sigma(x)Q_{k}^{(d)}(\sqrt{d}x)\tau_{d}(x),$
(61)

then we have the following equation holds in
$L^{2}([-\sqrt{d},\sqrt{d}],\tau_{d})$ sense

$\sigma(x)=\sum_{k=0}^{\infty}\lambda_{d,k}(\sigma)B(d,k)Q_{k}^{(d)}(\sqrt{d}x).$
(62)

###  F.3 Hermite polynomials

The Hermite polynomials $\\{{\rm He}_{k}\\}_{k\geq 0}$ form an orthogonal
basis of $L^{2}({\mathbb{R}},\mu_{G})$, where $\mu_{G}({\rm
d}x)=e^{-x^{2}/2}{\rm d}x/\sqrt{2\pi}$ is the standard Gaussian measure, and
${\rm He}_{k}$ has degree $k$. We will follow the classical normalization
(here and below, expectation is with respect to $G\sim{\sf N}(0,1)$):

$\displaystyle\mathbb{E}\big{\\{}{\rm He}_{j}(G)\,{\rm
He}_{k}(G)\big{\\}}=k!\,\delta_{jk}\,.$ (63)

As a consequence, for any function $\sigma\in L^{2}({\mathbb{R}},\mu_{G})$, we
have the decomposition

$\displaystyle\sigma(x)=\sum_{k=1}^{\infty}\frac{\mu_{k}(\sigma)}{k!}\,{\rm
He}_{k}(x)\,,\;\;\;\;\;\;\mu_{k}(\sigma)\equiv\mathbb{E}\big{\\{}\sigma(G)\,{\rm
He}_{k}(G)\\}\,.$ (64)

The Hermite polynomials can be obtained as high-dimensional limits of the
Gegenbauer polynomials introduced in the previous section. Indeed, the
Gegenbauer polynomials (up to a $\sqrt{d}$ scaling in domain) are constructed
by Gram-Schmidt orthogonalization of the monomials $\\{x^{k}\\}_{k\geq 0}$
with respect to the measure $\tau_{d}$, while Hermite polynomial are obtained
by Gram-Schmidt orthogonalization with respect to $\mu_{G}$. Since
$\tau_{d}\Rightarrow\mu_{G}$ (here $\Rightarrow$ denotes weak convergence), it
is immediate to show that, for any fixed integer $k$,

$\displaystyle\lim_{d\to\infty}{\rm
Coeff}\\{Q_{k}^{(d)}(\sqrt{d}x)\,B(d,k)^{1/2}\\}={\rm
Coeff}\left\\{\frac{1}{(k!)^{1/2}}\,{\rm He}_{k}(x)\right\\}\,.$ (65)

Here and below, for $P$ a polynomial, ${\rm Coeff}\\{P(x)\\}$ is the vector of
the coefficients of $P$. As a consequence, for any fixed integer $k$, we have

$\displaystyle\mu_{k}(\sigma)=\lim_{d\to\infty}\lambda_{d,k}(\sigma)(B(d,k)k!)^{1/2},$
(66)

where $\mu_{k}(\sigma)$ and $\lambda_{d,k}(\sigma)$ are given in Eq. (64) and
(61).