Optimal operating conditions of an entangling two-transmon gate
aa r X i v : . [ c ond - m a t . s up r- c on ] M a y Optimal operating conditions of an entanglingtwo-transmon gate
Antonio D’Arrigo
E-mail: [email protected]
Dipartimento di Fisica e Astronomia, Universit`a di Catania, c/o Viale A. Doria 6,Ed. 10, 95125 Catania, Italy and CNR - IMM - MATIS c/o DFA Via Santa Sofia 64,95123 Catania, Italy
Elisabetta Paladino
E-mail: [email protected]
Dipartimento di Fisica e Astronomia, Universit`a di Catania, c/o Viale A. Doria 6,Ed. 10, 95125 Catania, Italy and CNR - IMM - MATIS c/o DFA Via Santa Sofia 64,95123, Catania, Italy
Abstract.
We identify optimal operating conditions of an entangling two-qubit gaterealized by a capacitive coupling of two superconducting charge qubits in a transmissionline resonator (the so called ”transmons”). We demonstrate that the sensitivity of theoptimized gate to 1 /f flux and critical current noise is suppressed to leading order.The procedure only requires a preliminary estimate of the 1 /f noise amplitudes. Noadditional control or bias line beyond those used for the manipulation of individualqubits are needed. The proposed optimization is effective also in the presence ofrelaxation processes and of spontaneous emission through the resonator (Purcell effect).PACS numbers: 03.67.Lx, 85.25.-j,03.65.Yz ptimal operating conditions of an entangling two-transmon gate
1. Introduction
Superconducting circuits are a promising technology for the realization of quantuminformation on a solid state platform. Several types of qubits [1] have been developedrealizing high fidelity single qubit operations [2, 3]. Rapid progress has also beenmade towards the realization of robust and scalable universal two-qubit gates [4, 5, 6].The circuit quantum electrodynamics (cQED) [7] architecture demonstrated to beparticularly promising for scalable quantum information. In this scheme highlyentangled two [8, 9] and three qubits [10] have been generated and simple quantumalgorithms have been demonstrated [11, 12].The coherence times of the present generation of devices ( ∼ µ s) are about threeorders of magnitudes larger than the first implementations. A relevant step furthertoward this enhancement has been the elimination of linear sensitivity to low-frequency(1 /f ) noise by operating qubits at ”optimal” working points. After the first ”sweetspot” operation demonstrated in Ref. [3], a further boost of qubit performances hasbeen achieved in a cQED design named ”transmon” [13], which is almost insensitive tothe detrimental effect of 1 /f charge noise [14] at the price of reduced anharmonicity.However, cQED architectures share with other implementations the presence of 1 /f flux noise whose amplitude has a characteristic order of magnitude [15], and of 1 /f critical current noise [13]. Together with relaxation processes due to quantum noise,dephasing due to 1/f flux and critical current noise still limits the time scales overwhich phase coherence and entanglement are preserved. In fact, further improvementof the coherence times at least of one order of magnitude would be required to reachthe level for practical quantum error correction [16]. Recently in a new circuit-QEDarchitecture employing a three-dimensional resonator the error correction threshold hasbeen approached [17]. ”Optimization” is thus a key-word of the present generation ofsuperconducting nano-circuits. Clever circuit design and optimal tuning of multi-qubitarchitectures, supplemented by the use of improved materials, are two complementarystrategies currently exploited to address this problem.A major question currently unsolved is establishing the best strategy to maintainlong-enough a sufficient degree of entanglement. In the present article we address thisissue considering a universal two-qubit gate realized by a fixed capacitive coupling of twotransmons in a cQED architecture. The implementation of this scheme has been recentlyreported in Ref. [9] where a √ i − SWAP operation with individual single-shot non-destructive readout [18] and gate fidelity of 90%, partly limited by qubit decoherence,has been demonstrated. A similar system has been studied theoretically in [19, 20].Here we identify ”optimal” [21] operating conditions of a transmons √ i − SWAP gatetaking into account the multi-level nature of the nano-circuit. We find that an ”optimalcoupling” exists where the leading order effects of 1 /f flux and critical current noiseare eliminated. The amount of preserved entanglement is quantified by the concurrencebetween the two transmons, C ( t ), which we evaluate in analytic form. The efficiency ofthe ”optimal coupling” is demonstrated by the fact that, for typical 1 /f noise spectra ptimal operating conditions of an entangling two-transmon gate T ∗ SWAP2 & µ s (in the absence of other decay mechanisms). Inaddition, C ( t ) may attain values [23] guaranteeing violation of a Bell inequality until ∼ µ s and the gate fidelity is 99% up to ∼ µ s. Finally, we demonstrate thatthe optimization is effective also in the presence of relaxation processes due to fluxquantum noise. Similarly to other cQED systems [24], the gate efficiency can be limitedby spontaneous emission through the resonator. This limitation is likely to be overcomeby suitable Purcell filters or protected designs [25]. The optimization proposed in thepresent article can further improve the considerable performance of cQED two-qubitgates based on cavity-mediated interaction [12, 10] or on tunable effective interactionwith microwave control [26]. Remarkably, here effective elimination of omnipresent 1 /f noise sources is achieved even if one qubit does not operate at optimal bias and withoutadditional controls or bias lines beyond those used for the manipulation of individualqubits, an important feature for scalability.
2. Universal two-transmon gate
We consider two transmons with a fixed capacitive coupling, each qubit being embeddedin its superconducting resonator used for control and bit-wise readout [9, 18]. Theinteraction is effectively switched on/off by dynamically changing the qubits detuningusing single qubits control lines. For this reason one of the qubits does not operate atits sweet spot. In figure 1 (a) we report the circuit diagram of the considered system.Each transmon, denoted by the subfix α = 1 ,
2, consists of a Cooper-Pair-Box (CPB)characterised by the charging energy E Cα and Josephson energy E Jα = E Jα cos( φ α ),tunable via the magnetic flux threading the superconducting loop, φ α = π Φ α / Φ ,(Φ is the flux quantum). In the circuit-QED scheme each CPB is embedded in atransmission line resonator whose relevant mode is modeled as a LC oscillator [13].Thus the Hamiltonian of transmon α consists of the CPB Hamiltonian plus the dipole-like interaction with the LC oscillator [13] H α = E Cα (ˆ q α − q x,α ) − E Jα ( φ α ) cos ˆ ϕ α + ω rα a † α a α + 2 β α eV α ˆ q α ( a α + a † α ) , (1)where phase, ˆ ϕ α and charge, ˆ q α , are conjugate variables, [ ˆ ϕ α , ˆ q α ] = i . The resonatorenergy is ω rα = 1 / √ L α C α and a † α ( a α ) creates (annihilates) one photon in thetransmission line ( ~ = 1). V α = p ω rα / C α is the root-mean-square voltage of theoscillator and β α = C gα /C Σ α is the ratio between the gate capacitance coupling theCPB to the local mode and the CPB total capacitance.The transmon operates at E Cα ≪ E Jα . Under these conditions, values of thephases ϕ α close to zero are most favored. This motivates the neglect of the periodicboundary condition on the phases and the expansion of the cosine in Eq. (1). Within thisapproximation, the offset charge q x,α can be eliminated via a gauge transformation [13].Of course, the perturbative scheme cannot capture the non-vanishing charge dispersionof the transmon [27, 28]. In particular, the exponential decrease of the charge dispersion ptimal operating conditions of an entangling two-transmon gate Φ L (a) (b) C g C L Φ C | g i|−i| + i ω + − C g C c C B E C , E J C B E C , E J Figure 1. (a) Circuit diagram of the two transmon gate with capacitive couplingenergy E CC = (2 e ) C T /C Σ1 C Σ2 , where 1 /C T = 1 /C c + 1 /C Σ1 + 1 /C Σ2 and C Σ α = C Jα + C gα + C Bα . (b) Schematic level structure. with p E Jα /E Cα for E Jα /E Cα ≫ /f ) fluctuactions of the offset charge [14]. Here we relyon this well established result and eliminate q x,α from the outset. Expanding the cosinein Eq. (1) up to fourth order, the CPB Hamiltonian can be cast in the form of a weaklyanharmonic oscillator (Duffin oscillator) H D α = Ω α b † α b α − ( E Cα / b α + b † α ) , (2)where the bosonic operators b α , b † α are related to the charge operator via ˆ q α = − i ( E Jα / E Cα ) / ( b α − b † α ) / √ α ≡ Ω α ( φ α ) = p E Cα E Jα ( φ α ). Thetwo lowest eigenenergies of H D α identify the transmon- α qubit levels. Their splittingis ˜Ω α = Ω α − E Cα / α . The flux”sweet-spot” is at φ α = 0 [3, 13].The capacitive coupling between the CPBs, E CC (ˆ q − q x, )(ˆ q − q x, ), adds to P α H α leading to the Hamiltonian H = H D1 + H D2 + ¯ E CC b − b † )( b − b † ) , (3)where ¯ E CC = E CC ( E J E J / E C E C ) / is the effective coupling depending on thecontrol parameters φ α via the Josephson energies. Note that fluctuations of themagnetic fluxes affect the effective coupling between the qubits. Typical values [9]are E Cα ∼ E Jα ∼ −
30 GHz, leading to ¯ E CC = 10 − − − GHz.The coupled transmons eigenenergies and eigenstates are conveniently obtained bytreating in perturbation theory with respect to P α Ω α b † α b α both the anharmonic termsand the capacitive interaction included in V = ¯ E CC b − b † )( b − b † ) − X α E Cα
48 ( b α + b † α ) . (4)The level structure is schematically show in figure 1 (b). The splitting in thesubspace where the SWAP operation takes place (in short ”SWAP splitting”) reads ω + − = q ( ˜Ω − ˜Ω ) + ¯ E CC and the corresponding eigenstates spanning the ”SWAPsubspace” are |−i = − sin( η/ | i +cos( η/ | i and | + i = cos( η/ | i +sin( η/ | i , ptimal operating conditions of an entangling two-transmon gate η = ¯ E CC / ( ˜Ω − ˜Ω ) and | a, b i ≡ | a i | b i are eigenstates of P α Ω α b † α b α , ( a , b ∈ { , } ). The interaction is effectively switched on by tuning the single-qubit energyspacing to mutual resonance. The resonance condition is realized by tuning the flux biasuntil ˜Ω = ˜Ω , displacing one qubit from the sweet spot at φ α = 0. In the following wesuppose that φ = 0 and φ = 0. Under resonance conditions the √ i − SWAP operation | i → | ψ e i = [ | i − i | i ] / √ t E = π/ ω + − starting from a factorized initial state in the ”SWAP subspace”.
3. Optimal operating conditions: reduction of /f noise effects Since the two qubits do not operate at the same working point, the dominant sourceof dephasing is different for the two transmons. In particular, first order fluctuactionsof the transmon splittings are due to 1 /f critical current noise for transmon 1 andto 1 /f flux noise for transmon 2 [13]. These fluctuations can be treated in theadiabatic and longitudinal approximation [29] by replacing E Jα with E Jα (1 + x α ( t )).Here x α ( t ) represent stochastic fluctuations of the dimensionless critical current x ( t ) = δI c ( t ) = ∆ I c ( t ) /I c , and of the flux Φ , x ( t ) = tan( φ ) δφ ( t ). The leading ordereffect of adiabatic noise is defocusing, expressed by the ”static path” or static noiseapproximation (SPA) [29, 30] describing the average of signals oscillating at randomlydistributed effective frequencies (see Appendix A for the validity regimes of the SPAin the present problem). It is obtained by replacing x α ( t ) with statistically distributedvalues x α . In the SPA the coherence between the states |±i is h ρ + − ( t ) i = ρ + − (0) e − iω + − t h e − iδω + − t i , (5)where ρ ( t ) is the two-qubit density matrix and h ... i indicates the average over thefluctuations x α . Here we assume that they are uncorrelated random variables withGaussian distribution, zero mean and standard deviations Σ x α proportional to theamplitude of the 1 /f spectrum, S /fx α ( ω ) = π Σ x α [ln( γ Mα /γ mα ) ω ] − ( γ mα and γ Mα are thelow and the high frequency cut-offs of the 1 /f region). As demonstrated in Refs. [21, 22]the optimal operating condition is obtained imposing a minimum of the variance of thestochastic SWAP splitting, Σ = h ω − i − h ω + − i . This is simply understood consideringthe short-times expansion |h e − iδω + − t i| ≈ p − (Σ t ) , implying defocusing suppressionwhen Σ is minimal. Expanding ω + − around the fixed working point we getΣ ≈ X α (cid:18) ∂ω + − ∂x α (cid:19) Σ x α + 12 X α,β (cid:18) ∂ ω + − ∂x α ∂x β (cid:19) Σ x α Σ x β , (6)where all derivatives are evaluated at x α ≡
0. At resonance we find ∂ω + − /∂x α = ¯ E CC / ∂ ω + − /∂x α = − E CC /
16 + Ω / (4 ¯ E CC ), ∂ ω + − /∂x ∂x = − ¯ E CC / − Ω / (4 ¯ E CC ),where we put Ω α ≡ Ω, E Cα ≡ E C . The variance Eq. (6) is non-monotonic in thecoupling energy (figure 2 (a)) and its minimum depends on the noise variances Σ x α .For typical values of the amplitudes of 1 /f flux and critical current noise [32] thedominant effect is due to flux noise, Σ x ≫ Σ x and the optimal coupling is found at E opt CC ≈ E C (Σ x / √ / . Note that, since ¯ E CC depends on x α (via E Jα ), the differential ptimal operating conditions of an entangling two-transmon gate Figure 2.
Panel(a): SWAP-splitting variance Eq.(6) as a function of E CC /E C .The value of the minimum of Σ at the optimal point E opt CC = 2 E C (Σ x / √ / =1 . · − E C is one order of magnitude smaller than at E CC = 10 − E C . Panel (b):Dispersion branch δω + − ( x , x = 0) /E C for | x | ≤ x . The black line is for a genericcoupling E CC = 10 − E C , the red line is for the optimal coupling E opt CC . Parametersare E Cα = 1GHz, E Jα = 30 E Cα with φ = 0 .
64 and Σ x = 10 − . dispersion ∂ω + − /∂x α at x α = 0 is non vanishing unless the coupling is switched off.The condition of minimal variance effectively identifies an ”optimal” dispersion leadingto minimal defocusing, see figure 2(b).In addition we observe that, since E opt CC depends on E C but not on the Josephsonenergy, the optimized SWAP frequency, ω opt+ − ≈ ¯ E opt CC , can be engineered by appropriatelyfixing (within the experimental tolerances) the ratios E Jα /E Cα . This recipe can beconveniently applied even if an independent estimate of the flux noise amplitude, Σ x ,for the specific setup is not available. In fact, the variance of the stochastic SWAPsplitting, Σ , depends very smoothly on E CC (figure 2 (a)) allowing a practical estimateof E opt CC based on the characteristic value of Σ x observed in different flux and phasequbits. Alternatively, if different devices can be fabricated, one should select the samplewith the ratio E Jα /E Cα taking the right value for the given noise level of that particulardevice.The effectiveness against defocusing of operating at the optimal coupling is revealedby the concurrence [33], which we evaluate in the SPA. We assume the system is preparedin the state | i and freely evolves. In the adiabatic approximation populations areconstant thus C ( t ) ≈ | Im h ρ + − ( t ) i| . Evaluating the integral (5) we obtain C SPA ( t ) ≈ (cid:12)(cid:12)(cid:12) Im exp n −
12 Σ x (cid:16) ∂ω + − ∂x (cid:17) t i Σ x ( ∂ ω + − ∂x ) t oq i Σ x ( ∂ ω + − ∂x ) t (cid:12)(cid:12)(cid:12) . (7)A measure of the entanglement preservation is the ”SWAP decay time” [22] definedby the condition | C SPA ( T ∗ SWAP2 ) | = e − . At the optimal coupling T ∗ SWAP2 is one orderof magnitude larger than for a generic coupling, assuming remarkable values up to T ∗ SWAP2 ∼ µ s stable with increasing E J /E C , figure 3(a). In addition, a 99% fidelityto the Bell state | ψ ent i = [ | i + | i ] / √ T F ≈ µ s, about 4 times ptimal operating conditions of an entangling two-transmon gate
10 15 20 25 302050200500 E J (cid:144) E C T * S W A P H Μ s L H a L
10 15 20 25 30261020 E J (cid:144) E C T F H Μ s L H b L Figure 3. T ∗ SWAP2 (panel a) and T F (panel b) as a function of E J /E C for E Cα = 1GHz. The variances Σ x = 5 · − , Σ x = 10 − correspond to typicalvalues of 1 /f critical current and flux noise [32] rescaled to the present setup. Blacklines are obtained for E CC = 10 − E C , red lines correspond to E opt CC = 1 . · − E C . longer than for a generic coupling, figure 3(b). These results elucidate the capability ofthe proposed operating condition to drastically reduce defocusing due to 1 /f flux andcritical current noise. On the other hand, energy relaxation processes are expected tolimit the gate fidelity and the qubit relaxation times of the considered architecture [9].In the following Section we discuss the robustness of the optimal coupling condition torelaxation processes.
4. Optimal operating conditions: robustness to relaxation processes
We now discuss the robustness of the above optimization against relaxation processesdue to flux noise and to spontaneous emission through the resonator. Flux quantumnoise is due to the external magnetic flux bias through a mutual inductance M [13] andit enters the Josephson energies in H D and in ¯ E CC . It is included by adding to H , Eq.(3), the terms∆ H = − X α Ω α b † α b α ˆ x α − ¯ E CC b − b † )( b − b † ) X α ˆ x α . (8)For the transmon at the flux sweet spot it is ˆ x = δ ˆ φ /
2, for transmon 2 insteadˆ x = tan φ δ ˆ φ , where δ ˆ φ α are quantized phase fluctuations. ∆ H conserves the parityof the total number of the two transmons excitations. Thus it does not connectthe states |±i to the ground state which, to the first order in V takes the form | g i ∝ | i + a | i + P ij =1 , a j | , j i + a i | i, i . Disregarding thermal excitationprocesses to higher energy states, the only effect of flux quantum noise is inside thebi-dimensional subspace {|±i} . By solving a Bloch-Redfield master equation [34]relaxation and decoherence times in the SWAP subspace are given by the usual relation T SWAP2 = 2 T SWAP1 = { P α (sin η +( − α cos η ¯ E CC / Ω α ) S x α ( ω + − ) } − (a pure dephasingterm ∝ S x α (0) is disregarded with respect to defocusing due to 1 /f noise). ptimal operating conditions of an entangling two-transmon gate t H Μ s L C H t L Figure 4.
Envelope of the concurrence in the presence of 1 /f flux and critical currentnoise (as in figure 3) and flux quantum noise on qubit 2 with spectrum S x ( ω ) ≃ − ω .Black line is for E CC = 10 − E C ; the red line is for E opt CC = 1 . · − E C . The dashedgray line marks the value C = 1 / √
2, the dotted line C ( T SWAP2 ) = e − . The main contribution comes from linear phase fluctuactions of the transmondisplaced from the sweet spot, ˆ x . At low temperatures k B T ≪ Ω α , flux quantum noise is S x ( ω ) ≈ (Ω tan φ ) ( πM/ Φ ) ω/R . For typical parameters we estimate T SWAP2 ≈ M = 140Φ /A , R ∼ envelope of the concurrence, C ( t ), whichin the presence of 1 /f noise and flux quantum noise reads C ( t ) ≈ { [sin η ( ρ ++ ( t ) − ρ −− ( t )) + 2 cos η Re { ρ + − ( t ) } ] + 4 Im { ρ + − ( t ) } } / . (9)Here the population difference in the SWAP subspace is ρ ++ ( t ) − ρ −− ( t ) =(cos η − δ eq ) e − t/T SWAP1 + δ eq , with δ eq the thermal equilibrium value, and ρ + − ( t ) ≡h ρ + − ( t ) i e − t/T SWAP2 . We observe that for optimal coupling Bell inequality violation,guaranteed until [35] C ( t ) ≥ − / , occurs for times ∼ µ s, much longer than forgeneric coupling.In the cQED architecture each transmon is dispersively coupled to a resonatorused for control and readout. An important mechanism for T processes is spontaneousemission through the resonator (Purcell effect) [13, 24]. If each transmon operates atpositive resonator-transmon detunings, ∆ α = ω rα − Ω α ∼ ≫ g α ∼
50 MHz,where g α is the transmon-resonator coupling strength, the spontaneous emission rate ofthe coupled transmons is due to the ”single-mode” Purcell effect [24, 18]. To evaluateit we rewrite H , Eq. (3), in the basis of its perturbative eigenstates and perform therotating wave approximation eliminating terms describing the simultaneous excitation(de-excitation) of one resonator and the coupled-transmons system. The restriction tothe subspace {| g i , | σ = ±i} reads H + X α ω rα a † α a α + X α,σ = ± ( g α,σ | g ih σ | a † α + h . c . ) (10) ptimal operating conditions of an entangling two-transmon gate g α,σ = i √ β α eV α ( E Jα / E Cα ) / h g | ( b α − b † α ) | σ i . The eigenstates of (10) areobtained by treating the last term in first order perturbation theory. The groundstate is unmodified and reads | g, , i , where | m α i are Fock states of the α -thresonator, m α ∈ N . The corrections to the states | σ, m , m i read | σ, m , m i (1) = g σ √ m +1 E σ − E g − ω r | g, m +1 , m i + g σ √ m +1 E σ − E g − ω r | g, m , m +1 i , where E σ and E g are the unperturbedeigenenergies of (10). The spontaneous decay rate is obtained applying Fermi’s goldenrule to the interaction Hamiltonian of each resonator with its harmonic bath. Thetransition rate from the coupled transmons plus resonators state | σ, , i + | σ, , i (1) to the ground state | g, , i , is w σ = 2 X α κ α (cid:12)(cid:12)(cid:12) g α,σ E σ − E g − ω rα (cid:12)(cid:12)(cid:12) (11)where κ α is the spontaneous emission rate of oscillator α and we considered single photonlosses to each bath. The coupled transmons SWAP levels experience a Purcell inducedspontaneous emission rate reduced with respect to the sum of the resonators spontaneousemission rates. For identical transmons in cavities with a lifetime 1 /κ α ≈
160 ns weestimate 1 /w σ ≈ µ s (analogously to the transmon’s relaxation time predicted inRef. [13]), signaling a limitation to the optimized gate efficiency. We expect that therecently proposed Purcell filter or protection schemes [25] can be suitably extended to theconsidered two-qubit gate which is based on independent readout, possibly overcomingPurcell limitation.
5. Conclusions
In conclusion, we demonstrated optimization of a cQED entangling gate against anyrelevant 1 /f noise source while keeping the hardware simplicity of the fixed couplingand even if one qubit does not operate at optimal bias point. The estimated highperformance of the gate signals the effective elimination of leading order effects of 1 /f noise.Our analysis included all the relevant noise sources acting during the entanglementgenerating operation in the considered architecture. We have shown that the proposedscheme is robust with respect to relaxation processes due to quantum noise and it is likelyto foresee a design protected also from Purcell effect. Additional errors during readoutmay of course influence the overall gate fidelity of any specific implementation [9]. Theresponsible error sources need to be independently eliminated. However, the value ofthe optimal coupling is not affected by minimization of error sources acting before/afterthe coupled-qubits evolution. Similarly, for qubit-based quantum information [36],optimization of single and two qubits quantum operations is a key requirement, eventhough the overall quantum processor will suffer from error sources in between quantumoperations 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 of1 /f noise, on the other limitation of relaxation due to quantum noise. Concerning ptimal operating conditions of an entangling two-transmon gate t ( µ s) | < ρ + - ( t ) > / ρ + - ( ) |
300 320 340 360 t ( µ s) exp(-1) Figure A1.
Absolute value of the coherence h ρ + − ( t ) i /ρ + − (0) in the presence of 1 /f flux noise on qubit 2 with Σ x = 10 − . The (thick) red line is the result of the SPA,the (thin) blue line is the numerical evaluation of the adiabatic approximation (A.1)for γ m = 1 s − , γ M = 10 s − as in [31], the dashed blue line is for γ m = 1 s − , γ M = 10 s − : the smaller is the high-frequency cut-off γ M , the closer is theSPA to the adiabatic approximation. Inset: zoom around the time range where ρ + − ( t ) /ρ + − (0) ≈ e − . Other parameters are E Cα = 1 GHz, E Jα = 30 GHz, E CC = 1 . × Hz ( ¯ E CC ≈
65 MHz). In the simulations we considered an ensembleof ∼ random telegraph noise processes with switching rates distributed as ∝ /γ in[ γ m , γ M ]. The average is performed using 10 realizations of the stochastic process. intrinsic noise sources, like those responsible for 1 /f noise, material engineering at themicroscopic scale may be required in the near future. ”Passive” optimization startegiesmay be conveniently combined with ”active” control tools, like dynamical decouplingprotocols inspired to nuclear magnetic resonance which have been already applied tosuperconducting systems [31, 37]. On a longer time scale, imperfections in the coherentcontrol might represent the ultimate limit to computer performance. Acknowledgments
Discussions with G. Falci and information about ongoing experiments by D. Vion aregratefully acknowledged. This work was partially supported by EU through Grant No.PITN-GA-2009-234970 and by the Joint Italian-Japanese Laboratory on ”QuantumTechnologies: Information, Communication and Computation” of the Italian Ministryof Foreign Affairs.
Appendix A. Validity regime of the Static Path Approximation
The SPA is valid for times t < /γ Mα . Thus it applies to the considered √ i − SWAPoperation if t E = π/ ω + − ≈ − − − GHz < /γ Mα . Since flux noise is the mostrelevant 1 /f noise source in the considered setup, here we disregard critical currentfluctuations and consider the recent noise figures reported in Ref. [31]. In that article ptimal operating conditions of an entangling two-transmon gate /f flux noise extends up to ∼ t E < /γ Mα is satisfied.Moreover, we numerically verified that the SPA is a valid approximation also fortimes t > /γ M provided that γ M is smaller than the system oscillation frequency [29].In figure A1 we report the coherence between the states |±i in the SPA and the resultof the numerical evaluation of the adiabatic approximation ρ + − ( t ) = ρ + − (0) Z D [ { x α ( s ) } ] P [ { x α ( s ) } ] e − i R t ds ω + − ( { x α ( s ) } ) , (A.1)where ω + − ( { x α ( s ) } ) ≈ P α ∂ω + − ∂x α x α ( s ) + P α,β ∂ ω + − ∂x α ∂x β x α ( s ) x β ( s ) and the derivatives arereported in Section 3 below Eq. (6). In the figure we considered flux noise on qubit 2, x ( s ), distributed with Σ x = 10 − and γ m = 1 s − , γ M = 10 s − [31]. It is clearly seenthat the SPA is a reasonable approximation up to times ∼ /γ M . This legitimatesthe use of the SPA for the evaluation of the times T ∗ SWAP2 reported in figure 3. Theerror with respect to an estimate based on the adiabatic approximation is of ∼ References [1] Clarke J and Wilhelm F K 2008
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