We present a strategy to empirically determine the internal and control Hamiltonians for an unknown two-level system (black box) subject to various (piecewise constant) control fields when direct readout by measurement is limited to a single, fixed observable

Kolli, A

Oi, D K L

Schirmer, S G

English

arXiv

University of Strathclyde Institutional Repository

Strathprints Institutional RepositorySchirmer, S G and Kolli, A and Oi, D K L (2004) Experimental Hamiltonian identification for controlledtwo-level systems. Physical Review A, 69 (5). -. ISSN 1050-2947Strathprints is designed to allow users to access the research output of the University of Strathclyde.Copyright c© and Moral Rights for the papers on this site are retained by the individual authorsand/or other copyright owners. You may not engage in further distribution of the material for anyprofitmaking activities or any commercial gain. You may freely distribute both the url (http://strathprints.strath.ac.uk/) and the content of this paper for research or study, educational, ornot-for-profit purposes without prior permission or charge.Any correspondence concerning this service should be sent to Strathprints administrator:mailto:strathprints@strath.ac.ukhttp://strathprints.strath.ac.uk/arXiv:quant-ph/0311187v2  25 Apr 2004Experimental Hamiltonian identification for controlled two-level systemsS. G. Schirmer,1, 2, ∗ A. Kolli,1 and D. K. L. Oi11Department of Applied Maths and Theoretical Physics,University of Cambridge, Wilberforce Road, Cambridge, CB3 0WA, UK2Department of Engineering, Division F, Control Group,University of Cambridge, Trumpington Street, CB2 1PZ, UK(Dated: May 17, 2011)We present a strategy to empirically determine the internal and control Hamiltonians for anunknown two-level system (black box) subject to various (piecewise constant) control fields whendirect readout by measurement is limited to a single, fixed observable.PACS numbers: 03.65.Wj,03.67.LxAccurate determination of the dynamics of physicalsystems and their response to external (control) fieldsis crucial for many applications, especially quantum in-formation processing (QIP). Quantum process tomogra-phy (QPT) is a general procedure to identify the unitary(or completely positive) processes acting on a system byexperimentally determining the expectation values of acomplete set of observables [1].However, QPT is only a partial solution in many cases.One problem is the assumption that one can experimen-tally determine the expectation values of a complete setof observables, or at least perform arbitrary single qubitmeasurements for a register of n qubits. However, mostQIP proposals rely on a single readout process, i.e. mea-surement in one basis. For example, qubits encoded ininternal electronic states of trapped ions [2] or neutralatoms [3] are read out by quantum jump detection viaa cycling transition [4]. In solid-state systems based onCooper-pair boxes [5], Josephson junctions [6], or elec-trons in double-well potentials [7], final readout is viacharge localization using single electron transistors [8] orsimilar devices. Finally, solid state architectures basedon electron or nuclear spin qubits [9], are expected to belimited to σz measurements via spin-charge transfer.Generally, one assumes that a single projective mea-surement per qubit is sufficient since arbitrary one-qubit measurements can be realized by performing a lo-cal unitary transformation before measuring to achievea change of basis. However, implementing these ba-sis changes requires accurate knowledge of the dynam-ics of each individual qubit and its response to controlfields in the first place. Yet, in particular for solid-statequbits [5, 6, 7, 9, 10], it is difficult to predict the actualdynamics of a qubit precisely based on computer modelsand theory alone, since they may be sensitive to fabrica-tion variance and even vary from one qubit to the nextas a result. We address this problem of system identifi-cation with a single fixed measurement basis and presenta general strategy to identify the internal and controlHamiltonians for a qubit subject a number of piecewise∗Electronic address: sgs29@cam.ac.ukconstant controls using a single, fixed readout process.We can then bootstrap this process to implement QPT.Although the technique may be adapted to dissipativesystems, in this paper we assume Hamiltonian evolutionfor the purpose of identifying the dynamics relevant forthe implementation of local unitary operations, i.e. de-coherence times much greater than the gate operationtimes, which is crucial for systems of interest in QIP.The system evolution is thus governed by a unitary op-erator U(t, t0), via ρ(t) = U(t, t0)ρ(t0)U(t, t0)†, whereρ(t) is the system density operator, and U(t, t0) satisfiesthe Schro¨dinger equationih¯ddtU(t, t0) = H [f(t)]U(t, t0), (1)where H is the Hamiltonian of the system. The case ofinterest is whenH depends on external fields {fm}, whichwe can control. Assuming the fields are sufficiently weakand act independently, H has the formH [f(t)] = H0 +M∑m=1fm(t)Hm, (2)where H0 is the free evolution Hamiltonian and Hm form > 0 describes the interaction of the system with fieldfm. Each Hm can in turn be expanded in terms of theLie algebra generators; for a two-level system, the Paulimatrices σj for j ∈ {x, y, z}:2Hm = dm0I + dmxσx + dmyσy + dmzσz. (3)Thus we need to determine the real constants dmx, dmyand dmz for m = 0, 1, . . . ,M (the dm0 can be ignoredsince they result only in an unobservable global phasefactor).Geometrically, we can represent the state of the sys-tem by its Bloch vector s(t) = (sx, sy, sz)T where sj =Tr[σjρ(t)]. If the system satisfies Eqs.(1-3), its Blochvector evolves ass˙(t) =(R0 +M∑m=0fm(t)Rm)s(t) (4)2FIG. 1: Rotation about axis d and observable z(αt). Theexpectation value of the measurement operator will oscillatesinusoidally, amplitude and frequency are directly related tothe declination and length of the vector representing the ap-plied Hamiltonian.whereRm are the real anti-symmetric rotation generatorsRm = 0 dmz −dmy−dmz 0 dmxdmy −dmx 0 (5)and R = R0 +∑Mm=1 fmRm generates a rotation aboutd = d0 +M∑m=1fmdm (6)with dm = (dmx, dmy, dmz)T for m = 0, 1, . . . ,M . Iffm(t) vary in time then so does d. If they are piecewiseconstant, having fixed values fm for t0 ≤ t ≤ t1, thens(t) for t ∈ [t0, t1] rotates about the fixed axis d, where||d|| is the rotation frequency, and the unit vector dˆ =1||d||d specifies the rotation axis. This allows us to give anexplicit formula for the trajectory of s(t) with s(0) = s0:s(αt) = s0 cosαt + dˆ(s0 · dˆ)(1− cosαt) + (s0 × dˆ) sinαt= [I cosαt +A(1 − cosαt) +B sinαt]s0. (7)where the rotation angle is αt = α(t) = ||d||(t− t0) andA =12 2 sin2 θ cos2 φ sin2 θ sin(2φ) sin(2θ) cosφsin2 θ sin(2φ) 2 sin2 θ sin2 φ sin(2θ) sinφsin(2θ) cosφ sin(2θ) sinφ 2 cos2 θ ,B = 0 cos θ − sin θ sinφ− cos θ 0 sin θ cosφsin θ sinφ − sin θ cosφ 0 ,and the unit vector dˆ is expressed in spherical coordi-nates, dˆ = (sin θ cosφ, sin θ sinφ, cos θ)T .Without loss of generality we shall assume that we caninitialize the system in the state s0 = |0〉 = (0, 0, 1)T withrespect to the measurement basis, and that we can exper-imentally determine the value of the z-component of theFIG. 2: Bloch sphere and arrangement of rotation axes. Wecan define the relative angular displacement of the vectors dkrepresenting applied Hamiltonians with respect to a referencevector dr ( 6=z-axis).Bloch vector z = 〈σz〉 by repeated measurements. Insert-ing the expressions for A and B into Eq.(7), a rotationabout the axis d by the angle αt = ||d||(t−t0) transformsthe initial state s0 into s(αt) whose z-component isz(αt) = cos(αt) sin2 θ + cos2 θ. (8)z(αt) is constant exactly if θ = 0. Otherwise, it oscillatesand with minimum zmin = cos2 θ − sin2 θ = cos(2θ) forcos(αt) = −1 or αt = (2n + 1)π (for some integer n).Hence, if θ 6= 0 then we can experimentally find ||d|| andθ by determining the tmin =π||d|| and zmin (Fig. 1).For a single rotation it is sufficient to determine ||d||and θ and set φ = 0. For multiple rotations about differ-ent axes dk, however, we must also determine the relativeazimuthal angles φk = φ′k−φ′r with respect to a fixed ref-erence axis dr (Fig. 2). The reference axis should have avertical tilt angle θr ∈ [π4 ,3π4 ] and we shall focus on thiscase in this paper [11]. Having determined ||d|| and θ ofthe axes, and chosen a suitable reference axis dr, we caninitialize the system in the state s1 = (cosβ, sinβ, 0)T byrotating s0 about the axis dr by αr = cos−1(cos(2θr)+1cos(2θr)−1),where β = tan−1(−√−2 cos(2θr) sec θr). Rotating s1 byvarious angles α about the axis dk then gives z(α) =C(1 − cosα) + D sinα with C = 12 sin(2θk) cos(φk − β)and D = sin θk sin(φk − β), or equivalentlyz(α) = γ sin(α+ δ)− γ sin δ, (9)where C = γ sin δ and D = γ cos δ. Hence, we can obtainφ from experimental data for z(α) by finding δ and γ.To determine the evolution of the system subject to theseveral external fields we choose several field strengthsf(ℓ)m (ℓ = 1, . . . , L) for each control fm (m = 1, . . . ,M)and compute the rotation axesd(ℓ)0+m = d0 + f(ℓ)m dm (10)by finding θ(ℓ)0+m and ||d(ℓ)0+m|| for all control settings,choosing a reference axis, and determining all the rel-ative azimuthal angles φ(ℓ)0+m using the strategy outlined3above. Then we plot the x, y and z-components of d(ℓ)0+mfor ℓ = 1, 2, . . . , L versus the field strengths f(ℓ)m for eachfield, and fit a straight line to each set of data points.The vertical axis intercepts of the lines then give the x,y, and z-component of d0, and their slopes determine thex, y, and z-component of dm.To evaluate our strategy and find the best ways to ex-tract the information from noisy data we use computersimulations. We choose various sets of Hamiltonians, usethe proposed strategy to identify them and compare theseresults with the actual values. Experiments are simu-lated by computing individual σz measurements of s(t)according to Eq.(7), generating N pseudo-random num-bers rn ∈ [0, 1], where N is the number of times theexperiment is repeated, and taking the result of the nthmeasurement Mn to be 1 if rn < (1 + z)/2 and 0 other-wise. To account for measurement errors we introduce asymmetric error probability η ∈ [0, 1] indicating the fre-quency of erroneous measurement results, i.e., register-ing 1 when we should have measured 0, and vice versa.This is achieved by choosing another N random numbersen ∈ [0, 1], and changingMn to 1−Mn whenever en < η.Finally, we set 〈σz〉 = −1 +2N∑Nn=1 Mn.We illustrate our Hamiltonian identification strategyin detail with a specific test system:2H0 = 0.2σx + 0.1σz, 2H1 = σx + 0.9σy + 0.1σz,2H2 = 0.2σx + 0.9σz.(11)This Hamiltonian may arise, for instance, for a chargequbit [7] consisting of two quantum dots sharing a singleelectron, and two voltage gates B and S, intended to en-able us to change the potential barrier between them andinduce asymmetries in the double well potential. Themeasurement basis states |0〉 and |1〉 are the localizedground states of the charge in either of the two dots, andprojective measurements onto them can be performed viaa single electron transistor placed next to one of the dots.H0 indicates some tunneling between the dots when nocontrol voltages are applied to the gates and a slightpotential asymmetry, which might be due to imperfectplacement of the donor impurities. H1 induces tunnel-ing between the dots as desired but also induces a slightpotential asymmetry, which might occur if the electrodewas placed slightly off-center. H2 produces the desiredenergy level shift but also increases tunneling slightly.The first step towards identifying the HamiltoniansH0,H1 and H2 involves finding the rotation frequencies ||d0||as well as ||d(ℓ)0+1||, ||d(ℓ)0+2|| and the angles θ0, θ(ℓ)0+1, θ(ℓ)0+2for several gate voltages f(ℓ)1 and f(ℓ)2 . We do this byapplying each gate voltage for various periods of timetj and finding z(tj) = 〈σz(tj)〉 (Fig. 3). An estimate ofthe rotation frequency ω is obtained by taking the dis-crete Fourier transform (DFT) of z(tj) −1J∑Jj=1 z(tj)and identifying its maximum. We then estimate thetime t∗ =πωwhen z(αt) is expected to assume its firstminimum, acquire (circa 150) additional data points fort ∈ [0.85t∗, 1.15t∗] and determine the minimum zmin by0 50 100 1500.50.60.70.80.91Time (arb. units)<σz>0 0.5 1 1.500.010.020.030.040.05Frequency (arb. units)Fourier coeff. (an2 +b n2)||d0+2(10)|| = 0.634.4 4.6 4.8 5 5.2 5.4 5.60.450.50.550.6Time (arb. units)<σz>tmin=5.03, zmin=0.52FIG. 3: Identification of ||d|| and θ for d = d0 + f(10)2 d2.Shown is simulated data for d(10)0+2 for f(10)2 = 0.5. The topgraph is used to estimate the rotation frequency using theDFT, the bottom left plot shows the Fourier transform, andthe bottom right plot the magnification of the region of inter-est as well as the parabola that provides the best least-squaresfit to the data.0 1 2 3 4 5 6 7−0.500.51Angle α of 2nd rotation about  d0+2(10)<σz>zmin=−0.25αmin=1.12zmax=0.66αmax=4.23FIG. 4: Identification of φ for d = d0 + f(10)2 d2. The points∗, are a coarse sampling of the curve from which the firstaxis crossing can be estimated (red circle). This then givesestimates of the turning points αestmin and αestmax which can berefined by re-sampling (dots) in their vicinity and fitting aparabola (red curves). From this we determine the vertices(αmin, zmin) and (αmax, zmax).fitting a parabola to these data points; θ = 12 cos−1(zmin).This proved to be considerably more accurate for noisydata than fitting a function of the form Eq.(8) to thedata points and reading off the frequency and minimumfrom this curve directly.Next we determine the angles φ(ℓ)0+m relative to thereference axis, chosen to be d0 here. In all of the40 0.1 0.2 0.3 0.4 0.500.10.20.30.40.50.60.7f1 (arb. units) d 0+ f 1  d 1x−comp.y−comp.z−comp.0 0.1 0.2 0.3 0.4 0.500.20.40.6f2 (arb. units) d 0+f 2  d 2x−comp.y−comp.z−comp.FIG. 5: Identification of the components of d0 and dm.Typical simulated experimental run for the system definedby Eq.(11) and the estimated components of d for differentfield settings. Shown are the best straight line fit (dashedlines) and the actual values (solid line). Note the excel-lent agreement between the actual values and the best fitobtained. A typical run of the program yielded: d0 =(0.1987, 0.0064, 0.0992)T d1 = (0.9859, 0.9122, 0.1149)T , d2 =(0.2081, 0.0170, 0.8957)T , i.e., the Hilbert-Schmidt norm er-rors ||destm −dactm || were less than 3% (0.0066, 0.0238 and 0.0193for m = 0, 1, 2 respectively).following experiments the system is initialized in states1 = (cos β, sinβ, 0)T by applying a suitable rotationabout the reference axis. In our case the state s0 isrotated to s1 by letting it evolve freely for ≈ 8.1379time units yielding β ≈ −1.0489. We then apply eachfield f(ℓ)m for various times to achieve rotations by vari-ous angles α (Fig. 4). This yields the parameters γ =12 (zmax − zmin) and δ = π −12 (αmin + αmax) in Eq.(9)from which we can obtain φ, e.g., by setting D = γ cos δand φ = −β − sin−1(D/ sin(θ)) where θ is the verticaltilt angle of the rotation axis determined in the previousstep.Finally, having determined all the relevant parametersof the rotation axes, we convert the data to Cartesian co-ordinates, and plot the x, y and z components of d(ℓ)0+mfor each field fm as a function of f(ℓ)m (Fig. 5). From thestraight line fitting, we can then estimate the Hamiltoni-ans.In conclusion, we have presented a stepwise procedureto determine the dynamical response of a Hamiltonian2-level quantum system to control fields when we onlyhave access to measurement in a single fixed basis. Thisovercomes a weakness in the assumptions of QPT andthus enables us to bootstrap the procedure to estimateparameters for systems consisting of arrays of qubits. Nu-merical results indicate that the procedure is fairly robustfor Hamiltonians of physical interest. Extensions of theprocedure higher-dimensional or dissipative systems maybe possible.SGS and DKLO acknowledge funding from theCambridge-MIT Institute, EU grants RESQ (IST-2001-37559), TOPQIP (IST-2001-39215) and Fujitsu.[1] I. L. Chuang, M. A. Nielsen, J. Mod. Opt. 44, 2455(1997); J. F. Poyatos, J. I. Cirac, P. Zoller, Phys. Rev.Lett. 78, 390 (1997)[2] J. I. Cirac, P. Zoller, Phys. Rev. Lett. 74, 4091 (1995)& Nature 404, 579 (2000); D. Kielpinski, D. C. Monroe,D. J. Wineland, Nature 417, 709 (2000)[3] I. H. Deutsch, G. K. Brennen, P. S. Jessen, Prog. ofPhys. 48, 925 (2000); H. J. Briegel et al. , J. Mod. Opt.47, 415 (2000); C. Monroe, Nature 416, 238 (2002).[4] W. Nagourney, J. Sandberg, H. Dehmelt, Phys. Rev.Lett. 56, 2797 (1986); Th. Sauter et al., Phys. Rev. Lett.57, 1696 (1986); J. C. Bergquist et al., Phys. Rev. Lett.57, 1699 (1986)[5] Y. Maklin, G. Scho¨n, A. Shnirman, Nature 398, 305(1999); Y. Nakamura, Yu. A. Pashkin, J. S. Tsai, ibid. ,786.[6] Y. Maklin, G. Schon and S. Shnirman, Rev. Mod.Phys. 73, 357 (2001); Y. Nakamura, Y. A. Pashkin andJ. S. Tsai, Phys. Rev. Lett. 87, 246601 (2001)[7] L. C. L. Hollenberg et al., cond-mat/0306235; A. S. Dzu-rak et al., cond-mat/0306285 (2003)[8] M. H. Devoret, R. J. Schoelkopf, Nature 406, 1039(2000); A. A. Clerk et al., Phys. Rev. Lett. 89, 176804(2002); T. M. Buehler et al., cond-mat/0304384 (2003);J. Kinnunen, P. Torma, J. P. Pekola, cond-mat/0211154(2002).[9] D. Loss and D. P. DiVincenzo, Phys. Rev. A 57, 120(1998); B. E. Kane, Nature 393, 133 (1998)[10] A. Ekert, R. Josza, Rev. Mod. Phys. 68, 733 (1996)[11] If all axes have θ 6∈ [pi4, 3pi4] then all we can do is mapthe initial state s0 to the state closest to the equatorialplane by a pi2rotation about the most horizontal axisand proceed in a similar manner as above. However, theresulting expressions will be more complicated, and theaccuracy of φ will diminish when s1 is close to s0, butsuch a system would be a poor candidate for a qubit.

Experimental Hamiltonian identification for controlled two-level systems

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