Hamiltonian engineering for robust quantum state transfer and qubit readout in cavity QED

An exact solution is possible for SQUADD under the ideal conditions described above: sharp π-pulses, $g(t)=g$ for n(t) even, and $g(t)=0$ for n(t) odd. The time-evolution operator then breaks into segments associated with the intervals of duration τ between π-pulses. In the single-excitation subspace, ${N}_{\mathrm{ex}}={a}^{\dagger }a+{\sigma }_{+}{\sigma }_{-}=1$, the evolution operator for a single period of the decoupling sequence is

$$\begin{eqnarray}&&{U}_{1}={R}_{\hat{n}}({\rm{\Omega }}\tau ){R}_{\hat{z}}(-\xi \tau ){R}_{\hat{n}}({\rm{\Omega }}\tau ),\end{eqnarray} \tag{ 6 }$$

with

$$\begin{eqnarray}&&{\rm{\Omega }}=\sqrt{{g}^{2}+{\xi }^{2}/4},\end{eqnarray} \tag{ 7 }$$

$$\begin{eqnarray}&&\hat{n}=\displaystyle \frac{g}{{\rm{\Omega }}}\hat{x}+\displaystyle \frac{\xi }{2{\rm{\Omega }}}\hat{z}.\end{eqnarray} \tag{ 8 }$$

In equation (6), we have introduced the operator ${R}_{\hat{n}}(\theta )\equiv {{\rm{e}}}^{-{\rm{i}}\theta \hat{n}\cdot {\boldsymbol{\tau }}/2}$, which applies an SU(2) rotation by angle $\theta$ around the axis set by the unit vector $\hat{n}$ in the space spanned by the vector of pseudospins ${\boldsymbol{\tau }}=({\tau }_{x},{\tau }_{y},{\tau }_{z})$. These pseudospins are defined by

$$\begin{eqnarray}&&{\tau }_{x}=| g1\rangle \langle e0| +| e0\rangle \langle g1| ,\end{eqnarray} \tag{ 9 }$$

$$\begin{eqnarray}&&{\tau }_{y}={\rm{i}}(| g1\rangle \langle e0| -| e0\rangle \langle g1| ),\end{eqnarray} \tag{ 10 }$$

$$\begin{eqnarray}&&{\tau }_{z}=| e0\rangle \langle e0| -| g1\rangle \langle g1| .\end{eqnarray} \tag{ 11 }$$

In equations (9)–(11), g (e) labels the ground (excited) state of the qubit, while 0 or 1 is the number of photons in the cavity. The product of the three rotation matrices in equation (6) is itself a rotation matrix ${U}_{1}={R}_{\hat{v}}(\vartheta )$, where

$$\begin{eqnarray}&&\cos \displaystyle \frac{\vartheta }{2}=\sqrt{1-{A}^{2}-{B}^{2}},\end{eqnarray} \tag{ 12 }$$

$$\begin{eqnarray}&&\hat{v}=\displaystyle \frac{A}{\sqrt{{A}^{2}+{B}^{2}}}\hat{x}+\displaystyle \frac{B}{\sqrt{{A}^{2}+{B}^{2}}}\hat{z},\end{eqnarray} \tag{ 13 }$$

with

$$\begin{eqnarray}&&A=\displaystyle \frac{2g}{{\rm{\Omega }}}\left(\cos \displaystyle \frac{{\rm{\Omega }}\tau }{2}\cos \displaystyle \frac{\xi \tau }{2}+\displaystyle \frac{\xi }{2{\rm{\Omega }}}\sin \displaystyle \frac{{\rm{\Omega }}\tau }{2}\sin \displaystyle \frac{\xi \tau }{2}\right)\sin \displaystyle \frac{{\rm{\Omega }}\tau }{2},\end{eqnarray} \tag{ 14 }$$

$$\begin{eqnarray}&&B=\left(\displaystyle \frac{\xi }{{\rm{\Omega }}}\sin \displaystyle \frac{{\rm{\Omega }}\tau }{2}\cos \displaystyle \frac{\xi \tau }{2}-\cos \displaystyle \frac{{\rm{\Omega }}\tau }{2}\sin \displaystyle \frac{\xi \tau }{2}\right)\cos \displaystyle \frac{{\rm{\Omega }}\tau }{2}+\displaystyle \frac{{\xi }^{2}-4{g}^{2}}{4{{\rm{\Omega }}}^{2}}{\sin }^{2}\displaystyle \frac{{\rm{\Omega }}\tau }{2}\sin \displaystyle \frac{\xi \tau }{2}.\end{eqnarray} \tag{ 15 }$$

The evolution at the end of the full sequence of n_p pulses (and thus ${n}_{p}/2$ periods) is given by

$$\begin{eqnarray}&&U({t}_{f})={U}_{1}^{{n}_{p}/2}={R}_{\hat{v}}({n}_{p}\vartheta /2),\hspace{2cm}({n}_{p}\ \mathrm{even}).\end{eqnarray} \tag{ 16 }$$

Equation (16) gives a closed-form analytical expression for the evolution operator under SQUADD. Taking ${ \mathcal M }(| \psi \rangle \langle \psi | )=U({t}_{f})| \psi \rangle \langle \psi | {U}^{\dagger }({t}_{f})$ in equation (5), with $U({t}_{f})$ given by equation (16), we obtain the average state-transfer fidelity

$$\begin{eqnarray}&&F=\displaystyle \frac{1}{3}{\rm{E}}\left[1+{v}_{x}^{2}{\sin }^{2}\displaystyle \frac{{n}_{p}\vartheta }{4}+{v}_{x}\sin \displaystyle \frac{{n}_{p}\vartheta }{4}\right],\end{eqnarray} \tag{ 17 }$$

where ${\rm{E}}[\cdot ]$ is an ensemble average over the detuning ξ. Equation (17) is plotted as a function of n_p in figure 2 (purple solid line).

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To clarify the parametric dependences in equation (17), we set $\tau =\pi /({{gn}}_{p})$ for a complete state transfer (minimizing error to leading order in $\max (g,\xi )\tau$) and expand to leading order in $1/{n}_{p}$. This gives

$$\begin{eqnarray}&&1-F\simeq \displaystyle \frac{1}{6}\left[{\left(\displaystyle \frac{\pi }{4}\right)}^{2}{\left(\displaystyle \frac{{\rm{\Delta }}\xi }{g}\right)}^{4}+\displaystyle \frac{1}{3}{\left(\displaystyle \frac{{\rm{\Delta }}\xi }{g}\right)}^{2}\right]{\left(\displaystyle \frac{\pi }{2{n}_{p}}\right)}^{4},\end{eqnarray} \tag{ 18 }$$

valid for $\max (g,{\rm{\Delta }}\xi )\tau \ll 1$ (equivalently, ${n}_{p}\gg \pi \max (1,{\rm{\Delta }}\xi /g)$). The error ($1-F\propto 1/{n}_{p}^{4}$) is thus strongly suppressed with an increasing number of π-pulses, as shown in figure 2.

A small error can be reached when ${n}_{p}\gg 1$, even for strong dephasing, ${{gT}}_{2}^{* }=\sqrt{2}g/{\rm{\Delta }}\xi \lt 1$, where ${T}_{2}^{* }$ is the qubit free-induction decay time (dephasing time) due to inhomogeneous broadening ${\rm{\Delta }}\xi$. This result is apparent in figure 2, which gives the exact solution described above (solid purple line), along with the large-n_p expansion of equation (18) (dashed black line). Here, we have chosen ${{gT}}_{2}^{* }=1/10$. Even for this choice of parameters, placing the system in the strong-dephasing regime, errors smaller than 1% are reached with ${n}_{p}\sim 40$ pulses, at the onset of the validity criterion for equation (18): ${n}_{p}\gt \pi {\rm{\Delta }}\xi /g\sim 40$. Consequently, the usual weak-dephasing criterion ($1/{T}_{2}^{* }\ll g$) has been traded for a fast-control requirement ($\tau \ll {T}_{2}^{* }$)⁵ . Fast π-pulses in this limit have already been demonstrated with isolated spin qubits (not coupled to cavities) [13, 19, 37], and could in principle be made even faster for single spins by taking advantage of exchange coupling and the magnetic field gradient generated by a micromagnet [38, 39]. Since g(t) can be controlled electrically when these systems are coupled to cavities [5, 6], fast pulsing of g(t) may be possible in the very near future (we give an analysis of finite-bandwidth control for g(t) and the influence of counter-rotating terms in section 5.3).

When ${n}_{p}\to \infty$, inhomogeneous broadening becomes irrelevant and the fidelity will ultimately be limited by cavity damping at rate $\kappa$ (we neglect intrinsic qubit decay due to a homogeneous linewidth when $\kappa {T}_{2}\gt 1$). Accounting for finite cavity damping $\kappa \ne 0$ in the map ${ \mathcal M }$ in equation (5), and expanding for $\kappa /g\ll 1$, we find that the error saturates at

$$\begin{eqnarray}&&1-F=\displaystyle \frac{\pi }{6}\,\displaystyle \frac{\kappa }{g}+{ \mathcal O }\left(\displaystyle \frac{{\kappa }^{2}}{{g}^{2}}\right)\end{eqnarray} \tag{ 19 }$$

when ${n}_{p}\to \infty$. To establish the influence of cavity damping more generally as a function of $\kappa /g$ and n_p, we numerically solve the Lindblad master equation generated by a Liouvillian superoperator ${ \mathcal L }$ accounting for both Hamiltonian evolution under equation (4) and cavity damping. As shown in figure 2, cavity damping does indeed lead to a saturation of the error as a function of n_p at $1-F\sim \kappa /g$ (blue dots: $\kappa /g=1$, red triangles: $\kappa /g=1/100$).

As a concrete example, a coupling $g/2\pi \simeq 1\,\mathrm{MHz}$ has been predicted for spin qubits in GaAs double quantum dots [15], leading to ${{gT}}_{2}^{* }\simeq 0.05$ due to hyperfine coupling to nuclear spins [13]. Even in this case, SQUADD could enable coherent coupling between a single spin and a cavity. In addition, SQUADD could improve state transfer between a single spin confined in a carbon nanotube and a coplanar-waveguide resonator. In a recent experiment on this system, $g/2\pi =1.3\,\mathrm{MHz}$, $\kappa /2\pi =0.6\,\mathrm{MHz}$, and ${T}_{2}^{* }\simeq 60$ ns have been reported [5]. With these parameters, a large state-transfer error $1-F\simeq 0.42$ results from equation (5) without π-pulses. Using SQUADD, n_p = 10 ($\tau =\tfrac{\pi }{{{gn}}_{p}}\simeq 40$ ns) suffices to reduce the error from pure dephasing to 0.004. The total error is then $1-F\simeq 0.18$, limited by the large κ/g ratio in this experiment.

For $\kappa =0$, some insight into the dependence of the state-transfer error on n_p can be gained using average Hamiltonian theory, a standard tool in the analysis of open-loop Hamiltonian engineering protocols [24]. In average Hamiltonian theory, the evolution operator $U(t)={ \mathcal T }\exp [-{\rm{i}}{\int }_{0}^{t}{\rm{d}}{t}^{\prime} {H}_{{\rm{T}}}(t^{\prime} )]$ is recast in terms of a Magnus expansion [40, 41]:

$$\begin{eqnarray}&&U(t)=\exp \left[-{\rm{i}}{t}\displaystyle \sum _{k=0}^{\infty }{\overline{H}}^{(k)}\right].\end{eqnarray} \tag{ 20 }$$

Substituting ${H}_{{\rm{T}}}(t)$ (equation (4)) into the expressions for ${\overline{H}}^{(k)}$ [40, 41] gives the first few terms in the Magnus expansion for SQUADD

$$\begin{eqnarray}&&{\overline{H}}^{(0)}=\displaystyle \frac{1}{2\tau }{\int }_{0}^{2\tau }{{\rm{d}}{t}}_{1}{H}_{{\rm{T}}}({t}_{1})=\displaystyle \frac{g}{2}({a}^{\dagger }{\sigma }_{-}+a{\sigma }_{+}),\end{eqnarray} \tag{ 21 }$$

$$\begin{eqnarray}&&{\overline{H}}^{(1)}=-\displaystyle \frac{{\rm{i}}}{4\tau }{\int }_{0}^{2\tau }{{\rm{d}}{t}}_{1}{\int }_{0}^{{t}_{1}}{{\rm{d}}{t}}_{2}[{H}_{{\rm{T}}}({t}_{1}),{H}_{{\rm{T}}}({t}_{2})]=0,\end{eqnarray} \tag{ 22 }$$

$$\begin{eqnarray}{\overline{H}}^{(2)} & = & -\displaystyle \frac{1}{12\tau }{\displaystyle \int }_{0}^{2\tau }{{\rm{d}}{t}}_{1}{\displaystyle \int }_{0}^{{t}_{1}}{{\rm{d}}{t}}_{2}{\displaystyle \int }_{0}^{{t}_{2}}{{\rm{d}}{t}}_{3}\{[{H}_{{\rm{T}}}({t}_{1}),[{H}_{{\rm{T}}}({t}_{2}),{H}_{{\rm{T}}}({t}_{3})]]\\ & & +[[{H}_{{\rm{T}}}({t}_{1}),{H}_{{\rm{T}}}({t}_{2})],{H}_{{\rm{T}}}({t}_{3})]\}\\ & = & -\displaystyle \frac{g{\xi }^{2}{\tau }^{2}}{48}(a{\sigma }_{+}+{a}^{\dagger }{\sigma }_{-})-\displaystyle \frac{{g}^{2}\xi {\tau }^{2}}{24}\left({a}^{\dagger }a+\displaystyle \frac{1}{2}\right){\sigma }_{z}.\end{eqnarray} \tag{ 23 }$$

The leading term, ${\overline{H}}^{(0)}$, generates coherent vacuum Rabi oscillations between the qubit and the cavity. In equation (22), ${\overline{H}}^{(1)}$ vanishes because the toggling-frame Hamiltonian given by equation (4) has the following property: ${H}_{{\rm{T}}}(t)={H}_{{\rm{T}}}(2\tau -t)$, corresponding to a symmetric cycle with period $T=2\tau$ [35]. For such symmetric cycles, all odd orders vanish in the Magnus expansion, leading to ${\overline{H}}^{(1)}=0$ [42]. The leading source of error is then ${\overline{H}}^{(2)}$. In the subspace containing a single qubit or cavity excitation (${N}_{\mathrm{ex}}=1$), the first and second terms in ${\overline{H}}^{(2)}$ (equation (23)) lead to unwanted rotations by an angle $\propto \,{\tau }^{2}$ around the axes defined by ${\tau }_{x}$ (equation (9)) and ${\tau }_{z}$ (equation (11)), respectively. For small rotation angles, this gives a correction $\propto \,{({\tau }^{2})}^{2}$ to the fidelity (which involves an overlap between two state vectors at an angle $\propto \,{\tau }^{2}$), thus explaining the error $\propto \,{\tau }^{4}\propto 1/{n}_{p}^{4}$ obtained in equation (18).

A sufficient criterion for convergence of the Magnus expansion (equation (20)) for a periodic Hamiltonian ${H}_{{\rm{T}}}(t)$ with period $2\tau$ is [41]

$$\begin{eqnarray}&&{\int }_{0}^{2\tau }{\rm{d}}{t}\parallel {H}_{{\rm{T}}}(t)\parallel \lt \pi ,\end{eqnarray} \tag{ 24 }$$

where $\parallel O\parallel$ is the 2-norm of O, with O an arbitrary operator. Here, we have taken the detuning ξ to be Gaussian distributed over an infinite interval and the cavity mode is described by unbounded operators $a,{a}^{\dagger }$. Thus, $\parallel {H}_{{\rm{T}}}(t)\parallel$ is formally unbounded. However, realistically, we expect that the Gaussian distribution for ξ can be truncated at a few ${\rm{\Delta }}\xi$, and the cavity occupation can be truncated to include only a few quanta in Fock space, resulting in $| | a| | \sim | | {a}^{\dagger }| | \sim 1$, which gives ${\int }_{0}^{2\tau }{\rm{d}}{t}\parallel {H}_{{\rm{T}}}(t)\parallel \lesssim \max (g,{\rm{\Delta }}\xi )2\tau$ up to factors of order unity. Thus, for a fixed transfer time ${t}_{f}={n}_{p}\tau =\pi /g$, we expect convergence of the Magnus expansion for ${n}_{p}\gtrsim 2\max (1,{\rm{\Delta }}\xi /g)$.

Absolute convergence of the Magnus expansion does not guarantee that the first few terms of average Hamiltonian theory lead to an accurate description of the evolution operator. Since the terms of average Hamiltonian theory appear within the argument of an exponential in equation (20), correction terms that are small over a single period $2\tau$ may add up to produce large deviations over the entire evolution time ${t}_{f}={n}_{{\rm{p}}}\tau \gg \tau$. However, for SQUADD, the duration of the sequence is set to ${t}_{f}=\pi /g$ through the condition for complete state transfer. This sets a bound on the magnitude of deviations of a truncated Magnus expansion from the exact solution. Indeed, for $k\geqslant 2$, we have $\parallel {\overline{H}}^{(k)}\parallel {t}_{f}\lesssim \max (g{\rm{\Delta }}{\xi }^{k},{g}^{k}{\rm{\Delta }}\xi ){\tau }^{k}(\pi /g)$ $\sim \,\max [{({\rm{\Delta }}\xi /g)}^{k},{\rm{\Delta }}\xi /g]/{n}_{p}^{k}$, where we have used $\tau =\pi /{{gn}}_{p}$. Keeping only the first correction term in the Magnus expansion will then lead to an accurate description of the evolution operator (with $\parallel {\overline{H}}^{(k)}\parallel {t}_{f}\ll 1\ \forall \ k\geqslant 2$) in the limit

$$\begin{eqnarray}&&{n}_{p}\gg \max \left(1,\displaystyle \frac{{\rm{\Delta }}\xi }{g}\right).\end{eqnarray} \tag{ 25 }$$

In this limit, equation (24) already ensures that the expansion of average Hamiltonian theory is convergent. In this same limit, we find that the expansion given in equations (21)–(23) leads to the error given in equation (18), thus coinciding with the exact solution.

The ideal SQUADD sequence analyzed in section 3 assumes that the coupling can be tuned to vanish identically in the 'off' state. As a consequence, all terms in average Hamiltonian theory (equations (21)–(23)) commute with the total number of excitations ${N}_{\mathrm{ex}}$. If there were some residual coupling $g(t)\ne 0$ for n(t) odd, terms that do not commute with ${N}_{\mathrm{ex}}$ would appear in average Hamiltonian theory. Indeed, ${\overline{H}}^{(0)}$ would then contain a term $\propto \,g(a{\sigma }_{-}+{a}^{\dagger }{\sigma }_{+})$. In addition, for a sequence that is not a symmetric cycle $[{H}_{{\rm{T}}}(t)\ne {H}_{{\rm{T}}}(2\tau -t)$], ${\overline{H}}^{(1)}$ would contain, e.g., a cavity-squeezing term $\propto \,{{\rm{i}}{g}}^{2}\tau ({a}^{2}-{a}^{\dagger }\,{}^{2}){\sigma }_{z}$. This squeezing term may be useful, e.g., to enhance the fidelity of a qubit readout [44].

Given a finite off/on ratio ${g}_{\mathrm{off}}/g$ (where $g(t)={g}_{\mathrm{off}}$ for n(t) odd), the term $\propto g(a{\sigma }_{-}+{a}^{\dagger }{\sigma }_{+})$ in ${\overline{H}}^{(0)}$ generates a correction to the error given in equation (18) of order $\sim \,{({g}_{\mathrm{off}}/g)}^{2}$. This correction would ultimately limit the saturation fidelity at large n_p whenever ${g}_{\mathrm{off}}\ne 0$.

In general, over-rotation or under-rotation of the qubit due to imperfect control can lead to an accumulation of errors as the number of pulses n_p is increased. A simple way to avoid accumulation of these pulse errors is to use a phase-alternated sequence [35], in which the qubit rotation direction alternates from one π-pulse to the next. Consequently, the (fixed, deterministic) error $\varepsilon$ on the rotation angle of successive pulses cancels for n(t) even, but introduces a small over-rotation for n(t) odd. We evaluate the resulting correction $\delta F$ to the state-transfer fidelity of equation (18) by expanding $U(t)={ \mathcal T }\exp [-{\rm{i}}{\int }_{0}^{t}{\rm{d}}{t}^{\prime} {H}_{{\rm{T}}}(t^{\prime} )]$ to leading order in $\varepsilon$ and in τ. When $\max (g,{\rm{\Delta }}\xi )\tau \ll 1$, $\delta F$ is then well-approximated by this leading correction:

$$\begin{eqnarray}&&\delta F\simeq -\displaystyle \frac{1}{2}\left(\displaystyle \frac{4}{3}-\displaystyle \frac{1}{\sqrt{2}}\sin \displaystyle \frac{\pi }{\sqrt{2}}\right){\left(\varepsilon \displaystyle \frac{{\rm{\Delta }}\xi }{g}\right)}^{2}.\end{eqnarray} \tag{ 46 }$$

Thus, neglecting order-unity prefactors, pulse errors can be made negligible compared to the error given in equation (18) when $\varepsilon \ll \max [{({\rm{\Delta }}\xi /g)}^{2},1]/{n}_{p}^{2}$.

In this section, we evaluate the state-transfer fidelity using numerical simulations that take into account both the finite bandwidth of the coupling modulation g(t) and the counter-rotating terms in the Rabi Hamiltonian. In these simulations, we find the evolution of the system under the toggling-frame Hamiltonian

$$\begin{eqnarray}{H}_{{\rm{T}}}(t)=\left\{\begin{array}{ll}{H}_{{\rm{T}}}^{\mathrm{even}}(t), & n(t)\ \mathrm{even},\\ {H}_{{\rm{T}}}^{\mathrm{odd}}(t), & n(t)\ \mathrm{odd},\end{array}\right.\end{eqnarray} \tag{ 47 }$$

where

$$\begin{eqnarray}{H}_{{\rm{T}}}^{\mathrm{even}}(t) & = & \displaystyle \frac{1}{2}\xi {\sigma }_{z}+g(t)[{a}^{\dagger }{\sigma }_{-}+{a}^{\dagger }{\sigma }_{+}{{\rm{e}}}^{2{\rm{i}}{\omega }_{{\rm{q}}}t}+{\rm{h}}.{\rm{c}}.],\\ {H}_{{\rm{T}}}^{\mathrm{odd}}(t) & = & -\displaystyle \frac{1}{2}\xi {\sigma }_{z}+g(t)[{a}^{\dagger }{\sigma }_{+}+{a}^{\dagger }{\sigma }_{-}{{\rm{e}}}^{2{\rm{i}}{\omega }_{{\rm{q}}}t}+{\rm{h}}.{\rm{c}}.],\end{eqnarray} \tag{ 48 }$$

with ${\omega }_{{\rm{q}}}$ the qubit frequency (assumed to be tuned to equal the resonator frequency). In contrast with equation (4), equations (48) take into account the counter-rotating terms appearing in the Rabi Hamiltonian. These terms give rise to leakage outside the subspace of zero or one excitation when $g(\omega )\equiv {\int }_{-\infty }^{\infty }{\rm{d}}{t}\exp [{\rm{i}}\omega t]g(t)$ has significant weight at $\omega =2{\omega }_{{\rm{q}}}$.

To take into account the finite bandwidth of the coupling modulations, we convolve the ideal square-wave train of pulses with a Gaussian filter. In particular, we take the coupling to be given by

$$\begin{eqnarray}&&g(t)={{ \mathcal F }}^{-1}[{g}_{\mathrm{id}}(\omega )f(\omega )],\end{eqnarray} \tag{ 49 }$$

where

$$\begin{eqnarray}&&{g}_{\mathrm{id}}(\omega )={ \mathcal F }[{g}_{\mathrm{id}}(t)],\hspace{5mm}f(\omega )=\exp [-{\omega }^{2}/2{\sigma }_{{\rm{f}}}^{2}].\end{eqnarray} \tag{ 50 }$$

In equations (49) and (50), ${{ \mathcal F }}^{(-1)}$ is the (inverse) Fourier transform, ${g}_{\mathrm{id}}(t)$ is an ideal train of square-wave pulses having width ${\tau }^{\prime }$, period $2\tau$, and amplitude g. The function $f(\omega )$ is a Gaussian filter with standard deviation ${\sigma }_{{\rm{f}}}$ which eliminates high-frequency components of ${g}_{\mathrm{id}}(t)$. Evaluating equation (49) gives

$$\begin{eqnarray}&&g(t)=\displaystyle \sum _{j=0}^{{n}_{p}/2}\,{g}_{\mathrm{sq}}(t-2j\tau ),\end{eqnarray} \tag{ 51 }$$

where ${g}_{\mathrm{sq}}(t)$ describes a single filtered square pulse centered around t = 0,

$$\begin{eqnarray}&&{g}_{\mathrm{sq}}(t)=\displaystyle \frac{g}{2}\left\{\mathrm{erf}\left[\displaystyle \frac{{\sigma }_{{\rm{f}}}}{\sqrt{2}}\left(t+\displaystyle \frac{\tau ^{\prime} }{2}\right)\right]-\mathrm{erf}\left[\displaystyle \frac{{\sigma }_{{\rm{f}}}}{\sqrt{2}}\left(t-\displaystyle \frac{\tau ^{\prime} }{2}\right)\right]\right\}.\end{eqnarray} \tag{ 52 }$$

In equation (52), $\tau ^{\prime}$ may differ from the pulse interval τ (figure 4(a)). Neglecting corrections that are suppressed exponentially with ${\sigma }_{{\rm{f}}}\tau ^{\prime}$, the time required for the coupling to rise from 10% to 90% of its final value is, using equation (52),

$$\begin{eqnarray}&&{t}_{{\rm{r}}}\simeq \displaystyle \frac{2\sqrt{2}\ {\mathrm{erf}}^{-1}(4/5)}{{\sigma }_{{\rm{f}}}}\simeq \displaystyle \frac{2.563\,103}{{\sigma }_{{\rm{f}}}}.\end{eqnarray} \tag{ 53 }$$

When ${\sigma }_{{\rm{f}}}\lt {\omega }_{{\rm{q}}}$, the spectral weight of $g(\omega )$ is suppressed at $\omega =2{\omega }_{{\rm{q}}}$ and the influence of the counter-rotating terms becomes negligible. When ${\sigma }_{{\rm{f}}}\gt 1/{\tau }^{\prime }\sim 1/\tau$, the pulse train g(t) closely approximates a sequence of square-wave pulses. Crucially, there is always the possibility for a separation between ${\omega }_{{\rm{q}}}$, ${\sigma }_{{\rm{f}}}$, and $1/\tau$, allowing both conditions to be satisfied simultaneously:

$$\begin{eqnarray}&&{\omega }_{{\rm{q}}}\gt {\sigma }_{{\rm{f}}}\gt \displaystyle \frac{1}{\tau }.\end{eqnarray} \tag{ 54 }$$

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To verify the analysis given above, we numerically evaluate the fidelity of a state transfer using equation (5), considering evolution under the toggling-frame Hamiltonian given in equation (47), which fully accounts for counter-rotating terms. For a finite pulse rise time, ${t}_{{\rm{r}}}\ne 0$, g(t) will be non-zero even for n(t) odd. To suppress the resulting unwanted excitations of the qubit and cavity, we then take $\tau ^{\prime} =\tau -{t}_{{\rm{r}}}$ ($\tau ^{\prime}$ and ${t}_{{\rm{r}}}$ are illustrated in figure 4(a)). Consequently, the time-averaged coupling $\overline{g}={\int }_{0}^{2\tau }{\rm{d}}{t}\,g(t)/2\tau$ also decreases; the pulse interval that results in a complete state transfer is then obtained by numerically solving $\overline{g}{n}_{p}\tau =\pi /2$ for a given value of g and n_p. The resulting state-transfer error is shown in figure 4(b) (blue dots) as a function of ${\sigma }_{{\rm{f}}}/g$ for n_p = 100, ${{gT}}_{2}^{* }=1/10$, and $\kappa =0$. As ${\sigma }_{{\rm{f}}}/g$ increases, the error decreases, approaching the value given by equation (18) for perfectly square modulation of g(t) (dashed black line). Additional error due to a finite pulse rise time becomes negligible compared to the error already present for an ideal pulse (for this choice of parameters) for ${\sigma }_{{\rm{f}}}\gtrsim 500\,g$. Even when the bandwidth is too narrow for saturation to have occurred, ${\sigma }_{{\rm{f}}}\lt 500\,g$, figure 4(b) shows that the error due to inhomogeneous broadening can be suppressed substantially (without SQUADD, error would be of order 1 for ${{gT}}_{2}^{* }=1/10$). In the simulations, we have taken ${\omega }_{{\rm{q}}}=2000\,g$. This choice allows us to filter out the effect of the counter-rotating terms over the entire parameter range of the simulation, guaranteeing that ${\sigma }_{{\rm{f}}}\lt {\omega }_{{\rm{q}}}$ is satisfied.

To give a concrete example, taking $g/2\pi =1\,\mathrm{MHz}$ and ${\sigma }_{{\rm{f}}}=100\,g$, the parameters used in the simulation presented in figure 4(b) correspond to ${\sigma }_{{\rm{f}}}/2\pi =100\,\mathrm{MHz}$, ${\omega }_{{\rm{q}}}/2\pi =2\,\mathrm{GHz}$, and ${T}_{2}^{* }\simeq 16$ ns. Numerically solving $\overline{g}{n}_{p}\tau =\pi /2$ then gives $\overline{g}/2\pi \simeq 0.27\,\mathrm{MHz}$ and $\tau \simeq 9$ ns for n_p = 100. Even for this narrow bandwidth (which leads to ${t}_{{\rm{r}}}\simeq 4$ ns), our simulation yields a relatively small error, $1-F\simeq {10}^{-3}$.

In typical experimental settings, the finite duration of π-pulses poses an important practical limitation to the achievable fidelity of quantum operations [13, 45–49]. Therefore, in this section, we numerically evaluate the state-transfer fidelity under SQUADD as a function of the π-pulse duration.

With the cavity and the qubit on resonance, and working in the rotating frame, we describe qubit π-pulses with finite duration using the Hamiltonian

$$\begin{eqnarray}&&{H}_{{\rm{c}}}(t)=\displaystyle \frac{w(t)}{2}{\sigma }_{x},\end{eqnarray} \tag{ 55 }$$

valid under the rotating-wave approximation for the qubit drive. In equation (55), we have introduced the drive amplitude, w(t). Within the rotating-wave approximation for the qubit–cavity coupling (which is justified for ${\sigma }_{{\rm{f}}}\lt {\omega }_{{\rm{q}}}$, as explained above), substituting ${U}_{{\rm{c}}}(t)=\exp [-{\rm{i}}{\int }_{0}^{t}{{\rm{d}}{t}}_{1}w({t}_{1}){\sigma }_{x}/2]$ into equation (3) leads to the toggling-frame Hamiltonian

$$\begin{eqnarray}{H}_{{\rm{T}}}(t) & = & \displaystyle \frac{\xi }{2}[\cos \theta (t){\sigma }_{z}+\sin \theta (t){\sigma }_{y}]\\ & & +g(t)\left[\displaystyle \frac{1+\cos \theta (t)}{2}({a}^{\dagger }{\sigma }_{-}+a{\sigma }_{+})+\displaystyle \frac{1-\cos \theta (t)}{2}({a}^{\dagger }{\sigma }_{+}+a{\sigma }_{-})+{\rm{i}}\sin \theta (t)\displaystyle \frac{{a}^{\dagger }-a}{2}{\sigma }_{z}\right],\end{eqnarray} \tag{ 56 }$$

where we have introduced the qubit rotation angle

$$\begin{eqnarray}&&\theta (t)={\int }_{0}^{t}{{\rm{d}}{t}}_{1}\,w({t}_{1}).\end{eqnarray} \tag{ 57 }$$

We take w(t) to describe a train of identical pulses. For numerical evaluation, we consider perfect rectangular pulses of amplitude w₀ and duration ${t}_{{\rm{p}}}$ centered around $\left(m+\tfrac{1}{2}\right)\tau$, $m\ \in \ {\mathbb{N}}$ (figure 4(a)). For ${w}_{0}{t}_{{\rm{p}}}=\pi$, evolution under ${H}_{{\rm{T}}}(t)$ leads to π-pulses. We retrieve the toggling-frame Hamiltonian for instantaneous π-pulses (equation (4)) when taking the limit ${t}_{{\rm{p}}}\to 0$ in ${H}_{{\rm{T}}}(t)$ given by equation (56) while requiring ${w}_{0}{t}_{{\rm{p}}}=\pi$. In plots and numerical evaluations, we take g(t) to be given by equation (51), above. The strategies proposed here for error suppression with a finite pulse duration are, however, independent of the specific shape of w(t) and g(t). Importantly, the counter-rotating $[\propto {a}^{\dagger }{\sigma }_{+}+a{\sigma }_{-}]$ and cavity-displacement $[\propto ({a}^{\dagger }-a){\sigma }_{z}]$ terms in equation (56) both vanish for times t satisfying $\theta (t)=2j\pi$ (with $j\,\in \,{\mathbb{N}}$), at which the state-transfer is complete. These unwanted terms are, however, finite for $\theta (t)\ne 2j\pi$ (i.e., during π-pulses and for n(t) odd). The error that results can be suppressed by approximately turning off the coupling during the qubit π-pulses (by choosing, e.g., $\tau ^{\prime} =\tau -{t}_{{\rm{r}}}-{t}_{{\rm{p}}}$ as shown in figure 4(a)), in addition to turning the coupling off for n(t) odd. Any error arising from a finite pulse duration is then predominantly due to the term $\propto \,\xi \sin \theta (t){\sigma }_{y}$ in equation (56).

As in section 5.3, we numerically evaluate the fidelity F (equation (5)) of a quantum state transfer generated by the toggling-frame Hamiltonian given in equation (56). Figure 4(c) displays the resulting error $1-F$ as a function of ${t}_{{\rm{p}}}$ (blue dots) for n_p = 100, ${{gT}}_{2}^{* }=1/10$, ${\sigma }_{{\rm{f}}}/g=1000$, $\tau ^{\prime} =\tau -{t}_{{\rm{r}}}-{t}_{{\rm{p}}}$, and $\kappa =0$. To suppress the error below 1%, ${{gt}}_{{\rm{p}}}/2\pi \lt {10}^{-4}$ is required for these parameters. For, e.g., $g/2\pi =1\,\mathrm{MHz}$, this level of error suppression would require ${t}_{{\rm{p}}}\lt 0.1$ ns. Realizing such short pulses may be challenging experimentally.

To explain this poor suppression of error in the limit of small ${t}_{{\rm{p}}}$, we note that, for the pulse sequence described below equation (57) (top-left inset in figure 4(c)), ${H}_{{\rm{T}}}(t)$ does not result in a symmetric cycle (in the sense described in section 3.3). Indeed, for this sequence with period $T=2\tau$, $\theta (T-t)=2\pi -\theta (t)$, leading to $\sin \theta (T-t)=-\sin \theta (t)$ and thus to ${H}_{{\rm{T}}}(T-t)\ne {H}_{{\rm{T}}}(t)$ (see equation (56)). We thus expect the error to be finite at first order in average Hamiltonian theory [35], ${\overline{H}}^{(1)}\ne 0$ (see the discussion following equation (23)). This sequence only becomes a symmetric cycle (i.e., ${H}_{{\rm{T}}}(T-t)={H}_{{\rm{T}}}(t)$, leading to ${\overline{H}}^{(1)}=0$) in the limit ${t}_{{\rm{p}}}\to 0$.

A symmetric cycle (leading to ${\overline{H}}^{(1)}=0$) may be obtained using the phase-alternated sequence described in section 5.2, in which the qubit rotation direction (i.e. the sign of $w(t)$) alternates from one π-pulse to the next. However, for this sequence, ${\int }_{0}^{T}{{\rm{d}}{t}}_{1}\sin \theta ({t}_{1})\ne 0$, leading to finite error at zeroth order (${\overline{H}}^{(0)}$) for any pulse with finite ${t}_{{\rm{p}}}$ due to the term $\propto \,\sin \theta (t){\sigma }_{y}$ in equation (56). A sequence that is a symmetric cycle (leading to ${\overline{H}}^{(1)}=0$) and for which ${\int }_{0}^{T}{{\rm{d}}{t}}_{1}\sin \theta ({t}_{1})=0$ (leading to vanishing error at zeroth order) is obtained when:

• (i)  
  w(t) is periodic and odd, $w(t)=-w(-t);$
• (ii)  
  w(t) describes identical π-pulses that come in pairs with common phase, corresponding to successive rotations about $+\hat{x},\ +\hat{x},\ -\hat{x},\ -\hat{x}$, ... , with no specific assumption about the pulse shape except that $w(t)\gt 0\ \forall \ t\in \ [0,\tau ]$.

Error due to finite pulse duration then arises entirely from terms of order ${\overline{H}}^{(2)}$ or higher. A sequence that fulfills the above criteria is shown in the bottom-right inset in figure 4(c), in which we consider the specific example of square pulses.

As expected from the above discussion, the phase-alternated sequence shown in the bottom-right inset in figure 4(c) leads to an error (red triangles in figure 4(c)) that is significantly reduced compared with the original fixed-phase sequence (blue dots in figure 4(c)). The error is also significantly more strongly suppressed in the limit of small ${t}_{{\rm{p}}}$ for the phase-alternated sequence, relative to the fixed-phase sequence. For the same parameters as above, an error $\lt 1 \%$ is obtained for ${{gt}}_{{\rm{p}}}/2\pi ={10}^{-3}$, corresponding to ${t}_{{\rm{p}}}=1$ ns for $g/2\pi =1\,\mathrm{MHz}$. For ${{gt}}_{{\rm{p}}}/2\pi \lt {10}^{-3}$, the error quickly decreases and plateaus near the value given by equation (18). Finally, as explained in section 5.2, phase-alternated sequences have the additional advantage of suppressing an accumulation of deterministic pulse errors.

When a qubit is successively prepared and measured $m\gg 1$ times to estimate an expectation value, the measurement statistics describing the mean of many independent repeated measurements (a so-called 'soft average' [54, 55]) become Gaussian due to the central limit theorem. The performance of the readout is then well-characterized by the SNR. For a qubit being measured through a cavity, as considered here, the SNR compares the first two moments of the measurement operator

$$\begin{eqnarray}&&M={\rm{i}}\sqrt{\kappa }{\int }_{0}^{{t}_{f}}{\rm{d}}{t}[{a}_{\mathrm{out}}^{\dagger }(t)-{a}_{\mathrm{out}}(t)],\end{eqnarray} \tag{ 60 }$$

which gives the integrated homodyne-detection signal for a measurement time t_f, with ${a}_{\mathrm{out}}(t)$ the output field leaking from the cavity with damping rate $\kappa$. These first two moments of M are quantified by the measurement signal X and noise Ξ,

$$\begin{eqnarray}&&X=| \langle M{\rangle }_{+}-\langle M{\rangle }_{-}| ,\end{eqnarray} \tag{ 61 }$$

$$\begin{eqnarray}&&{\rm{\Xi }}={({\rm{\Delta }}{M}_{+}^{2}+{\rm{\Delta }}{M}_{-}^{2})}^{1/2},\end{eqnarray} \tag{ 62 }$$

$$\begin{eqnarray}&&{\rm{\Delta }}{M}_{\pm }^{2}=\langle {M}^{2}{\rangle }_{\pm }-\langle M{\rangle }_{\pm }^{2},\end{eqnarray} \tag{ 63 }$$

where $\langle O{\rangle }_{\pm }\equiv \mathrm{tr}[O({t}_{f}){\rho }_{\pm }(0)]$ and ${\rho }_{\pm }(0)\equiv | \pm \rangle \langle \pm {| }_{{\rm{q}}}\otimes | 0\rangle \langle 0{| }_{{\rm{c}}}$. The SNR is then simply

$$\begin{eqnarray}&&\mathrm{SNR}=X/{\rm{\Xi }}.\end{eqnarray} \tag{ 64 }$$

In this section, we evaluate X and Ξ for the readout scheme described below equation (59), accounting for the first two nonvanishing orders in the Magnus expansion for the time-periodic Liouvillian ${ \mathcal L }$ (defined explicitly in equation (68), below): $\overline{{ \mathcal L }}={\overline{{ \mathcal L }}}^{(0)}+{\overline{{ \mathcal L }}}^{(2)}$. We will show that while ${\overline{{ \mathcal L }}}^{(0)}$ generates the required conditional coherent-state displacement, ${\overline{{ \mathcal L }}}^{(2)}$ results in qubit switching at a rate ${\rm{\Gamma }}\simeq {g}^{2}{\tau }^{2}\kappa /24$ in the basis $\{| \pm {\rangle }_{{\rm{q}}}\}$. This qubit switching acts as a source of telegraph noise in the Langevin equation for the cavity field a(t) [56]. We will evaluate the SNR including noise fom qubit switching and show that for a sufficiently short pulse interval, $\kappa \tau \lt 1$, the readout considered here can result in a large SNR $(\mathrm{SNR}\gt 1)$ even in the weak-coupling regime ($g\lt \kappa$).

To evaluate the SNR for a given measurement scheme, it is useful to relate M to the input field ${a}_{\mathrm{in}}(t)$ and to the field a(t) inside the cavity. This relation is given by the input–output formula [56]

$$\begin{eqnarray}&&{a}_{\mathrm{out}}(t)={a}_{\mathrm{in}}(t)+\sqrt{\kappa }\,a(t).\end{eqnarray} \tag{ 65 }$$

Assuming that the input is vacuum, substitution of equation (65) into (60) gives

$$\begin{eqnarray}&&\langle M{\rangle }_{\pm }={\rm{i}}\kappa {\int }_{0}^{{t}_{f}}{\rm{d}}{t}[\langle {a}^{\dagger }(t){\rangle }_{\pm }-\langle a(t){\rangle }_{\pm }],\end{eqnarray} \tag{ 66 }$$

$$\begin{eqnarray}&&\langle {M}^{2}{\rangle }_{\pm }=\kappa {t}_{f}+2{\kappa }^{2}{\int }_{0}^{{t}_{f}}{{\rm{d}}{t}}_{1}{\int }_{0}^{{t}_{f}-{t}_{1}}{{\rm{d}}{t}}_{2}[\langle {a}^{\dagger }({t}_{1}+{t}_{2})a({t}_{1}){\rangle }_{\pm }-\langle a({t}_{1}+{t}_{2})a({t}_{1}){\rangle }_{\pm }+{\rm{h}}.{\rm{c}}.].\end{eqnarray} \tag{ 67 }$$

Equations (66) and (67) relate $\langle M{\rangle }_{\pm }$ and $\langle {M}^{2}{\rangle }_{\pm }$—and thus the SNR—to simple expectation values and autocorrelation functions of the cavity field a(t). Employing standard formulas [56], these expectation values and autocorrelation functions are easily calculated knowing the time-evolution superoperator $V(t,{t}_{0})$, generating evolution for the qubit–cavity density matrix. We find $V(t,{t}_{0})$ by solving the (time-inhomogeneous) master equation

$$\begin{eqnarray}&&\dot{V}(t,{t}_{0})={ \mathcal L }(t)V(t,{t}_{0}).\end{eqnarray} \tag{ 68 }$$

In equation (68), we have introduced the Liouvillian ${ \mathcal L }(t)$ describing cavity damping at rate $\kappa$ and unitary evolution under the qubit–cavity toggling-frame Hamiltonian, ${H}_{{\rm{T}}}(t)$,

$$\begin{eqnarray}&&{ \mathcal L }(t)\cdot =\,-{\rm{i}}[{H}_{{\rm{T}}}(t),\cdot ]+\kappa { \mathcal D }[a]\cdot ,\end{eqnarray} \tag{ 69 }$$

$$\begin{eqnarray}&&{ \mathcal D }[a]\cdot =\,a\cdot {a}^{\dagger }-\displaystyle \frac{1}{2}({a}^{\dagger }a\cdot +\,\cdot \,{a}^{\dagger }a),\end{eqnarray} \tag{ 70 }$$

where the centerdot ('·') represents an arbitrary operator upon which the relevant superoperator is applied. In equation (69), ${H}_{{\rm{T}}}(t)$ is given by equation (4), taking $\xi =0$ and $g(t)=g\,\forall \,t$.

To evaluate $V(t,{t}_{0})$ analytically, we assume that $t-{t}_{0}=2{n}_{p}\tau$. We then use the Magnus expansion

$$\begin{eqnarray}&&V(t,{t}_{0})=\exp \left[\displaystyle (t-{t}_{0})\sum _{\,k=0}^{\infty }{\overline{{ \mathcal L }}}^{(k)}\right].\end{eqnarray} \tag{ 71 }$$

As in average Hamiltonian theory, the terms ${\overline{{ \mathcal L }}}^{(k)}$ are time-independent because ${ \mathcal L }(t)$ is periodic, ${ \mathcal L }(t+2\tau )={ \mathcal L }(t)\forall t$. The first few terms of the expansion in ${\overline{{ \mathcal L }}}^{(k)}$ are obtained by replacing ${\overline{H}}^{(k)}\to {\rm{i}}{\overline{{ \mathcal L }}}^{(k)}$ and ${H}_{{\rm{T}}}(t)\to {\rm{i}}{ \mathcal L }(t)$ in equations (21)–(23) [41].

To gain insight into the problem, we evaluate the SNR to leading order in the Magnus expansion, equation (71). In this first approach, we neglect any qubit decay that may arise from higher-order terms in the expansion. We then find

$$\begin{eqnarray}&&V(t,{t}_{0})=\exp [(t-{t}_{0}){\overline{{ \mathcal L }}}^{(0)}],\end{eqnarray} \tag{ 72 }$$

$$\begin{eqnarray}&&{\overline{{ \mathcal L }}}^{(0)}\cdot =\,-{\rm{i}}[{\overline{H}}^{(0)},\cdot ]+\kappa { \mathcal D }[a]\cdot ,\end{eqnarray} \tag{ 73 }$$

$$\begin{eqnarray}&&{\overline{H}}^{(0)}=\displaystyle \frac{g}{2}(a+{a}^{\dagger }){\sigma }_{x}.\end{eqnarray} \tag{ 74 }$$

According to equations (72) and (73), the qubit forever remains in its initial state $| \pm \rangle \langle \pm {| }_{{\rm{q}}}$. For $\kappa {t}_{f}\gg 1$, the cavity correspondingly settles in the coherent state $| \alpha \rangle \langle \alpha {| }_{{\rm{c}}}=| \mp {\rm{i}}{g}/\kappa \rangle \langle \mp {\rm{i}}{g}/\kappa {| }_{{\rm{c}}}$. Since the cavity field leaks from the output port at a rate $\kappa /2$, this steady state leads to $X\propto \kappa {t}_{f}\times g/\kappa \propto {{gt}}_{f}$. In addition, noise in the output field then entirely comes from shot noise: ${\rm{\Delta }}{M}_{\pm }^{2}=\kappa {t}_{f}$, giving [44]

$$\begin{eqnarray}&&X=4{{gt}}_{f},\hspace{3mm}{\rm{\Xi }}=\sqrt{2\kappa {t}_{f}}\hspace{2.5mm}\Rightarrow \hspace{2.5mm}\mathrm{SNR}\propto \sqrt{{t}_{f}}.\end{eqnarray} \tag{ 75 }$$

Therefore, in this ideal scenario, signal always accumulates faster than noise, making it possible to achieve arbitrarily large SNR simply by increasing t_f.

In practice, qubit relaxation leads to a saturation of the signal and to an enhancement of the noise, thus limiting the achievable SNR. Qubit relaxation can be intrinsic, coming from coupling of the qubit to a decay channel independent of the cavity. Higher-order corrections to the leading-order Magnus expansion taken here also lead to qubit decay via the cavity. This can be seen by means of a short-time expansion of $\langle {\sigma }_{x}(t){\rangle }_{\pm }$. Indeed, the term of order ${ \mathcal O }(t)$ in this short-time expansion gives decay at a rate analogous to that of Purcell decay:

$$\begin{eqnarray}&&{\rm{\Gamma }}\equiv {\left|\displaystyle \frac{{\rm{d}}}{{\rm{d}}{t}}\langle {\sigma }_{x}(t){\rangle }_{\pm }\right|}_{t\to 0}\simeq | \mathrm{tr}\{[{{\overline{{ \mathcal L }}}^{(2)}}^{\dagger }{\sigma }_{x}]{\rho }_{\pm }(0)\}| =\displaystyle \frac{{g}^{2}{\tau }^{2}}{24}\kappa .\end{eqnarray} \tag{ 76 }$$

The term of order ${ \mathcal O }(t)$ in the above expansion of $\langle {\sigma }_{x}(t){\rangle }_{\pm }$ dominates over the correction term of order ${ \mathcal O }({t}^{2})$ when $\kappa t\lt 256/[3{(\kappa \tau )}^{2}]$.

To take qubit relaxation into account in the calculation of the SNR, we employ the Langevin equation for the cavity field a(t), considering the average Hamiltonian ${\overline{H}}^{(0)}$ in equation (74). This gives

$$\begin{eqnarray}&&\dot{a}(t)+\displaystyle \frac{\kappa }{2}a(t)=-{\rm{i}}\displaystyle \frac{g}{2}{\sigma }_{x}(t)-\sqrt{\kappa }{a}_{\mathrm{in}}(t).\end{eqnarray} \tag{ 77 }$$

Equation (77) has the form of the equation of motion of a Brownian particle with mass m, momentum p, and friction coefficient $\gamma$: $\dot{p}+(\gamma /m)p=\eta (t)$ [56]. In equation (77), the fluctuating force $\eta (t)$ comes from a combination of shot noise from the input field ${a}_{\mathrm{in}}(t)$ and telegraph noise from the qubit through the Heisenberg-picture operator ${\sigma }_{x}(t)$. We assume that the qubit–cavity coupling is turned on at time t = 0 and that the cavity interacts with its environment starting in the distant past, at $t\to -\infty$. The solution to equation (77) is then

$$\begin{eqnarray}&&a(t)=-{\rm{i}}\displaystyle \frac{g}{2}{\int }_{0}^{t}{\rm{d}}{t}^{\prime} {{\rm{e}}}^{-\kappa (t-t^{\prime} )/2}{\sigma }_{x}(t^{\prime} )-\sqrt{\kappa }{\int }_{-\infty }^{t}{\rm{d}}{t}^{\prime} {{\rm{e}}}^{-\kappa (t-t^{\prime} )/2}{a}_{\mathrm{in}}(t^{\prime} ).\end{eqnarray} \tag{ 78 }$$

For a qubit undergoing simultaneous excitation and relaxation at equal rates ${\rm{\Gamma }}/2$ in the eigenbasis of ${\sigma }_{x}$, we have

$$\begin{eqnarray}&&\langle {\sigma }_{x}(t){\rangle }_{\pm }=\pm \exp (-{\rm{\Gamma }}t),\end{eqnarray} \tag{ 79 }$$

$$\begin{eqnarray}&&\langle {\sigma }_{x}(t){\sigma }_{x}(t^{\prime} ){\rangle }_{\pm }=\exp (-{\rm{\Gamma }}| t-t^{\prime} | ).\end{eqnarray} \tag{ 80 }$$

Substituting equations (78)–(80) into equations (66) and (67), we evaluate the signal and noise using equations (61) and (62). We find

$$\begin{eqnarray}&&X=\displaystyle \frac{2g\kappa }{\kappa /2-{\rm{\Gamma }}}\left(\displaystyle \frac{1-{{\rm{e}}}^{-{\rm{\Gamma }}{t}_{f}}}{{\rm{\Gamma }}}-\displaystyle \frac{1-{{\rm{e}}}^{-\kappa {t}_{f}/2}}{\kappa /2}\right),\end{eqnarray} \tag{ 81 }$$

$$\begin{eqnarray}&&{\rm{\Xi }}=\sqrt{2\kappa {t}_{f}+\displaystyle \frac{4{g}^{2}{\kappa }^{2}}{({\kappa }^{2}/4-{{\rm{\Gamma }}}^{2}){{\rm{\Gamma }}}^{2}}f({t}_{f})-\displaystyle \frac{{X}^{2}}{2}},\end{eqnarray} \tag{ 82 }$$

where we have introduced

$$\begin{eqnarray}f(t) & = & {\rm{\Gamma }}t-\displaystyle \frac{{\rm{\Gamma }}}{{\rm{\Gamma }}+\kappa /2}\left(1+\displaystyle \frac{2{\rm{\Gamma }}}{\kappa }\right)[1-{{\rm{e}}}^{-({\rm{\Gamma }}+\kappa /2)t}]-\left(1-\displaystyle \frac{2{\rm{\Gamma }}}{\kappa }{{\rm{e}}}^{-\kappa t/2}\right)(1-{{\rm{e}}}^{-{\rm{\Gamma }}t})\\ & & +\displaystyle \frac{2{\rm{\Gamma }}}{\kappa }(1-{{\rm{e}}}^{-\kappa t/2})\left[{{\rm{e}}}^{-{\rm{\Gamma }}t}+\displaystyle \frac{{\rm{\Gamma }}}{\kappa }(1-{{\rm{e}}}^{-\kappa t/2})\right]-\displaystyle \frac{4{{\rm{\Gamma }}}^{2}}{{\kappa }^{2}}\left[{\rm{\Gamma }}t-\displaystyle \frac{{\rm{\Gamma }}}{\kappa }(3-4{{\rm{e}}}^{-\kappa t/2}+{{\rm{e}}}^{-\kappa t})\right].\end{eqnarray} \tag{ 83 }$$

To simplify the above expressions, we expand X and $f({t}_{f})$ to leading order in ${\rm{\Gamma }}{t}_{f}$ around ${\rm{\Gamma }}{t}_{f}=0$. We also assume that the cavity has reached its steady state; we thus have ${\rm{\Gamma }}{t}_{f}\ll 1\ll \kappa {t}_{f}$. Therefore, in equations (81) and (83), we drop corrections that are exponentially small for $\kappa {t}_{f}\gg 1$. In equation (81), we also drop terms of order ${ \mathcal O }({\rm{\Gamma }}/\kappa )$ or higher, which do not change the dependence of X and Ξ on t_f. However, in equation (83), we keep the terms of order ${ \mathcal O }({\rm{\Gamma }}/\kappa )$, since they grow faster than linearly with t_f, but drop corrections of order ${ \mathcal O }({{\rm{\Gamma }}}^{2}/{\kappa }^{2})$ or higher. We then find

$$\begin{eqnarray}&&X\simeq 4{{gt}}_{f},\hspace{5mm}{\rm{\Xi }}\simeq \sqrt{2\kappa {t}_{f}+\displaystyle \frac{16}{3}{g}^{2}{\rm{\Gamma }}{t}_{f}^{3}}.\end{eqnarray} \tag{ 84 }$$

Equation (84) shows that ${{\rm{\Xi }}}^{2}$ contains two terms: one from photon shot noise, $\propto \,\kappa {t}_{f}$, and an additional contribution from qubit switching, $\propto \,{g}^{2}{\rm{\Gamma }}{t}_{f}^{3}$. Therefore, including qubit switching, the noise grows faster than the signal ($\propto \,{t}_{f}$) for sufficiently large t_f. This is visible in figure 5(a), in which we plot $X({t}_{f})$ and ${\rm{\Xi }}({t}_{f})$ resulting from an exact numerical solution of the master equation given by equation (68). In figure 5(a), $X({t}_{f})$ and ${\rm{\Xi }}({t}_{f})$ are represented by the solid black line and the dotted red line, respectively. Using the dashed blue line, we also plot ${\rm{\Xi }}({t}_{f})=\sqrt{2\kappa {t}_{f}}$, expected for pure photon shot noise, equation (75). Clearly, excess noise due to qubit decay determines the optimal measurement time ${t}_{\mathrm{opt}.}$ that maximizes the SNR (shown by the double arrow in figure 5(a)). We evaluate ${t}_{\mathrm{opt}.}$ analytically by maximizing $\mathrm{SNR}=X/{\rm{\Xi }}$, with X and Ξ given by equation (84). We find

$$\begin{eqnarray}&&{\rm{\Gamma }}{t}_{\mathrm{opt}.}\simeq \displaystyle \frac{1}{2}\sqrt{\displaystyle \frac{3}{2}}\displaystyle \frac{\sqrt{\kappa {\rm{\Gamma }}}}{g},\end{eqnarray} \tag{ 85 }$$

$$\begin{eqnarray}&&\mathrm{SNR}\simeq {\left(\displaystyle \frac{6{g}^{2}}{\kappa {\rm{\Gamma }}}\right)}^{1/4},\hspace{1.5cm}({t}_{f}={t}_{\mathrm{opt}.}).\end{eqnarray} \tag{ 86 }$$

Equation (86) provides a simple relationship between the maximal SNR and the cooperativity $C\equiv {g}^{2}/\kappa {\rm{\Gamma }}$, $\mathrm{SNR}\simeq {(6C)}^{1/4}$.

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Equation (86) gives the maximal SNR when the qubit undergoes switching in the eigenbasis of ${\sigma }_{x}$. As seen above, this can be due to the subleading term ${\overline{{ \mathcal L }}}^{(2)}$ in the Magnus expansion, equation (71), which leads the qubit to decay at the rate given in equation (76). When this mechanism is the dominant source of decay in the eigenbasis of ${\sigma }_{x}$, we substitute equation (76) into (86) to find the corresponding optimal SNR,

$$\begin{eqnarray}&&\mathrm{SNR}\simeq \displaystyle \frac{2\sqrt{3}}{\sqrt{\kappa \tau }},\hspace{1.5cm}({t}_{f}={t}_{\mathrm{opt}.})\end{eqnarray} \tag{ 87 }$$

valid for ${\rm{\Gamma }}{t}_{f}\ll 1\ll \kappa {t}_{f}\ll 256/[3{(\kappa \tau )}^{2}]$. The last inequality arises from the short-time expansion performed in equation (76). Equation (87) implies that $\mathrm{SNR}\gt 1$ is achievable even in the weak-coupling regime, $g\lt \kappa$. This result is shown in figure 5(b), in which we plot the maximal SNR obtained from an exact numerical solution of equation (68) as the black dots for $g/\kappa =1/10$. This numerical result is in good agreement with the optimal SNR given in equation (87), displayed as the dashed blue line.

We now discuss conditions under which the Magnus expansion used here (equation (71)) converges. The Magnus expansion converges when ${\int }_{0}^{2\tau }{\rm{d}}{t}\parallel { \mathcal L }(t)\parallel \lt \pi$ [41]. For $g\lt \kappa$, the steady-state cavity population is small: $\langle {a}^{\dagger }a{(t)\rangle =(g/\kappa )}^{2}\lt 1$ for $\kappa {t}_{f}\gg 1$. In this situation, we can represent the operators a and ${a}^{\dagger }$ by truncated matrices of small dimension, making $\parallel {a}^{(\dagger )}\parallel \sim 1$. This implies that $\parallel { \mathcal L }(t)\parallel \sim \kappa$, and we conclude that the Magnus expansion converges for $\kappa \tau \lesssim \pi /2$ under the assumption that $g\lt \kappa$. This statement is consistent with figure 5(b), which shows excellent agreement between the exact numerical solution (black dots) and equation (87) (dashed blue line) for $\kappa \tau \lt \pi /2$.

In contrast to the case of many repeated measurements (described above), for a single-shot readout, the measurement statistics are typically non-Gaussian. Indeed, while the conditional probability distribution describing the integrated signal $\langle M{\rangle }_{\pm }$ would simply describe a displaced Gaussian in the absence of switching, random switching events (e.g. qubit decay due to the mechanism described above) lead to significant bimodality [57]. To characterize readout, the full probability distribution of the measurement outcomes is then needed; the first and second moments (characterized by equations (61) and (62)) are typically not sufficent. A good measure of quality that accounts for the full probability distribution is the single-shot fidelity. To evaluate the fidelity, we use a readout model that takes into account qubit switching at symmetric rates ${\rm{\Gamma }}/2$, where ${\rm{\Gamma }}$ is given by equation (76) [57, 58]. In the same regime as above ($g\ll \kappa$ and $\kappa \tau \ll 1$), this leads to a single-shot fidelity that converges asymptotically to

$$\begin{eqnarray}&&{F}_{1}=1-\displaystyle \frac{{(\kappa \tau )}^{2}}{192}\left[\mathrm{log}\displaystyle \frac{96}{{(\kappa \tau )}^{2}}+{ \mathcal O }(| \mathrm{log}\kappa \tau {| }^{-1/2})\right]\end{eqnarray} \tag{ 88 }$$

as $\kappa \tau \to 0$. For $\kappa \tau =0.1$, this yields a single-shot fidelity of 99.95%, showing that the error due to the first correction term in the Magnus expansion is rapidly suppressed in the limit of short pulse intervals.

Equation (88) also shows that the readout proposed here can have a high single-shot fidelity even in the weak-coupling regime ($g\lt \kappa$). This readout may then be useful in several novel experimental settings where it is challenging to achieve strong coupling. For example, a spin qubit in a carbon nanotube has recently been successfully coupled to a microwave resonator, but the coupling achieved is marginal, $g/\kappa \sim 1$ [5]. Alternative setups for semiconductor spin qubits in quantum dots or at single donor impurities coupled to microwave cavities have predicted couplings $g/2\pi \lesssim 1\,\mathrm{MHz}$ [15, 18, 43], typically smaller than the damping rate $\kappa /2\pi =2\mbox{--}10$ MHz [59, 60].