Optimal operating conditions of an entangling two-transmon gate
We identify optimal operating conditions of an entangling two-qubit gate realized by a capacitive coupling of two superconducting charge qubits in a transmission line resonator (the so called ”transmons”). We demonstrate that the sensitivity of the optimized gate to flux and critical current noise is suppressed to leading order. The procedure only requires a preliminary estimate of the noise amplitudes. No additional control or bias line beyond those used for the manipulation of individual qubits are needed. The proposed optimization is effective also in the presence of relaxation processes and of spontaneous emission through the resonator (Purcell effect).
Superconducting circuits are a promising technology for the realization of quantum information on a solid state platform. Several types of qubits  have been developed realizing high fidelity single qubit operations [2, 3]. Rapid progress has also been made towards the realization of robust and scalable universal two-qubit gates [4, 5, 6]. The circuit quantum electrodynamics (cQED)  architecture demonstrated to be particularly promising for scalable quantum information. In this scheme highly entangled two [8, 9] and three qubits  have been generated and simple quantum algorithms have been demonstrated [11, 12].
The coherence times of the present generation of devices (s) are about three orders of magnitudes larger than the first implementations. A relevant step further toward this enhancement has been the elimination of linear sensitivity to low-frequency () noise by operating qubits at ”optimal” working points. After the first ”sweet spot” operation demonstrated in Ref. , a further boost of qubit performances has been achieved in a cQED design named ”transmon” , which is almost insensitive to the detrimental effect of charge noise  at the price of reduced anharmonicity. However, cQED architectures share with other implementations the presence of flux noise whose amplitude has a characteristic order of magnitude , and of critical current noise . Together with relaxation processes due to quantum noise, dephasing due to 1/f flux and critical current noise still limits the time scales over which phase coherence and entanglement are preserved. In fact, further improvement of the coherence times at least of one order of magnitude would be required to reach the level for practical quantum error correction . Recently in a new circuit-QED architecture employing a three-dimensional resonator the error correction threshold has been approached . ”Optimization” is thus a key-word of the present generation of superconducting nano-circuits. Clever circuit design and optimal tuning of multi-qubit architectures, supplemented by the use of improved materials, are two complementary strategies currently exploited to address this problem.
A major question currently unsolved is establishing the best strategy to maintain long-enough a sufficient degree of entanglement. In the present article we address this issue considering a universal two-qubit gate realized by a fixed capacitive coupling of two transmons in a cQED architecture. The implementation of this scheme has been recently reported in Ref.  where a operation with individual single-shot non-destructive readout  and gate fidelity of , partly limited by qubit decoherence, has been demonstrated. A similar system has been studied theoretically in [19, 20]. Here we identify ”optimal”  operating conditions of a transmons gate taking into account the multi-level nature of the nano-circuit. We find that an ”optimal coupling” exists where the leading order effects of flux and critical current noise are eliminated. The amount of preserved entanglement is quantified by the concurrence between the two transmons, , which we evaluate in analytic form. The efficiency of the ”optimal coupling” is demonstrated by the fact that, for typical noise spectra measured in superconducting nanocircuits, the concurrence is predicted to decay on a time scale  s (in the absence of other decay mechanisms). In addition, may attain values  guaranteeing violation of a Bell inequality until s and the gate fidelity is up to s. Finally, we demonstrate that the optimization is effective also in the presence of relaxation processes due to flux quantum noise. Similarly to other cQED systems , the gate efficiency can be limited by spontaneous emission through the resonator. This limitation is likely to be overcome by suitable Purcell filters or protected designs . The optimization proposed in the present article can further improve the considerable performance of cQED two-qubit gates based on cavity-mediated interaction [12, 10] or on tunable effective interaction with microwave control . Remarkably, here effective elimination of omnipresent noise sources is achieved even if one qubit does not operate at optimal bias and without additional controls or bias lines beyond those used for the manipulation of individual qubits, an important feature for scalability.
2 Universal two-transmon gate
We consider two transmons with a fixed capacitive coupling, each qubit being embedded in its superconducting resonator used for control and bit-wise readout [9, 18]. The interaction is effectively switched on/off by dynamically changing the qubits detuning using single qubits control lines. For this reason one of the qubits does not operate at its sweet spot. In figure 1 (a) we report the circuit diagram of the considered system. Each transmon, denoted by the subfix , consists of a Cooper-Pair-Box (CPB) characterised by the charging energy and Josephson energy , tunable via the magnetic flux threading the superconducting loop, , ( is the flux quantum). In the circuit-QED scheme each CPB is embedded in a transmission line resonator whose relevant mode is modeled as a oscillator . Thus the Hamiltonian of transmon consists of the CPB Hamiltonian plus the dipole-like interaction with the oscillator 
where phase, and charge, , are conjugate variables, . The resonator energy is and () creates (annihilates) one photon in the transmission line (). is the root-mean-square voltage of the oscillator and is the ratio between the gate capacitance coupling the CPB to the local mode and the CPB total capacitance.
The transmon operates at . Under these conditions, values of the phases close to zero are most favored. This motivates the neglect of the periodic boundary condition on the phases and the expansion of the cosine in Eq. (1). Within this approximation, the offset charge can be eliminated via a gauge transformation . Of course, the perturbative scheme cannot capture the non-vanishing charge dispersion of the transmon [27, 28]. In particular, the exponential decrease of the charge dispersion with for only results from the exact diagonalization of the CPB Hamiltonian in the phase basis . It leads to exponential suppression of sensitivity to low-frequency () fluctuactions of the offset charge . Here we rely on this well established result and eliminate from the outset. Expanding the cosine in Eq. (1) up to fourth order, the CPB Hamiltonian can be cast in the form of a weakly anharmonic oscillator (Duffin oscillator)
where the bosonic operators , are related to the charge operator via and we put . The two lowest eigenenergies of identify the transmon- qubit levels. Their splitting is and it can be tuned by changing the magnetic flux . The flux ”sweet-spot” is at [3, 13].
The capacitive coupling between the CPBs, , adds to leading to the Hamiltonian
where is the effective coupling depending on the control parameters via the Josephson energies. Note that fluctuations of the magnetic fluxes affect the effective coupling between the qubits. Typical values  are GHz and GHz, leading to GHz.
The coupled transmons eigenenergies and eigenstates are conveniently obtained by treating in perturbation theory with respect to both the anharmonic terms and the capacitive interaction included in
The level structure is schematically show in figure 1 (b). The splitting in the subspace where the SWAP operation takes place (in short ”SWAP splitting”) reads and the corresponding eigenstates spanning the ”SWAP subspace” are and , where and are eigenstates of , (, ). The interaction is effectively switched on by tuning the single-qubit energy spacing to mutual resonance. The resonance condition is realized by tuning the flux bias until , displacing one qubit from the sweet spot at . In the following we suppose that and . Under resonance conditions the operation is realized by free evolution for a time starting from a factorized initial state in the ”SWAP subspace”.
3 Optimal operating conditions: reduction of noise effects
Since the two qubits do not operate at the same working point, the dominant source of dephasing is different for the two transmons. In particular, first order fluctuactions of the transmon splittings are due to critical current noise for transmon 1 and to flux noise for transmon 2 . These fluctuations can be treated in the adiabatic and longitudinal approximation  by replacing with . Here represent stochastic fluctuations of the dimensionless critical current , and of the flux , . The leading order effect of adiabatic noise is defocusing, expressed by the ”static path” or static noise approximation (SPA) [29, 30] describing the average of signals oscillating at randomly distributed effective frequencies (see A for the validity regimes of the SPA in the present problem). It is obtained by replacing with statistically distributed values . In the SPA the coherence between the states is
where is the two-qubit density matrix and indicates the average over the fluctuations . Here we assume that they are uncorrelated random variables with Gaussian distribution, zero mean and standard deviations proportional to the amplitude of the spectrum, ( and are the low and the high frequency cut-offs of the region). As demonstrated in Refs. [21, 22] the optimal operating condition is obtained imposing a minimum of the variance of the stochastic SWAP splitting, . This is simply understood considering the short-times expansion , implying defocusing suppression when is minimal. Expanding around the fixed working point we get
where all derivatives are evaluated at . At resonance we find , , , where we put , . The variance Eq. (6) is non-monotonic in the coupling energy (figure 2 (a)) and its minimum depends on the noise variances . For typical values of the amplitudes of flux and critical current noise  the dominant effect is due to flux noise, and the optimal coupling is found at . Note that, since depends on (via ), the differential dispersion at is non vanishing unless the coupling is switched off. The condition of minimal variance effectively identifies an ”optimal” dispersion leading to minimal defocusing, see figure 2(b).
In addition we observe that, since depends on but not on the Josephson energy, the optimized SWAP frequency, , can be engineered by appropriately fixing (within the experimental tolerances) the ratios . This recipe can be conveniently applied even if an independent estimate of the flux noise amplitude, , for the specific setup is not available. In fact, the variance of the stochastic SWAP splitting, , depends very smoothly on (figure 2 (a)) allowing a practical estimate of based on the characteristic value of observed in different flux and phase qubits. Alternatively, if different devices can be fabricated, one should select the sample with the ratio taking the right value for the given noise level of that particular device.
The effectiveness against defocusing of operating at the optimal coupling is revealed by the concurrence , which we evaluate in the SPA. We assume the system is prepared in the state and freely evolves. In the adiabatic approximation populations are constant thus . Evaluating the integral (5) we obtain
A measure of the entanglement preservation is the ”SWAP decay time”  defined by the condition . At the optimal coupling is one order of magnitude larger than for a generic coupling, assuming remarkable values up to s stable with increasing , figure 3(a). In addition, a fidelity to the Bell state is maintained up to s, about times longer than for a generic coupling, figure 3(b). These results elucidate the capability of the proposed operating condition to drastically reduce defocusing due to flux and critical current noise. On the other hand, energy relaxation processes are expected to limit the gate fidelity and the qubit relaxation times of the considered architecture . In the following Section we discuss the robustness of the optimal coupling condition to relaxation processes.
4 Optimal operating conditions: robustness to relaxation processes
We now discuss the robustness of the above optimization against relaxation processes due to flux noise and to spontaneous emission through the resonator. Flux quantum noise is due to the external magnetic flux bias through a mutual inductance  and it enters the Josephson energies in and in . It is included by adding to , Eq. (3), the terms
For the transmon at the flux sweet spot it is , for transmon 2 instead , where are quantized phase fluctuations. conserves the parity of the total number of the two transmons excitations. Thus it does not connect the states to the ground state which, to the first order in takes the form . Disregarding thermal excitation processes to higher energy states, the only effect of flux quantum noise is inside the bi-dimensional subspace . By solving a Bloch-Redfield master equation  relaxation and decoherence times in the SWAP subspace are given by the usual relation (a pure dephasing term is disregarded with respect to defocusing due to noise).
The main contribution comes from linear phase fluctuactions of the transmon displaced from the sweet spot, . At low temperatures , flux quantum noise is . For typical parameters we estimate s at optimal coupling (, ) . Thus the efficiency of the optimized gate on the SWAP time scale is not limited by relaxation processes due to flux noise. This is illustrated in figure 4 where we plot the envelope of the concurrence, , which in the presence of noise and flux quantum noise reads
Here the population difference in the SWAP subspace is , with the thermal equilibrium value, and . We observe that for optimal coupling Bell inequality violation, guaranteed until  , occurs for times s, much longer than for generic coupling.
In the cQED architecture each transmon is dispersively coupled to a resonator used for control and readout. An important mechanism for processes is spontaneous emission through the resonator (Purcell effect) [13, 24]. If each transmon operates at positive resonator-transmon detunings, MHz, where is the transmon-resonator coupling strength, the spontaneous emission rate of the coupled transmons is due to the ”single-mode” Purcell effect [24, 18]. To evaluate it we rewrite , Eq. (3), in the basis of its perturbative eigenstates and perform the rotating wave approximation eliminating terms describing the simultaneous excitation (de-excitation) of one resonator and the coupled-transmons system. The restriction to the subspace reads
where . The eigenstates of (10) are obtained by treating the last term in first order perturbation theory. The ground state is unmodified and reads , where are Fock states of the -th resonator, . The corrections to the states read , where and are the unperturbed eigenenergies of (10). The spontaneous decay rate is obtained applying Fermi’s golden rule to the interaction Hamiltonian of each resonator with its harmonic bath. The transition rate from the coupled transmons plus resonators state to the ground state , is
where is the spontaneous emission rate of oscillator and we considered single photon losses to each bath. The coupled transmons SWAP levels experience a Purcell induced spontaneous emission rate reduced with respect to the sum of the resonators spontaneous emission rates. For identical transmons in cavities with a lifetime ns we estimate s (analogously to the transmon’s relaxation time predicted in Ref. ), signaling a limitation to the optimized gate efficiency. We expect that the recently proposed Purcell filter or protection schemes  can be suitably extended to the considered two-qubit gate which is based on independent readout, possibly overcoming Purcell limitation.
In conclusion, we demonstrated optimization of a cQED entangling gate against any relevant noise source while keeping the hardware simplicity of the fixed coupling and even if one qubit does not operate at optimal bias point. The estimated high performance of the gate signals the effective elimination of leading order effects of noise.
Our analysis included all the relevant noise sources acting during the entanglement generating operation in the considered architecture. We have shown that the proposed scheme is robust with respect to relaxation processes due to quantum noise and it is likely to foresee a design protected also from Purcell effect. Additional errors during readout may of course influence the overall gate fidelity of any specific implementation . The responsible error sources need to be independently eliminated. However, the value of the optimal coupling is not affected by minimization of error sources acting before/after the coupled-qubits evolution. Similarly, for qubit-based quantum information , optimization of single and two qubits quantum operations is a key requirement, even though the overall quantum processor will suffer from error sources in between quantum operations or at preparation/readout.
Eliminating decoherence remains the biggest challenge for superconducting systems. Further optimization may require on one side suppression of higher order effects of noise, on the other limitation of relaxation due to quantum noise. Concerning intrinsic noise sources, like those responsible for noise, material engineering at the microscopic scale may be required in the near future. ”Passive” optimization startegies may be conveniently combined with ”active” control tools, like dynamical decoupling protocols inspired to nuclear magnetic resonance which have been already applied to superconducting systems [31, 37]. On a longer time scale, imperfections in the coherent control might represent the ultimate limit to computer performance.
Appendix A Validity regime of the Static Path Approximation
The SPA is valid for times . Thus it applies to the considered operation if . Since flux noise is the most relevant noise source in the considered setup, here we disregard critical current fluctuations and consider the recent noise figures reported in Ref. . In that article flux noise extends up to MHz, thus we can reasonably expect that the condition is satisfied.
Moreover, we numerically verified that the SPA is a valid approximation also for times provided that is smaller than the system oscillation frequency . In figure 5 we report the coherence between the states in the SPA and the result of the numerical evaluation of the adiabatic approximation
where and the derivatives are reported in Section 3 below Eq. (6). In the figure we considered flux noise on qubit 2, , distributed with and , . It is clearly seen that the SPA is a reasonable approximation up to times . This legitimates the use of the SPA for the evaluation of the times reported in figure 3. The error with respect to an estimate based on the adiabatic approximation is of (inset of figure 5).
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