Flux Qubit Readout in the Persistent Current Basis at arbitrary Bias Points
aa r X i v : . [ qu a n t - ph ] J a n Flux Qubit Readout in the Persistent Current Basis at arbitrary Bias Points
M. Sch¨ondorf, A. Lupa¸scu, and F. K. Wilhelm Theoretical Physics, Saarland University, 66123 Saarbr¨ucken, Germany Institute for Quantum Computing, Department of Physics and Astronomy,and Waterloo Institute for Nanotechnology, University of Waterloo, Waterloo, Ontario, Canada N2L 3G1
Common flux qubit readout schemes are qubit dominated, meaning they measure in the energyeigenbasis of the qubit. For various applications, measurements in a basis different from the energyeigenbasis are required. Here we present an indirect measurement protocol, which is detector domi-nated instead of qubit dominated, yielding a projective measurements in the persistent current basisfor arbitrary bias points. We show, that with our setup it is possible to perform a quantum nonde-molition measurement in the persistent current basis at all flux bias points with fidelities reachingalmost 100%.
I. INTRODUCTION
The measurement postulate is fundamental in the for-mulation of quantum mechanics [1]. To obtain informa-tion about the quantum state of a closed system oneneeds to employ an interaction with an additional read-out system (meter). It is possible to design this inter-action such that the measured observable is an integralof motion during the readout process. This is calleda quantum non-demolition (QND) measurement. QNDmeasurements enable repeated measurements to have thesame outcome and were originally proposed to exceed thestandard quantum limit in connection with the detectionof gravitational waves [2–4]. The interest in QND mea-surement methods has increased with the development ofquantum information, where they play an important rolein various aspects, e.g. error correction [5] or initializa-tion by measurement [6].Superconducting flux qubits [7] are especially interest-ing for the field of quantum annealing [8–15], where theintrinsic possibility for inductive coupling and the ratherlarge anharmonicity deliver a big advantage. However,for flux qubits QND measurements in the persistent cur-rent basis have only been performed far away from theflux degeneracy point [16–20]. At the degeneracy pointthe expectation value of the persistent current, whichis the measurement variable, is zero for the qubit en-ergy eigenstates. Measurement in the energy eigenba-sis at the degeneracy point is possible by coupling thequbit transversely to a resonator, leading to a measure-ment of the quantum inductance [21–24], or by using amore complicated scheme based on modulated coupling[25]. The ability to perform measurements in the fluxbasis at an arbitrary operation point is especially inter-esting in quantum annealing. To be able to measure dur-ing the anneal process without first driving the qubit faraway from the degeneracy point would yield huge advan-tages, e.g. avoid quenches in anealing schedules, whichlimit success probability [13, 26, 27] or realize quantumspeedup with only stoquastic interactions [28]. In addi-tion, state tomography would benefit from such a readoutscheme, since measurements in canonical conjugated ba-sis are necessary [29, 30]. Φ Φ ^ c ML I ,C | | I p FIG. 1. Circuit for the measurement protocol. The qubit(yellow) is coupled to the large SQUID loop of the quantumprobe, here the cjj-SQUID (green). The large loop is cou-pled to an additional flux readout loop (blue) and an externalcontrol flux Φ c (red) is applied to the small SQUID loop. Here we present a method to measure the state of aflux qubit for arbitrary biases, ranging from the symme-try point to points far from the symmetry point, whichis both projective and high fidelity. In contrast to usualflux qubit measurements [16–20], we measure in the per-sistent current basis at all bias points and not in theenergy eigenbasis of the qubit.The paper is organized as follows. In Sec. II we presentour setup the corresponding Hamiltonian. The four dif-ferent steps of the measurement protocol are discussed indetail in Sec III. In Sec. IV we present our results and inSec. V we give the conclusion.
II. SETUP AND HAMILTONIAN
The proposed indirect measurement protocol includesa quantum probe in between the flux qubit we want toread out and the actual readout resonator (e.g. SQUID).This probe is a compound Josephson junction SQUID(cjj-SQUID) [31–33]. The cjj-SQUID is coupled induc-tively to the superconducting flux qubit we want to mea-sure, leading to the Hamiltonian for the coupled qubit-probe system (for setup see Fig. 1)ˆ H = φ L (cid:18) ξ ˆ q ϕ β cjj (Φ c ) cos ˆ ϕ − g √ ξ ˆ ϕ ˆ σ z (cid:19) + ˆ H qb , (1)where ξ = e/φ p L/C Σ , φ = Φ / π and g = √ ξM I p /φ , with mutual inductance M , persistent cur-rent I p , inductance of the large cjj-SQUID loop L , sum ofthe two junction capacitances C Σ and Φ the flux quan-tum. The quantum variable of the probe is the averagephase of the junctions ˆ ϕ = 2 π ˆΦ / Φ and ˆ q is the conju-gated variable. ˆ H qb denotes the flux qubit Hamiltonianrepresented in the persistent current basis {| (cid:9) i , | (cid:8) i} ˆ H qb = ǫ σ z + ∆2 ˆ σ x , (2)with energy spacing ǫ and tunneling energy ∆. Note thatwe do not include the Hamiltonian of the readout loop inEq. (1), since it is decoupled during the whole dynamicsof interest and only used after the protocol is performedto readout the persistent current state of the probe.A special property of the cjj-SQUID is that thescreening parameter depends on the additional controlflux Φ c applied to the small loop, i.e. β cjj (Φ c ) =(2 I L/φ ) cos(Φ c / φ ) [33], with critical current of theSQUID junctions I . The measurement starts with thecjj-SQUID operated in a regime where the potential isparabolic, and centered at a value that depends on thestate of the qubit. Next, the control flux Φ c is used totransform the potential into a double well barrier poten-tial, leading to states localized in one of the wells, incorrespondence with the two qubit states. In contrastto usual measurement schemes, here we present a detec-tor dominated measurement by choosing strong or evenultrastrong coupling [34] between the qubit and the quan-tum probe, such that the measured observable is deter-mined by the eigenbasis of the operator coupled to theprobe. Here this is the persistent current basis, as op-posed to the qubit energy eigenbasis. We show that ourmeasurement protocol enables an almost perfect QNDmeasurement at the degeneracy point and can achievemeasurement fidelities close to 100%. Note that the pro-tocol also works in a similar way for ǫ = 0. III. MEASUREMENT PROTOCOL
The measurement protocol is schematically shown inFig. 2. It has four main steps: the initialization, the pre-measurement, the effective decoupling and the readoutof the probe.In the initialization step we prepare the qubit in anarbitrary initial state α | (cid:9) i + β | (cid:8) i and the cjj-SQUIDin the ground state | g i . Here the qubit and the probeare decoupled and the screening parameter β cjj of thecjj-SQUID is zero, meaning it is described by a harmonicoscillator potential. After intialization we start the premeasurement. Forthis, we turn on the coupling between the qubit and thecjj-SQUID. During this step, the external bias on thesmall coupler loop is still choosen as Φ c = Φ /
2, suchthat the barrier is zero and the cjj-SQUID potential ispurely quadratic. By turning on the coupling betweenthe cjj-SQUID and the qubit, an entangled state betweenthe qubit and the pointer states of the probe is created,performing the premeasurement. Since the cjj-SQUIDstarts in the ground state | g i , the coupling term shifts thecenter of the Gaussian distribution of the phase. For zerocoupling, the cjj-SQUID state is centered around h ˆ ϕ i = 0,until the coupling shifts the mean value to h ˆ ϕ i = ϕ p h ˆ σ z i ,where ϕ p denotes the absolute value of the cjj-SQUIDpotential minimum. Note that the shift depends on thequbit state as follows from (1).Here we want to choose parameters such that the inter-action does not induce any excitation of the cjj-SQUIDsinitial ground state, meaning we require a perfect adia-batic time evolution of the system [35]( α | (cid:9) i + β | (cid:8) i ) | g i −→ α eff | (cid:9) , g − i + β eff | (cid:8) , g + i , (3)where | g i is the coupler ground state centered aroundzero and | g ± i are the corresponding displaced groundstates. α eff and β eff include the time evolution under thebare qubit Hamiltonian [35].There are two factors thatchange the probability amplitudes α and β . On the onehand if the state is not an eigenstate of ˆ H qb , it evolvesunder the bare qubit Hamiltonian. However as we willsee later, this is strongly depressed by the ramping of thebarrier. On the other hand the coupling to the quantumprobe leads to a measurement induced dephasing, mean-ing the phase information encoded in α and β gets lostduring the measurement, as we will also see in more de-tail in the next section (for more details see App. A 2).The effective coupling energy ∆ eff gets rescaled due tothe interaction with the cjj-SQUID [35]. To make theadiabatic approximation applicable, the timescale of theinteraction must satisfy the adiabatic theorem [36]. Thisyields the condition max t ˙ g ( t ) √ ξ ≪ Ω , (4)with characteristic frequency of the quadratic part of thecjj-SQUID, Ω = 1 / √ LC . Violating this condition leadsto transitions between cjj-SQUID pointer states whichdestroys the distinguishability, since there is no longera clear map between direction of persistent current andqubit state.Besides the fact that we want the measurement to dis-criminate between the qubit states, we additionally wantthe measurement to be QND. A QND measurement isachieved, when the measured observable is an integralof motion during the measurement, meaning successivemeasurements of the qubit yield the same result [37].This is achieved by the third step of our protocol, the ef-fective decoupling. Especially in the case ǫ ≪ ∆, the non- FIG. 2. Principle of the measurement scheme. Color code analog to Fig 1. 1.) Initialization: The qubit (yellow) and thecjj-SQUID (green) initial state are prepared. 2.) Premeasurement: The coupling between the qubit and the cjj-SQUID isramped up, such that the qubit states get entangled with corresponding pointer states. 3.) Effective decoupling: The c-jjSQUID potential is turned from a single well to a double well potential, resulting in an exponential decrease of the effectivecoupling ∆ eff . 4.) Readout: The cjj-SQUID persistent current state is read out with an additional flux readout device (e.grf-SQUID). commuting part of the system and the interaction Hamil-tonian is crucial, hence severe backaction would appearduring the macroscopic readout of the probe. Thereforein the effective decoupling step, we use the external biasΦ c to ramp the barrier of the cjj-SQUID potential froma single well harmonic potential to a double well poten-tial with a high barrier. This exponentially decreases theeffective coupling energy ∆ eff , resulting in a reduction ofthe non-commuting part. Here ∆ eff means the tunnelingmatrix element between the two compound states | (cid:8) , g − i and | (cid:9) , g + i . With this we freeze the dynamics of thequbit, yielding an effective decoupling of the qubit andthe probe, necessary for a QND measurement [37]. Notethat the tuning of the barrier also has to be adiabaticallyon the cjj-SQUID timescale to again avoid excitations tohigher modes, such that we have to modify condition (4)and include the time derivative of the screening parame-ter β cjj ( t ) max t (cid:20) ˙ g ( t ) √ ξ , ˙ β cjj ( t ) (cid:21) ≪ Ω . (5)In a last step we can measure the probe state usingthe additional persistent current readout with indicating almost no backaction, since ∆ eff ( T ) ≈
0, where T denotesthe time for the overall protocol.Because of the non-commuting nature of the interac-tion and the system Hamiltonian, there is also a backaction induced during the premeasurement and the ef-fective decoupling. Therefore one needs to perform thesetwo steps fast with respect to the characteristic qubittimescale T p ∆ + ǫ ≪ h, (6)where h is the Planck constant. However, this generalcondition is too strict in our case. On the one hand thewhole point of the third step is to decrease the effectivedecoupling rate to almost zero and on the other hand inthe case ǫ ≫ ∆ the backaction is negligible, since systemand interaction Hamiltonian almost commute. Includingthese facts, the QND condition for our system is givenby Z T ∆ eff ( t )d t ≪ h. (7)Note that the effective tunneling rate is time dependent,since it is influenced by the interaction with the probe. -0.5 0 0.5 1 1.5 / | () | FIG. 3. Results of the numerical simulations of the measure-ment protocol for ∆ /h = 0 .
1Ω and linear time schedule with g max / Ω = 1 and β max cjj = 2 and ξ = 0 . C Σ = 15 fF and L = 600 pH). The initial qubit state ischosen to be | i = 1 / √ | (cid:9) i + | (cid:8) i ). Shown is the probabilitydistribution of projection to qubit left (red) and right (blue)persistent current state. Numerical (solid) and analytical re-sults (dotted). Because of the entanglement of the pointer states andthe qubit states after the premeasurement, a high barrierof the cjj-SQUID potential also frustrates a tunnelingbetween the qubit states. This leads to the fact that(7) is even satisfied for measurement times larger thanthe qubits characteristic time, as we will see in the nextsection.At the end of the decoupling step, it is important thatthe two pointer states are statistically distinguishable,meaning that the maximal coupling strength g max = g ( T ), needs to be chosen such that the condition [38] h ϕ ( T ) i − h ϕ ( T ) i ≥ σ ( T ) + σ ( T )] , (8)is satisfied at the end. Condition (8) is a qualitative mea-sure for statistical distinguishability, but does not quan-tify measurement fidelity. Here h ϕ ( T ) i i is the expecta-tion value of the pointer state if the qubit is in state i and σ ( T ) i is the respective standard deviation. Both aretaken at the end of the measurement protocol.The distinguishability criterion gives a lower bound forthe necessary maximal coupling strength g max . The mea-surement fidelity is limited by the overlap of the pointerstates (see Fig . 3) and transitions between different cjj-SQUID states during the interaction process. Thereforethe most general expression for the measurement fidelityis given by F meas = F (cid:8) + F (cid:9) , (9)where F i denotes the probability to get the right mea-surement result if the qubit is prepared in the energy t c (t)g(t)/ eff (t) (linear) FIG. 4. Time evolution of the screening parameter β cjj , thecoupling g and the coupling energy ∆ eff . eigenstate | i i . The state fidelities read F i = Z b i a i (cid:18)(cid:12)(cid:12)(cid:12) h ϕ, (cid:9) | ˆ U ( t ) | g, i i (cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12) h ϕ, (cid:8) | ˆ U ( t ) | g, i i (cid:12)(cid:12)(cid:12) (cid:19) d ϕ (10)where i ∈ { (cid:9) , (cid:8) } , ˆ U ( t ) is the time evolution operatorwhich describes the time dynamics of the measurementprocess and { a (cid:9) , b (cid:9) } = {−∞ , } , { a (cid:8) , b (cid:8) } = { , ∞} . IV. RESULTS
We want to quantitatively study the measurement pro-tocol. The most important point here is to quantify theright time scales and system parameters to obtain highmeasurement fidelities and prove the QNDness of the pro-tocol. To solve the time dependent Hamiltonian (1) nu-merically, we evolve the cjj-SQUID part in harmonic os-cillator modes. Here we truncate the Hamiltonian after100 excitations. Since g ( t ) and β cjj ( t ) are time dependentwe have to solve a time dependent Schr¨odinger equation.For this we use a standard Runge-Kutta method. In thesimulations we assume that the coupling and the non-linearity are turned on simultaneously instead of succes-sively. We simulate the full Hamiltonian (1) includingthe qubit dynamics.We choose the simplest possible time dependence here,where we tune up the coupling and the barrier linearly.Here the maximal value of the coupling is g max / Ω = 1and the maximal screening parameter β max cjj is 2. Theoverall time interval in which we ramp up both param-eters is chosen to be 10 / Ω and Ω is ten times the qubitfrequency. The time evolution of the coupling, the screen-ing parameter and the linear approximation of the effec-tive coupling strength (see App. A 2 for more details onhow to calculate this) are shown in Fig. 4. We studythe measurement protocol at the flux degeneracy point ǫ = 0 and choose as initial state the qubit eigenstate g max / m ea s u r e m en t f i de li t y = 0.05= 0.1= 0.15 -5 FIG. 5. Measurement fidelity depending on g max for differentvalues of m for the same parameters as in Fig. 3 and fordifferent values of ξ . The inset shows the infidelity in log scaleto illustrate how the fidelity tends to one for large couplings. | i = ( | (cid:9) i + | (cid:8) i ) / √
2. After modeling the time evolutionof the pointer states using (1), we calculate the measure-ment fidelity given by Eq. (9). Fig. 3 proves that thepointer states of the cjj-SQUID nicely resolve the qubitstates. The measurement error is given by the overlap ofthe two probability distributions. For the chosen param-eters this results in a measurement fidelity of 1, since theoverlap of the states is zero.To quantify this, we also show the measurement fidelitydependence on the maximal coupling strength g max fordifferent values of ξ in Fig 5. We see that the measure-ment fidelity strongly increases for larger values of g max until it reaches a plateau at fidelity 1. For smaller ξ , thefidelities are slightly lower, but do not vary significantlyin the range of realistic system parameters [33, 39–41].Even though the ultrastrong coupling regime is accessi-ble in flux qubit architectures ([42–44]), it is more feasi-ble to work in the strong coupling regime. However, evenin this regime which corresponds to g max / Ω ≈ .
1, themeasurement fidelities are quite high. E.g. for ξ = 0 . .
8% for the same measurementtime as in Fig. 3. Since the coupling is weaker the in-teraction time needed for a resolving premeasurement isalso longer, hence it is supporting to choose longer mea-surement times. For an increased measurement time of T = 40 / Ω the fidelity for ξ = 0 . g = 0 . T . This means that there is a tradeoff between couplingand measurement time. One can choose a smaller cou-pling when at the same hand the measurement time isincreased, but the coupling is roughly lower bounded bycondition (8). Note that a measurement time of 10 / Ω inour choice of parameters corresponds to the characteris-tic time scale of the qubit. For flux qubits this would bein the ns regime.Here we model measurement at the flux degeneracypoint, but the protocol can lead to perfect fidelities for ǫ = 0, e.g. for the same parameters as in Fig. 3 but FIG. 6. Decay of excited state population during measure-ment for initial state | (cid:9) i and the same parameters as in Fig3. for the case ǫ/h = ∆ /h = Ω /
10, our simulations showan almost perfect measurement fidelity of 0 . ǫ = 0.In Fig. 3, also the time evolution of the density matrixelements for the initial qubit state | i = 1 / √ | (cid:9) i + | (cid:8) i )at the degeneracy point is studied. The parameters arethe same as before. We see that the measurement inducesa strong dephasing in the measurement basis (persistentcurrent basis). This is what one expects since entan-gling the qubit with the respective pointer states meanstransferring qubit information to the probe system ([45]or [37]). The fact that the measurement induces a de-phasing in the persistent current basis proves that themeausrement protocol does not measure in the energyeigenbasis of the qubit, but in the eigenbasis of the probe.The diagonal elements on the other hand stay constant,meaning the population in the persistent current basis isconserved.As mentioned before, a way to determine the QNDnessof a measurement is the comparison of repeated succes-sive readouts. Since here the measurement observableis the persistent current, we have to study the decay ofthe corresponding states | (cid:8) i , | (cid:9) i of the qubit to check forQNDness. The QNDness in our system can be quantifiedas the probability that the qubits initial persistent cur-rent state is preserved after premeasurement, irrespectiveof the measurement outcome [37, 46], yielding the expres-sion F QND = h (cid:8) | ˆ U QB ( t ) | (cid:8) i + h (cid:9) | ˆ U QB ( t ) | (cid:9) i , (11)where ˆ U QB ( t ) = Tr probe { ˆ U ( t ) } denotes the effective timeevolution of the qubit during the premeasurement. Thismeans nothing else than successive measurements givingthe same results which is the textbook definition of QND-ness. As mentioned, to ensure QNDness of the protocolwe ramp up the barrier and effectively discriminate thetime evolution of the system, which leads to the fact that FIG. 7. Decay of excited state population during measure-ment for initial state | (cid:9) i for the same parameters as in Fig.3. only in the beginning of the protocol the qubit suffers asmall rotation. Fig. 3 shows the decay of the diagonaldensity matrix element during the measurement proto-col, which is in the order of 10 − for the chosen param-eters. Expression (11) can be determined numericallyand yields a QND fidelity of F QND = 99 . /f -flux noise [47–49].In App. A 3 we model the influence of 1 /f -noise actingon the cjj-SQUID of our measurement circuit. It can beseen that neither the measurement fidelity is changed sig-nificantly (error in the order of 10 − ) due to the noise,nor additional back action is induced on the qubit. Thisshows that the presented protocol is not more susceptibleto noise than conventional flux qubit readout architec-tures.Using a Gaussian approximation for the pointer statesand assuming a completely adiabatic time evolution, itis also possible to derive an analyitcal expression forthe probabiltiy amplitude of the pointer states. A com-parison between analytical and numerical results can befound in Fig. 3. We see that both results match very well,allthough there is a small deviation arising from higherorder potential terms, i.e. the numerical distributionsare slightly asymmetric and shifted towards Φ / Φ = 0 . F meas = Φ (cid:18) ϕ p ( T ) σ ( T ) (cid:19) , (12)where ϕ p is the position of the minima of the cjj-SQUIDdouble well potential at the end of the protocol, σ ( T ) = (2 m Ω p − β cjj ( T ) cos ϕ p ( T )) − / is the standard devia-tion and Φ( x ) denotes the normal cumulative distributionfunction. The detailed calculations can be found in theSupplement, where we additionally study the backactionanalytically and show a qualitative agreement with thenumerically found back action results.For the sense of completeness, we want to point outthat D-wave also uses the cjj-SQUID as a qubit but thatthe measurement method differs from the one presentedhere. They use a quench to first tune the qubits intothe regime ǫ ≫ ∆ and then perform a persistent currentreadout (see e.g. [26]). V. CONCLUSION
In conclusion we have presented an indirect measure-ment protocol to perform fast read out a flux qubit atevery bias point in the persistent current basis, with pos-sible measurement fidelities close to 100%. Further themeasurement is also shown to be QND, which increasesthe possibility for applications in fundamental flux qubitexperiments as well as in the perspective of quantum an-nealing even more. A special feature is that the readoutat the flux degeneracy point is performed in the persis-tent current basis, being potentially useful in terms ofquantum annealing but also for other applications suchas quantum state tomography.
ACKNOWLEDGMENTS
This material is based upon work supported by the In-telligence Advanced Research Projects Activity (IARPA)and the Army Research Office (ARO) under Contract No.W911NF-17-C-0050. Any opinions, findings and conclu-sions or recommendations expressed in this material arethose of the authors and do not necessarily reflect theviews of the Intelligence Advanced Research Projects Ac-tivity (IARPA) and the Army Research Office (ASO).Further we thank Simon J¨ager for fruitful discussions.
Appendix A: Analytical Results1. Measurement fidelity
In this section we will analytically describe the setuppresented in Sec I, especially giving approximate expres-sion for the success probability.Since the regime of interest is Ω ≫ ǫ, ∆, we considerthe qubit Hamiltonian as the perturbation of the system V = ǫ σ z + ∆2 σ x . (A1)As shown in the main part, the phase-charge space rep-resentation of the unperturbed Hamiltonian reads H ( t ) = q m + m Ω ϕ m Ω β c ( t ) cos( ϕ ) − m Ω λ ( t ) σ z ϕ. (A2)with effective mass L/ (2 ξφ ) . Without the cosine term,this yields a shifted harmonic oscillator where the shiftdepends on the qubit state. To include the contributionof the non-harmonic cosine part, we will approximate thepotential around its minimum. It is U ′ ( ϕ ) = m Ω ϕ − m Ω β c ( t ) sin( ϕ ) − m Ω λ ( t ) σ z . (A3)The condition U ′ ( ϕ ) = 0 leads to an equation for thepotential minimum, depending on σ z . Since σ z = ± ϕ ± ( t ) = ± ϕ p ( t ) , (A4)where ϕ p ( t ) denotes the positive valued minimum. Theeffective potential up to second order then reads U ( ϕ ) ≈ ϕ p ( t ) σ z + m Ω − β c ( t ) cos( ϕ p ( t ))] ( ϕ − ϕ p ( t )ˆ σ z ) (A5)= ϕ p ( t ) σ z + m ˜Ω( t ) ϕ − m ˜Ω( t ) ϕ p ( t ) ϕ ˆ σ z . (A6)with time dependent frequency ˜Ω( t ) =Ω p − β c ( t ) cos( ϕ p ). Note that the frequency doesnot depend on the qubit state, because of the symmetryof the cosine. This leads to the effective Hamiltonianˆ H ( t ) ≈ q m + m ˜Ω( t ) ϕ − m ˜Ω( t ) ϕ p ( t ) ϕ ˆ σ z (A7)= ˜Ω( t ) a † a − ˜Ω( t ) s m ˜Ω( t )2 ϕ p ( t )( a † + a )ˆ σ z . (A8)The last part implies a qubit dependent shift of the har-monic oscillator, such that we can diagonalize this Hamil-tonian with the displacement operatorˆ˜ H ( t ) = D † ( ˜ ϕ p ( t )ˆ σ z ) HD ( ˜ ϕ p ( t )ˆ σ z ) (A9)= ˜Ω( t ) a † a (A10)where ˜ ϕ p ( t ) = ϕ p ( t ) q m ˜Ω( t ) /
2. The time dependenceof the transformation induces an additional inertia term.As mentioned before, we choose time scales to be diabaticon the qubit and adiabatic on the coupler time scale.Hence in zeroth order we assume the SQUID state tofollow the minimum adiabatically, so we ignore the termproportional to ˙˜ ϕ p (inertia part) for now. Additionallywe ignore the contribution arising from the zeroth orderof the Taylor expansion, since it only acts as a correctionof the bare qubit Hamiltonian (for more details see App.A 2). We can directly write down the solution to (A10) in theposition space which is a Gaussian distribution aroundthe minimum of the potential ϕ ( t ) = (cid:0) πσ ( t ) (cid:1) / e (cid:16) ϕ − ϕp h ˆ σz i ( t )2 σ ( t ) (cid:17) + ip ϕ | ϕ i , (A11)with standard deviation σ ( t ) = 1 / q m ˜Ω( t ) and p be-ing the average momentum. Let us now assume the qubitstarts in a superposition state and the cjj-SQUID in itsground state (centered around ϕ = 0). The time evolu-tion reads( α | (cid:9) i + β | (cid:8) i ) | g i ˆ U → α eff | (cid:9) , ϕ − ( t ) i + β eff | (cid:8) , ϕ + ( t ) i , (A12)with | ϕ ± ( t ) i = (cid:0) πσ ( t ) (cid:1) − / e (cid:16) ϕ ∓ ϕp σ ( t ) (cid:17) + ip ϕ | ϕ i (A13)and where α eff and β eff include the time evolution in-duced by the bare qubit Hamiltonian, i.e when the stateis not an eigenstate (see [35] for more details). We are es-pecially interested in the probabilities for the SQUID tobe in the left or right persistent current state, dependingon the qubit state. E.g. the probability to get the rightmeasurement result if the qubit starts in the | (cid:9) i state(equivalent to F (cid:9) of the main text) is given by F (cid:9) ( t ) = 1 p πσ ( t ) Z −∞ e ( ϕ + ϕp ) σ ( t )2 d ϕ (A14)= Φ (cid:18) ϕ p ( T ) σ ( t ) (cid:19) , (A15)with Φ( x ) = √ π R x −∞ e − t d t denoting the normal cu-mulative distribution function. In the same manner wecan write down the probability to get the right measure-ment result when the qubit starts in state | iF (cid:8) ( T ) = − Φ (cid:18) − ϕ p ( T ) σ ( T ) (cid:19) . (A16)This expressions correspond to the two contributions thatappear in the expression for the fidelity, hence in theGaussian approximation F can be written as F ( T ) = Φ (cid:18) ϕ p ( T ) σ ( T ) (cid:19) , (A17)where we used the fact that Φ( t ) is an odd function. For-tunately, Gaussians are among the simplest special func-tions and the expectation value is completely determinedby the standard deviation σ ( T ), hence the fidelity is fullydetermined by σ ( T ) and ϕ p . This fact can be used to e.g.put a lower bound on the measurement fidelity and de-termine the corresponding system parameter intervals toreach this fidelity. Here the main parameters that can bevaried are λ max and β maxcjj . One could also optimize the / | () | FIG. 8. Comparison of the numerical (solid) and analytical(dotted) results for the same parameters as in the main text. schedule, i.e. find an optimal pulse for the time dynamicsof the coupling and the barrier to optimize both, mea-surement fidelity and back action. However, this wouldyield an optimal control problem and can be tracked byfuture work. A lower bound for the respective system pa-rameters is given by the distinguishability condition (Eq.(3)). Since the distributions are symmetric, the conditionhas the simplified form ϕ p ( T ) ≥ σ ( T ) (A18)In Fig. 8 the distribution of the cjj-SQUID state (de-pending on the qubit state) is compared to the numericalresults. We basically see what we expect; the two resultsqualitatively coincide but there are corrections comingfrom the higher order potential terms. Since we modelthe double well potential of the cjj-SQUID with two har-monic potentials, the two actual expectation are slightlyshifted compared to the Gaussian ones. Additionally thewidth of the actual distribution is also slightly smaller.In App. A 2 we use the same strategy to calculate anexpression for the time evolution of the density matrixin the cjj-SQUID ground subspace. Since the calcula-tions are rather involved we put them into the appendix.The analytics show the right qualitative and long timebehavior but differ quantitatively from the numerical re-sults caused by different approximations made during thecalculation.All in all this section shows, that the intuitive pictureof the system dynamics, we gave when we described themeasurement scheme in the main text can be quantifiedwith the given analytical results assuming an adiabatictime evolution of the pointer states. Since the analyti-cal results also give a good agreement with the numerics,the adiabatic approximation is satisfied for the chosentime scale, avoiding any induced transitions between dif-ferent cjj-SQUID states. Further the given results couldbe used to optimize system parameters for real world ap-plications.
2. Backaction
Here we will try to analytically approximate the backaction of the measurement on the qubit. For this we firsttransform the Hamiltonian into an interaction frame (i.ethe displaced oscillator frame) such that we can writedown the time dependent Hamiltonian as a tensor sumof two dimensional matrices (within the adiabatic ap-proximation). Then we can study the time evolution ofthe qubit subspace density matrix and with this makestatements about the back action.As shown in (A 1) we can diagonalize H approxi-mately by applying the displacement operatorˆ D ( ˜ ϕ p ( t )ˆ σ z ) . (A19)This leads to a diagonal Hamiltonian plus an additionalinertia term coming from the time dependence of thetransformation and a correction of the bare qubit Hamil-tonian arising from the fact that the two minima of thetilted double well potential are not at the same potentiallevel˜ H = ˜Ω( t )ˆ a † a − i ˙˜ ϕ p ( t )( a † − a ) − λϕ ( t )ˆ σ z (A20)= ˜Ω( t )ˆ a † a − i ˙˜ ϕ ( t ) p ˙ ϕ p + 14 ˙˜Ω( t )˜Ω( t ) ϕ p ( t ) ! − λϕ ( t )ˆ σ z (A21)where p ( t ) is the average momentum at time t , whichcan be rewritten using the correspondence principle p ( t ) = m ˙ ϕ p ( t ). The last term arises from the zerothorder of the Taylor expansion. Hence we need to takeinto account two correction terms. We also have to checkwhat is the effect of the transformation on the bare qubitHamiltonian˜ V = ˆ D † ( ˜ ϕ p ( t )ˆ σ z ) [ ǫ ˆ σ z + ∆ˆ σ x ] ˆ D ( ˜ ϕ p ( t )ˆ σ z ) (A22) ≈ ǫ ˆ σ z + ∆ˆ σ x + 2 ˜ ϕ p ( t ) p ( t )∆ˆ σ y , (A23)where we only kept the first order term of the Baker-Campbell-Hausdorff formula. Since we assume ∆ ≪ Ω,the σ y correction is assumed to be rather small com-pared to the σ z correction arising from ˆ˜ H , hence willbe ignored in the following. With this we can write theHamiltonian in the transformed basis as a tensor sum˜ H ( t ) = ⊕ ∞ N =0 ˜ H N ( t ) , (A24)where H N ( t ) is the Hamiltonian in the N excitation sub-space {| , N − i , | , N + i} and has the form˜ H N ( t ) = (cid:18) N ˜Ω( t ) − γ ( t ) ∆2 h N + | N − i ∆2 h N − | N + i N ˜Ω + γ ( t ) (cid:19) , (A25)with γ ( t ) = − m ˙ ϕ p ( t ) + 14 ˙˜Ω( t )˜Ω( t ) ϕ p ( t ) + λϕ p ( t ) ! (A26)˜Ω N ( t ) = N ˜Ω( t ) . (A27)Here | N ± i refer to the N excitation states of a shiftedharmonic oscillator, where the sign depends on the qubitstate and the shift is given by (A4) (for more details onthe shifted harmonic oscillator we refer to [50]). Here weassumed an adiabatic time evolution of the cjj-SQUIDdynsmics by setting h N ± | M ± i = h N ± | M ∓ i = 0 if N = M . Note that h N + | N − i = h N − | N + i , hence ˜ H ( t ) is her-mitian as demanded. The overlapp between the shiftedoscillator vacuum states is given by [50] h − | + i = e − ϕ p / (A28)Because of the block diagonal structure of the Hamilto-nian, we can also write down the time propagator U ( t ) ina block diagonal structure. For this we need the followingexpressions U N ( t ) = exp (cid:18) i T Z t d t ′ ˜ H N ( t ′ ) (cid:19) , (A29) with the time ordering operator T . Since we assume thetime evolution to be diabatic on the qubit subspace andwe are interested in the dominating back action effects,we use first order Magnus expansion to calculate the timepropagators V N ( t ) V N ( t ) ≈ exp (cid:18) i Z t d t ′ H ( t ′ ) (cid:19) . (A30)Defining the parameters Γ( t ) = R t d t ′ γ ( t ′ ) and ˜∆ N ( t ) =∆ / R t d t ′ h N + | N − i ( t ′ ) the propagator of the N excita-tion subspace can be written as VN ( t ) = e R t t ′ ˜Ω N ( t ′ ) cos (cid:18)q Γ( t )2 + ˜∆ N ( t )2 (cid:19) − i Γ( t ) q Γ( t )2+ ˜∆ N ( t )2 sin (cid:18)q Γ( t )2 + ˜∆ N ( t )2 (cid:19) i ˜∆ N q Γ( t )2+ ˜∆ N ( t )2 sin (cid:18)q Γ( t )2 + ˜∆ N ( t )2 (cid:19) i ˜∆ N q Γ( t )2+ ˜∆ N ( t )2 sin (cid:18)q Γ( t )2 + ˜∆ N ( t )2 (cid:19) cos (cid:18)q Γ( t )2 + ˜∆ N ( t )2 (cid:19) + i Γ( t ) q Γ( t )2+ ˜∆ N ( t )2 sin (cid:18)q Γ( t )2 + ˜∆ N ( t )2 (cid:19) . (A31) Since the back action tends to be strongest at the degen-eracy point, we choose ǫ = 0 in the following, such that H QB = ∆2 σ x . We want to study the time evolution of anarbitrary qubit state, when we prepare the SQUID in theground state ( h N i = 0), leading to the following densitymatrix at t = 0ˆ ρ (0) = (cid:18) | α | αβ ∗ α ∗ β | β | (cid:19) ⊗ | i h | (A32)The time evolution of this state can then be calculatedusing V ( t ). We are especially interested in the densitymatrix of the qubit at time t , so we trace out the cjj-SQUID degrees of freedom ρ QB ( t ) = Tr cjj { ρ ( t ) } (A33)= | α ( t ) | | (cid:9) i h (cid:9) | + α ( t ) β ∗ ( t ) | (cid:9) i h (cid:8) | e − ˜ ϕ p ( t ) + α ∗ ( t ) β ( t ) | (cid:8) i h (cid:9) | e − ˜ ϕ p ( t ) + | β ( t ) | | (cid:8) i h (cid:8) | . (A34)Here we clearly see the measurement induced dephasingappearing as an exponential damping of the off diago-nal elements, depending on the displacement between thetwo pointer states. The time evolution of the prefactors α and β can be calculated using the time propagator. Forthe initial state | + i we have also chosen the main text, itis α = 1 / √ β = 1 / √ ρ QB00 ( t ) = 12 − ( t )Γ( t ) κ ( t ) sin κ ( t ) ! (A35) ρ QB01 ( t ) = 12 (cid:18) − t ) κ ( t ) sin κ ( t ) − i Γ( t ) κ ( t ) sin κ ( t ) cos κ ( t ) (cid:19) exp (cid:0) − ˜ ϕ p ( t ) (cid:1) (A36) ρ QB10 ( t ) = (cid:16) ρ QB01 ( t ) (cid:17) ∗ (A37) ρ QB11 ( t ) = 1 − ρ QB00 ( t ) , (A38)where we defined κ ( t ) = p ∆ ( t ) + Γ( t ) . In Fig. 4 wesee the time evolution of the parameters Γ( t ) and ˜∆( t ).We see that for t T , Γ gets much larger than ∆ leadingthe oscillating term of the diagonal elements to go tozero, such that at the end of the measurement process thepopulation is the same as in the beginning, proving themeasurement to be QND. The long time behavior of theoff diagonal elements are dominated by the measurementinduced dephasing, i.e. the exponential part. Thereforethe offdiagonal elements completely decay for t T ,what we also see in the numerical results.However, even though the analytical results predict theright qualitative behavior and the right long time behav-ior, there are deviations between the analytical and nu-merical results. E.g. the predicted damped osicllations0 t/ -10-8-6-4-2024
Sample E rr o r -3 away from fluxdegeneracy point FIG. 10. Error of the measurement result at the flux degen-eracy point(left) and way from it for ǫ = ∆ (right) and 100different generations of 1 /f -noise. The yellow line shows alinear fit to visualize the average error. of the diagonal elements around 1 / T is in the order of the qubit timeevolution for the chosen parameters, it is not completelyreasonable to assume a diabatic time evolution on thequbit time scale. Hence to get more rigorous results onehas to include higher orders of the Magnus expansion.Second, we ignored the contributions coming from noncommutating character of the interaction and the qubitHamiltonian. Even though the studied contributions arethe leading back action terms, for T comparable to thequbit time scale, the other contributions also start tomatter.
3. Flux noise acting on the cjj-SQUID
Here we will study an effect which is more specific forour setup, i.e. flux noise acting on the quantum probeduring the measurement. We assume 1 /f flux noise,which typically appears in superconducting flux qubitarchitectures. To generate the noise trajectories, we usethe matlab inbuilt object dsp.ColoredNoise , which cre-ates 1 / | f | α noise, with α to choose from [ − , t without noisewith noise FIG. 11. Left:Evolution of the density matrix element ρ (cid:9)(cid:9) ( t )for a vacuum environment (solid blue) and a 1 /f -noise envi-ronment (dotted yellow). t -15-10-5051015 N o i s e ( ) FIG. 12. Example of a 1 /f -noise signal generated by samplingGaussian random processes and adding a variance to accountfor low frequency shifts. erate one noise trajectory. Since dsp.ColoredNoise givesnoise trajectories with zero mean value, we additionallyadd a constant offset to every noise trajectory, to ac-count for low frequency shifts. For every noise trajectorythis offset is again obtained by a Gaussian sampling withmean 0 and variance A log( T exp /T wf ). Here A is the am-plitude of the power spectral density S ( f ) = A/ | f | , T wf is the duration of the actual readout process (here 10ns) and T exp = N r ( T wf + T reset ) is the total experimentaltime. This time results from the sum of the actual mea-surement time plus the reset time T reset , multiplied bythe number of repetitions N r necessary to obtain goodmeasurement statistics. In our simulations we assume T reset = 1 ms, N r = 100 and A = (2 µ Φ ) for the smallloop and A = (10 µ Φ ) for the larger loop. These valuesare good upper bounds for realistic flux qubit experi-ments (e.g. [51]).In Fig. 10 we show the measurement fidelity for the 1001
20 22 24 26 28 30 log (Hz) -150-145-140-135-130-125-120 d B PSD generated noiseTheoretical PSD
FIG. 13. Corresponding PSD to the 1/f noise signal shown inFig. 12. different 1 /f -noise generations. The results are shownfor a qubit at (left) and away from the symmetry point(right). We see that in both cases the measurement fi-delity is only changed in a very small amount ( ≈ − ). This means that the additional ingredient of our scheme,i.e. the quantum probe, does not make the system moresusceptible for flux noise.Besides a direct change of the measurement results,flux noise induced in the cjj-SQUID could also lead toback action on the qubit itself. To prove that this alsohas no significant effect in our case, we study the timeevolution of the qubit density matrix elements for one1 /f -noise generation. As an input state we choose thepersistent current state | (cid:9) i and the parameters are thesame as in Fig. 3 of the main text. 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