Time-optimal performance of Josephson charge qubits: A process tomography approach
TTime–optimal performance of Josephson charge qubits: A process tomographyapproach
Robert Roloff ∗ and Walter P¨otz † Fachbereich Theoretische Physik, Institut f¨ur Physik,Karl–Franzens Universit¨at Graz, Universit¨atsplatz 5, 8010 Graz, Austria (Dated: April 16, 2019)A process tomography based optimization scheme for open quantum systems is used to determinethe performance limits of Josephson charge qubits within current experimental means. The qubitis modeled microscopically as an open quantum system taking into account state leakage, as well asenvironment–induced dephasing based on experimental noise spectra. Within time–optimal controltheory, we show that the competing requirements for suppression of state leakage and dephasingcan be met by an external control of the effective qubit–environment interaction, yielding minimalgate fidelity losses of around ∆ F ≈ − under typical experimental conditions. PACS numbers: 03.67.Lx, 03.67.Pp, 85.25.Cp, 02.30.Yy
I. INTRODUCTION
Within the circuit model of quantum computing, thefundamental building block of a quantum computer isthe quantum bit or qubit . Our ability to precisely exe-cute unitary operations within qubits is an essential pre-requisite for the implementation of quantum algorithmsand for harnessing the full computational power of morecomplex quantum system. Among the various proposalsfor a physical realization of a qubit, specially designedsuperconducting circuits have been identified as promis-ing candidates. Their potential is mostly founded onthe high technological standards by which SQUIDS canbe fabricated and controlled, as well as their promise re-garding array scalability. Several types of Josephson–junction–based qubit designs, such as charge, phase, andflux qubits, have been proposed and explored in the lab-oratory.
Conditional gate operations have alreadybeen performed experimentally.
While preliminary results indeed look promising, con-siderable improvement in gate performance will be nec-essary to make these structures useful in larger arrays ofquantum gates. Similar to all solid–state–based qubits,Josephson qubits suffer from two main shortcomings:they are quantum two–level systems only within approx-imation and there is non–negligible coupling to the en-vironment. The former may result in leakage to non–computational basis states which affects the fidelity ofquantum gates.
The latter results in an un-wanted population relaxation and destruction of statesuperpositions, both being detrimental for quantum com-putation.
In this work we study Josephson charge qubits andshow how the conflicting requirements for suppression ofboth state leakage and decoherence can be accomplishedwithin optimal control theory. In the first part, we out-line the model for the Josephson charge qubit includ-ing leakage and environmental interaction on which webase our study. Then, we formulate a cost functionalfor state–independent optimization of open quantum sys-tems based on the Kraus representation and process to- mography. This method can be readily applied to anyopen quantum system, including other qubit implemen-tations as well as multi–qubit gates and is not necessar-ily restricted to solid–state realizations. In the remain-der of the paper, we demonstrate the approach for theHadamard gate implementation.
II. JOSEPHSON CHARGE QUBIT
The basic structure of a Josephson charge qubit isshown in Fig. 1.
The characteristic energy scales arethe charging energy E C of the superconducting island,the Josephson coupling energy E J and the superconduct-ing energy gap ∆. If ∆ is the largest energy, the problem V C Φ E J , C J E J , C J FIG. 1: (Color online) Superconducting charge qubit with 2Josephson junctions. The two junctions are realized by smallinsulating layers which divide a superconducting ring into asmall island (red/gray) and a reservoir (thick black). Thequbit is driven by an external flux Φ and a gate voltage V .The voltage is coupled capacitively ( C ) to the island. TheJosephson junctions are characterized by a capacitance C J and the Josephson coupling energy E J . can be reduced to a situation where no quasiparticle ex-citation is found on the island and only Cooper pairs cantunnel through the Josephson junctions. Cooper–pair re-laxation on the island, as well as quasiparticles tunneling a r X i v : . [ c ond - m a t . s up r- c on ] J un from the reservoir onto the island, contribute, in princi-ple, to relaxation and decoherence. The latter takes placeat time scales of the order ( T qp ) − ∼ g T δ r π (cid:126) N qp , where g T denotes the conductance of the Josephson junctions (inunits of e /h ), δ r is the quasiparticle level spacing in thereservoir and N qp is the number of quasiparticles. Fortypical charge–qubit architectures, T qp is of the order of10 − s to 10 − s and, thus, on a time scale much largerthan we are investigating in the present work. Cooper–pair relaxation within the island is even weaker and maybe neglected, too.Within the charge basis { | n (cid:105)} , the Hamiltonian of thesystem, H S = 4 E C [ n − n c ( V )] − E J cos (cid:18) π ΦΦ (cid:19) cos Θ , (1)may be written as, H S = (cid:88) n (cid:110) E C [ n − n c ( V )] | n (cid:105) (cid:104) n | + E J (Φ) ( | n + 1 (cid:105) (cid:104) n | + | n (cid:105) (cid:104) n + 1 | ) (cid:111) , (2)with Θ (the phase difference across a Josephson junc-tion) being the conjugate variable to the number n ofadditional Cooper pairs on the island. E J (Φ) = − E J cos ( π Φ / Φ ) (3)and n c ( V ) = V C/ (2 e ) contain two independent physicalcontrol fields, V = V g + V p and Φ, where V g and V p denote a dc and a pulse gate voltage, respectively. Φ isthe external magnetic flux. The latter can be used totune E J (Φ) between − E J and 0. By proper choice ofthe dc gate voltage V g , one can set the qubit “workingpoint” to the charge degeneracy point, where the qubitis insensitive to charge fluctuations up to the first order.Using, next to the computational basis { | (cid:105) , | (cid:105)} , twoadjacent leakage states | − (cid:105) and | (cid:105) , the effective “leakyqubit” Hamiltonian at the charge degeneracy point reads, H S = E C E J (Φ) 0 0 E J (Φ) 0 E J (Φ) 00 E J (Φ) 0 E J (Φ)0 0 E J (Φ) 8 E C + 4 E C n p ( t ) X,X = − − , n p = n p ( V p ) . (4)For our computation we consider a typical charge qubitwith energies E C = 150 µ eV and E J = 35 µ eV from theexperiment. The qubit can be controlled by tuningthe external magnetic flux [Φ = Φ( t )] and the appliedpulse gate voltage [ V p = V p ( t )].In the derivation of the Hamiltonian [Eq. (1)], theloop–inductance of the superconducting ring has been ne-glected. For the parameters used in the present work, this approximation is justified if L (cid:28) Φ / (cid:0) π E J (cid:1) ≈
20 nH.We also assume that the Josephson coupling energiesof the two junctions are identical. However, if E J → E J + δE J for one of the junctions, Eq. (3) turns into E J (Φ) = − E J (cid:12)(cid:12)(cid:12) (1 + η ) cos (cid:16) π ΦΦ (cid:17) + iη sin (cid:16) π ΦΦ (cid:17)(cid:12)(cid:12)(cid:12) , with η ≡ δE J E J , see e.g. Ref. 20. In order to avoid such anasymmetry, it is possible to substitute one of the Joseph-son junctions in Fig. 1 by a SQUID, which enables oneto tune the Josephson energies to be equal. In superconducting qubits the dominant dephasingmechanism (at low temperature) is attributed to noisewhich changes from 1 /f to Ohmic behavior at frequen-cies of typically k B T / (cid:126) . The microscopic origin of the1 /f noise is not fully understood yet but it is believedthat it originates from background charge fluctuations.Ohmic contributions may result from intrinsic sources orfrom gate lines. We map these fluctuations ontoa bath of harmonic oscillators coupling linearly to thequbit.
The system is thus modeled by a Hamiltonianof the form, H ( t ) = H S ( t ) + H B + H I , H B = (cid:88) k (cid:126) ω k b † k b k ,H I = (cid:126) X ⊗ (cid:88) k g k (cid:16) b † k + b k (cid:17) ≡ X ⊗ Γ , (5)where g k is the effective coupling constant of the spin–boson interaction and b † k and b k are the bosonic creationand annihilation operators for the mode with frequency ω k . The dynamics of system, bosonic bath, and the in-teraction between the two, respectively, is governed bythe Hamiltonians H S , H B , and H I , whereby the exter-nal control field (cid:15) ( t ) to be optimized is contained in H S only, thus assuming that there is no direct control over H B and H I .Apart from correlation functions, noise has to be clas-sified by its amplitude distribution (AD). For the spin–boson model with a thermal bath the AD is normal–distributed (Gaussian noise), which agrees quite well withexperiment, especially at the short time scales with whichwe are concerned. Guided by the experimental resultsof Ref. 15, we construct the spectral density by choosing J ( ω ) = J /f ( ω )+ J f ( ω ) consisting of a 1 /f and an Ohmiccontribution, J /f = α /f / (4 k B T ) , J f ( ω ) = α f / (2 (cid:126) ) ω. (6)For thermal equilibrium, the noise spectrum S Γ ( ω ) asso-ciated with the operator Γ in Eq. (5) in terms of J ( ω ) isthen (a tilde denoting the interaction picture), S Γ ( ω ) = (cid:68)(cid:110) ˜Γ( t ) , ˜Γ( t (cid:48) ) (cid:111)(cid:69) ω = ∞ (cid:90) −∞ dτ (cid:110) ˜Γ( t ) , ˜Γ( t (cid:48) ) (cid:111) e iτω = 2 (cid:126) J ( ω ) coth (cid:20) (cid:126) ω k B T (cid:21) , τ ≡ t − t (cid:48) . (7)For (cid:126) ω (cid:28) k B T and (cid:126) ω (cid:29) k B T , Eqs. (6) and (7)give the 1 /f noise spectrum, S /f ( ω ) ≈ α /f /ω , andthe Ohmic spectrum, S ω ( ω ) ≈ α f ω , respectively. Thestrength of the charge fluctuations (proportional to α /f ), has been determined in experiment to saturate at ∼ (10 − e ) for temperatures lower than 200 mK. Theslope of the Ohmic contribution is given in Ref. 15. Aplot of the specific noise spectrum which we use numeri-cally can be seen in the inset of Fig. 3a.Inspection of Eqs. (4) reveals that simple controlstrategies for leakage suppression and minimization ofdecoherence are in conflict with one another. The mostconvenient way to minimize dephasing within the com-putational subspace would be a large absolute value ofthe Josephson coupling energy E J (Φ) because it sets theminimum time ( t op ) needed to perform unitary transfor-mations which incorporate rotations around the x –axisof the Bloch sphere (as needed for the Hadamard gate).The faster t op , the lesser the effect of decoherence onthe gate. However, in the present superconducting qubitarchitecture, E J (Φ) is also responsible for coupling tothe non–computational basis states | − (cid:105) and | (cid:105) . Dueto the structure of X in Eq. (5), the net decoherencerate is enhanced when they participate in the dynamics.Thus, if decoherence–effects are present, the transfer ofcoherence to the leakage subspace is highly undesirable.In the unitary case, coherence can be transferred back tothe computational subspace without loss (given sufficientcontrol). These conflicting requirements, as well as thecomplexity of an open quantum system, make the designof pulses which maximize the fidelity of the quantum gatea nontrivial task. III. STATE–INDEPENDENT OPTIMALCONTROL
In the context of quantum information processing, co-herent control of unitary operations within the qubit isof fundamental interest. In contrast to the optimiza-tion of state–selective transitions, e.g. to maximize theprobability of a certain pathway in a chemical reaction, here we are interested in the optimization of the wholedynamical map. Hence, the qubit should perform a de-sired unitary operation, irrespective of its initial state.We call this task “state–independent” optimal control.We consider a Hilbert space which is a tensor productof the Hilbert space of the open quantum system (fromnow on denoted as system ) H S and the Hilbert spaceof the environment (which we will call bath ) H B , H = H S ⊗ H B = ( H S ⊕ H S ) ⊗ H B . Within the system wedistinguish between computational states on which thegate operation is specified, spanning H S , and “leakagelevels” spanning H S . The task of state–independent optimization can bestated as follows. The density matrix of the system, ρ S ( t ), evolves in time according to the map, ρ S (0) (cid:55)→ ρ S ( t ) = E t { ρ S (0) } , for which the superoperator E t is functionally dependent upon externally applied controlfields (cid:15) ( t ), i.e. E t = E t [ (cid:15) ]. In order to maintain the pos-itivity of ρ S ( t ), the map E has to be completely positive and thus can be represented by Kraus operators K m , ρ S ( t ) = E t [ (cid:15) ] { ρ S (0) } = (cid:88) m K m [ (cid:15) ]( t ) ρ S (0) K † m [ (cid:15) ]( t ) , (8)with K m [ (cid:15) ]( t ) depending on the propagator of the com-posite system, and hence on (cid:15) ( t ). We now want to find acontrol field (cid:15) ∗ ( t ) for which, at some final time t f , E t f [ (cid:15) ∗ ]approaches the desired mapping E D as closely as possible.For quantum gate operations, for example, one would set E D { . } = U D ( t f ) ( . ) U † D ( t f ) , (9)where U D ( t f ) is the unitary operation to be executedwithin the gate. In the context of quantum informa-tion theory it is useful to formulate a cost functionalwithin the language of process tomography (see e.g. Refs. 27,29,30,31) to define a measure of how well sucha, not necessarily unitary, desired operation has been ac-complished. We rewrite the mapping E by expanding theKraus operators, K m ( t ) = (cid:80) n α mn ¯ K n , with α mn ∈ C and ¯ K n ∈ A , where A denotes a complete basis set of M × M matrices. M = M C + M L is the total numberof orthonormal basis states, consisting of M C compu-tational and M L leakage levels. The final state of thequantum system, starting out in ρ (0), now reads, ρ ( t f ) = E t f { ρ (0) } = (cid:88) m,n ¯ K m ρ (0) ¯ K † n χ mn ( t f ) , (10)with χ mn = (cid:80) k α km α ∗ kn . For the example of the Joseph-son charge qubit described above, one has M C = 2, and M L = 2 with basis states { | (cid:105) , | (cid:105)} and { | − (cid:105) , | (cid:105)} ,respectively. Hence, we are dealing with a M × M =16 ×
16 representation of χ [see Figs. 2(a) and 2(d)]. Thematrix χ in Eq. (10) is usually termed process tomographymatrix and in order to compute its elements, we choose afixed set of operators { σ j } = B (for simplicity we choose B = A ) for which we determine the time–evolution withrespect to the mapping E , i.e. , σ j ( t f ) ≡ E t f { σ j } . (11)For one–qubit operations with 2 leakage levels or two–qubit gates, the tensor product of Pauli matrices τ i ⊗ τ j with i, j ∈ { x, y, z, } and τ = A and B , which we use in present work. In ex-periment, Eq. (11) corresponds to preparation in andsubsequent time–evolution of suitable different initialstates ( ∼ σ j ) of the quantum system. Recently, process-tomography methods have been applied to superconduct-ing qubits. Because Eq. (8) is a linear mapping, onecan rewrite Eq. (11) by E t f { σ j } = (cid:88) k c jk σ k . (12)The mapping E t f is described completely in terms of co-efficients c jk , whose experimental determination involvesquantum state tomography. The last step in order tocalculate the process tomography matrix relates c jk to χ mn . Therefore, we note that, combining Eqs. (10)–(12), (cid:88) k c jk σ k = (cid:88) m,n σ m σ j σ n χ mn ( t f ) . (13)To extract the coefficients c jk we apply the scalar product (cid:104) σ i , σ k (cid:105) = δ ik and get from Eqs. (11)–(13), c ji = (cid:104) σ i , σ j ( t f ) (cid:105) , (14)= (cid:88) m,n (cid:104) σ i , σ m σ j σ n (cid:105) χ mn ( t f ) ≡ (cid:88) m,n B imjn χ mn ( t f ) .χ mn ( t f ) can now be calculated by solving the systemof linear equations (cid:104) σ i , σ j ( t f ) (cid:105) = (cid:80) m,n B imjn χ mn ( t f ),which includes inversion of B .Defining the operator, ˆ χ = (cid:88) m,n ( ¯ K ∗ n ⊗ ¯ K m ) χ mn , (15)we formulate a simple cost functional, J ≡ (cid:12)(cid:12)(cid:12)(cid:12) P ˆ χ − ˆ χ D (cid:12)(cid:12)(cid:12)(cid:12) = tr (cid:110)(cid:2) P ˆ χ − ˆ χ D (cid:3) (cid:2) P ˆ χ − ˆ χ D (cid:3) † (cid:111) , (16)where P denotes the projector onto the M C -dimensionalcomputational Hilbert space and 0 ≤ J ≤ J max = 2 M C .For the charge qubit example, P ij = (cid:88) k =6 , , , δ i,k δ j,k with i, j ∈ (cid:8) , , ..., M = 16 (cid:9) . The transformation given in Eq. (15) corresponds tostacking the columns of the density matrix ρ S from leftto right on top of one another, i.e. ρ S → col ( ρ S ), whichyields a M dimensional single–column vector, termedcol ( ρ S ). For the present work, this transformation is ofthe following form, ρ S = ρ ρ ρ ρ ρ ρ ρ ρ ρ ρ ρ ρ ρ ρ ρ ρ → col ( ρ S ) = ρ ρ ρ ρ ρ ρ ρ ρ ρ ρ ρ ρ ρ ρ ρ ρ , (17) where framed density matrix elements belong to the com-putational subspace. The process tomography matrix χ changes accordingly, χ → ˆ χ , as given in Eq. (15). Forfurther details see Ref. 28. J measures the norm dis-tance between the target operation ˆ χ D and the actualoperation ˆ χ executed at time t f for control field (cid:15) . Forthe Hadamard gate with 2 leakage levels we set,ˆ χ D = U ∗ D ⊗ U D , with U D = 1 √ − . A similar approach based on a super–operator formalismis given in Ref. 34.
IV. QUBIT DYNAMICS
Starting with the von–Neumann equation for the fullsystem dynamics, ddt ρ ( t ) = − i (cid:126) [ H ( t ) , ρ ( t )], the time evo-lution of the operators ˜ σ j ( t ) (with tilde denoting the in-teraction picture) is computed within a non–Markovianmaster equation in Born approximation, ddt ˜ σ j ( t ) = (18) − (cid:126) (cid:90) t dt (cid:48) tr B (cid:110)(cid:104) ˜ H I ( t ) , (cid:104) ˜ H I ( t (cid:48) ) , ˜ σ j ( t (cid:48) ) ⊗ ˜ ρ B (0) (cid:105)(cid:105)(cid:111) , using a numerical method described in Ref. 36 to dealwith the kernel, which is non–local with respect to time.tr B denotes the partial trace over all bosonic degrees offreedom. For ˜ ρ B (0) we assume a bath in thermal equilib-rium at T = 50 mK. We also introduce a sharp infraredand a continuous ultraviolet cutoff for the spectral den-sity, i.e. , J ( ω ) → (cid:0) J /f ( ω )Θ ( ω − Λ ir ) + J f ( ω ) (cid:1) e − ω/ Λ uv . (19)We choose Λ ir = 2 π ×
100 Hz and Λ uv = 2 π ×
100 GHz (Ohmic noise in Josephson charge qubits hasbeen measured up to frequencies of 100 GHz). Thelower bound for the infrared cutoff is determined by thedata–acquisition time of the experiment. We have testedour optimal pulses with respect to different infra- andultraviolet cutoff frequencies, see Figs. 3(a) and 3(b).When evaluating the trace of the double commutator inEq. (18), one has to calculate bath correlation functionsof the form (cid:68) ˜Γ( t )˜Γ( t (cid:48) ) (cid:69) = tr B (cid:110) ˜Γ( t )˜Γ( t (cid:48) )˜ ρ B (0) (cid:111) , which can be derived analytically using the spectral den-sity given in Eq. (19) and (6). V. RESULTS
Our strategy is to use short control pulses of high am-plitude which move the qubit away from the degeneracypoint only briefly and are within current experimentalcapabilities. We employ a time–optimal control strat-egy which incorporates the final time of the gate oper-ation ( t f ) as an additional control parameter, compati-ble with experimental control field strength. The controlfields are of the form, n c ( t ) = 12 + g ( t ) A c sin ( ω c t + ϕ c ) e − γ c ( t − t c ) , Φ( t )Φ = g ( t ) (cid:26) − A Φ { ω Φ t + ϕ Φ ) } × e − γ Φ ( t − t ) (cid:105)(cid:111) , with (cid:110) A c/ Φ , ω c/ Φ , ϕ c/ Φ , γ c/ Φ , t c/ Φ (cid:111) representing free pa-rameters. Additionally, we restrict { n c ( t ) , Φ( t ) } to startand end at the degeneracy point ( n c = 1 / , Φ = 0)by using an envelope function g ( t ) [dashed, gray line inFig. 2(b)] and to satisfy other constraints imposed by thedesign of the superconducting circuit. There are othermethods, such as driving the qubit by NMR–like tech-niques and adiabatic pulses, within which fidelities of ∼ . − . Gate fidelities of ∼ . Recently, single qubit op-erations with gate errors of 1 ∼
2% have been realizedwithin phase and transmon qubits.
To optimize the cost functional Eq. (16) we use a par-allelized, constrained version of a differential evolutionalgorithm with 300 individuals per generation and 3000generations per optimization run. The algorithm findsan optimum for control fields given in Figs. 2(b) and 2(c).We typically choose crossing probabilities ( C ) and scal-ing parameters ( S ) within a range of C ∈ [0 . , .
96] and S ∈ [0 . , . t f ≈
50 ps, gate fidelity ( F ) losses ∆ F = 1 − F ≡ ( J [ (cid:15) ∗ ] /J max ) / as low as ≈ − are achievable. Aplot of the associated matrix–element deviations fromthe Hadamard operation, (cid:12)(cid:12)(cid:0) ˆ χ − ˆ χ D (cid:1) mn (cid:12)(cid:12) , is shown inFig. 2(a). Losses from decoherence (mainly included inred/dark gray bars) dominate over those from leakage(mainly included in white bars) by about a factor of 15,largely due to Ohmic contributions in the noise spec-trum. To demonstrate the importance of including ad-ditional Cooper pair occupation on the superconductingisland within the model, we optimized the gate withoutleakage and then used these control fields to steer thequbit subjected to leakage. The value of the cost func-tional increases by about 4 orders of magnitude corre-sponding to an increase in ∆ F by 2 orders of magnitude.Incorporation of leakage states | − (cid:105) and | (cid:105) does notadd significant fidelity losses for optimal pulses given inFigs. 2(a) and 2(b). Therefore, expanding the leakagestate–space beyond | − (cid:105) and | (cid:105) is not necessary.We have also explored longer pulses ( t f = 200 ps) and Short pulses ∼ ps (a) (b) Abs (cid:64)(cid:72)
Χ(cid:96) (cid:45)Χ(cid:96) D (cid:76) mn (cid:68) m n t (cid:144) ps Φ (cid:72) t (cid:76) (cid:144) Φ (c) t (cid:144) ps n c (cid:72) t (cid:76) Longer pulses ∼ ps (d) (e) Abs (cid:64)(cid:72)
Χ(cid:96) (cid:45)Χ(cid:96) D (cid:76) mn (cid:68) m n t (cid:144) ps Φ (cid:72) t (cid:76) (cid:144) Φ (f) t (cid:144) ps n c (cid:72) t (cid:76) FIG. 2: (Color online) (a,d) Deviation of the Hadamard pro-cess tomography matrix elements ( ˆ χ ) mn for optimized controlfields [as given in Figs. 2(b,e) and (c,f)] from desired values ` ˆ χ D ´ mn . Black areas denote matrix elements which are ir-relevant and, hence, arbitrary. They neither alter the qubitdynamics nor induce transitions out of the qubit subspace, i.e. leakage. For simplicity, they are plotted as zero [see alsoEq. (17)]. Red (dark gray) columns show deviations of matrixelements which represent the gate operation within the qubitsubspace, whereas white bars display leakage. (b,e) and (c,f)The strength of the optimal control fields lies within realisticvalues and the optimal solutions have a simple shape. do not start at the degeneracy point with respect to Φ[the working point is now set at ( n c = 1 / , Φ = Φ / F for 200 pspulses are about 6 times higher than those for 50 ps ones(∆ F ≈ − ).Time–optimal control inevitably leads to short pulses(typically of the order of 50 ps for present field intensi- (a) - - - L ir (cid:144) s - L u v (cid:144) H s - L J H L ir , L uv L - - Ω (cid:144)H s - L S H Ω L (cid:144) H Ñ s L - - - - - (b) (c) (cid:45) (cid:76) ir (cid:144) s (cid:45) J (cid:72) (cid:76) i r (cid:76) (cid:144) (cid:45) J (cid:72) (cid:76) ir (cid:76) at (cid:76) uv (cid:61)
512 GHz 2 (cid:45) (cid:45) (cid:76) uv (cid:144)(cid:72) s (cid:45) (cid:76) J (cid:72) (cid:76) u v (cid:76) (cid:144) (cid:45) J (cid:72) (cid:76) uv (cid:76) at (cid:76) ir (cid:61) s (cid:45) FIG. 3: (Color online) (a) Variation in the cost func-tional with respect to changes in the infrared and ultravio-let cutoff frequency. We used the pulse sequences shown inFigs. 2(b) and 2(c). Inset, solid line: Noise spectrum em-ployed for our calculations. Dashed and dotted–dashed linesrepresent 1 /f and f noise contributions, respectively. (b) Fora given ultraviolet cutoff, the cost functional shows a log-arithmic dependence on Λ ir , i.e. J (Λ ir ) ∼ log (1 / Λ ir ), asexpected (see Ref. 23). (c) For J (Λ uv ) we obtain a logarith-mic dependence over a broad regime. However, when Ohmiccontributions become relevant the cost functional increasessignificantly. ties) whose generation in experiment is a demanding task. However, picosecond electrical pulses can be producede.g. by optoelectronic devices, such as photocunductiveswitches or by optical rectification of ultrashort opticalpulses using nonlinear media (e.g. LiTaO ). Ultrafastpulse–shaping methods have been discussed extensivelyin Ref. 12.
VI. CONCLUSION
By applying a newly developed, process–tomography–based optimal control theory for open quantum systemsto a Josephson charge qubit, we have shown that one–qubit gates, such as the Hadamard gate, can be real-ized with remarkably high fidelity. The strategy hasbeen to keep deviations of the control fields with respectto the degeneracy point as short as possible while per-forming the desired unitary operation in time–optimizedfashion compatible with experimentally available con-trol field strength. A fully quantum mechanical descrip-tion based on experimental noise spectra has been em-ployed to model dephasing effects and additional non–computational basis states have been included to accountfor unwanted Cooper–pair occupation on the supercon-ducting island. Depending on the gate operation time,which has been treated as a variable, we could achieve fi-delities of the order of F ≈ − − to 1 − − . In termsof the process tomography matrix this corresponds to er-rors of the order Abs (cid:2)(cid:0) ˆ χ − ˆ χ D (cid:1) mn (cid:3) ≈ − − − . Thuswe find that charge qubits can be made to perform at anequal level with current realizations of other Josephsonqubits (such as phase, flux or transmon qubits) whichhave been tested in experiment so far. VII. ACKNOWLEDGMENT
The authors wish to acknowledge financial support ofthis work by FWF under Project No. P18829, as well ashelpful discussions with M. Wenin. ∗ Electronic address: robert.roloff@uni-graz.at † Electronic address: [email protected] A. Shnirman, G. Sch¨on, and Z. Hermon, Phys. Rev. Lett. , 2371 (1997). Y. Makhlin, G. Sch¨on, and A. Shnirman, Rev. Mod. Phys. , 357 (2001). J. Q. You and F. Nori, Phys. Rev. B , 064509 (2003). Y. Nakamura, Y. A. Pashkin, and J. S. Tsai, Nature ,786 (1999). J. M. Martinis, S. Nam, J. Aumentado, and C. Urbina,Phys. Rev. Lett. , 117901 (2002). T. P. Orlando, J. E. Mooij, L. Tian, C. H. van der Wal,L. Levitov, S. Lloyd, and J. J. Mazo, Phys. Rev. B ,15398 (1999). J. Clarke and F. K. Wilhelm, Nature , 1031 (2008). T. Yamamoto, Y. A. Pashkin, O. Astafiev, Y. Nakamura,and J. S. Tsai, Nature , 941 (2003). J. H. Plantenberg, P. C. de Groot, C. J. P. M. Harmans,and J. E. Mooij, Nature , 836 (2007). R. Fazio, G. M. Palma, and J. Siewert, Phys. Rev. Lett. , 5385 (1999). S. Montangero, T. Calarco, and R. Fazio, Phys. Rev. Lett. , 170501 (2007). A. Sp¨orl, T. Schulte-Herbr¨uggen, S. J. Glaser,V. Bergholm, M. J. Storcz, J. Ferber, and F. K.Wilhelm, Phys. Rev. A , 012302 (2007). P. Rebentrost and F. K. Wilhelm, Phys. Rev. B ,060507(R) (2009). F. Motzoi, J. M. Gambetta, P. Rebentrost, and F. K. Wil-helm, arXiv:0901.0534v2 (2009). O. Astafiev, Y. A. Pashkin, Y. Nakamura, T. Yamamoto,and J. S. Tsai, Phys. Rev. Lett. , 267007 (2004). G. Ithier, E. Collin, P. Joyez, P. J. Meeson, D. Vion, D. Es-teve, F. Chiarello, A. Shnirman, Y. Makhlin, J. Schriefl,et al., Phys. Rev. B , 134519 (2005). Y. Makhlin, G. Sch¨on, and A. Shnirman, Nature , 305(1999). R. Lutchyn, L. Glazman, and A. Larkin, Phys. Rev. B ,014517 (2005). R. M. Lutchyn, L. I. Glazman, and A. I. Larkin, Phys.Rev. B , 064515 (2006). A. Barone and G. Patern`o,
Physics and Applications of theJosephson Effect (Wiley, 1982). Y. Nakamura, Y. A. Pashkin, T. Yamamoto, and J. S.Tsai, Phys. Rev. Lett. , 047901 (2002). A. J. Leggett, S. Chakravarty, A. T. Dorsey, M. P. Fisher,A. Garg, and W. Zwerger, Rev. Mod. Phys. , 1 (1987). Y. Makhlin, G. Sch¨on, and A. Shnirman, J. Chem. Phys. , 315 (2004). O. Astafiev, Y. A. Pashkin, Y. Nakamura, T. Yamamoto,and J. S. Tsai, Phys. Rev. Lett. , 137001 (2006). J. P. Palao and R. Kosloff, Phys. Rev. Lett. , 188301(2002). D. J. Tannor and S. A. Rice, J. Chem. Phys. , 5013(1985). M. Nielsen and I. Chuang,
Quantum Computationand Quantum Information (Cambridge University Press,2002). T. F. Havel, J. Math. Phys. , 534 (2003). J. B. Altepeter, D. Branning, E. Jeffrey, T. C.Wei, P. G.Kwiat, R. T. Thew, J. L. O’Brien, M. A. Nielsen, andA. G.White, Phys. Rev. Lett. , 193601 (2003). J. F. Poyatos, J. I. Cirac, and P. Zoller, Phys. Rev. Lett. , 390 (1997). M. A. Nielsen, Phys. Lett. A , 249 (2002). M. Neeley, M. Ansmann, R. C. Bialczak, M. Hofheinz,N. Katz1, E. Lucero, A. O’Connell, H. Wang, A. N. Cle-land, and J. M. Martinis, Nature Physics , 523 (2008). J. M. Chow, J. M. Gambetta, L. Tornberg, J. Koch, L. S.Bishop, A. A. Houck, B. R. Johnson, L. Frunzio, S. M.Girvin, and R. J. Schoelkopf, Phys. Rev. Lett. , 090502(2009). M. Wenin and W. P¨otz, Phys. Rev. B , 165118 (2008). H. Carmichael,
Statistical Methods in Quantum Optics 1:Master Equations and Fokker–Planck Equations (Springer,2002). M. Wenin and W. P¨otz, Phys. Rev. A , 022319 (2006). E. Collin, G. Ithier, A. Aassime, P. Joyez, D. Vion, andD. Esteve, Phys. Rev. Lett. , 157005 (2004). E. Lucero, M. Hofheinz, M. Ansmann, R. C. Bialczak,N. Katz, M. Neeley, A. D. O’Connell, H. Wang, A. N.Cleland, and J. M. Martinis, Phys. Rev. Lett. , 247001(2008). URL . J. F. Holzman, F. E. Vermeulen, and A. Y. Elezzabi, Appl.Phys. Lett. , 134 (2000). A. Nahata and T. F. Heinz, Opt. Lett. , 867 (1998). S. E. Sklarz and D. J. Tannor, J. Chem. Phys. , 87(2006).43