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1 [ ] 4 Oct 2011 Decoherence times of universal two-qubit gates inthe presence of broad-band noiseE. Paladino,Dipartimento di Fisica e Astronomia, Universit`a di Catania and CNR IMM MATIS, Catania , C/O Viale Andrea Doria 6, , 95125 Catania , D Arrigo,Dipartimento di Fisica e Astronomia, Universit`a di Catania and CNR IMM MATIS, Catania , C/O Viale Andrea Doria 6, , 95125 Catania , Mastellone, Centro Italiano Ricerche Aerospaziali - Via Maiorise snc - 81043 Capua,Caserta, FalciDipartimento di Fisica e Astronomia, Universit`a di Catania and CNR IMM MATIS, Catania , C/O Viale Andrea Doria 6, , 95125 Catania , generation of entangled states of two quantum bits is afundamental step toward the implementation of a quantum information nano-devices this operation is counteracted by the solid-stateenvironment,characterized by broadband and non-monotonic power spectrumoften 1/fat lowfrequencies.

2 For single qubit gates, incoherent processes due tofluctuations actingon different time scales result in peculiar short- and long-time behaviors. Markoviannoise originates exponential decay with relaxation and decoherence times,T1andT2, simply related to the symmetry of the qubit-environment coupling with 1/fpower spectrum at low frequencies is instead responsible for defocusingprocesses and algebraic short-times behavior. In this article we identify the relevantdecoherence times of an entangling operation due to the different decoherence channelsoriginated from solid state noise. Entanglement is quantified by the concurrence, whichwe evaluate in analytic form employing a multi-stage approach. Optimal operatingconditions of reduced sensitivity to noise sources are identified. Weapply this analysisto a superconducting i SWAP gate for experimental noise numbers: , , , to:New J.

3 Times of universal two-qubit gates in the presence of broad-band noise21. IntroductionThe implementation of a universal two-qubit gate involving an entanglement operationon two quantum bits represents a necessary step toward the construction of a scalablequantum computer [1]. Intense research on solid state nano-devices during thelast decade has established the possibility to combine quantum coherent behaviorwith the existing integrated-circuit fabrication technology. In particular, based onsuperconducting technologies, a variety of high-fidelity single qubitgates are nowadaysavailable [2, 3, 4], two-qubit logic gates [5, 6] and violations of Bell s inequalities [7] havebeen demonstrated, high-fidelity Bell states generated [8]. The recent demonstrations ofsimple quantum algorithms [9] and three-qubit entanglement [10] arefurther importantsteps toward a practical quantum computation with superconducting requirements for building an elementary quantum processor are however quitedemanding on the efficiency of the protocols.

4 This includes both a severe constrainton readout and a sufficient isolation from fluctuations to reduce decoherence noise sources are often characterized by broad-band and non-monotonicpower spectrum. Similar noise characteristics have been reportedin implementationsbased on Cooper-pair-boxes (CPB) [11, 12, 13], in persistent current [14] and phasequbits [15, 16]. Usually, the spectrum of at least one of the noise sources is 1/fat low-frequencies [12, 14, 17, 18, 19, 20]. At the system s eigen-frequencies instead(5 15 GHz) indirect measurements indicate white or ohmic spectrum [12,13, 14].Sometimes spurious resonances of various physical origin have been observed [15, 16, 21].At the single-qubit level, the effects of the environmental degreesof freedomresponsible for the various parts of the spectrum have been clearly identified leading toa convenient classification in terms ofquantum noiseandadiabatic noiseeffects [13, 22].

5 Understanding how these mechanisms affect an entanglement-generating two-qubit gateis a relevant issue not yet investigated and it is the subject of the present picture for a single qubit can be summarised as follows. Noise at frequenciesof the order of the system s splittings may induce incoherent energy exchanges betweenqubit and environment (quantum noise). Relaxation processes occur only if the qubit-environment interaction induces spin flips in the qubit eigenbasis, for transversenoise. Weakly-coupled Markovian noise can be treated by a Born-Markov masterequation [23]. It leads to relaxation and decoherence times denotedrespectivelyT1andT2in Nuclear Magnetic Resonance (NMR) [24]. For transverse noise they are relatedbyT2= 2T1. Longitudinal noise does not induce spin flips, but it is responsible forpure dephasing with a decay-time denotedT 2[24].

6 In general, both relaxation and puredephasing processes occur and the resulting decoherence time isT2= [1/(2T1)+1/T 2] quantum measurements require averages of measurementsruns, the maineffect of fluctuations with 1/fspectrum is defocusing, similarly to inhomogeneousbroadening in NMR [24]. Fluctuations with large spectral componentsat low frequenciescan be treated as stochastic processes in the adiabatic approximation (adiabatic noise).The short-times decay of qubit coherences depends on the symmetry of the qubit-Decoherence times of universal two-qubit gates in the presence of broad-band noise3environment coupling Hamiltonian. For transverse noise, the time dependence isalgebraic [1 +at2] 1/4, for longitudinal noise it is exponential quadratic exp( bt2)( static-path [22] or static-noise [13] approximation).The simultaneous presence of adiabatic and quantum noise can be treated in amulti-stage approach [22].

7 In simplest cases, the effects of the twonoise componentsadd up independently in the coherences time-dependence. Defocusing is minimizedwhen noise is transverse with respect to the qubit Hamiltonian [11]. The qubit is saidto operate at an optimal point characterised by algebraic short-times behavior followedby exponential decay on a scale the present article we perform a systematic analysis of the effects and interplayof adiabatic and quantum noise on a universal two-qubit gate, extending the multi-stageelimination approach introduced in ref. [22]. Understanding these effects is crucial in theperspective of implementing solid-state complex architectures. Our system consists oftwo coupled qubits each affected by transverse and longitudinal noise with broad-bandand non-monotonic spectrum. Such a general situation has not being studied in theliterature.

8 Previous studies concentrated on harmonic baths withmonotonic spectrumrelying on master equation and/or perturbative Redfield approach[25], or on numericalmethods [26], or on formal solutions for selected system observables [27].We quantify entanglement via the concurrence [28]. To compare withbit-wisemeasurements, single qubit switching probabilities are also evaluated. Our analysisis based on approximate analytic results and exact numerical simulations. Our mainresults are: (i) The identification of characteristic time scales of entanglement decaydue to adiabatic noise, quantum noise and their interplay; (ii) The characterizationof relaxation and dephasing for an entanglement operation via the time scalesTR,TSW AP1,TSW AP2andTSW AP 2. We point out the dependence of these scales on thesymmetry of the Hamiltonian describing the interaction between each qubit and thevarious noise sources; (iii) The demonstration that a universal two-qubit gate can beprotected against noise by operating at an optimal coupling , extending the conceptof single-qubit optimal point.

9 The article is organized as follows. In Section 2 we introduce the Hamiltonian modelfor two-qubit entanglement generation in the presence of independent noise sourcesaffecting each unit. In Section 3 the general features of the power spectra of thesefluctuations, as observed in single qubit experiments, are summarized. The relevantdynamical quantities are introduced and the multi-stage approachto eliminate noisevariables and obtain a reduced description of the two-qubit systemis illustrated. InSections 4 and 5 we derive separately the effect of quantum noise within a masterequation approach and the leading order effect (quasi-static approximation) of adiabaticnoise. Finally, in Section 6 we discuss their interplay and introduce therelevant timescales characterizing loss of coherence and entanglement of a universal two-qubit gatein a solid-state environment.

10 Results are summarized in table 5 and in table 6. InAppendix A the entanglement generating model is derived for capacitive coupled CooperPair Boxes (CPBs) including fluctuations of all control parameters. In Appendix BDecoherence times of universal two-qubit gates in the presence of broad-band noise4 Table and eigenvectors ofH0expressed in the computational basis| i | i1 | i2, , {+, }with z| i = | i and tan = c/(2 ).i i|ii0 2+ ( c/2)2 (sin /2)|+ +i+ (cos /2)| i1 c/2( |+ i+| +i)/ 22 c/2(|+ i+| +i)/ 23 2+ ( c/2)2(cos /2)|+ +i+ (sin /2)| ithe effect of selected impurities strongly coupled to the device is pointed out andwe speculate on the possibility to extend the optimal coupling scheme under Universal entangling gateEntanglement-generating two qubit gates have been implemented based on differentcoupling strategies.