Shortcuts to adiabaticity for open systems in circuit quantum electrodynamics

Shortcuts to adiabaticity are powerful quantum control methods, allowing quick evolution into target states of otherwise slow adiabatic dynamics. Such methods have widespread applications in quantum technologies, and various shortcuts to adiabaticity protocols have been demonstrated in closed systems. However, realizing shortcuts to adiabaticity for open quantum systems has presented a challenge due to the complex controls in existing proposals. Here, we present the experimental demonstration of shortcuts to adiabaticity for open quantum systems, using a superconducting circuit quantum electrodynamics system. By applying a counterdiabatic driving pulse, we reduce the adiabatic evolution time of a single lossy mode from 800 ns to 100 ns. In addition, we propose and implement an optimal control protocol to achieve fast and qubit-unconditional equilibrium of multiple lossy modes. Our results pave the way for precise time-domain control of open quantum systems and have potential applications in designing fast open-system protocols of physical and interdisciplinary interest, such as accelerating bioengineering and chemical reaction dynamics.


SUPPLEMENTARY NOTE 1. EXACT INPUT-OUTPUT THEORY
Here we derive the exact input-output formula used to simulate the output signal of the system shown in Fig. 1. Our analysis follows that of Ref. [1], although we additionally account for the distance l from the input capacitor C in to the filter a, which is necessary for the theory to match experimental observations. For wavenumber k, the phase accumulated after passing through this distance is θ = k ×l. We find that this phase has a profound effect on the final output signal. Our goal is to determine how the output field r o responds to the input field c i and its interaction with the system, including filter mode a, the resonator mode b and the qubit state. To simplify the calculation, instead of directly including the qubit state, we will account for its effect by modifying other system mode frequencies.
To build the mode network, we start from the most left input port and consider the transition and reflection of C in as: where Γ = Z l −Z 0 Z l +Z 0 is the reflection coefficient, Z 0 is the impedance of the line and the loaded impedance of C in is Z l = 1 iωC in . As a second step, we consider the effect of the microwave length from C in to the system as: After this, the microwave reaches the T connection between the filter and the feedline. Its scattering matrix is: Here we assume the impedance of the line connecting the capacitor is also Z 0 . Part of the wave in the feedline will drive the filter mode a, which satisfies the input-output formula: where κ a is the leakage rate of the mode a. Assuming no output reflection (r i = 0), the relation between the input field c i , the output field r o and the filter mode a is: To determine the relation between the input c i and the output r o it suffices to deduce how a depends on c i . Under the rotating wave approximation (RWA), the equations of motion in the drive frequency (ω s ) rotating frame are: where ∆ a(b) = ω a(b) − ω s is the detuning of mode a(b) frequency ω a(b) relative to the drive frequency ω s , and J is the coupling strength between modes a and b. It follows from Eq. 1, 2, 3 and 4 that : where the effective detuning, leakage rate, and input field of mode a are ∆ a = ∆ a + Im(Γe i2θ )κ a /4, andκ a = κ a [1 + Re(Γe i2θ )]/2, andã i = √ κ a (1 − Γ)e iθ c i /(2 √κ a ) respectively. Note that Eq. 8 is equivalent, up to redefining various parameters, to Eq. 3 and Eq. 4 of the main text. According to the designed values of l and C in , we estimate θ ∼ 0.05 and Γ ∼ 0.98 − 0.17i. To simulate the dynamics of modes a and b, we use a Lindblad master equation with the Hamiltonian: and the Lindblad operator √κ a a. The effective driving field (t) follows from Eq. 8 and 9 as which is averaged in the simulation, given the classical (coherent) input field c i . Substituting this into Eq. 5 gives our final input-output formula: To account for the weak nonlinearity of the resonator and the uncertainty in the estimated design parameters, in the simulation we multiply the √ κ a a term on the right-hand side of Eq. 11 by a complex coefficient, chosen to fit the simulation to experimental data.
Physical interpretation. Because the filter mode a driven by (t) can be solved using the Lindblad master equation, we can determine the output mode r o once we know the driving waveform (t). The physical meaning of Eq. 11 can be interpreted as follows. The factor 2 before (t) means only half of the input mode c is used to drive mode a. The signal that finally reaches the output port is twice the driving. The e i2θ Γ term means half of the leakage of a directly goes to the output port, and the other half will go to the input side and be reflected by C in . Finally, these two branches interfere with each other and contribute a complex factor between the input and the system leakage.

SHORT DERIVATION
Here we give a simple, short derivation of the counterdiabatic (CD) driving (Eq. 1 in the main text) for a single driven bosonic mode coupled to a cold bath, based on a mean-field approximation. We leave a rigorous derivation to Supplementary Note (SN) 3.
As we will see in SN 3, the bosonic mode under consideration can be well approximated by a coherent state, and thus we can use a mean-field approximation for the Heisenberg picture bosonic field a(t), i.e. α(t) = a(t) . Following SN 1, the dynamics are given by the Langevin equation in the drive frame of frequency ω s : where ∆ r ≡ ω r − ω s is the cavity-drive detuning, κ is the damping rate due to coupling to the readout line, and (t) is the effective drive field. The instantaneous equilibrium state is obtained by settingα = 0. We denote this instantaneous equilibrium state as: If the drive field is varied slowly enough, the adiabatic theorem guarantees thatᾱ(t) be the solution of Eq. 12. Define the instantaneous diabatic excitation δ(t) = α(t)−ᾱ(t). It follows from Eq. 12, and Eq. 13 and its time derivative, that the dynamics of δ(t) satisfies: where CD is the (new) CD driving. From the boundary conditions δ(0) = 0 andδ(0) = 0, we obtain the desired CD driving as: Then, for an arbitrarily drive (t) , the instantaneous equilibrium stateᾱ(t) is always the exact dynamic solution of Eq. 12.

OPEN QUANTUM DYNAMICS APPROACH
In this section, we give rigorous derivations of CD driving for a single driven-dissipative bosonic mode, based on two approaches: (i) Lindblad dynamics [2] and (ii) an adiabatic shortcut of the decoherence free subspace (DFS) [3]. These results justify the mean-field approximation assumed in SN 2, and give additional insight into the adiabatic dynamics of our system.
In what follows, we set = 1. After rotating wave approximation, the Hamiltonian in the driving frame is: where the cavity-drive detuning ∆ r = ω r − ω s depends on the qubit state in the dispersive regime, and (t) is the effective drive amplitude. The transmission line is viewed as a channel for both driving and dissipation, so the dynamics for the cavity density matrix ρ is described by the master equation:ρ Here, only photon decay is considered, since at the effective temperature T mxc = 75 mK, the average photon population is N ≈ 0.015 1 at readout frequency ω s ≈ 2π × 6.5 GHz.
Lindblad dynamics approach. In the adiabatic approximation for open systems [4], in the limit where the Liouvillian L(t) is slowly varying, the density matrix ρ evolves independently in each generalized eigenspace of L(t). In other words, ρ can be decomposed into a direct sum of components, one for each independently evolving Jordan block of L(t).
The adiabaticity can be made exact by adding a CD Hamiltonian H CD which suppress the inertial part of L(t) that causes transitions between different Jordan blocks [2].
To determine H CD , we first we find a superoperatorÔ(t) that transforms L(t) into Jordan canonical form (JCF). That is, with respect to a certain (not necessarily Hermitian) basis for the density matrix B = {ρ 1 , ρ 2 , . . .}, we havê where J i (t) are the Jordan blocks of size n i × n i . Second, we transfer to the adiabatic frame defined by ρ (t) =Ô(t) −1 ρ(t) and show that the non-JCF part of the new Lindblad superoperator L (t), i.e.ρ (t) = L (t)ρ (t), can be exactly cancelled by adding a specific CD driving Hamiltonian H CD (t) to the system.
In the first step, we chooseÔ(t) to take the form of a displacement superoperator is the displacement operator [5]. Using the fact thatD(α(t))a = a − α(t), it is straightforward to show that Choosing precisely the instantaneous equilibrium state of SN 2, Eq. 13 -eliminates the time-dependent driving term in H J (t). Thus, in the adiabatic frame defined by D(ᾱ(t)), the Liouvillian L J (t) = L J is time-independent, so a fixed basis B can be chosen in which L J is in JCF.
In the second step, in the adiabatic frame ρ (t) =D(ᾱ(t)) −1 ρ(t) we havė i.e. the dynamics are exactly in JCF except for an inertial Hamiltonian which mixes the Jordan blocks of L J . Adding an additional CD term consistent with SN 2, Eq. 15.
Decoherence free subspace approach. The time-dependent decoherence free subspace (DFS) is a subspace of the full system Hilbert space, in which the open system dynamics is unitary and quasi-steady, i.e. its instaneous motion is generated by an effective Hamiltonian H eff defined within the DFS. By identifying the time-dependent DFS of our system, we derive the CD driving (Eq. 24) and compare it to the Lindblad dynamics approach.
Following the definition in [3], for system dynamics described by a Lindblad master H eff acting on a state in the DFS results in a state in the DFS. For a single lossy mode described by Eq. 17, the DFS exists and is spanned by the single state |ᾱ(t) forᾱ(t) = i (t)/(∆ r − iκ/2), and H eff takes the form of a displaced oscillator H eff (t) = ∆ r (a † −ᾱ * (t))(a −ᾱ(t)).
Suppose evolution of the DFS is given by the unitary transformation U (t), i.e. U (t)|φ j (0) = |φ j (t) . By direct analogy with closed system CD driving, we can transform to the adiabatic frame defined by U (t) and cancel diabatic excitations out of the DFS by adding a CD Hamiltonian H CD (t) = iU (t)U (t) † . In our example, the natural choice for U (t) is the displacement D(ᾱ(t)), from which we can derive the CD Hamiltonian H CD = iḊ(ᾱ(t))D(ᾱ(t)) −1 , equivalent to Eq. 24 obtained from the Lindblad dynamics approach.

Comments.
Steady states and adiabatic timescale. In the adiabatic frame, the Liouvillian L J in Eq.

takes the form
Its eigenvalues can be found by observing where |n are the Fock states, indicating L J is upper triangular in the subspace spanned by {|0 j|, |1 j + 1|, |2 j + 2|, . . .} (or their Hermitian conjugates) for natural numbers j.
Hence, L J has non-degenerate eigenvalues e j,k = i∆ r j − κ(j/2 + k), k = 0, 1, 2, . . . (or their complex conjugates) in each subspace and can be exactly diagonalized. We note that the only steady state of L J , i.e. the eigenstate of L J with zero eigenvalue, is given by j = k = 0, which is the vacuum state |0 in the adiabatic frame or the coherent state |α(t) in the lab frame.
We also comment on the timescale required for the adiabatic approximation to hold in open systems, following the results of [4]. Analogously to closed quantum systems, a sufficient condition for adiabatic evolution of an open system is where t f is the total evolution time, u, v ≡ tr(u † v) defines the inner product, ρ j,k (t) are (labframe) eigenstates of L(t) with eigenvalues e j,k , andρ j,k (t) are eigenstates of L(t) † . Here the adjoint L(t) † is defined as the superoperator that satisfy The LHS is hard to evaluate in practice, and a crude estimate is obtained by setting ρ j ,k (s), dρ j,k (s)/ds ∼ 1 for normalized time s ≡ t/t f . The adiabatic condition for the total time t f is then derived as .

(27)
For ∆ r comparable to κ, which is a usual experimental scenario, the adiabatic condition is t f κ −1 , making STA useful for fast protocols operating within unit lifetimes. This adiabatic condition is verified in Fig. 10, where sin 2 -shaped pulses are applied for different durations t f and sin 2 -shaped output signals are observed only for t f > 10κ −1 ≈ 600 ns.
DFS from Lindblad dynamics. The derivation of CD driving from both Lindblad dynamics and the DFS approach relies on switching to the adiabatic frame defined by D(ᾱ(t)), i.e. |φ = D(ᾱ(t)) † |φ . We note that the DFS of our system (i.e. the coherent state |ᾱ(t) ) is the vacuum state |0 in the adiabatic frame -the only steady eigenstate (i.e. having an eigenvalue with a non-negative real part) of L J . As a result, the steady eigenspace of the Liouvillian L(t) is equivalent to the DFS, whereas this is not true in general since purity of these steady states requires a zero-temperature approximation or negligible thermal photon number N (ω) 1 in the frequency band of interest. For bosonic modes in the high temperature regime (N (ω) ∼ 1 or N (ω) 1), CD driving is still possible by the Lindblad dynamics approach even though the pure-state DFS does not exist. In this case, although CD driving does not prevent heating into the steady thermal state in the adiabatic frame, it ensures fast transport of this steady state, which is still of practical interest.
Mean Field Approximation. Here we show that the single driven-dissipative mode remains in a coherent state, which justifies the mean-field approximation used in SN 2. For open quantum systems, the coherent state is known to be the consequence of the zerotemperature approximation of the environment [6]. Specifically, in the frame defined by a general displacement D(α(t)), the dynamics in Eq. 22 can be rewritten aṡ Choosing α(t) that satisfies the Langevin dynamics (Eq. 12) thus eliminates the driving term. Consequently, the system stays in the vacuum state in the displaced frame, corresponding to the coherent state |α(t) in the lab frame.
where φ i(f ) is the initial(final) state and ∆H eff = H eff − H eff . Geometrically, the LHS of Eq. 29 is the Bures length s Bures of the geodesic joining the initial and final state and the RHS is the integrated total length of the system trajectory whose velocity is given by [10]. Here, F Q (t) is the quantum Fisher information. We define the quantum efficiency of our protocol to be: As shown in Eq. 28, for a general driving (t) and the system initialized in the ground state, the dynamics is described by the displacement operator, i.e.
28). We note that U (t) can generate arbitrary dynamics in the space orthogonal to the system state |φ(t) = |α(t) , but the extra freedom can be shown to have no contribution to the uncertainty ∆H 2 eff (t) . With this choice of U (t) it is straightforward to show that H eff (t) = iα(t)a † + h.c. and ∆H 2 eff (t) = |α(t)|. For CD driving, ∆H 2 eff (t) is simply the added drive | CD (t) − (t)|, which provides the resource for adiabatic speedup in view of the energy-time uncertainty principle. Identifying |φ i,f = |α i,f and applying the triangle inequality, we obtain with equality achieved by straight-line trajectories -made possible by CD driving -in phase space (see Fig. 2 a). The spiral trajectory α(t) in Fig. 2 a is calculated from Eq.
12 with parameters ∆ r /2π = 3 MHz, κ −1 = 62.88 ns. Fig. 2  In this section, we derive the multi-mode optimal control (MMOC) protocol used in the main text, which takes the hybrid frequencies of multiple oscillators as input, and generates a single-port waveform that puts these lossy bosonic modes into thermal equilibrium at a desired final time t f . We first present a general framework which can be applied to multiple port driving, and then analyse the simpler single port case which is analytically and experimentally more tractable, and sufficient for our needs in the main text.
General multiple-port framework. Consider n linear bosonic modes {a i } n i=1 with frequencies ω i and linear couplings J ij between modes a i and a J . Each mode is coupled to a feedline with strength κ i and driven by an input field c i at frequency ω d . In general, the fields c i can be linearly dependent if they come from the same feedline. In the rotating frame with frequency ω d and after rotating wave approximation (RWA), the system Hamiltonian has the form where ∆ i = ω i − ω s is the ith detuning. In the Heisenberg picture, following the inputoutput formalism [11], the Langevin dynamics for the ith mode is given byȧ Adopting the mean-field approximation α i ≡ a i for all bosonic modes and defining the effective drive i ≡ √ κ i c i , we can rewrite the Langevin dynamics in matrix form: where Ω is the (complex) frequency matrix, α = (α 1 , . . . , α n ) T is the column vector of the where the i0 , ij , if are constants. Our goal is to put α(t) into the target equilibrium state α f = iΩ −1 f at final time t f , starting from initial equilibrium state α 0 = iΩ −1 0 . The propagator and general solution of differential equation Eq. 33 are where θ is the step function. Using Ω = O −1 Ω D O and Eq. 36, our goal can be achieved by for ij , where∆ k = ∆ k − iκ k /2 is the complex hybrid detuning. Treating the piece-wise driving ij as a vector l of dimension n×m and defining the n×mn matrix M kl ≡ O ki G kj (l = 1, 2, . . . , mn), Eq. 37 reduces to the linear equations If complete information of O or Ω are given, the general solution of Eq. 39 can be found by performing a singular value decomposition (SVD) of the matrix M. We concentrate instead on the case of single-port driving, which is considerably simpler.
Single port driving. In the special case of single-port driving, all drivings i (t) are linearly dependent, and Eq. 34 reduces to for constant coefficients c i , a single-port driving vector j , and boundary conditions 0 , f .
In this case the i O ki c i terms in Eq. 37 cancel, to give which takes the form of a linear constraint on .
Eq. 41 can be similarly solved via SVD of the n × m matrix G, i.e. G = U · D · V for unitary matrices U, V and diagonal matrix D = (diag(s 1 , . . . , s n ), 0 n×(m−n) ) (m ≥ n, 0 is the zero matrix). This gives the general form of as where x n+1 , . . . , x m are free complex parameters and V −1 i is the ith column of V −1 , which can be chosen to optimize a user-defined objective function such as the maximum power output of the pulse (see SN 6). We note that in the single-port driving case, the only input to the protocol is the complex detuning∆ i , which is simpler to measure experimentally than the multi-port driving case where full information of Ω is required.
Experimental implementation. Our single-port driving experiment in the main text corresponds to where a i , b i are the Purcell filter and readout resonator field conditioned on qubit state Energy consumption. The total energy consumption (up to an overall constant) of our pulse in Eq. 42 is, due to unitarity of V −1 , where u, v denotes the inner product. From Eq. 44 we see that the minimum energy solution E min is obtained by setting x i = 0.
Minimizing the maximum power output. Given the output power limitations of the microwave devices, it is desirable to minimize the maximum output power P max ({x i }) ≡ max i (| i ({x i })| 2 ) of the pulse. To achieve this, we numerically minimize P max (as a function of free parameters x i from Eq. 42) using a differential evolution algorithm. The resulting optimized MMOC pulses for both the ring-up and reset stage are those used in the main text. Fig. 3 shows the numerical results of P max (in dB) in units of the steady power P 0 after t f , plotted for the ring-up stage with different protocol times t f . For comparison, we also plot a lower bound on P max , which follows from Eq. 44 and the fact that Given the protocol time t f = 60 ns ≈ 1/κ 0 r used in the main text, we find P max = 14.5 dB after numerical optimization, which is a 4.1 dB reduction from that of the minimum energy pulse. For speedup beyond unit resonator lifetime 1/κ 0 r , P max grows rapidly and may induce unwanted qubit transitions, which sets a speed limit for the MMOC protocol, as discussed in SN 7.
Computational complexity. For single-port MMOC, the number of total qubit-state-

SUPPLEMENTARY NOTE 7. INFLUENCES OF THE LARGE DRIVE
We observe that the output signal drifts with a large driving power, which sets a limit on the steady-state driving power of our protocols. This can be explained by the nonlinearity of the resonator [12]. At the same time, according to the previous study [13], higher transmon levels are excited due to the non-RWA part of the qubit-resonator Hamiltonian, which becomes on-resonant as the photon number in the resonator increases through a Raman-like process. These two observations are shown to be closely related in theoretical simulations [14]. Here, we conduct two different experiments to confirm this point and find limitations of our protocol when applied to the transmon-resonator cQED system.
In the first experiment (Fig. 4), we compare the output signal of a small pulse of strength 1 a.u., and another larger pulse of strength 2.66 a.u.. Each point is averaged over 3 × 10 4 measurements and moving averaged with a Savitzky-Golay filter (width 21, order 3). IQ traces of the output signal in Fig. 4(c) show a clear drift even long after 5κ −1 a , which can be qualitatively explained by the nonlinearity of the cavity mode. In Fig. 5 and 6, another experiment is conducted to test the impact on the transition out of the |0 state of different pulse amplitudes and durations. A significant drop in P 0 is observed above amplitude 1 a.u.. At this drive amplitude we estimate the steady-state cavity photon number (via qubit spectroscopy) to be roughly the critical photon number n c ≡ (∆/2g) 2 ≈ 18 [15].

LOW-PASS FILTER IN THE AWG DRIVING LINE
Here we show that corrections to the MMOC pulses imposed by the low-pass fourth-order Chebyshev filter are negligible. In our experiments, the MMOC pulse (Eq. 42) from the arbitrary wave generator (AWG) has a carrier driving frequency ω s /(2π) of 200 to 250 MHz, which passes through the filter with a cutoff frequency ω c /(2π) = 750 MHz. The piece-wise constant pulse causes the Gibbs phenomenon, a potential source of error.
Here we give a qualitative evaluation of this error. To simplify our calculations, we assume that the passband's transfer function g is 1, and is 0 outside the passband. The waveform after the filter (t) is described by a convolution F of the pre-filter waveform (t) with the   filter function with frequency cutoffs ω 0 = ω s − ω c and ω 1 = ω s + ω c . The calculation is done in the rotating given by G = G , which satisfies where [t 0 , t 1 ] is the finite time window in which the corrected pulse is integrated over, and θ j (t) is the jth square function which equals 1 for t j−1 ≤ t ≤ t j and 0 elsewhere. The single integral Eq. 49 is easy to evaluate numerically and should be compared to the original matrix elements G ij . We find that the relative difference between G ij and G ij is less than 10 −6 and is generally independent of the choice of window time t 0 , t 1 .

SUPPLEMENTARY NOTE 9. ADDITIONAL EXPERIMENTAL DATA
In this section, we show additional experimental data we have collected. In Fig. 7, we test the widely used square wave driving with an initial amplitude twice as large as the remaining waveform, and see little decrease in the equilibrium time. In Fig. 8, we apply the CD pulse designed according to the set of parameters for the |0 and |1 states, and confirm that in the single-port driving situation, CD is only able to accelerate one mode at one time.
The main text shows the 4-mode MMOC protocol controlling all four modes with the same pulse. In Fig. 9, we design the MMOC for two modes corresponding to only one specific qubit state, then apply it on both qubit states. In this case, the method only works when the qubit is in the correct state. In Fig. 10, we compare the output amplitudes of the sin 2 and the corresponding CD ringup drives of different durations. Fig. 11 and Fig. 12 show the output signal's IQ trajectories for the sin 2 and the corresponding CD ringup drives. In Fig. 13, a large-amplitude and far off-resonant sin 2 drive , and corresponding CD drive are applied. The far-detuning guarantees the cQED system will not be excited. According to the input-output theory, the output signal will be a simple rescaling of the input signal.
The designed input ringup duration t f is 30 ns. However, we see the output reaches the designed stable value around t = 65 ns. This tail indicates the filtering effect of some low-Q and energy-storing microwave components in the feedline. This effect is more noticeable when the input signal is larger and can be simulated by a convolution with a low-pass filter transfer function.
To understand the effect of MMOC on qubit readout, we also apply MMOC to the rising edge of the readout pulse and compare the resulting readout fidelity with that corresponding to the regular rectangular pulse. Results are shown in Fig. 14 and Fig. 15. We find that MMOC improves the readout fidelity of |0 but decreases the readout fidelity of |1 due to T1 decay that occurs during the MMOC pulse. Longer T1 and a Bayesian-based algorithm will mitigate this error by correcting the bit flipping during the readout pulse [16] (experiment in progress).
Finally, by setting the decay rate κ 0 r = 0, we compare open system CD with closed system CD. Results are shown in Fig. 16. As expected, we find that only the open system CD pulse accelerates the transition to equilibrium. That is because κ 0 r and the drive detuning are of a similar magnitude. and (c), we see that the initial larger driving pulse does not make the equilibrium process much faster.
Supplementary Figure 8. Apply the CD driving designed for |0 on |1 state. The driving pulse for |0 is the same as the one used in Fig. 2 in the main text. The same pulse applied when the qubit is prepared in the |1 state will not accelerate the evolution to equilibrium.  We see a huge spike during the ringup for a short CD driving, which does not mean the system undergoes highly non-equilibrium dynamics. According to the input-output theory, the monitored IQ is the coherent superposition of the input driving and the leakage of the system. When the ringup is only 30 ns, the large amplitude will excite the untargeted filter mode, which is indicated by the spiral collapse to the equilibrium after 30 ns.
Supplementary Figure 12. I-Q plots of the sin 2 drive for different ringup durations. To see how long it takes to reach quasi-equilibrium dynamics for the corresponding bare sin 2 ringup, we measure the IQ trajectories for different t f . Until t f = 800 ns, the trajectory will not change in shape, which indicates the system has reached the quasi-equilibrium dynamics. Compared this with the < 100 ns in the case of CD driving. large-amplitude drivings. When we detune the drive frequency by −2π × 100 MHz, the lossy cQED system will not be excited, and the output signal closely follows the input signal. However, we see a delay in the output signal before reaching equilibrium after ringup, and an extended ringdown after the drive has concluded. This is particularly pronounced when the driving power is large. to improved readout fidelity of |0 but decreased readout fidelity of |1 . This can be explained as follows. In the case of |0 , after the MMOC ringup, the IQ plot is more concentrated because the resonator is already in a steady state. If we apply the rectangular pulse, the natural charging duration (5/κ) is about 350 ns, which means the IQ trajectory keeps drifting during the readout.
In the case of |1 , the main limitation is the T1 decay that occurs during the 180 ns MMOC pulse.
One possible solution to this is to reduce the duration of the MMOC pulse, taking care to ensure that the pulse is not so short so as to lead to a maximum amplitude large enough that causes unwanted qubit excitations. open CD response, and we can reduce this with a smaller driving amplitude. The closed system CD drive does not accelerate the transition to equilibrium (see Fig. 2 in the main text which shows that it takes roughly 300 ns for a bare 100 ns sin 2 pulse to cause the resonator to reach equilibrium).
Measurements are repeated 30,000 times.