Abstract
We propose an efficient qubit initialization protocol based on a dissipative environment that can be dynamically adjusted. Here, the qubit is coupled to a thermal bath through a tunable harmonic oscillator. Ondemand initialization is achieved by sweeping the oscillator rapidly into resonance with the qubit. This resonant coupling with the engineered environment induces fast relaxation to the ground state of the system, and a consecutive rapid sweep back to off resonance guarantees weak excess dissipation during quantum computations. We solve the corresponding quantum dynamics using a Markovian master equation for the reduced density operator of the qubitbath system. This allows us to optimize the parameters and the initialization protocol for the qubit. Our analytical calculations show that the groundstate occupation of our system is well protected during the fast sweeps of the environmental coupling and, consequently, we obtain an estimate for the duration of our protocol by solving the transition rates between the lowenergy eigenstates with the Jacobian diagonalization method. Our results suggest that the current experimental state of the art for the initialization speed of superconducting qubits at a given fidelity can be considerably improved.
Introduction
Preparation of a qubit into a welldefined initial state is one of the key requirements for any quantum computational algorithm.^{1, 2} The conventional passive initialization protocol relies on the relaxation of the qubit to a thermal state determined by the residual coupling to the environment. This protocol is inherently slow because the relaxation rate has to be minimized to decrease the probability of errors in a coherent quantum computation.
In the context of errorfree quantum computing, a long initialization time would not present a problem since the quantum register has to be prepared only once in the beginning of the computation. However, realistic quantum computational devices also suffer from gate errors which have to be corrected with quantumerrorcorrection codes.^{3} Such codes rely on logical qubits which consist of several physical qubits. The codes initiate from a predetermined state for the physical qubits and are being constantly executed during a computation. They also have strict requirements for the initialization and gate error thresholds for individual qubits, of the order of 10^{−5} for the conventional concatenated codes.^{4, 5} In the more refined topological quantum error correction codes,^{6} the logical error can be suppressed with stabilizing measurements, which increase the thresholds up to 10^{−2} for the physical qubit operations and lead to improved protection of quantum information during the computation. Nevertheless, stabilizer codes such as surface^{7, 8} and color^{9, 10} codes still require a continuous supply of measurement qubits in a known lowentropy state. Thus, initialization time is also an issue in largescale quantum computing.
Fast and accurate qubit initialization remains a technological challenge in the superconducting qubit implementations that have shown great potential for scalability.^{8} Major developments have been made with active protocols, such as initialization by successive projective measurements,^{11,12,13} by Purcellfiltered cavity,^{14} or by cooling with a coherent microwave drive.^{15,16,17,18} The typical figure of merit of a protocol is the time τ _{10} required for a qubit excitation to decay by an order of magnitude. In the reported experiments,^{14} a thermalequilibrium fidelity of 99.9% with τ _{10} = 40 ns has been obtained with a protocol for a Purcellfiltered superconducting qubit with transition frequency ω _{q}/(2π) = 5.16 GHz and intrinsic relaxation time T _{1} = 540 ns. However, the method uses a tuned qubit that is not desirable since it reserves a broad frequency band, and a high groundstate fidelity using a longlived qubit remains to be demonstrated with this method. An initialization protocol based on coherent driving has reached a 99.5% groundstate fidelity with τ _{10} = 1.4 μs, ω _{q}/(2π) = 5 GHz, and T _{1} = 37 μs,^{17} without relying on qubit tuning. Even though this protocol meets the error thresholds of the topological codes, largescale quantum computing calls for shorter initialization times. Furthermore, initialization fidelities greater than 99.99% are preferred in largescale computations to reduce the number of physical qubits needed for a logical qubit.
Qubit initialization can also be achieved with feedbackbased schemes,^{11, 19,20,21,22} which rely on a singleshot measurement of the qubit state. In recent experiments,^{19, 20} the discrimination of the qubit state is predominantly limited by T _{1}, with the current benchmark for the average assignment fidelity being 99.8% in 700ns readout time.^{19} However, an increase of the fidelity by each order of magnitude requires yet another reduction of the readout time or an increase of T _{1} by an order of magnitude, which still remains an experimental challenge. Furthermore, the fidelity and speed of this method may not exceed that of the quantum measurement, and hence significantly contribute to the total error budget. Therefore, the search for a more efficient initialization scheme is of great importance.
In solidstate systems, an initialization protocol can potentially be realized by strongly coupling the qubit to a cold dissipative reservoir for fast initialization and by switching off the coupling for the actual computation.^{23, 24} If the bath has a lower effective temperature than the qubit, the groundstate fidelity of the qubit is increased. This type of setup is a part of environmental quantumstate engineering by dissipation, where one aims to drive the system into a desired steady state by using a carefully tailored environment.^{25,26,27,28,29,30} Such engineering has already been used in generation of coherent superposition states,^{31,32,33} in creation of entanglement,^{34,35,36} and in simulations of open quantum systems.^{37, 38}
In this paper, we focus on a groundstate initialization proposal,^{39, 40} where a superconducting qubit and a lowtemperature resistive bath are coupled indirectly through two resonators as shown in Fig. 1. The effective inductance of the left resonator, which is capacitively coupled to the bath, can be dynamically adjusted, allowing control over its bare resonance frequency. If the left resonator is sufficiently detuned from the qubit, it shunts the noise of the resistive bath at the qubit frequency, and hence the decoherence of the qubit is dictated by its slow intrinsic relaxation rate. If the left resonator is in resonance with the qubit, the qubit couples strongly to the bath leading to an increased relaxation rate. We use two resonators because a single resonator cannot be easily detuned from the qubit by several linewidths due to the required strong coupling with the cold bath and because the second resonator, coupled directly to the qubit, can be used for qubit readout using the standard dispersive methods.^{41}
Previous calculations of static transition rates in this scenario indicate that the lifetime of the qubit can be controlled over several orders of magnitude.^{39, 40} However, an actual initialization protocol and its dynamics, speed, and fidelity have not been reported. Here, we develop a qubit initialization protocol and solve its corresponding quantum dynamics by using a Markovian master equation. We show analytically that the ground state is protected during the sweeps of the left resonator to and from the resonance, implying that the speed of the protocol is determined by the strong resonant coupling with the dissipative bath. At the resonance, we find an approximative analytic solution for the lowenergy eigenproblem using Jacobian diagonalization.^{42, 43} It yields a useful lower bound for the duration of the protocol at a given fidelity. Optimization of the physical circuit parameters suggests that, with presentday technology, the current benchmarks for fidelity and protocol speed^{17} can be considerably improved with the help of our protocol.
Results
System
The abovediscussed tunableenvironment qubit can be conveniently studied using a lumpedelement circuit model shown in Fig. 1a. Here, the superconducting qubit with the transition energy ħω _{q} is coupled to a bosonic heat bath through two LC resonators. Both resonators are formed by a lumped capacitance C _{ k } and an inductance L _{ k }, where k = L,R refer to the left and right resonators, respectively. The bath arises from the resistance R at temperature T. The left resonator is coupled directly to the bath and to the right resonator through capacitances C _{E} and C _{c}, respectively. The inductance L _{L}(t) of the left LC resonator is made tunable using a SQUID, the Josephson inductance of which is controlled by an external magnetic flux. As a consequence, the bare angular frequency of the left resonator \({\omega _{\rm{L}}}(t) = 1{\rm{/}}\sqrt {{L_{\rm{L}}}(t){C_{\rm{L}}}} \) can also be adjusted with the external flux. The right resonator has a fixed bare angular frequency \({\omega _{\rm{R}}} = 1{\rm{/}}\sqrt {{L_{\rm{R}}}{C_{\rm{R}}}} \) and is coupled to the qubit through the capacitance C _{q}.
The Hamiltonian of the circuit can be written as^{40}
where the three terms describe the system, the resistive environment, and their interaction, respectively. The system Hamiltonian can be expressed as
where
and
Above, \({\hat a_{\rm{L}}}\), \({\hat a_{\rm{R}}}\), and \({\hat \sigma _  }\) are the annihilation operators for the left and right resonators and the qubit, respectively. In the following, we denote the eigenstates of the unperturbed Hamiltonian (3) with \(\left {{n_{\rm{L}}},{n_{\rm{R}}},{n_{\rm{q}}}} \right\rangle \), where the occupation numbers of the left and right resonators can have values n _{L}, n _{R} = 0, 1, 2, … and that of the qubit assumes values n _{q} = g,e, which stand for the ground and excited state, respectively. When the rotating wave approximation (RWA) applied in Eq. (5) is accurate, the total occupation number N = n _{L} + n _{R} + n _{q} is a conserved quantity during a unitary time evolution. The Hamiltonian \({\hat H_{\rm{S}}}(t)\) describes a tripartite system formed by two harmonic resonators and a qubit. The right resonator is coupled bilinearly to the left resonator and to the qubit with the respective coupling frequencies \({g_{{\rm{LR}}}}(t) = g_{{\rm{LR}}}^{\rm{0}}\sqrt {{\omega _{\rm{L}}}(t){\rm{/}}{\omega _{\rm{R}}}} \) and \({g_{{\rm{Rq}}}} = \frac{{e{C_{\rm{q}}}\sqrt {\hbar {\omega _{\rm{R}}}/{C_{\rm{R}}}} }}{{\hbar \left( {{C_{\rm{q}}} + {C_{\rm{J}}}} \right)}}{\left( {\frac{{{E_{\rm{J}}}}}{{{E_{\rm{C}}}}}} \right)^{1/4}}\), where \(g_{{\rm{LR}}}^{\rm{0}} = {\omega _{\rm{R}}}{C_{\rm{c}}}{\rm{/}}\left( {2\sqrt {{C_{\rm{L}}}{C_{\rm{R}}}} } \right)\) is the resonant coupling strength between the left and right resonators, C _{J} is the capacitance of the Josephson junction, and E _{J} and E _{C} = e ^{2}/[2(C _{q} + C _{J})] are the Josephson coupling energy and the charging energy per electron for the superconducting island, respectively. Coupling between the qubit and the left resonator is mediated by the right resonator and is, therefore, of second order in coupling frequencies \(g_{{\rm{LR}}}^{\rm{0}}\) and g _{Rq}. Consequently, the right resonator acts as an additional filter for the thermal noise of the bath. In our analytic considerations, we apply the RWA for both of the coupling terms, cf. Eq. (5). The interaction with the bath is also bilinear and described by
where \({\hat V_{\rm{L}}} = \sqrt {\frac{{\hbar {\omega _{\rm{L}}}(t)}}{{2{C_{\rm{L}}}}}} \left( {\hat a_{\rm{L}}^\dag + {{\hat a}_{\rm{L}}}} \right)\) is the voltage over the left resonator, and \(\delta {\hat V_{{\rm{res}}}}\) describes the voltage fluctuations over the resistor. The resistor Hamiltonian \({\hat H_{\rm{E}}}\) is given by that of a bosonic thermal bath^{44}. We do not express it explicitly since we are only interested in the transition rates, which are determined in thermal equilibrium by the Johnson–Nyquist spectrum of the voltage fluctuations:
Furthermore, we neglect any filtering of this spectrum owing to the resistor itself by assuming that the frequencies relevant for the system obey \(\omega \ll 1\)/(RC _{R})^{40}.
In typical realizations of superconducting qubits, the temperature of the circuitry is below 100 mK and the resonator and qubit frequencies lie in the 5–10GHz range. As a consequence, thermal occupations of the one and two excitation states for a detuned system are roughly below 10% and 1% relative to the groundstate occupation, respectively. Furthermore, during the protocol, nonadiabatic transitions between instantaneous eigenstates with different total occupation numbers N are suppressed owing to negligible matrix elements for such transitions. This allows the development of an approximative analytic model in the subspace spanned by the four lowestenergy instantaneous eigenstates of the Hamiltonian (2). We emphasize that the following discussion is not specific to the lumpedelement model or superconducting qubits, but can be used rather generally for qubits with indirect adjustable coupling to a thermal bath, as shown in ref. 40 with the distributed circuit elements which are frequently used in contemporary circuit QED.
Transition rates
The transition rates from the mth instantaneous eigenstate of the Hamilonian \({\hat H_{\rm{S}}}(t)\) to the nth state can be calculated from Fermi’s golden rule as
where ω _{ mn }(t) = ω _{ n }(t) − ω _{ m }(t), and ω _{ k }(t) are the eigenfrequencies corresponding to the eigenstates \(\left {{\Psi _k}(t)} \right\rangle \) of the Hamiltonian \({\hat H_{\rm{S}}}(t)\), i.e., \({\hat H_{\rm{S}}}(t)\left {{\Psi _k}(t)} \right\rangle = \hbar {\omega _k}(t)\left {{\Psi _k}(t)} \right\rangle \), \({\Gamma _0} = {\left( {{C_{\rm{E}}}{\rm{/}}\sqrt {{C_{\rm{L}}}{C_{\rm{R}}}} } \right)^2}\left( {R{\rm{/}}{Z_{\rm{R}}}} \right){\omega _{\rm{R}}}\), and \({Z_{\rm{R}}} = \sqrt {{L_{\rm{R}}}{\rm{/}}{C_{\rm{R}}}} \). We thus observe that positivefrequency fluctuations in the environment cause emission in the small system.^{45} Note, that in Eq. (8) we express the transition rates in units of Γ_{0}, which equals the bare zerotemperature leftresonator transition rate for ω _{L}(t) = ω _{R}. Clearly, we can maximize the transition rates by maximizing C _{E} and R with respect to C _{R} and Z _{R}, respectively. We further note that in principle C _{E} should be added to C _{L} to obtain the effective leftresonator capacitance, but if \({C_{\rm{E}}} \ll {C_{\rm{L}}}\) its effects on the eigenstates and eigenfrequencies of the system are negligible.^{40}
In Methods, we analytically solve the transition rates for the first three excited eigenstates employing the RWA. We first truncate the Hamiltonian (2) to the lowenergy subspace spanned by the ground state and singleexcitation states. Then, we diagonalize the static Jaynes–Cummings coupling in Eq. (5) between the right resonator and the qubit. As a result, the first three eigenstates of the Hamiltonian (2) are the eigenstates of the effective Hamiltonian matrix
which is written in the basis formed by 1, 0, g〉 and the Jaynes–Cummings eigenstates  ± 〉 with the corresponding eigenfrequencies \({\omega _ \pm } = \omega _{{\rm{Rq}}}^{{\rm{av}}} \pm {\Omega _{{\rm{Rq}}}}\). We use the shorthand notations \(\omega _{ij}^{{\rm{av}}} = \left( {{\omega _i} + {\omega _j}} \right){\rm{/}}2\), \({\Omega _{ij}} = \frac{{\left {{\delta _{ij}}} \right}}{2}\sqrt {1 + 4\lambda _{ij}^2} \), δ _{ ij } = ω _{ i } − ω _{ j }, and λ _{ ij } = G _{ ij } /δ _{ ij }. Above, we have G _{Rq} = g _{Rq} and, thus, λ _{Rq} is the usual perturbation parameter of the Jaynes–Cummings model that characterizes the strength of the coupling. Similarly, the coupling strengths between the left resonator and the states  ± 〉 are determined by λ _{L ± }(t) where \({G_{{\rm{L}} \pm }}(t) = {g_{{\rm{LR}}}}(t)\sqrt {\left[ {1 \pm {\delta _{{\rm{Rq}}}}{\rm{/}}\left( {2{\Omega _{{\rm{Rq}}}}} \right)} \right]{\rm{/}}2} \).
We treat the coupling to the left resonator perturbatively using Jacobian diagonalization.^{42, 43} Jacobian diagonalization of a hermitian matrix consists of a sequence of similarity transformations designed to annihilate one offdiagonal matrix element at a time. Optimal convergence is achieved by forming the subspace at each step of the basis states with the strongest coupling. Depending on the value of the control parameter ω _{L}(t), we have either λ _{L+}(t) ≥ _{ λL−}(t) or λ _{L+}(t) < λ _{L−}(t) and, thus, the Jacobian diagonalization is most efficiently done in a temporally piecewise manner. As a result, we obtain the transition rates
where \({\omega _1}(t) = \omega _{{\rm{L}}  }^{{\rm{a}}v}(t)  {\Omega _{{\rm{L}}  }}(t)\), \({\omega _2}(t) = \omega _{{\rm{L}} \pm }^{{\rm{a}}v}(t) \mp {\Omega _{{\rm{L}} \pm }}(t)\), and \({\omega _3}(t) = \omega _{{\rm{L}} + }^{{\rm{a}}v}(t) + {\Omega _{{\rm{L}} + }}(t)\). For transition rate (11), we use the top signs when λ _{L+}(t) ≥ λ _{L−}(t), and bottom signs when λ _{L+}(t) < λ _{L−}(t). Note that the transition rates between the first three excited states are zero in the RWA due to the selection rules for our environmental coupling term. However, we include all transition rates to the numerical solution, within the truncation. Furthermore, the principle of detailed balance \({\Gamma _{mn}} = {\rm{exp}}\left[ {  \hbar {\omega _{mn}}{\rm{/}}\left( {{k_{\rm{B}}}T} \right)} \right]{\Gamma _{nm}}\) holds, which implies that the excitation rates are strongly suppressed in the lowtemperature limit. In addition to the engineered environment described by the resistor R, the qubit and the right resonator typically dissipate energy to their own intrinsic environments with the transition rates \(\kappa _{\rm{q}}^{\rm{I}}\) and \(\kappa _{\rm{R}}^{\rm{I}}\), respectively.
We compare the analytic eigenfrequencies and transition rates with the corresponding numerical results in Fig. 2. For large detuning \(\left( {g_{{\rm{LR}}}^{\rm{0}},{g_{{\rm{Rq}}}} \ll \left {{\omega _{\rm{R}}}  {\omega _{\rm{q}}}} \right} \right)\), we find that already the first diagonalization step in the Jacobian diagonalization procedure results in a very good agreement with the numerically obtained frequencies and rates. We also observe that when the left resonator is in resonance with either the right resonator or the qubit [δ _{L ± }(t) ≈ 0], the transition rates from the resonant eigenstates to the ground state are equal. We take advantage of this fact in our protocol below. In the vicinity of the resonances, there exist regions where the transition rates from the nearly resonant states change by several orders of magnitude. The widths of these regions are directly proportional to the coupling term \(g_{{\rm{LR}}}^0\), and also to g _{Rq } for the case of ω _{L}(t) = ω _{−}.
Protocol
Our proposed initialization protocol is depicted in Fig. 2c and proceeds as follows. In the beginning of the protocol (t = 0), the parameters of the setup follow the hierarchy
In particular, the qubit frequency is chosen to be the smallest in order to minimize the effects of the possible multiphoton processes in the qubit caused by the rightresonator at any stage of the protocol. For t = 0 and t = τ, where τ is the duration of the protocol, the coupling to the engineered environment should be minimal so that the intrinsic sources of dissipation are dominating the qubit decoherence. Therefore, we refer to the bare leftresonator frequency ω _{L}(0) = ω _{L}(τ) as the ‘‘off’’ state of our setup, and choose its value such that \({g_{{\rm{LR}}}}(0) \ll {\omega _{\rm{L}}}(0)  {\omega _{\rm{R}}}\). This guarantees that (see Figs. 1 and 2)
since the widths of the regions for enhanced transition rates are proportional to \(g_{{\rm{LR}}}^{\rm{0}}\). This way the added dissipative channel does not create excess decoherence for the qubit. We also choose \({g_{{\rm{Rq}}}} \ll {\omega _{\rm{R}}}  {\omega _{\rm{q}}}\), which implies only weak hybridization between the qubit and the rightresonator. In our numerical simulations, we use the values shown in Table 1 for the relevant parameters.
In the first stage of the protocol, the left resonator is swept fast to resonance with the effective right resonator at \({\omega _{\rm{L}}}({t_1}) = {\omega _ + } \approx {\omega _{\rm{R}}} + g_{{\rm{Rq}}}^{\rm{2}}{\rm{/}}{\delta _{{\rm{Rq}}}}\) _{,} which is only slightly hybridized with the qubit due to the dispersive coupling. As a consequence, the right resonator becomes strongly coupled to the cold bath resulting in an increase in its relaxation rate to the ground state by orders of magnitude (see Fig. 2). This operation point (denoted in Fig. 2 with ‘‘op 1’’) guarantees equal relaxation rates for both resonators, which is important as the relative occupations are typically not known in the beginning of the protocol and, also, because nonadiabatic transfer of occupation between the resonators can occur during the fast sweep. Any occupation in either resonator is then dissipated to the resistor during t : t _{1} → t _{2}. The wait time Δt _{2} depends on the required protocol error α = 1 − P _{0}(τ), where we use the notation Δt _{ i } = t _{ i } − t _{ i−1} for the relevant protocol time intervals with t _{0} = 0 and t _{5} = τ, and define P _{0}(τ) as the desired groundstate occupation and the end of the protocol.
After the first relaxation step (t ≥ t _{2}), the resonators are in the ground state within the protocol error α. Then, the left resonator is swept into resonance with the effective qubit at \({\omega _{\rm{L}}}\left( {{t_3}} \right) = {\omega _  } \approx {\omega _{\rm{q}}}  g_{{\rm{Rq}}}^2{\rm{/}}{\delta _{{\rm{Rq}}}}\). At this operation point (denoted in Fig. 2 with ‘‘op 2’’), the relaxation rate of the qubit is increased by several orders of magnitude and is equal to the leftresonator rate. The latter is important since at this operation point, the qubit is also hybridized with the left resonator and, consequently, any initial occupation in the qubit is partly transferred to the left resonator during the sweep. By waiting for t : t _{3} → t _{4}, the hybridized qubit and left resonator dissipate their energy to the resistor. The second wait time Δt _{4} is also determined by the desired value of α. Finally, the leftresonator frequency is swept back to its high initial value ω _{L}(τ) = ω _{L}(0). In addition to the wait times Δt _{2} and Δt _{4}, the protocol duration \(\tau = \mathop {\sum}\nolimits_{i = 1}^5 {\Delta {t_i}} \) is determined by the three sweep times Δt _{1}, Δt _{3,} and Δt _{5}. For simplicity, we will assume in our discussions that the sweep times are equal, i.e., t _{ s } = Δt _{1} = Δt _{3} = Δt _{5}.
The speed of the protocol for a given fidelity should be maximized for efficient use in quantum information processing. In general, a good initialization protocol should have \(\kappa _{\rm{q}}^{\rm{I}}\tau \ll 1\). This way one can perform multiple initializations during a coherent quantum computation. In our protocol, this requires the minimization of the combined duration of the three sweeps of the control parameter ω _{L}(t) and the relaxation intervals, during which the actual initialization takes place. The length of the relaxation intervals is set by the relaxation rates and the desired fidelity, implying that after they are optimized, the duration of the protocol can be shortened only by faster sweeping. However, it is well known that fast changes in parameters can induce nonadiabatic transitions between the instantaneous eigenstates of the system.^{46,47,48,49} Our requirements for the optimal operation points (ω _{L} ≈ ω _{ ± }) guarantee that our protocol is robust with respect to changes between the relative occupations of the instantaneous eigenstates during the first two sweeps. After the second relaxation process, the system lies in the ground state within the desired error α. Thus, the essential sweep is the last one starting from the qubit resonance (ω _{L}(t _{4}) = ω _{−}) and the ground state Ψ_{0}(t _{4})〉, and ending to the far offresonant ground state Ψ_{0}(τ)〉. The relevant question is the following: how much of the ground state is excited during the final fast sweep?
We can calculate the transition probabilities P _{ mn } ≡ P _{ m → n }(τ ; t _{4}) = 〈Ψ_{ n }(τ)Ψ_{ m }(t _{4})〉^{2} between the lowenergy eigenstates in the sudden approximation using the RWA results obtained in the previous section. However, since we are interested in the groundstate sweep fidelity P _{S} = P _{00} and since in the RWA the ground state is unaffected by the change of parameters, we have to include the contributions arising from the counterrotating terms. In Methods, we derive the worstcase estimate for the groundstate sweep fidelity in the sudden approximation. We obtain
where we assume that the sweep duration Δt _{5} → 0. Thus, the deviation from the perfect sweep fidelity is given by the difference between the perturbation parameters g _{LR}(t)/[ω _{L}(t) + ω _{R}] before and after the sweep. Orderofmagnitude estimates for typical superconducting circuit parameters give g _{LR}/(2π) ~ 100 MHz, ω _{R}/(2π),ω _{q}/(2π) ~ 10 GHz and, accordingly, we have that the deviation from full fidelity is 1 − P _{S} ~ [g _{LR}/(ω _{q} + ω _{R})]^{2} ~ 10^{−4}. Typically, the deviation is a couple of orders of magnitude smaller since the difference between the perturbation parameters is very small. We can, therefore, assume in our analytic considerations that the ground state is well protected during the sweeps of parameters in the Hamiltonian, and that the protocol duration can be estimated solely in terms of the relaxation rates of the static stages of the protocol, i.e., τ ≈ Δt _{2} + Δt _{4}. In the experimental implementation of the protocol, any cross coupling between the qubit and the flux line, used to adjust the leftresonator frequency, should be considered together with excitations of the SQUID. However, these issues seem not to significantly affect the achievable speed of our initialization protocol.
In Fig. 3, we compare the analytically obtained sweep fidelity from Eq. (15) with the sweep fidelity obtained by solving the instantaneous ground state Ψ_{0}(t)〉 numerically. We observe that if the RWA is valid, i.e., for \(g_{{\rm{LR}}}^{\rm{0}} \ll {\omega _{\rm{R}}}\), the analytic result closely follows the numerical solution. For increasingly strong coupling, the secondorder perturbation theory becomes insufficient which is visible as a large deviation between the analytic and numerical results. Even for \(g_{{\rm{LR}}}^{\rm{0}} = 0.1{\omega _{\rm{R}}}\), however, the deviation from the perfect fidelity is of the order of 10^{−7}, which indicates that the effects of the fast sweep on the groundstate fidelity of the protocol can be neglected.
Markovian master equation for the timedependent system
In order to obtain more quantitative understanding of the protocol, we study its dynamics with the help of a Markovian master equation for the reduced system density operator \({\hat \rho _{\rm{S}}}(t) = {\rm{T}}{{\rm{r}}_{\rm{E}}}\left\{ {\hat \rho (t)} \right\}\), where the total density operator \(\hat \rho (t)\) obeys the von Neumann equation
where \(\hat H(t)\) is defined in Eq. (1). We first diagonalize the system Hamiltonian \({\hat H_{\rm{S}}}(t)\), as defined in Eq. (2), in a timeindependent basis {n〉} with the timedependent unitary transformation \(\hat D(t) = \mathop {\sum}\nolimits_n {\left {{\Psi _n}(t)} \right\rangle \left\langle n \right} \). After the transformation, the timeevolution of \(\hat \rho '(t) = {\hat D^\dag }(t)\hat \rho (t)\hat D(t)\) is governed by the effective system Hamiltonian
where the latter term causes nonadiabatic transitions. Such term always appears if one wishes to preserve the form of the von Neumann equation in a timedependent unitary transformation. After the transformation, the derivation of the master equation proceeds in a conventional manner^{50,51,52}: We assume that the initial state of the total system is uncorrelated, i.e., \(\hat \rho (0) = {\hat \rho _{\rm{S}}}(0) \otimes {\hat \rho _{\rm{E}}}(0)\), and that the bath is in a thermal state, described by \({\hat \rho _{\rm{E}}}(0)\), throughout the temporal evolution. We consider only weak coupling to the environment and apply the standard Born and Markov approximations in the interaction picture, and subsequently trace over the environmental degrees of freedom. In the Markov approximation, the correlation time of the environment is negligibly short and we can express the master equation for \(\hat \rho _{\rm{S}}^\prime (t)\) in the secular approximation as
Above, we have defined the ladder operators between the static basis states as \({\hat \pi _{nm}} = \left n \right\rangle \left\langle m \right\). The instantaneous transition rates Γ_{ mn }(t) are identical to those obtained with Fermi’s golden rule in Eq. (8).^{53} Similar to the case of a static Hamiltonian,^{50, 51} the environmental decoherence is included in the Lindblad terms on the last three rows of the master equation. The first term models the transitions between the adiabatic states that dissipate energy to the environment, the second term represents absorption from the environment and the last term induces dephasing of the adiabatic states. We note that the secular approximation made above is justified if the relaxation rates are small compared to the minimum separation between the eigenfrequencies of the system,^{50} i.e., we have
We note that the adiabatic master equation above can be improved by making successive diagonalizing transformations for \({\hat H_{{\rm{eff}}}}(t)\). As a result, the master equation is represented in the basis of the socalled superadiabatic states, the timedependence of which is typically suppressed after each diagonalizing transformation. Such adiabatic renormalization was first described for a general timedependent quantum system by Berry,^{54} and later applied to studies of dissipation in driven superconducting qubits.^{55,56,57} The lowestorder superadiabatic correction was studied in refs. 58, 59.
Protocol speed
Let us study the decay of an excitation in our system by solving the master equation (18) for an initial state spanned by the lowenergy adiabatic states as \(\left {\Psi (0)} \right\rangle = \mathop {\sum}\nolimits_{n = 0}^4 {{a_n}(0)\left {{\Psi _n}(0)} \right\rangle } \). We assume that the environment is so cold that thermal excitations of the system are negligible. This guarantees that the quantum state \({\hat \rho _{\rm{S}}}(t)\) of the system remains in the lowenergy subspace during the temporal evolution. In the beginning of a realistic initialization procedure, we may have no knowledge on the distribution of the occupations P _{ n }(0) = a _{ n }(0)^{2}. Therefore, we choose the operation points of our protocol such that
and
which is the case for \({\omega _{\rm{L}}}({t_2}) \approx {\omega _ + } \approx {\omega _{\rm{R}}} + g_{{\rm{Rq}}}^{\rm{2}}{\rm{/}}{\delta _{{\rm{Rq}}}}\) and \({\omega _{\rm{L}}}\left( {{t_4}} \right) \approx {\omega _  } \approx {\omega _{\rm{q}}}  g_{{\rm{Rq}}}^{\rm{2}}{\rm{/}}{\delta _{{\rm{Rq}}}}\), respectively.
Before discussing the numerical results, we present a simple analytic estimate for the initialization fidelity. The general form for the excitedstate occupations can be solved from the master equation (18) and written as
By neglecting the small contributions of the fast sweeps, these can be readily written in terms of Eqs. (20) and (21). We find that the deviation from the perfect fidelity, \(\alpha (\tau ) = 1  {P_0}(\tau ) = \mathop {\sum}\nolimits_{i = 1}^3 {{P_i}(\tau )} \), can be written as
where the relative initial occupations of the excited states have been defined as \({\overline P _i}(0) = {P_i}(0){\rm{/}}\mathop {\sum}\nolimits_{j = 1}^3 {{P_j}(0)} \) for i = 1, 2, 3. The tolerable deviation α(τ) from perfect fidelity at the end of the protocol is fixed in the beginning of the protocol. Since we do not know the relative occupations in the beginning of the initialization protocol, we have to wait the times Δt _{2} ≈ ln[α(0)/α(τ)]/Γ_{2} and Δt _{4} ≈ln[α(0)/α(τ)]/Γ_{1} so that any excitation in the system is decayed down to the desired accuracy. As a consequence, we obtain an upper bound for the total wait time \(\overline \tau = \Delta {t_2} + \Delta {t_4}\) of the protocol as
Above, we denote α(τ) = 10^{−β} α(0) where β ≥ 0, and τ _{10} = 2ln(10)[1 + (ω _{R}/ω _{q})^{2}]/Γ_{0}. For ω _{R} ≳ ω _{q}, we have that τ _{10} ≈ 4ln(10)/Γ_{0,} which sets the time scale for the decrease of α(τ) by an order of magnitude. We note that the secular approximation used in the derivation of the master equation requires through Eq. (19) that
where the equality holds since in our case \(0 < {g_{{\rm{Rq}}}} \ll {\delta _{{\rm{Rq}}}}\) and ω _{q} < ω _{R}. Additionally, we required that in the off state of the protocol, the intrinsic rates dominate over those of the engineered environment, i.e., \({\Gamma _{10}}(0) = {\alpha _{\rm{q}}}\kappa _{\rm{q}}^{\rm{I}}\) and \({\Gamma _{20}}(0) = {\alpha _{\rm{R}}}\kappa _{\rm{R}}^{\rm{I}}\), where \({\alpha _{\rm{q}}},{\alpha _{\rm{R}}} \ll 1\). We show in Methods that these lead to the condition \({\Gamma _0} \le {\omega _{\rm{R}}}\root 3 \of{{{{\gamma ^2}{\alpha _{\rm{q}}}\kappa _{\rm{q}}^{\rm{I}}\delta _{{\rm{Lq}}}^2(0){\rm{/}}\left\{ {{{\left[ {{\delta _{{\rm{Lq}}}}(0) + {\omega _{\rm{q}}}} \right]}^2}{\omega _{\rm{R}}}} \right\}}}}\), where \(\gamma = {\Gamma _0}{\rm{/\Gamma }}_{\rm{0}}^{{\rm{max}}} \ll 1\) and δ _{Lq}(0) = ω _{L} − ω _{q} is the detuning between the leftresonator and the qubit in the off state of the protocol. This sets a lower bound for the decay time:
where the numerical estimate is made for typical superconducting circuit parameters ω _{R}/(2π) = 10 GHz, ω _{q}/(2π) = 9.5 GHz, δ _{Lq}(0)/(2π) = 2 GHz, and \(\kappa _{\rm{q}}^{\rm{I}}{ = 10^4}\) s^{−1}. We have used β = 0.5 and confirmed with a classical calculation of the transition rate Γ_{2} that the secular approximation holds within a relative error of 4%. We also set α _{q} = 0.1. The lower bound for the decay time above would represent a significant improvement to that of the current experimental benchmark for qubit groundstate initialization protocol.^{17} By choosing the off state infinitely far from the qubit (δ _{Lq} → ∞), the decay time can be improved to τ _{10} ≳ 90 ns. In the following, we set α(0) = 1 in order to obtain an estimate for the total wait time for initialization of a qubit excitation. For example, if one wishes to obtain a ground state fidelity of α(τ) = 10^{−3} with our protocol, the wait time should be of the order of \(\overline \tau = 3{\tau _{10}}\).
Qubit temperature
Above, we calculated the deviation α(τ) from the perfect fidelity by neglecting the intrinsic dissipation in the qubit. However, in addition to the temperature T of the resistor, the fidelity is reduced by the intrinsic temperature T _{q} of the qubit. We estimate this effect by first defining the effective qubit temperature T _{eff} in terms of the excitedstate occupation \(P_{{\rm{ex}}}^{\rm{q}}\) of the qubit at the end of the protocol as
By using the principle of detailed balance, we can solve \(P_{{\rm{ex}}}^{\rm{q}}\) in our fourstate model from
where \(P_{{\rm{ex}}}^{\rm{q}} = \frac{1}{2}\left( {{P_1} + {P_2}} \right)\) and we have included the coupling \(\kappa _{\rm{q}}^{\rm{I}}\) between the qubit and its intrinsic environment. The occupation at the end of the protocol can be calculated at the second operation point, where we use Γ_{10} = Γ_{20} = (ω _{q}/ω _{R})^{2}Γ_{0}/2 = Γ_{1}, Γ_{30} = 0, and the detailed balance relation \({\Gamma _{0i}} = {\Gamma _1}\,{\rm{exp}}\left[ {  \hbar {\omega _{0i}}{\rm{/}}\left( {{k_{\rm{B}}}T} \right)} \right]\) for the excitation and absorption rates. Above, we have neglected the intrinsic excitation of the right resonator. We note that with the above assumptions P _{3} = 0 and, thus, \({P_0} = 1  2P_{{\rm{ex}}}^{\rm{q}}\). Thus, we can write the excited state population for the qubit as
We show in Fig. 4 the above analytic excited qubit state occupation \(P_{{\rm{ex}}}^{\rm{q}}\) and the corresponding effective temperature T _{eff}. For our example parameters (see Table 1) and for the intrinsic qubit temperature T _{q} = 100 mK, the data show a saturation of the qubit excitation and the effective qubit temperature to values \(P_{{\rm{ex}}}^{\rm{q}} \approx 4 \times {10^{  6}}\) and T _{eff} ≈ 36 mK, respectively, for T ≲ 30 mK. In our simulations, we use the value T = 10 mK. In the remaining calculations, we have neglected the intrinsic qubit dissipation in order to keep our results independent on the temperature of the intrinsic environment of the qubit.
Numerical results
Let us compare the analytic model above with the numerical solution of the master equation (18). To this end, we solve the master equation by truncating to the subspace spanned by the five lowest adiabatic energy eigenstates which are obtained by diagonalizing the instantaneous Hamiltonian \({\hat H_{\rm{S}}}(t)\), defined in Eq. (2). In particular, the transition rates in Eq. (8) are calculated with the numerically obtained instantaneous eigenfrequencies and eigenstates. We have confirmed that the relative errors caused by the truncation in the groundstate occupation are of the order of 10^{−7} or smaller. In Fig. 5, we present the dynamics of the occupations P _{ i }(t) for the four lowestenergy states. We study the decay of a single excitation with three different initial occupation probabilities. Our choice for the offstate (see Fig. 2) guarantees that the three degrees of freedom are initially weakly coupled and the states Ψ_{1}(0)〉, Ψ_{2}(0)〉, and Ψ_{3}(0)〉 can be well approximated by the first excited states of the qubit, the right resonator, and the left resonator, respectively.
In the numerical simulations, we choose the wait times at the operation points equal to those presented in the analytic model. If the excitation belongs initially to the qubit, the numerical results are in very good agreement with the analytic occupations obtained in Eq. (22). During the sweep to the qubit resonance, part of the initial qubit occupation is transferred nonadiabatically to the left resonator. However, the final numerical fidelity is close to the analytic estimate since at the second operation point the qubit and the left resonator become hybridized and the relaxation rates for the resulting two states are equal. This is clearly visible in Fig. 6 where we show the dependence of the protocol error α(τ) on the total duration τ. Similarly, if the right resonator has a nonvanishing initial occupation, part of it is transferred to the leftresonator during the first sweep. Again, this does not lead to major deviations from the analytic fidelity due to the equal relaxation rates at the first operation point.
We note that one has to be careful when tuning to the first operation point. If the operation point is above the resonance, i.e., ω _{L}(t _{1}) > ω _{+}, any occupation remaining in the state Ψ_{2}(t)〉 at t = t _{2} will be transferred to the state Ψ_{3}(t)〉 in a Landau–Zenertype process when the system is subsequently swept across the avoided crossing. This leads to a decrease in the protocol fidelity, since at the second operation point the relaxation rate Γ_{30} ≈ 10^{−2}Γ_{0} (see Fig. 2).
If the desired fidelity P _{0}(τ) = 1 − α(τ) is close to unity, the wait times have to be long and, as a consequence, the choice for the initial location of the excitation does not have a large influence on the protocol time. The analytic estimate (24) serves as an upper bound for the total wait time \(\overline \tau \), and we approach the upper bound in the case of the full qubit excitation. Fortunately, the numerically obtained fidelity is always higher than that given by the analytic upper bound for the wait time, since there always exist some residual relaxation to ground state from the nonresonant states. If the excitation is partly or completely in the rightresonator, the desired fidelity is reached faster than in the analytic prediction, as depicted in Fig. 6. This occurs because the part of the initial rightresonator occupation that has not decayed at the first operation point can continue the decay at the second operation point due to the hybridization with the left resonator, which causes occupation transfer during the first sweep. We also observe that for our choice of parameters the relatively strong coupling between the qubit and the right resonator, g _{Rq}/δ _{Rq} ≈ 0.14, causes oscillations in the protocol fidelity as a function of the protocol time τ.
Discussion
In summary, we have proposed and modeled a qubit initialization protocol where the coupling between a superconducting qubit and an engineered environment can be externally controlled with a tunable resonator. Using experimentally realizable parameters, we have solved the timedependent Markovian master equation for the protocol and shown that the tunable resonator can be used for fast and precise reset of a qubit excitation. We have also demonstrated that fast changes of the bare angular frequency of the tunable resonator do not reduce the final groundstate fidelity of the protocol. As a result, the duration of the protocol for a given fidelity can be estimated in terms of the decay rate of the tunable resonator, Γ_{0}, with a simple analytic model. We also found that the present experimental stateoftheart decay time^{17} for groundstate qubit initialization may be decreased almost by an order of magnitude. Moreover, we observed that at dilution refridgerator temperatures (T≲30 mK) the effective qubit temperature can be reduced to one third of its intrinsic temperature of 100 mK, resulting in an excited qubit state occupation of roughly 10^{−6}.
In the offmode of the protocol, the coupling to the engineered environment has to be weak enough such that the decoherence during quantum computation is determined mainly by the intrinsic environment of the qubit. This sets a limitation for the protocol speed since the decay rate Γ_{0} has to be lower than the intrinsic dissipation rates in the system. Furthermore, the assumption of weak coupling with the environment restricts the possible values of decay rates. In future work, the evolution of the reduced density operator should be solved in the regime of strong environmental coupling, which may lead to further improvements in the protocol speed. Also, the possibilities of combining our protocol with driven reset method^{17} should be investigated for further improvements on the protocol duration and speed.
Methods
Jacobian diagonalization
We perturbatively solve the eigenproblem of the instantaneous Hamiltonian (2). In the RWA, the occupation number \(\hat N \equiv \hat a_{\rm{L}}^\dag {\hat a_{\rm{L}}} + \hat a_{\rm{R}}^\dag {\hat a_{\rm{R}}} + {\hat \sigma _ + }{\hat \sigma _  }\) is a conserved quantity because \(\left[ {{{\hat H}_S},\hat N} \right] = 0\). This means that \({\hat H_{\rm{S}}}(t)\) and \(\hat N\) have joint eigenstates. Eigenvalues of the occupation number are N = n _{L} + n _{R} + n _{q} = 0, 1, 2, …, where n _{L} and n _{R} are the occupation numbers of the left and right resonators, respectively, and assume values n _{L}, n _{R} = 0, 1, 2, …. The number n _{q} denotes the qubit occupation and assumes values 0 and 1. Each occupation number N belongs to (2N + 1) degenerate eigenstates of the form n _{L}, n _{R}, n _{q}〉. These are also the eigenstates of the unperturbed Hamiltonian \({\hat H_0}(t)\). We note that if the counterrotating terms neglected in the RWA are included, the occupation number is no longer conserved and the above arguments do not hold. In the numerical simulations we employ the nonRWA Hamiltonian.
We are interested in the transitions between the lowenergy eigenstates. The N = 1 manifold consists of three states: {0, 0, e〉, 0, 1, g〉, 1, 0, g〉} where, for clarity, we use the symbols g ↔ 0 and e ↔ 1 for the qubit occupation number n _{q} in the ground and excited state, respectively. As a consequence, the eigenstates of the RWA Hamiltonian with occupation number N = 1 are linear combinations of these three basis states. Therefore, it is enough to find eigenstates for the truncated 3 × 3 Hamiltonian matrix
We rely on approximative methods for an intuitive analytic solution for an arbitrary value of ω _{L}(t). The challenge is to find the solution for nearly resonant states, for which nondegenerate perturbation theory fails. The coupling between the leftresonator and the qubit is of second order in the coupling frequencies \(g_{{\rm{LR}}}^{\rm{0}}\) and g _{Rq}, which causes slow convergence in the conventional nearly degenerate and Brillouin–Wigner perturbation theories. Instead, we first diagonalize the Hamiltonian H _{S}(t) in the N = 1 manifold of the static subspace formed by the right resonator and the qubit, resulting in the Hamiltonian in Eq. (9) which was written as
where \({\omega _ \pm } = \omega _{{\rm{Rq}}}^{{\rm{av}}} \pm {\Omega _{{\rm{Rq}}}}\). We also recall the shorthand notations \(\omega _{ij}^{{\rm{av}}} = \left( {{\omega _i} + {\omega _j}} \right){\rm{/}}2\), \({\Omega _{ij}} = \frac{{\left {{\delta _{ij}}} \right}}{2}\sqrt {1 + 4\lambda _{ij}^2} \), δ _{ ij } = ω _{ i } − ω _{ j }, and λ _{ ij } = G _{ ij } /δ _{ ij }, with G _{Rq} = g _{Rq} and \({G_{{\rm{L}} \pm }}(t) = {g_{{\rm{LR}}}}(t)\sqrt {\left[ {1 \pm {\delta _{{\rm{Rq}}}}{\rm{/}}\left( {2{\Omega _{{\rm{Rq}}}}} \right)} \right]{\rm{/}}2} \). We then study the corrections caused by the coupling to the left resonator employing the Jacobian diagonalization^{42, 43} for the eigenproblem. We are especially interested in the transition rates when the left resonator is in resonance either with the right resonator or the qubit.
In Eq. (31), Hamiltonian \({\hat H_{\rm{S}}}\) is written in the basis spanned by {−〉, +〉, 1, 0, g〉}, where
At this point, the left resonator is coupled to the hybridized qubit and right resonator states  ± 〉, which in the dispersive regime \(\left( {{g_{{\rm{Rq}}}} \ll \left {{\delta _{{\rm{Rq}}}}} \right} \right)\) have the Jaynes–Cummings eigenfrequencies
where we have anticipated our choice of parameters and assumed that δ _{Rq} > 0. If the left resonator is far detuned, the perturbation parameters \({\lambda _{{\rm{L}} \pm }}(t) \ll 1\) and hence the above basis states accurately approximate the true eigenstates. In the vicinity of resonances with \({\lambda _{{\rm{L}} + }}(t) \gg 1\) or \({\lambda _{{\rm{L}}  }}(t) \gg 1\), one has to take effects arising from the coupling terms into account by making subsequent diagonalizations in the resonant subspaces. However, if \({G_{{\rm{L}} \pm }} \ll \left {{\omega _ + }  {\omega _  }} \right\) one can neglect the contributions caused by the offresonant coupling term G _{∓}.
When λ _{L+}(t) ≥ λ _{L−}(t), we make a diagonalizing Jacobian transformation for the Hamiltonian (31) in the subspace spanned by +〉 and 1, 0, g〉 (see Fig. 7). We obtain
where \({\omega _{ \uparrow / \downarrow }}(t) = \omega _{{\rm{L + }}}^{{\rm{av}}}(t) \pm {\Omega _{{\rm{L}} + }}(t)\) and \({G_{ \uparrow {\rm{/}} \downarrow  }}(t) = {G_{{\rm{L}}  }}(t)\sqrt {\left\{ {1 \pm {\delta _{{\rm{L}} + }}(t){\rm{/}}\left[ {2{\Omega _{{\rm{L}} + }}(t)} \right]} \right\}{\rm{/}}2} \). Furthermore,
are the states corresponding to the approximative eigenvalues \(\hbar {\omega _{ \uparrow / \downarrow }}(t)\). Thus in Eq. (34), the Hamiltonian \({\hat H_{\rm{S}}}(t)\) is represented in the basis \(\left\{ {\left  \right\rangle ,\left { \downarrow (t)} \right\rangle ,\left { \uparrow (t)} \right\rangle } \right\}\). For all values of the control parameter ω _{L}(t) of our protocol, the perturbation parameter \({\lambda _{ \uparrow  }}(t) \ll 1\) and, thus, further Jacobian transformations cause only minor corrections to the third excited eigenfrequency \({\omega _3}(t) \approx {\omega _ \uparrow }(t)\) and eigenstate \(\left {{\Psi _3}(t)} \right\rangle \approx \left { \uparrow (t)} \right\rangle \).
In the vicinity of the qubit resonance, δ _{L−}(t) ≈ 0, one has λ _{L+}(t) < λ _{L−}(t). Consequently, we diagonalize the Hamiltonian (31) in the subspace spanned by the states −〉 and 1, 0, g〉 and obtain
where have neglected the small couplings between the new approximate eigenstates and defined the eigenfrequencies
and the corresponding eigenstates
Using Eq. (8), we obtain the transition rates (10)–(12). For the eigenfrequency (38) and eigenstate (41) of the second excited state, we use the top signs when λ _{L+}(t) ≥ λ _{L−}(t) and bottom signs when λ _{L+}(t) < λ _{L−}(t). Note also that the eigenstates (37)–(39) are approximately orthogonal since \(\left\langle {{\Psi _1}(t){\rm{}}{\Psi _2}(t)} \right\rangle = \sqrt {1  {\delta _{{\rm{L}}  }}(t){\rm{/}}\left[ {2{\Omega _{{\rm{L}}  }}(t)} \right]} \sqrt {1  {\delta _{{\rm{L}} + }}(t){\rm{/}}\left[ {2{\Omega _{{\rm{L}} + }}(t)} \right]} \ll 1\) for λ _{L+}(t) ≥ λ _{L− }(t), and \(\left\langle {{\Psi _2}(t){\rm{}}{\Psi _3}(t)} \right\rangle = \sqrt {1 + {\delta _{{\rm{L}}  }}(t){\rm{/}}\left[ {2{\Omega _{{\rm{L}}  }}(t)} \right]} \sqrt {1 + {\delta _{{\rm{L}} + }}(t){\rm{/}}\left[ {2{\Omega _{{\rm{L}} + }}(t)} \right]} \ll 1\) for λ _{L+}(t) < λ _{L− }(t). The Jacobian iteration could be continued further but by comparing these results with the numerical solution of the eigenproblem, we observe that the couplings between the states above are already so small that the Hamiltonian is accurately diagonalized.
Groundstate fidelity during sudden sweeps
The groundstate sweep fidelity \({P_{\rm{S}}} = {\left {\left\langle {{\Psi _0}(\tau ){\rm{}}{\Psi _0}({t_4})} \right\rangle } \right^2}\) can be calculated in the sudden approximation by finding the corrections to the RWA ground state \(\left {\Psi _{\rm{0}}^{{\rm{RWA}}}(t)} \right\rangle = \left {0,0,{\rm{g}}} \right\rangle \) caused by the counterrotating terms \({\hat H_2}(t) = \hbar {g_{{\rm{LR}}}}(t)\left( {\hat a_{\rm{L}}^\dag \hat a_{\rm{R}}^\dag + {{\hat a}_{\rm{L}}}{{\hat a}_{\rm{R}}}} \right) + i\hbar {g_{{\rm{Rq}}}}\left( {{{\hat a}_{\rm{R}}}{{\hat \sigma }_  }  \hat a_{\rm{R}}^\dag {{\hat \sigma }_ + }} \right)\). Since these change the occupation number by two, we should expand our lowenergy basis to cover the eigenstates of the occupation numbers N = 0, 1, 2, 3, which gives altogether 16 basis states. However, the perturbation divides the eigenspace to two uncoupled sets, one of which is formed by the evenoccupation states and the other one by the oddoccupation states. In conclusion, we do not expect the groundstate sweep fidelity to depend on the matrix elements between the ground state and any eigenstate with odd parity (N = 1, 3, 5, …). Thus, the counterrotating terms break the symmetry in the RWA Hamiltonian that leads to the conservation of the occupation number, and replace it with a weaker requirement of parity conservation.
We analytically calculate the correction arising from the counterrotating term \({\hat H_2}(t)\) only for the ground state, which is created in our lowenergy subspace by the offresonant coupling between 0, 0, g〉 and the N = 2 eigenstates. Since we have \({g_{{\rm{LR}}}}(t){\rm{/}}\left[ {{\omega _{\rm{L}}}(t) + {\omega _{\rm{R}}}} \right] \ll 1\) and \({g_{{\rm{Rq}}}}{\rm{/}}\left( {{\omega _{\rm{q}}} + {\omega _{\rm{R}}}} \right) \ll 1\) during the whole protocol, we can treat the effects arising from \({\hat H_2}(t)\) in the secondorder nondegenerate perturbation theory. Since \({H_2}(t)\left {0,0,{\rm{g}}} \right\rangle = {g_{{\rm{LR}}}}(t)\left {1,1,{\rm{g}}} \right\rangle  i{g_{{\rm{Rq}}}}\left {0,1,{\rm{e}}} \right\rangle \), we obtain
where the zero of energy is set by the RWA groundstate energy and in the ground state we have neglected the secondorder terms, which are outside our lowenergy subspace. The deviation from zero energy is caused by the Bloch–Siegerttype^{60} nonresonant corrections to the RWA result. The normalization of the state is given by
where the latter equality is written to second order in the small parameters g _{LR}(t)/[ω _{L}(t) + ω _{R}] and g _{Rq}/(ω _{q} + ω _{R}). Thus, the ground state can be approximately expressed as
As a consequence, the sweep fidelity can be expressed as
where the sweep is assumed to start from ω _{L}(t _{4}) = ω _{−} ≈ ω _{q}. Thus, the maximum deviation from the perfect survival of the ground state is given by the square of the difference between the perturbation parameters.
Parameter optimization and the numerical method
In this section, we optimize the parameters for our protocol. According to our analytic model, the decay time τ _{10} determines the transition rate as
Furthermore, we require that in the off mode the transition rates are much smaller than the intrinsic rates \(\kappa _{\rm{R}}^{\rm{I}}\) and \(\kappa _{\rm{q}}^{\rm{I}}\). Thus, we obtain upper bounds for the transition rates in Eqs. (10)–(12) by writing them in the limit where \({\omega _{\rm{L}}}(0) \gg {\omega _{\rm{R}}},{\omega _{\rm{q}}}\) as^{40}
where we have employed the zerotemperature limit and δ _{Lq}(0) determines the maximum sweep range in our protocol. If ω _{L}(0) is large, the lowest three excited states can be approximated as \(\left {{\Psi _1}(0)} \right\rangle \approx \left {0,0,{\rm{e}}} \right\rangle \), \(\left {{\Psi _2}(0)} \right\rangle \approx \left {0,1,{\rm{g}}} \right\rangle \), and \(\left {{\Psi _3}(0)} \right\rangle \approx \left {1,0,{\rm{g}}} \right\rangle \). We also note that \({\Gamma _{20}}(0) = \Gamma _{10}^ \uparrow (0)\frac{{{\omega _{\rm{R}}}\delta _{{\rm{Lq}}}^{\rm{2}}(0)\delta _{{\rm{Rq}}}^{\rm{2}}}}{{g_{{\rm{Rq}}}^2{\omega _{\rm{q}}}{{\left( {{\delta _{{\rm{Lq}}}}(0)  {\delta _{{\rm{Rq}}}}} \right)}^2}}} < \Gamma _{10}^ \uparrow \), since the parameters of our protocol satisfy ω _{q} < ω _{R} and g _{Rq} < δ _{Rq}.
Another restriction for the transition rates comes from the fact that the master equation is valid only if
where the latter approximations hold for our protocol since \({g_{{\rm{LR}}}}(t) = g_{{\rm{LR}}}^{\rm{0}}\sqrt {{\omega _{\rm{L}}}(t){\rm{/}}{\omega _{\rm{R}}}} \). The first condition above determines the maximum Γ_{0}, since in our protocol g _{Rq} /δ _{Rq} < 1 and ω _{q}/ω _{R} < 1. In the following, we write
and assume that \(\gamma \ll 1\) so that inequalities (53) hold. We can optimize the decay time τ _{10} by defining the tolerances \({\alpha _{\rm{q}}} = \Gamma _{10}^ \uparrow {\rm{/}}\kappa _{\rm{q}}^{\rm{I}}\) and \({\alpha _{\rm{R}}} = {\Gamma _{20}}{\rm{/}}\kappa _{\rm{R}}^{\rm{I}}\). The inequalities (50) and (51) hold for \({\alpha _{\rm{q}}},{\alpha _{\rm{R}}} \ll 1\). We thus obtain from Eqs. (50) and (51) that
We wish to minimize our decay time, which means that we need to maximize \(g_{{\rm{LR}}}^{\rm{0}}\). The largest allowed value for \(g_{{\rm{LR}}}^{\rm{0}}\) is given by the prefactor in the inequality above. This can be achieved when the qubit–rightresonator coupling is chosen as
In this case, an upper bound for the rate coefficient follows from Eqs. (54) and (55), and can be written as
which is independent of g _{Rq}. In the latter inequality above, we have assumed that \({\omega _{\rm{L}}}(0) \gg {\omega _{\rm{q}}}\). For typical experimental parameter values δ _{Lq}(0)/(2π) = 2 GHz, ω _{R}/(2π) = 10 GHz, ω _{q}/(2π) = 9.5 GHz, δ _{Rq}/(2π) = 0.5 GHz, \(\kappa _{\rm{q}}^{\rm{I}}{ = 10^4}\) s^{−1}, \(\kappa _{\rm{R}}^{\rm{I}}{ = 10^6}\) s^{−1}, and for α _{R} = α _{q} = 0.1, we obtain
Above, we have set γ = 0.5 and confirmed with a classical calculation of the transition rate Γ_{20}(t _{1}) that in the case of g _{Rq} = 0, the secular approximation holds up to a relative error of 4%. The lower bound for the decay time is obtained from Eq. (49) to be
The numerically computed transition rates Γ_{ n0}(t) and eigenfrequencies ω _{ n }(t) are obtained by solving the eigenproblem for \({\hat H_{\rm{S}}}(t)\) in the eigenbasis of the unperturbed Hamiltonian \({\hat H_0}(t)\), where we have included the eigenstates \(\left {{n_{{\rm{L/R}}}}} \right\rangle \) of the left and rightresonators up to n _{L/R} = 5, and solved the master equation in the static basis consisting of five energetically lowest states. We have confirmed that the relative error in the final groundstate occupation caused by the truncation of the basis is of the order of 10^{−7} or smaller.
Data availability
The numerical code and the datasets analyzed during the current study are available from the corresponding author on reasonable request.
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Acknowledgements
This work was supported by the Academy of Finland through its Centers of Excellence Programme under grant numbers 251748 and 284621, and through grant number 305306. We also acknowledge funding from the European Research Council under Consolidator Grant number 681311 (QUESS) and from the Vilho, Yrjö, and Kalle Väisälä Foundation.
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J.T. carried out the analytic and numerical calculations based on the initial idea of M.M., and was mainly responsible for writing the manuscript. M.P. made the classical calculation of the transition rates. T.A.N. and M.M. provided feedback and supervised the project. All authors discussed the results and commented on the manuscript.
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Correspondence to Jani Tuorila.
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Tuorila, J., Partanen, M., AlaNissila, T. et al. Efficient protocol for qubit initialization with a tunable environment. npj Quantum Inf 3, 27 (2017). https://doi.org/10.1038/s4153401700271
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