Observation of topological transitions in interacting quantum circuits
P. Roushan, C. Neill, Yu Chen, M. Kolodrubetz, C. Quintana, N. Leung, M. Fang, R. Barends, B. Campbell, Z. Chen, B. Chiaro, A. Dunsworth, E. Jeffrey, J. Kelly, A. Megrant, J. Mutus, P. O'Malley, D. Sank, A. Vainsencher, J. Wenner, T. White, A. Polkovnikov, A. N. Cleland, J. M. Martinis
OObservation of topological transitions in interacting quantum circuits
P. Roushan , ∗ C. Neill , ∗ Yu Chen , ∗ M. Kolodrubetz , C. Quintana , N. Leung , M. Fang , R. Barends , B.Campbell , Z. Chen , B. Chiaro , A. Dunsworth , E. Jeffrey , J. Kelly , A. Megrant , J. Mutus , P. O’Malley , D.Sank , A. Vainsencher , J. Wenner , T. White , A. Polkovnikov , A. N. Cleland , and J. M. Martinis † Department of Physics, University of California, Santa Barbara, CA 93106-9530, USA and Department of Physics, Boston University, Boston, MA 02215, USA
The discovery of topological phases in con-densed matter systems has changed the modernconception of phases of matter [1, 2]. The globalnature of topological ordering makes these phasesrobust and hence promising for applications [3].However, the non-locality of this ordering makesdirect experimental studies an outstanding chal-lenge, even in the simplest model topological sys-tems, and interactions among the constituent par-ticles adds to this challenge. Here we demon-strate a novel dynamical method [4] to exploretopological phases in both interacting and non-interacting systems, by employing the exquisitecontrol afforded by state-of-the-art superconduct-ing quantum circuits. We utilize this method toexperimentally explore the well-known Haldanemodel of topological phase transitions [5] by di-rectly measuring the topological invariants of thesystem. We construct the topological phase dia-gram of this model and visualize the microscopicevolution of states across the phase transition,tasks whose experimental realizations have re-mained elusive [6, 7]. Furthermore, we developeda new qubit architecture [8, 9] that allows simul-taneous control over every term in a two-qubitHamiltonian, with which we extend our studiesto an interacting Hamiltonian and discover theemergence of an interaction-induced topologicalphase. Our implementation, involving the mea-surement of both global and local textures ofquantum systems, is close to the original ideaof quantum simulation as envisioned by R. Feyn-man [10], where a controllable quantum system isused to investigate otherwise inaccessible quan-tum phenomena. This approach demonstratesthe potential of superconducting qubits for quan-tum simulation [11, 12] and establishes a power-ful platform for the study of topological phases inquantum systems.
Since the first observations of topological ordering inquantum Hall systems in the 1980s [1, 2], experimentalstudies of topological phases have been primarily lim-ited to indirect measurements. The non-local nature oftopological ordering renders local probes ineffective, andwhen global probes, such as transport, are used, inter-pretations [13] are required to infer topological propertiesfrom the measurements. Topological phases are charac-
Figure 1.
Dynamical measurement of Berry curvatureand C h . In this schematic drawing, brown arrows representthe ground states (adiabatic limit) for given points on a closedmanifold S (green enclosure) in the parameter space, and theblue arrows are the measured states during a non-adiabaticpassage. According to (2), the Berry curvature B can becalculated from the deviation from adiabaticity. Integrating B over S gives the Chern number C h , which corresponds tothe total number of degeneracies enclosed. terized by topological invariants, such as the first Chernnumber C h , whose discrete jumps indicate transitions be-tween different topologically ordered phases [14, 15]. Fora quantum system, C h is defined as the integral over aclosed manifold S in the parameter space of the Hamil-tonian as C h ≡ π I S B · dS, (1)where B is the Berry curvature [16–18]. As illustrated inFig. 1 and discussed in the supplement, B can be viewedas an effective magnetic field with points of groundstate degeneracy acting as its sources, i.e. magneticmonopoles [17, 19]. Using Gauss’s law for the Berry cur-vature (magnetic field), C h simply counts the number ofdegenerate energy eigenvalues (magnetic monopoles) en-closed by the parameter manifold S . C h , which is invari-ant under perturbations to the shape of S , is a topologicalnumber that reflects a property of the manifold of statesas a whole and not a local property of parameter space.In previous works, topological properties of highlysymmetric quantum systems have been measured [20–22].However, since these earlier studies relied on interferenceto evaluate the accumulated phase, these methods arenot readily generalizable. To circumvent this, Gritsev et a r X i v : . [ qu a n t - ph ] J u l al. [4] proposed a general method to directly measure thelocal Berry curvature. The underlying physics of theiridea is that motion in a curved space will be deflectedfrom a straight trajectory; in other words, curvature re-veals itself as an effective force. For example, a chargedparticle moving in a magnetic field experiences the well-known Lorentz force. Similarly, Gritsev et al. showedthat in a region of the parameter space with Berry cur-vature B , if we "move" a quantum system by changing aparameter of its Hamiltonian with rate v , then the stateof the system feels a force F given by F ∝ v × B + O ( v ) . (2)This force leads to deviations of the trajectory from theadiabatic path which can be detected through measure-ments of the observables in the quantum system (seeFig. 1 and [17]). Therefore, as long as the ramping of pa-rameters is done slowly, but not necessarily adiabatically,the deviation is directly proportional to B . As the adia-batic limit is generally hard to achieve, this relation hasthe important advantage of needing only a moderatelyslow change of state and only requires that the linearterm in (2) dominates the dynamics.This dynamical method suggests a way to directlymeasure B , from which C h can be calculated using (1).This provides an alternative means to study topologi-cal phases, significantly different from conventional ap-proaches. Admittedly, implementing this procedure re-quires the ability to continually change the system Hamil-tonian, which is difficult to do in most experimental situ-ations. However, in a fully controllable quantum system,this provides a powerful means to probe the topologicalproperties of the ground state manifold through dynami-cal measurements. Here we demonstrate an implementa-tion of this type of measurement using a quantum circuitbased on superconducting qubits [11, 12, 23].We first demonstrate a basic implementation of thedynamical method. The quantum state of a single su-perconducting qubit [17, 24] is equivalent to a spin-1/2particle in a magnetic field. Its Hamiltonian can be writ-ten as H S = − ~ H · σ , (3)where σ = ( σ x , σ y , σ z ) are the Pauli matrices, and H = ( H X , H Y , H Z ) is analogous to a control magneticfield. Full control over the parameters of this Hamilto-nian is achieved by microwave pulses that control H X and H Y , and an applied flux through the qubit’s SQUID loopwhich controls H Z . To illustrate the dynamical method,we measure C h for a spherical ground state manifold in H -parameter space (Fig. 2). We use θ and φ as sphericalcoordinates and consider the parameter trajectory start-ing at the north pole at t = 0 and ramps along the φ = 0 meridian ( H Y = 0 ) with constant velocity v θ = dθ/dt until it reaches the south pole at t = T f . To realize Figure 2.
Dynamical measurement of C h . a. Asimultaneous microwave pulse H X ( t ) = H r sin( πt/T f ) anddetuning pulse H Z ( t ) = H r cos( πt/T f ) are applied to con-struct a parameter space trajectory. The pulse sequence re-sults in a parameter space motion along the φ = 0 meridian( H Y = 0 plane) on S . b. The state of the qubit during thisramp ( H r / π = 10 MHz and T f = 600 ns) is determined us-ing tomography [17], and shown (blue dots) on the surface ofthe Bloch sphere. motion on a spherical manifold, the control sequences of H Z and H X are chosen such that the control magnitude | H | = H r is constant [17]. In the adiabatic limit, thewavefunction would remain in the instantaneous groundstate of H S , with the Bloch vector parallel to the direc-tion of the control field, following the meridian. Instead,for non-adiabatic ramps, a deviation from the meridianis observed, as shown in Fig. 2(b). Here the Bloch vec-tor is measured at each point in time by interruptingthe ramp and performing state tomography. Note thatthis deviation is not due to noise, but rather is the ex-pected non-adiabatic response [17]. For this trajectory,the force F takes the form f φ = ~ H r h σ y i sin θ , where h σ y i is the expectation value of σ y . Integrating over theresulting deflection (shaded light red in Fig. 2(b)) gives C h = 1 ± . . Note that given the symmetry of theHamiltonian, a line integral is sufficient for measuring thesurface integral of C h (see (1)) [17, 25]. A value of unityis expected, as the qubit ground state has a single degen-eracy at H = 0 , corresponding to an effective monopole,the enclosing parameter sphere S should yield C h = 1 . Inthe supplement we demonstrate the robustness of C h bydeforming the surface manifold S and discuss the sourcesof error,[26].Using our controllable quantum circuit, we can explorewhat is perhaps the simplest model of topological behav-ior in condensed matter, the Haldane model [5, 17]. Thismodel serves as a foundation for other topological insu-lator models [27–29], yet its experimental realization hasremained elusive [6, 7]. To show that the quantum Halleffect could be achieved without a global magnetic field,Haldane introduced a non-interacting Hamiltonian on a Figure 3.
Dynamic measurement of the topologicalphase diagram and adiabatic visualization of phases.a.
Dynamical determination of the phase diagram. first h σ y i was measured during ramps similar to those in Fig 2(a), andthen C h was calculated. The dashed line is the expected phaseboundary at H = H r . b , c. With adiabatic state prepara-tion, the state of the qubit was prepared and measured over agrid on the surface of the parameter sphere and then mappedto the hexagonal momentum-space plane. The ground statesare presented as Bloch vectors, whose colors indicate their h σ z i values. H /H r = 1 . for b and H /H r = 0 for c . Thegray lines show the FBZ of the honeycomb lattice and highsymmetry points K and K are marked. d. The measured C h from the adiabatic and dynamical (white arrow in a ) methodsare plotted vs. H /H r . honeycomb lattice [5] given by H G ( k x , k y ) = ~ v F ( k x σ x + k y σ y ) + ( m − m t ) σ z , (4)where v F is the Fermi velocity, m is the effective mass,and m t corresponds to a second-neighbor hopping in alocal magnetic field. The key prediction of the Haldanemodel is that if m /m t > the system is in a trivial insu-lating phase, and otherwise in a topological phase, where edge states and quantized conductance appear. Using aconfocal mapping [17] one can recast Eq. (4) into thesingle-qubit Hamiltonian (3). If we consider sphericalmanifolds S of radius H r displaced from the origin inthe z direction by H , then H /H r in the qubit systemplays the same role as m /m t in the Haldane model. InFig. 3(a) we plot the results of this measurement, showing C h as a function of H r and H , which shows plateaus atvalues 0 and 1 separated by a phase transition boundaryline at H r = H . This transition can be easily under-stood: when H < H r the degeneracy at H = 0 lieswithin the sphere giving C h = 1 , whereas for H > H r itlies outside the sphere giving C h = 0 .The nature of the topological and trivial phases canbe further revealed by probing their microscopic struc-ture with a conventional adiabatic method. Accordingto Haldane, each phase has its own signature spin tex-ture in momentum space. We again consider sphericalsurfaces S and adiabatically ramp the control parame-ters to their final values on S . The resulting Bloch vec-tors are then tomographically measured [17], and ideallypoint in the same direction as the final H . With a con-focal mapping (see [17]), S can be mapped to the firstBrillouin zone (FBZ) of the honeycomb lattice. There-fore, the adiabatically measured ground state vectors on S can be depicted in the FBZ. Fig. 3(b) and (c) showthe results for two manifolds with H /H r = 1 . and 0,corresponding to trivial and topological phases, respec-tively. By following the orientation of the state-vectoralong any path starting at K and moving to K (cor-ners of the FBZ) and back to K one can see that in thetopological case the state vector makes one full rotation,while in the trivial case and only tilts away from verticaland then returns, without completing a rotation. Thesespin texture maps can be used to extract local Berry cur-vature [17]. As shown in Fig. 3(c), the resulting C h fromthis adiabatic method shows good agreement with thedynamical method of measurement.Moving beyond the realm of non-interacting systems,we now study the topological phase diagram for an in-teracting Hamiltonian, obtained by measuring C h in acoupled two-qubit system. The intriguing physics of thetopological properties of this kind of interacting systemhas to date been mostly unexplored, due to experimentalchallenges. One significant source of challenge is that oneneeds full control over the entire parameter space, includ-ing over any coupling terms in the Hamiltonian. Here weachieve this kind of full control by using a new design forour superconductong qubit, which includes the ability tocontinuously vary the inter-qubit coupling strength g (weterm this new type of qubit the "gmon" [8, 9]).The Hamiltonian of this system is given by H Q = − ~ H σ z + H · σ + H · σ − g ( σ x σ x + σ y σ y )] , (5)where 1 and 2 refer to qubit 1 (Q1) and qubit 2 (Q2) Figure 4.
Topological phase diagram of an interacting system. a.
The position of the monopoles in H -space forthe points A through F shown in panel c , with a spherical manifold of radius H r / π = 10 MHz. b,c.
The topological phasediagram of Eq. (5). In panel b , C h was measured for two fixed g/ π values of 0 and 4 MHz. In panel c , C h was measured forfixed H r / π =
10 MHz. Dashed lines are topological transitions calculated analytically. d. The analytically calculated phasediagram showing three distinct C h volumes and the separatrix plane. The phase diagram cuts in b, c are indicated by coloredslices. respectively, and the biasing field H is now only appliedto Q1.There are equivalent condensed matter systems towhich this system can be mapped, as with the Haldanemodel, as discussed in [17].However, in the absence of anyexperimental realization of these models, our experimentis perhaps closer to Feynman’s original idea of quantumsimulation [10], where a controllable quantum system isused to investigate otherwise inaccessible quantum phe-nomena.Using the tunable inter-qubit coupling, we can ac-cess all regions of the 7-dimensional parameter space ofour Hamiltonian. Here we explore spherical manifoldswith fixed ( H , | H | , | H | , g ) , analogous to the singlequbit experiment.We perform experiments where both H = H = H r are ramped simultaneously with magni-tude | H r | = H r , while H and g are zero except duringthe time t = 0 to T f , as illustrated in the supplement [17].The measured C h is shown in Fig. 4(b) and (c) for threedistinct cuts though this parameter space, as shown bycolored rectangles in Fig. 4(d). For g = 0 [panel (b)], thetwo qubits behave independently and the physics is thesame as for the single qubit case. Since only Q1 is subjectto H , C h changes by 1 through the transition H = H r . A new phase with C h = 0 emerges when the coupling g is large. In Fig. 4(b) for g/ π = 4 MHz, the C h = 0 phase(blue) is seen at small H r when H r . g . In Fig. 4(c) thisphase also appears when g & H r , showing that the transi-tion is interaction-driven and appears when the coupling g becomes dominant. Because (5) is not SU(2) symmet-ric, the results do not simply reflect the total spin of thesystem. However, an intuitive understanding of thesephases and transitions can be attained in certain limits:at large H r , the spins align paramagnetically with thefield and add up to give C h = 2 . At large g , the spinsform an entangled singlet which does not respond to theapplied field, giving C h = 0 . Away from these limitingcases, these simple arguments are not applicable, but C h remains quantized.Analytic solutions predicting the phase diagram canbe obtained by calculating when points with degenerateground states cross the spherical manifold [17]. Thesephase boundaries are depicted in Fig. 4(d) and show threedistinct regions. As discussed above, the region where g dominates (blue) has C h = 0 , while where H r domi-nates (red) C h = 2 . There is a direct 0 to 2 transitionwhen H = 0 , but at finite values the system first goesthrough the green C h = 1 region. This latter behavioris seen in Fig. 4(c). The dashed lines in panels (b) and(c) are from this analytic solution, which uses no freeparameters, and are in good agreement with the mea-surements. The deviations are mainly systematic errors,due to crosstalk between simultaneous control pulses. Asshown in Fig. 4(a), the points of ground state degeneracyare located on the z -axis of the H r -space [17]. Sub-figuresA, B and C correspond to the dots on Fig. 4(c), where g is small. In this limit, H affects the energy of onlyone qubit, and increasing it moves only one monopolepast the surface (C). For D, E, and F where instead H is small, increasing g furthers the monopole separation,eventually moving both monopoles outside the surface(F).An important benefit of working with a fully control-lable Hamiltonian, as here, is that a number of differ-ent condensed matter systems can be mapped onto thismodel system. For our 2-qubit system, we show [17] thatthe system can be mapped to either an interacting model,or alternatively a 4-band non-interacting electron modelthat is a non-trivial extension of the two-band Haldanemodel. In general, with n qubits one can study topolog-ical phases in non-interacting n -band models, an other-wise daunting experimental task. Perhaps more interest-ing will be to use qubit systems to study the topologicalphases of interacting spin- / systems, where tantalizingevidence for fractionalization has been found [30]. Acknowledgments:
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1. The gmon qubits 31.1. The gmon coupling architecture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.2. Basic design principle of the gmon . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.3. Coherence of the gmon . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42. Mapping the Single-Qubit Hamiltonian to the Haldane Model and Adiabatic Measurement of the Chernnumber 42.1. Haldane model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42.2. Adiabatic measurement, confocal mapping, and direct measurement of the Chern number . . . . . 52.3. Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63. Loci of Monopole Singularities for the Two-Qubit System 74. Hitchhiker’s Guide to the Chern number and Berry Curvature 84.1 Berry connection, phase, curvature and all that . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84.1.1 Berry connection and phase . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84.1.2 Geometric interpretation of Berry connection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84.1.3 Berry connection as a vector potential: deriving an observable field . . . . . . . . . . . . . . . . . 84.1.4 Geometric interpretation of the Berry curvature field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94.1.5 From local to global properties: the Chern number C h . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94.2 Topological interpretation of C h in terms of enclosed degeneracies . . . . . . . . . . . . . . . . . . . . . . . . . . . 104.2.1 Degeneracy as a source of Berry curvature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104.2.2 Sources of degeneracy as magnetic monopoles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114.3 Choice of coordinate system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114.3.1 Cartesian coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124.3.2 Spherical coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125. Mapping the Two-Qubit Hamiltonian to Electronic Band Structure 136. Experimental protocols, calibration, and analysis 17References 22 Figure S1.
Device architecture. (a)
Optical image of the device showing two coupled gmon transmons on the top halfof the chip and the two coupled gmons used in this work on the lower half (zoomed-in view in inset). (b)
The layout of thetwo-qubit gmon system. We supply bias currents using the lower blue lines to tune the inductance of the coupler junction(middle) and the qubit frequencies (left, right). We apply microwave pulses to each qubit via the gray trace. We read out thestate of the qubits dispersively via readout resonators: each qubit is capacitively coupled to a resonator (green lines; meanderedlines in inset of (a)).
1. THE GMON QUBITS1.1. The gmon coupling architecture
In this work we implemented an adjustable inductive coupling between two qubits. Adjustable coupling has typicallybeen difficult with superconducting qubits, as fixed capacitive coupling may only be modified by detuning, so it hasthe problems of limited on/off range and crosstalk. Here we use a novel qubit design called the gmon, which allowsa continuous variation of the inter-qubit coupling strength g over nanosecond time scales without any degradation inthe coherence of the constituent Xmon qubits [1, 2]. The adjustable inductive coupling between the transmons allows g/ π to be varied between − MHz and MHz, including zero, without changing the bare qubit frequencies. Thedevice was fabricated using standard optical and e-beam lithography techniques, discussed in recent works of ourgroup such as [3]. The experiment was performed at the base temperature of a dilution fridge ( ∼
20 mK).
As shown in Fig. S1, the gmon design is based on the Xmon qubit design. One important feature of the Xmondesign [4] is the single-ended ground in contrast to differential or floating grounds. In the absence of adjustablecoupling, the SQUID loops (Fig. S1(b)) would be directly connected to the ground plane. This design feature givesus the ability to capacitively couple qubits with elements such as the drive lines, the readout resonators, and nearbyqubits. In the gmon architecture, instead of immediately terminating the qubit SQUID to ground, we add a linearinductor (the meandering CPW element colored in purple and labeled "tapping inductance") between the SQUIDand ground(CPW stands for coplanar waveguides). This creates a node (where the purple CPW meets the horizontalblue CPW) that allows us to couple the two qubits. The two qubits then can be connected with a CPW line. Thisconnecting line is interrupted with a Josephson junction, which acts as a tunable inductor that can be used to tunethe inter-qubit coupling strength g , hence the name gmon.The basic operation of the gmon can be understood from a simple linear circuit model. An excitation created in Q1will mostly flow to ground through its tapping inductance, but a small fraction will flow to the tapping inductance ofQ2, generating a flux in Q2. The mutual inductance resulting from the flux in Q2 due to an excitation current in Q1can be calculated from simple current division, and the coupling strength is proportional to this mutual inductanceto high accuracy [1, 2]. The current division ratio, which sets the coupling strength g , can be varied by changing thesuperconducting phase difference across the tunable inductance. This is done by flux biasing its junction, using thecurrent line labeled "coupler tuning" in panel (b).An important advantage of this architecture is that it prevents crosstalk, a serious hurdle for many other experi-mental works. Coupling the qubits at the nodes between the SQUID and the tapping inductance allows a DC currentto set the coupling strength. Because of the open loop of each qubit (due to capacitance) this DC current cannot flowto the SQUID and change the qubit frequencies. Thus, the two capacitors act as DC blocks. This key ingredient ofthe gmon design minimizes the crosstalk between the qubits and the coupler. One major concern of the coupler circuit is degradation of the qubit coherence. The gmon architecture requiredadding CPW lines to connect qubits, ground plane cross-overs which involve dielectrics, and a tunable inductance. Ifthe capacitive loss due to these elements are not properly considered, the coherence of the system could be substantiallydegraded. With the gmon design, the fraction of the qubit energy stored in these coupling elements scales as thesquare of the ratio of the tapping inductance to the qubit SQUID inductance, which is / . Therefore theseelements do not affect the qubit coherence [2]. Furthermore, to avoid inductive loss, we used a relatively small mutualcoupling of around 1 pH to the coupler tuning line. This coupling places an upper bound of 200 µ s on the energyrelaxation time T . The average measured T for our device was around 10 µ s, independent of the coupling strength.This is comparable to the performance of Xmon qubits with the same geometry and material. As demonstrated withXmon qubits [3], the coherence can be improved by widening the capacitor and using MBE-grown aluminum films.
2. MAPPING THE SINGLE-QUBIT HAMILTONIAN TO THE HALDANE MODEL AND ADIABATICMEASUREMENT OF THE CHERN NUMBER2.1. Haldane model
To show that the quantum Hall effect could be achieved without a global magnetic field, Haldane introduced anon-interacting Hamiltonian[5], which served as the cornerstone of future topological band studies. He introduced amassive Dirac Hamiltonian with different mass terms at the two non-equivalent corners of the Brillouin zone K , K .Near these points the Hamiltonian is given by H ± G ( k ± x , k ± y ) = ~ v F ( k ± x σ x ± k ± y σ y ) + ( m ∓ m t ) σ z , (1)where v F is Fermi velocity, and k + x ( k − x ) , k + y ( k − y ) are measured from two non-equivalent corners of the Brillion zone K ( K ) . m is the mass associated with inversion symmetry breaking, and m t corresponds to a second-neighborhopping in a local magnetic field. The key prediction of the Haldane model is that if m /m t > the system is in atrivial insulating phase, and otherwise in a topological phase, where edge states and quantized conductance appear.Using a confocal mapping, discussed below, one can recast Eq. (1) into the single qubit Hamiltonian of the maintext (eqn. (3)). For convenience we re-parameterize that equation in terms of a field H along the z -axis, and a radialfield H r with orientation given by θ, φ , such that H = ( H r sin θ cos φ, H r sin θ sin φ, H − H r cos θ ) . Then, for θ valuesclose to 0 and π the single qubit Hamiltonian becomes H ± S ( H , H r , θ, φ ) = − ~ H r (sin( θ ) cos( φ ) σ x + sin( θ ) sin( φ ) σ y ) + ( H ± H r ) σ z ) . (2)By comparing H + S ( H − S ) to H + G ( H − G ) , it becomes evident that H /H r in the qubit system plays the same role as m /m t in the Haldane model. The fact that the topological phase transition occurs at H /H r =1 is consistent withthe Haldane model, where the transition takes place at m /m t =1. Similar to the Haldane model, where k x and k y span a manifold of states in the Brillouin zone, θ and φ span a manifold in the parameter space of the qubit system.With this mapping, the two distinct phases observed in Fig. 3(a) of the main text correspond to the topological andtrivial phases in the Haldane model. Figure S2.
Experimental visualization of the topological phases and their evolution across the transition.(a)
In the Haldane model of graphene, in addition to the nearest neighbor hopping ( t ), a second neighbor hopping ( t ) isalso considered with a variable phase φ controlled by the locally-varying flux through the plaquette, as well as a sublattice"mass" m corresponding to a difference in chemical potentials between the sublattices. This system is topologically trivialif | m | > | m t = 3 √ t sin φ | and non-trivial otherwise. (b) A color-assisted representation of the mapping from a sphereparameterized by H , H r , θ, φ to the hexagonal Brillouin zone of graphene. b. With adiabatic state preparation, the stateof the qubit was prepared and measured over a grid on the surface of the parameter space spheres. Selected adiabatic Blochsphere vectors are shown for H /H r = 0 and 1.2. (d) With adiabatic state preparation, the state of the qubit was prepared andmeasured along the φ = 0 meridian for various H /H r values. The Bloch sphere states are presented with arrows whose colorsindicate their h σ z i values. The topological and trivial phase each has its own signature textures. By following the orientation ofthe state-vector along any path starting from K to K and back to K one can see that in the topological case the state-vectorfully winds around; however, for the trivial phase it only tilts away from the north pole of the Bloch sphere and comes backwithout winding around. To visualize how the qubit and Haldane model are topologically related, we now explicitly construct a mappingbetween the single qubit parameter space and momentum space in the Haldane model. We use this to map the qubitBloch vector measured by adiabatic state preparation to the first Brillouin zone of graphene and then compute itsChern number, thus completing the analogy with the Haldane model. By using a confocal mapping, the parameterspace points can be mapped to the hexagonal Brillouin zone of the honeycomb lattice of graphene (from H -space tok-space). The mapping places the points on the northern part of the spherical manifold of radius unity around the K point of the Brillouin zone and the southern hemisphere points around the K point. For the northern hemisphere ρ = r ( φ ) tan θ ϕ = φ (3)where θ and φ are the spherical coordinates of the northern hemisphere of the sphere in the parameter space, ρ and ϕ are the polar coordinates of the mapped circle, and r is given by r ( φ ) = b sin( π/ π/ − φ ) for ≤ φ < π/ b sin( π/ φ for π/ ≤ φ < π/ b sin( π/ φ − π/ for π/ ≤ φ < π, (4)where b = | K − K | . The mapping of the southern hemisphere takes a similar form. This mapping is illustrated in Fig.S2(b), and covers only one third of the first Brillouin zone (FBZ). To cover the entire FBZ, the mapping was repeatedthree times. As the colors in Fig. S2(b) show, the north pole in the parameter space maps to three equivalent K points at the corners of FBZ and the south pole to the three K points. With the mapping principle explained, nowwe can "move" the ground states that are measured adiabatically on the spherical surfaces in the parameter spaceand place them in the FBZ of the honeycomb lattice. Fig. S2(c) and (d) show the results for H /H r = 1 . , and 0,corresponding to trivial and topological phases, respectively.Knowing the ground state of the system for each k x and k y point in the FBZ, C h can be calculated directly from C h = 14 π Z B.Z σ ( k ) · (cid:20) ∂ σ ( k ) ∂k x × ∂ σ ( k ) ∂k y (cid:21) d k . (5)Using this relation C h numbers shown in Fig. 3(d) of the main text as well as the ones on panels (c) and (d) of Fig.S2 are calculated. While without any mapping, C h could be calculated by modifying (5) to make it appropriate forspherical coordinates, mapping from the sphere to a 2D plane allowed us to use (5) directly. It is interesting to notethat due to the topological nature of the phases, the details of the mapping do not matter, and other mappings couldhave worked as well. We note that the Haldane model consists of a half filled lattice of non-interacting spins, while we constructed amanifold of ground states by measuring the qubit over a closed surface. This difference is resolved by considering amapping of the ground state manifold to the valence band of graphene, while the excited state manifold maps to theconduction band. Therefore, by probing the entire parameter manifold of the qubit ground state, we are probing theentire valence band. This distinction is why C h of the electronic system can be measured with a single measurement,since all of the electronic states at different momenta are filled and hence probed simultaneously. On the other hand,the qubit can be measured only at a single point of the qubit system’s parameter space at a given time, which is whyall the parameter angles must be probed separately and integrated to give C h .
3. LOCI OF MONOPOLE SINGULARITIES FOR THE TWO-QUBIT SYSTEM
We have used the electromangetic analogy extensively in the main text to plot the (theoretical) locations of themagnetic monopoles in parameter space; we now show how one can identify their locations. We begin by pointing outthat the magnetic monopole density can also be written as ρ m = π ∇ · ( ∇ × A ) , a seeming contradiction given thatthe divergence of a curl is known to be zero! The resolution to this seeming contradiction is that ∇ · ( ∇ × A ) is indeedzero whenever the function A is smooth. However, near ground state degeneracies one cannot pick a smooth choiceof gauge (even locally), since the ground state undergoes a sharp change. Therefore ρ m is allowed to be non-zero ifand only if the ground state is degenerate. Thus, one can reduce the problem of finding the magnetic charge densityto the simpler one of locating the ground state degeneracies.For general two qubit Hamiltonians, degeneracies can be readily located numerically using conventional techniquesto minimize the ground state energy gap. However, for the specific case of our cylindrically symmetric two-qubitHamiltonian, we can solve the problem analytically. As a reminder, the Hamiltonian of interest is H = −
12 [ H r ˆ n ( θ, φ ) · ( σ + σ ) + H σ z + g ( σ x σ x + σ y σ y )] . (6)This Hamiltonian has U (1) invariance, meaning that the Hamiltonian at φ = 0 can be mapped to arbitrary φ using H ( θ, φ ) = e iφσ z tot / H ( θ, e − iφσ z tot / , (7)where σ z tot = σ z + σ z . While this invariance is computationally useful, it does not lead to any additional conservationlaws, so on general grounds one does not expect to find degeneracies of our Hamiltonian (a × matrix) in theabsence of symmetry. However, at θ = 0 and π , the U (1) invariance becomes a U (1) symmetry, σ z tot is a conservedquantity, and this enables ground state degeneracies.The values θ = 0 and π lie along the z -axis, so we reparameterize the Hamiltonian along this axis as H = −
12 [ H z ( σ z + σ z ) + H σ z + g ( σ x σ x + σ y σ y )] . (8)Since total spin along the z axis is now conserved, there are two obvious eigenstates: with energies E ↑↑ / ↓↓ = ± ( H z + H / . Within the s z tot = 0 sector, the Hamiltonian reduces to H ↑↓ = − (cid:18) H g g − H (cid:19) , (9)which has eigenenergies E ↑↓ = ± p H / g .The ground state energy levels of these two sectors are degenerate when | H z + H / | = p H / g , from whichwe find H deg z = − H ± p H + 4 g . (10)Having located these degeneracies, we can identify their magnetic monopole charges as Q m = 1 based on the jump C h that we find (experimentally and theoretically) at the topological transitions.This U (1) symmetry at the poles was useful for our analysis, but it is not likely to exist for more complicated casesin which the Hamiltonians are not quite so exquisitely tunable. Therefore, by our above logic, we might argue thatthe degeneracies should go away if there are no longer any symmetries protecting them. However, our topologicalproperties are robust against any perturbation, so despite the loss of symmetry, the degeneracies may drift around,but they do not disappear! This is a situation in which the degeneracies are protected not by a global symmetry, butrather by an emergent topological protection [6]. Breaking the U (1) invariance of the model – for example by addinga σ x term to the Hamiltonian – would disrupt the measurement. In our case, such a symmetry-breaking term doesnot present any fundamental challenges. It would simply add φ dependence to the Berry curvature, now requiringdata to be taken for ramps of θ at multiple values of φ to allow integration over this direction as well. While it iscertainly more time consuming to take this extra data, fundamentally it is no more difficult.
4. HITCHHIKER’S GUIDE TO THE CHERN NUMBER AND BERRY CURVATURE
In the main paper, we defined C h in terms of something called the “Berry curvature,” which may have seemedmysterious. Here, we introduce the concept of Berry curvature in the context of the more familiar Berry phasestudied in adiabatic quantum mechanics. This will allow us to understand both the geometric and “electromagnetic”interpretations of Berry physics in an intuitive but quantitative way, which will in turn lead to the topologicalinterpretation used in the main paper of C h as a count of the number of degeneracies enclosed by a ground statemanifold. We note that our pedagogical treatment of degeneracies as sources of a curvature field largely follows theoriginal exposition of the Berry phase and curvature by M. V. Berry [7]. Suppose we have a Hamiltonian that depends on a set of external parameters, which we describe by the parameterspace vector R , with corresponding ground states | ψ ( R ) i ; i.e., H ( R ) | ψ ( R ) i = E | ψ ( R ) i . An example would bethe three-dimensional (3D) parameter space associated with a single-qubit Hamiltonian in a rotating frame, H ( R ) = − ~ ( H X σ x + H Y σ y + H Z σ z ) , with R = ( H X , H Y , H Z ) . Alternatively, if we take the rotating field in sphericalcoordinates, the natural parameters are magnitude H r ≡ | H r | and angles θ and φ (as in the main text). The Berryconnection (from which the Berry curvature is defined) associated with the ground state manifold is then A = i h ψ |∇ R | ψ i , (11)which when integrated around a closed path C in parameter space yields the celebrated geometric Berry phaseassociated with that path [7, 8] γ ( C ) = I C A · d R . (12)This fact can be derived from the Schrödinger evolution of a quantum state as C is traversed in parameter space inthe adiabatic limit, and is independent of whatever dynamical phase is accumulated throughout the closed trajectory.However, it is not necessary to understand the phenomena of Berry phase from the perspective of the time-evolutionof adiabatic systems – one can simply view it as a consequence of the geometry of an eigenstate manifold, which willsoon become apparent in our discussion. The Berry connection A is an interesting construct because the meaning of the expression ∇ R | ψ i is ambiguouswhen only H ( R ) is given: unlike the coordinates X of real space, where a state | ψ i can be expanded as a wavefunctionof spatial coordinates and ∇ X is a natural operator on these wavefunctions, here it is instead the Hamiltonian itselfthat is a function of the parameter space coordinates R . A manifold of ground states can be associated with amanifold in parameter space via the defining eigenvalue condition H ( R ) | ψ ( R ) i = E ( R ) | ψ ( R ) i ; however, althoughthe states | ψ ( R ) i all live in the same Hilbert space, this eigenvalue condition does not tell us the phase of | ψ ( R ) i at different R . In other words, we must specify what is essentially a choice of gauge when it comes to relative phaserelations, and since this choice can be made arbitrarily, we cannot expect it to have any intrinsic physical meaning.Once a choice is made for these phases however, the name “connection” for A signifies that A encodes a way toequate (or “connect”) ground state vectors at two nearby points R and R + d R in parameter space, analogous to thedifferential geometric notion of parallel transport of tangent vectors along a manifold [9]. In light of its dependence on ∇ R , A is therefore gauge-dependent . The remarkable fact however, as realized byBerry, is that its integral around a loop is actually gauge- independent (modulo π ), and can therefore be measured.This is easily seen: suppose we change our definition of | ψ ( R ) i by an arbitrary local phase factor, | ψ ( R ) i → e i Γ( R ) | ψ ( R ) i . Then by equation (11), A ( R ) is modified by the addition of the term −∇ R Γ( R ) , which integratesto zero around a closed path. The observant reader will notice that this takes the same form as the change of themagnetic vector potential under a gauge transformation. Recalling that the magnetic field is a gauge-invariant (i.e.,directly measurable) quantity derivable from the magnetic vector potential, this motivates us to follow our experiencewith classical electromagnetism and define the analogue of the magnetic field, B ≡ ∇ × A . This will allow us torewrite the integral (12) defining the Berry phase in terms of an observable integrand B . We will see that this “Berryfield” has the interpretation of intrinsic curvature of the ground state manifold. In addition, this endeavor will exposesome interesting physics, including the main topic of our work: topological transitions (jumps in C h ) associated withdegeneracy points.Continuing the analogy, where for simplicity we consider a 3D parameter space, we obtain the Berry field from theBerry connection: B ( R ) ≡ ∇ R × A ( R ) . (13)The Berry curvature field B is the vector form of what is known as the Berry curvature tensor, defined for generaldimensionality and coordinate parametrizations by the antisymmetric tensor B ≡ ∂ µ A ν − ∂ ν A µ generalizing the curl: B = B xx B xy B xz B yx B yy B yz B zx B zy B zz = B z − B y − B z B x B y − B x ; (14)that is, B = ( B x , B y , B z ) = ( B yz , B zx , B xy ) . In our case, for short we will simply call B the Berry curvature. The Berry phase associated with a closed path can now be calculated from (12) using Stokes’ theorem by integratingthe Berry curvature over a bounding surface, γ ( C ) = Z Z S B ( R ) · d S , (15)where S is a surface manifold in parameter space whose boundary is C . This is the direct analogue of a chargedparticle acquiring an Aharanov-Bohm phase when its path encloses a magnetic flux. However, the Berry curvature isa local geometric property, and for 2D manifolds can be physically measured through equation (2) of the main text.Intuitively, the Berry curvature at R is equal to the ratio of the geometric phase accumulated over a loop surrounding R to the parameter space area enclosed by that loop, in the limit that the size of the loop goes to zero; in otherwords, it locally measures the noncommutativity of parallel transport, which manifests itself as a local “twisting andturning” of the state vector in parameter space via the accumulation of Berry phase. This is analagous to the factthat carrying a tangent vector on a geodesic triangle on the surface of a sphere causes the tangent vector to changedirection when the triangular path returns to its starting point, even though locally the vector is always transportedin a parallel fashion. The analogue of the Lorentz force [equation (2) of the main text] for the “magnetic field” B isrelated to this geometry-induced “deflection.” C h One of the main points of this work is that through the analogy to electromagnetism, we can understand howto relate these geometric properties to topological properties of the ground state manifold as a whole. The naturalquestion is then what generates the field B – is it the current of “charged particles,” or an analogue to the magneticmonopole? Consideration of this question leads us to a Gauss’s law interpretation of C h , whose definition we repeathere [equation (1) of the main text]: C h ( {| ψ i} ) ≡ π I S B · d S . (16)This is an integral of B over a closed (meaning no boundary curve C ) ground state manifold in parameter space, andgives nonlocal information about this manifold in the form of a discrete integer through the Chern theorem [10]. To0deduce the quantization of C h , we will use an argument similar to Dirac’s argument [11] showing that the magneticmonopole charge is quantized. After that, we will explicitly relate this quantized value to the number of enclosed“magnetic monopoles” in parameter space. As usual, we restrict ourselves to 2D surfaces in a 3D parameter space.The astute reader may wonder, given the definition of B = ∇ × A , why C h is not simply zero – after all, a simpleapplication of Stokes’ theorem shows that the integral of the curl of a function over any closed surface must vanish:imagine forming an arbitrary closed path C on the surface manifold S , and let S and S be the two surfaces intowhich C divides S . Taking into account the relative orientation of the two surfaces we then have C h = 12 π (cid:18)Z S B · d S − Z S B · d S (cid:19) . (17)A naive application of Stokes’ theorem would say that each term is equal to the line integral of A around the samepath, but with opposite signs, leading to C h = 0 . However, this assumes that a single Berry connection (i.e., vectorpotential) can be defined over the entire manifold with some sufficient smoothness condition. Since Stokes’ theoremcan be intuitively understood by dividing the surface of integration into infinitely many infinitesimal circulationintegrals of A and noting that neighboring circulations cancel everywhere except along the surface boundary C , ifthere is a singularity in A then Stokes’ theorem will break down. It then becomes a topological constraint on anyvector potential covering S that there must be a singularity in A somewhere on the surface, which allows for thepossibility of non-zero C h . The interesting fact is that the location of this singularity depends on the choice of vectorpotential (i.e., is gauge-dependent), but its existence does not depend on the choice of gauge. We note that a similarargument with what is now known as the Aharanov-Bohm phase associated with a physical magnetic field leads toDirac’s quantization condition for real magnetic monopoles [11].However, there is still a constraint on the possible values of C h . Looking again at equation (17), since the geometricphase (12) accumulated by traversing C is physically observable (modulo π ), using Stokes’ theorem for each surfacewith its own vector potential it must be the case that the flux of B through S differs from the flux of B through S by a factor of πN , where N = C h is an integer. C h , which is a property of the entire ground state manifold and cannot be probed locally, is therefore an example ofa discrete topological invariant. In particular, C h is robust to perturbations to the parameter space manifold, and itis reasonable to expect that it can only undergo transitions between different quantized values when there is singularbehavior on the surface S . In the next section, we will see that in our experiment, these singularities are preciselythe locations of ground state degeneracy in the Hamiltonian, and will show that when the degeneracies consideredare two-fold, N is in fact precisely equal to the number of two-fold degeneracies enclosed by the surface. C h in terms of enclosed degeneracies What determines this mysterious integer N , and how can we observe it? The concept of C h as a topologicalinvariant is reminiscent of the Gauss-Bonnet theorem from differential geometry, which relates the integral of theGaussian curvature over a closed surface to its topological genus. In the case of the Gauss-Bonnet theorem, thetopological genus is equal to the number of “holes” it has, for example, 0 for a sphere and 1 for a torus (a “donut”).Just as the number of “holes” of a torus cannot be determined by local probing, C h is a global, “topological” property ofa ground state manifold. To understand what determines topological transitions between its different integral values,we must consider that there are other energy levels above the ground state energy level E , and include the possibilityof degeneracies where for example E ( R ) = E ( R ) . In 4.2.1 we will see that degeneracies behave analogously tomagnetic monopoles as the “sources” for B and, through the familiar Gauss’s law, see in 4.2.2 that for well-behavedtwo-fold degeneracies C h simply counts the number of degeneracies enclosed by the manifold. In this work, measurements are made of the ground state manifold | ψ ( R ) i , but states of higher energy must beconsidered to understand the important role of degeneracy points in topological transitions. Namely, let | ψ n ( R ) i denote the eigenstate corresponding to the n th energy level. To relate Berry curvature to degeneracy, we first usethe fact that the curl of a gradient is zero along with the definitions of A and B [equations (11) and (13)] to write B = i [ ∇ R h ψ | ] × [ ∇ R | ψ i ] . We can then use the common trick of inserting the identity, expanded in terms ofthe energy eigenstates | ψ n ( R ) i , in between the bra and the ket: B ( R ) = i P n =0 [ ∇ R h ψ | ] | ψ n i × h ψ n | [ ∇ R | ψ i ] ,1where we have excluded the n = 0 term because it vanishes (this is easily seen as a consequence of normalization, h ψ | ψ i = 1 ). We can replace h ψ n | [ ∇ R | ψ i ] with the equivalent expression h ψ n | [ ∇ R H ] | ψ i / ( E − E n ) for n = 0 (thisis a straightforward consequence of differentiating the defining eigenvalue equation H ( R ) | ψ n ( R ) i = E n ( R ) | ψ n ( R ) i with respect to R and rearranging terms), arriving at the equation B ( R ) = i X n =0 h ψ | [ ∇ R H ] | ψ n i × h ψ n | [ ∇ R H ] | ψ i ( E n − E ) . (18)From this, we can see that degeneracies (where E n = E ) can act as sources for the Berry curvature field B . This alsoexplicitly shows that B can be written without using phase-ambiguous derivatives ∇ R | ψ i of kets with respect to R [as in the definition of A ( R ) ], but instead in terms of more natural derivatives ∇ R H of H with respect to R , meaningthat it does not matter what phase we assign to eigenstates corresponding to different R . Furthermore, we see thatunder certain assumptions about the behavior of H and E n near degeneracy, the singularities in Berry curvature areprecisely the points of degeneracy. We also note that equation (18) relates the Berry curvature to the generalized forceoperator −∇ R H , which connects this discussion to formula (2) of the main text for the Lorentz force. A derivationof this force in terms of B µν using perturbation theory can be found in [12]. Finally, we make the analogy between degeneracies and magnetic monopoles concrete. If we consider a closed 2Dsurface manifold S which bounds a 3D manifold in parameter space that possibly contains two-fold degeneracies,we can straightforwardly derive the interpretation of C h as the number of source “magnetic monopole” singularitiesenclosed by the ground state manifold. Note that we can assume in a 3D space that degeneracies will occur at isolatedpoints [13], and are therefore the magnetic monopoles that we seek. When only two energy levels E and E areinvolved in a two-fold degeneracy, we only need to consider one term from the sum (18) and can restrict ourselvesto the relevant two-level subspace. It can be shown that C h is invariant under manifold perturbations as long asthose perturbations don’t cause a degeneracy to cross S , so to extract the contribution to C h from a single encloseddegeneracy at R we are free to shrink the manifold down to a small sphere centered around R [so that only the ( E − E ) term contributes] and shift the origin of our coordinates to R . With an appropriate rescaling of parameterspace coordinates, following [7] we can then write a general hermitian two-level Hamiltonian as H = (cid:18) Z X − iYX + iY − Z (cid:19) , (19)where X , Y , and Z are the rescaled coordinates in the Pauli basis, i.e., R = ( X, Y, Z ) (the exact nature of thisscaling is unimportant). In terms of this parametrization the energies are E / = ±√ X + Y + Z = ± R , so thatthe degeneracy is at the origin. We can immediately suspect that this leads to a monopole distribution for B because / ( E − E ) ∝ /R . The precise calculation is dealt with in Berry’s original paper [7] using basic Pauli matrixalgebra, resulting in the ground state Berry curvature field for a two-fold degeneracy at the origin, B = − R R . (20)We note that this is the same answer obtained for the Berry field for the specific case of a spin- particle subjected to aphysical magnetic field [8, 14]. This is (up to a sign) the same expression for the magnetic field generated by a magneticmonopole of magnetic charge / , and therefore by Gauss’s law leads to a contribution to C h of (4 π ) / (2 π ) × / ,as we claimed. Gauss’s law then immediately yields for our experiment C h = Q encm . (21) Here we clarify the choice of coordinate system used throughout the main work. There is some ambiguity in howwe define the Berry connection in spherical coordinates. One way is to close our eyes and pretend that we don’t2know that θ and φ are spherical angles, instead simply treating them as Euclidean parameters. We will call this the“Cartesian” choice, which gives for example the φ -component A Cφ = i h ψ | ∂ φ | ψ i . (22)Alternatively, we could explicitly take into account the non-Euclidean metric associated with spherical coordinates,using ∇ f = ∂f∂r ˆ r + r ∂f∂θ ˆ θ + r sin θ ∂f∂φ ˆ φ [15] to yield the “spherical” definition A Sφ = i r sin θ h ψ | ∂ φ | ψ i = A Cφ r sin θ . (23)This difference may have confused the reader. Below, we will show that either method works, and both of them canbe used to arrive at Eq. (4) of the main text. This is arguably the simpler method, though it is a bit harder to justify. As above, for the ground state manifoldwe define for φ or θ A Cφ/θ ≡ i (cid:10) ψ (cid:12)(cid:12) ∂ φ/θ (cid:12)(cid:12) ψ (cid:11) . (24)Since we are integrating over a spherical surface ( r fixed), we will not need to take any derivatives with respect to r .In these pseudo-Cartesian coordinates then, the non-trivial component of the Berry curvature is B Cθφ ≡ ∂ θ A Cφ − ∂ φ A Cθ . (25)We can then perform the surface integral by noting that, in Cartesian coordinates, the surface element is just dS = dθdφ , so that for a spherical manifold C h = 12 π Z B · d S = 12 π Z π dφ Z π dθB Cθφ , (26)which is the expression we expected. If we take the spherical version of the gradient, then the φ and θ components are A Sφ = i r sin θ h ψ | ∂ φ | ψ i = A Cφ r sin θ ,A Sθ = i r h ψ | ∂ θ | ψ i = A Cθ r . (27)The Berry curvature vector is given by B S = ∇ × A S , which in general is a complicated expression. However, for ourspherical surface of integration, the Chern integral is given by C h = 12 π Z B · d S = 12 π Z B Sr dS r , (28)since the surface element is strictly radial: d S = ˆ rdS r = ˆ r ( r sin θdθdφ ) , (29)where we have used the standard form of a spherical surface element. Taking the curl in spherical coordinates, theradial component of B S is B Sr = 1 r sin θ (cid:2) ∂ θ (sin θA Sφ ) − ∂ φ A Sθ (cid:3) = 1 r sin θ (cid:2) ∂ θ A Cφ − ∂ φ A Cθ (cid:3) = B Cθφ r sin θ . (30)3Plugging Eqs. (29) and (30) into (28), we again get the Cartesian expression for C h (26).Finally, we note that for our case, the Hamiltonian is cylindrically invariant: we can get the Hamiltonian at arbitrary φ from the Hamiltonian at φ = 0 by just rotating the spins by an angle φ around the z -axis. Accordingly, the Berrycurvature must be cylindrically symmetric, meaning that B θφ ( θ, φ ) = B θφ ( θ ) is independent of φ . Therefore, if weplug into the expression for the Chern number, we find C h = 12 π Z π dφ | {z } =1 Z π dθB θφ ( θ ) = Z π B θφ ( θ ) dθ . (31)We now show how the equation (4) of the main text was derived. Starting with the Hamiltonian of a single qubitor equivalently spin-1/2 particle in a magnetic field: H S = − ~ H X σ x + H Y σ y + H Z σ z ) , (32)and re-parameterizing it for spherical coordinates, it becomes H S ( H r , θ, φ ) = − ~ H r (sin θ cos φ σ x + sin θ sin φ σ y + cos θ σ z ) . (33)Therefore, F φ = − h ∂ φ H ( φ = 0) i = ~ H r sin θ h σ y i . (34)Using equation (2) of the main text, ~ B θφ dθ = ~ H r sin θ h σ y i dt, (35)which is used in the main text in computing C h from the measured values of H r and h σ y i .
5. MAPPING THE TWO-QUBIT HAMILTONIAN TO ELECTRONIC BAND STRUCTURE
As in the main text, we consider the two-qubit Hamiltonian H = − ~ h H σ z + H r ˆ n ( θ, φ ) · ( σ + σ ) − g ( σ x σ x + σ y σ y ) i (36)for fixed H , H r , and g . For this section we assume ~ = 1 . At a given value of θ and φ , this Hamiltonian is a × matrix; a general N -qubit Hamiltonian would similarly be N × N . To help understand the topology of thisHamiltonian, we wish to map it to a more conventional electronic Hamiltonian, as we did in mapping the single qubitto the Haldane model of graphene. In this supplement, we show that (36) can be mapped to either a four-bandmodel of non-interacting electrons in the spirit of the Haldane mapping or a four-band interacting electron modelwith interactions that are short-range in momentum space. Finally, we comment on the extension of these mappingsto higher numbers of qubits.For both non-interacting and interacting electron mapping, we again utilize the idea that a given angle ( θ, φ ) of the rotating field H r corresponds to a point in momentum space (see the single qubit Haldane supplement): k = ( k x , k y ) ↔ ( θ, φ ) . Then the simple idea which worked for mapping the single qubit to the Haldane model is to“fermionize” the spin: σ αj → X ss c † js σ αss c js , (37)where α = x, y, z , j = 1 , specifies the qubit, and s, s = {↑ , ↓} iterate through the spin states. For example, thismapping gives σ x → c † ↑ c ↓ + c † ↓ c ↑ . Performing these replacements we get4 Figure S3. Illustration of the four-band non-interacting lattice model to which we map our two-qubit model. (a)
The modelconsists of three stacked triangular lattices (A, B, and C), the middle of which (B) contains two spin/orbital states. In additionto nearest neighbor hopping ( t ) and on-site hybridization the B sublattice ( t ), electrons on the A and C sublattices experiencemagnetic field that adds phase to the hopping ( t e iϕ ). Finally, an effective Zeeman field splits the spin/orbital states on allsublattices. (b) to (e) Energy dispersions for this model along a cut containing the K and K corners of the first Brillouin zone.We fix energy by setting t = 1 . At the K and K points, the sublattices decouple; we label the sublattice that is occupied inthe ground state at these points. C h then counts the number of times the wavefunction “twists” between the sublattices. H ( k ) = ( H r cos θ k + H )[( c k ↑ ) † c k ↑ − ( c k ↓ ) † c k ↓ | {z } σ z ] + H r cos θ k [( c k ↑ ) † c k ↑ − ( c k ↓ ) † c k ↓ | {z } σ z ] + H r sin θ k cos φ k [( c k ↑ ) † c k ↓ + ( c k ↓ ) † c k ↑ | {z } σ x +(1 → H r sin θ k sin φ k [ − i ( c k ↑ ) † c k ↓ + i ( c k ↓ ) † c k ↑ | {z } σ y +(1 → g c k ↑ ) † c k ↓ ( c k ↓ ) † c k ↑ + ( ↑↔↓ )] . (38)The last term of this Hamiltonian contains a four-fermion operator, so this is an interacting fermionic Hamiltonianwith four flavors of fermion ( c ↑ , c ↓ , c ↑ , c ↓ ). To maintain one spin per qubit, we want the many-body ground stateat half-filling and without double occupancy on “site” j = 1 , . However, the interaction remains short-range inmomentum space, meaning the electronic Hamiltonian is still separable into momentum sectors: H = P k H k . Suchmodels are similar to the mean-field BCS Hamiltonian [16], in this case with the additional wrinkle of being local inmomentum space. momentum space.While this first mapping is true to the interacting nature of the qubit, it gives little physical insight into thetopological transition. To try to understand this better, we now discuss how the same system can be mapped toa four-band Haldane-like model of non-interacting electrons. Unlike the interacting case, we present a microsopicmodel that will realize this topology. The model is shown schematically in Fig. S3a . The basic idea is to considerelectrons hopping on stacked triangular lattices with a single internal degree of freedom (spin/orbital/etc.) that cantake one of two values, which we denote ↑ and ↓ . The middle layer of the stack, which we call B, supports both ↑ and ↓ states, while the upper (lower) layer supports only ↑ ( ↓ ). This could be realized, for example, by a lattice wherethe middle layer has two orbital states (e.g. p x and p y orbitals), while the outer layers have only one orbital state(e.g. s orbitals). We assume there is a magnetic field gradient as in Fig. S3a, which gives zero field layer B and yieldsopposite magnetic field at A and C. We then consider four quadratic terms in the Hamiltonian:1. Nearest neighbor hopping t , which connects the B sublattice to the A and C sublattices. The matrix element isassumed to be equal for spin up hopping to either up or down, which make senses for orbital degrees of freedomor if ↑ and ↓ represent real spins, but with different quantization axes on A/C than on B.52. Second neighbor hopping on the A and C sublattices, which picks up a phase due to the magnetic flux. Forsimplicity, we consider a flux of Φ = 3Φ / − Φ / per plaquette, where Φ = h/e is the quantum of flux; thisgives phase ϕ = π/ on the hopping, removing a diagonal shift in the energy bands of the A and C sublattices(see [5]).3. On-site hybridization t between the effective spin states on the B sublattice.4. An effective Zeeman shift h z s z , where s z is the internal spin/orbital degree of freedom.Let us examine these terms in the language of the Haldane Hamiltonian. First note that the A and B (or equivalentlyC and B) sublattices have the exact structure as the sublattices of monolayer graphene. Therefore, the first Brillouinzone of this non-interacting electron model is equivalent to that of graphene, and we can naturally expand theHamiltonian in small deviations of the momenta around the non-identical zone corners K and K (see [cite Haldanesupplement]). There are four states in each unit cell: B ↑ , B ↓ , A ↑ , and C ↓ . The nearest neighbor hopping t is the only term that connects the sublattices, so it is responsible for producing a graphene-like dispersion relation.However, to get the Chern number of 2, this is slightly different from the Haldane model of graphene. To see this,consider quantizing the spins along the x -axis. It is easy to see that the state |↑ x i = √ ( |↑ z i + |↓ z i hops freely and willgive the dispersion of a graphene lattice with hopping amplitude t . However, a state with spin |↓ x i = √ ( |↑ z i − |↓ z i is annihilated by this hopping term, so in addition to the graphene dispersion, there are two flat bands at energy zeroif only the t term is considered (see Fig. S3(b)).The remaining terms then determine the topology by breaking the degeneracies at momenta K and K . For instance,the t e iϕ hopping only occurs on the A and C sublattices, so at the K and K points (where the t hopping vanishes),the electrons only live on the A or the C sublattice. As the momentum is varied from K ( θ = 0 ) to K ( θ = π ),the electronic ground state winds from sublattice A to C, which results in C h = 2 (see Fig. S3(c)). This is preciselythe action of the probe field, so we see that t ∼ H r . Similarly, the t term hybridizes the orbitals on the B lattice,causing the energy of the symmetric state on the B lattice to go down. For strong enough t , this can push the energyof the symmetric state on B lattice below the A and C energies throughout the Brillouin zone, resulting in Chernnumber zero (i.e., no wrapping of wave function, see Fig. S3(d)). This is the same role as the qubit interactions, sonot surprisingly t ∼ g . Finally, if we again consider t = 0 , then a large positive Zeeman field h z will push the energyof the spin down state below that of the spin up. In the presence of t hopping, this gives a ground state windingfrom the A sublattice to the B sublattice as momentum goes from K to K (Fig. S3(e)), yielding Chern number one.Not surprisingly, this gives that h z ∼ H .More explicitly, the Hamiltonian described above can be written H = X r (cid:2) − t X j ( c † r ↑ + c † r ↑ )( c r + a j ↓ + c † r + a j ↓ ) − t X j ( A sublattice z }| { e iϕ c † r ↑ c r + b j ↑ + C sublattice z }| { e − iϕ c † r ↓ c r + b j ↓ ) + t c † r + a ↑ c r + a ↓ + ( ↑↔↓ )) − h z c † r ↑ c r ↑ + c † r + a ↑ c r + a ↑ − ( ↑→↓ )) (cid:3) + h.c. , (39)where r are the sites on the A/C sublattice, a j are the nearest neighbor displacements, and (following Haldane’sconvention), b j are the next-nearest-neighbor displacements along directions with positive hopping phase on the Asublattice. Diagonalizing with phase ϕ = π/ , this gives Bloch Hamiltonian H k = − t P j sin( k · b j ) − h z − t P j cos( k · a j ) − t P j cos( k · a j ) 0 − t P j cos( k · a j ) − h z t − t P j cos( k · a j ) − t P j cos( k · a j ) t h z − t P j cos( k · a j )0 − t P j cos( k · a j ) − t P j cos( k · a j ) 2 t P j sin( k · b j ) + h z , (40)where the columns denote A ↑ , B ↑ , B ↓ , and C ↓ in that order. For comparison, the two-qubit Hamiltonian in the basis ↑↑ , ↑↓ , ↓↑ , ↓↓ H Q = 12 − H r cos θ − H − H r sin θ − H r sin θ − H r sin θ − H − g − H r sin θ − H r sin θ − g H − H r sin θ − H r sin θ − H r sin θ H r + H . (41)By inspecting these two Hamiltonians, we see that they map to each other under the identification k ↔ ( θ, φ ) , t √ H r , − t = g , h z = H , (42)6where we used the fact that P j sin( k · b j ) = ± √ / at the corners of the first Brillouin zone. Therefore, thetopology of the ground band of this four-band electronic model is equivalent to that of the two-qubit system that weexperimentally investigate.It is clear from the above discussion that a system of L qubits with L eigen states would map to a non-interactingmodel with L bands. While 2 or 4 band models are not so crazy, an eight band model with only a singled filled band– as would be needed for L = 3 qubits – is starting to get physically less realistic. Clearly the scaling of the numberof bands with the number of qubits is such that these non-interacting Haldane-like models will become exponentiallymore difficult to engineer as the system becomes larger. Working instead with the interacting model helps quite abit; simple counting requires only L flavors of fermion (spin up and down for each qubit) at half-filling and with nodouble-occupancy. However, this model has no obvious microscopic interpretation, so for the time being we considerit less physical. Therefore, we conclude that as the qubit number is increased (and restricted to the above mappingmethods), it becomes increasingly unworkable to think of the system in terms of electrons on a lattice. For large spinlattices, we really should think of our measurement as simply probing the topology of the spin manifold, a problemwhich is interesting in and of itself. It is also worth pointing out that the two mapping we have described aboutonly work for our choice of parameter manifolds, namely fixed external field strength with a rotating angle appliedequally to each qubit. By using different choices of manifold, even within the same two-qubit system, we can engineerdifferent effective condensed matter models, demonstrating the flexibility of these two-qubit systems.7 Figure S4.
Control sequence used for the single qubit experiments. (a)
The pulse sequence used to obtain the phasediagram shown in Fig. 3(a) of the main text. Every control sequence began with preparing the qubit in its ground state,which was achieved by waiting for times much longer than the qubit relaxation time (a few tens of microseconds). In the phasediagram measurements, h σ y i was measured at time steps during the ramps, where the first data point was measured at t = 0 and the last one at t = T f = 1000 ns. To measure each data point the sequence was repeated times. This -point h σ y i profile as a function of time from to T f was then multiplied by a sine profile (see equation (4) in the main text) andintegrated to give C h . h σ y i was measured by inserting a R X ( π/ pulse before the h σ z i measurement. The microwave pulsewith a sine profile and detuning pulse with a cosine profile constitute a semi-circular ramp in the parameter space, and giventhe symmetry of the single-qubit Hamiltonian, this is sufficient to calculate the curvature over the entire spherical manifold. (b) Control sequence for adiabatic state preparation and measurement. In contrast to the dynamic method (equation (4) of themain text), in the adiabatic state preparation process, the qubit needs to remain close to the instantaneous eigenstate of thesystem during the ramp. To evolve to the ground state of the Hamiltonian with parameters ( H X , H Y , H Z ) , we start from theorigin of parameter space, where all pulses are zero, and gradually turn ( H X , H Y , H Z ) to their final values in 500 ns. The pulsesthen remain at their target values for 500 ns (hence T A = 1000 ns ). Over this fixed pulse regions at 100 points (distributeduniformly from 500 ns to 1000 ns) the state of the qubit was measured with tomography and the results are averaged to presenta single Bloch vector data corresponding to given ( H X , H Y , H Z ) values. To visualize the ground states over the entire sphericalmanifold S , the process was repeated for different values of ( H X , H Y , H Z ) to form a grid over this parameter space sphere.
6. EXPERIMENTAL PROTOCOLS, CALIBRATION, AND ANALYSIS
Here we provide the outline of the experiment and its basic protocols. The first step is the calibration of thepulses so we know H X / π and H Z / π with good accuracy. An important aspect of calibration is also finding thecompensating pulse such that when we only H X / π , the state of the qubit remains in the YZ plane. The detailsof these steps are explained in Fig. S5. Next, one needs to find a proper ramp speeds to be sure the higher ordererrors in equation 2 of main text remain small; another words, how much non-adiabatic a ramp can be and still yielda good result. This is shown in figure 6, where we explored the three parameters that set the non-adiabaticity of aramp: H X / π , H Z / π , and T f . After finding that one needs to set all the ramps such that the adiabaticity measure A remains acceptable. After finding proper ramp speeds and calibrations, one can do the single qubit experiment,which involves applying H X / π with sine envelope and H Z / π with a cosine profile, as discussed in Fig. S4. The twoqubit experiment requires additional calibrations to what is mentioned in here and is discussed in [2].8 Figure S5.
Pulse Calibration.
Single qubit microwave and detuning pulses were calibrated separately before applying themsimultaneously to the qubits. (a)
To calibrate the microwave pulse, the qubit was prepared in its groundstate | + ˆ z i and amicrowave pulse with a sine envelope of amplitude H r was applied to the qubit. The state of the qubit (the Bloch vector) wasmeasured at each point in time by interrupting the ramp and performing full state tomography. As shown in panel (a) , forinstance, to measure h σ y i , a rotation of π/ around the X axis was performed before the h σ z i measurement. This pulse resultsin a cyclic motion of the Bloch vector in the Y-Z plane, with a non-zero out of plane component. The out of plane componentis mainly due to leakage to other states due to finite inharmonicity of the qubit system. Therefore, the measured out of planecomponent ( h σ x i , orange points in panel (c) ) needs to be calibrated, which was done by adding a compensating microwavepulse on the Y-axis, with a variable amplitude during the pulse sequence such that it keeps h σ x i close to zero. A typical resultbefore and after calibration is shown in (c) . Fitting the h σ z i and h σ y i with a single fitting parameter can be done using theSchrödinger equation. The resulting value in this case is H r / π = 44 . MHz. The dark blue and green solid lines are the resultof the fitting. During the calibration since full state tomography was performed, we normalized the measured values of h σ x i , h σ y i , h σ z i such that h σ x i + h σ y i + h σ z i = 1 . The detuning pulse was applied and measured similarly. This was done bybringing the qubit to the equator of the Bloch sphere with a π/ -pulse first, and then applying the detuning in the absence ofmicrowave pulse, and fitting the result with the Schrödinger equation. There was no compensation pulse to be considered inthis case. Figure S6.
Adiabaticity required to measure C h . Although equation (2) of the main text does not require adiabaticity,it does require the ramp in parameter space to be done slowly, such that O ( v θ ) remains negligible. One needs to operate withramp velocities for which deviation from adiabaticity varies linearly with the ramp speed. Slower ramps are more adiabatic andhence better in this regard, but they have a small deviation from adiabaticity, which would be hard to measure experimentally.On the other hand, ramps that are too fast also contain non-adiabatic errors that are not linearly proportional to ramp speed,and hence should be avoided. In this figure, the top row shows the experimental results of measuring C h by making variouselliptical ramps and traversing them with different velocities. A microwave pulse of X ( t ) = H X sin( πt/T f ) and detuning of Z ( t ) = H Z cos( πt/T f ) are used, with H X / π and H Z / π varied from 0 to 10 MHz. Five different speeds are used, which areset by T f , where T f is the time it takes to ramp from the north pole of the manifold to its south pole. The lower row shows thenumerical results using the same ramps, obtained from the time dependent Schrödinger equation. In this example, we seek tomeasure C h over a manifold of ground states that encloses the origin of the parameter space. The theoretical value of C h in thiscase is 1 [8]. From left to right, as T f becomes longer, the ramps are more adiabatic and the measured value for C h approachesone. In each panel, moving from lower left to upper right, adiabaticity increases, since A = T f H r / π = T f p H X + H Z / π .For T f = 400 ns or longer, a good estimate of C h can be achieved, as almost the entire plot is red, regardless of the shape ofthe manifold. The method yields a good estimate of C h for A > . . To provide a visual guide, the deformation of the sphericalmanifolds to ellipsoids, by keeping H X fixed and increasing H Z (horizontal axis below figure), and by keeping H Z fixed andincreasing H X (vertical axis left of figure) are shown. Figure S7.
Control sequence used for the 2-qubit experiments. (a)
The pulse sequence used on individual qubitsand the coupler elements are shown. The microwave pulse applied to each qubit had a sine envelope. The detuning of Q2has a cosine profile, and the detuning of Q1 has a cosine profile plus an offset defining H . With a rectangular pulse, thecoupling between the two qubits is turned on during the active part of other pulses. The synchronization of the pulses as wellas finding the flux value corresponding to g = 0 were done [2] prior to running the sequence. In addition, a calibration matrixto take various types of crosstalk into account was measured and implemented. This included both microwave and flux-biasingcrosstalk [2]. Using equation (4) of the main text, the C h for 2-qubit manifolds is the summation of individual ones. Therefore,each pulse sequence was run twice, once to measure h σ y i and again to measure h σ y i and the results were added to give thephase diagram plots shown in the Fig. 4 of the main text. Figure S8. "Decomposition" of the topological phase diagram obtained with 2 qubits.
To demonstrate thefluctuation at each C h plateau and avoid obscuring it with the color map, here we replot the topological phase diagram shownin Fig. 4(c) of the main text ( H r / π = 10 MHz ). Each panel shows the data in a given interval of C h values. The same orderof color tones is used in each panel, but the limits of the color scale for each panel is different. The black dashed lies are thefit using the analytical solution based on finding the loci of the monopoles(degeneracies of the Hamiltonian in this case). Thedeviations from the expected values have several sources: the crosstalk between the two qubits is likely the primary source,as the individual qubits were calibrated accurately. While the pulse length T f was kept an order of magnitude smaller thanthe decoherence time in the system, decoherence and measurement errors also contribute to the error. Understanding theseerror mechanism is currently under way. The sharpness of the transition from one C h plateau to another is mainly relatedto the speed of the ramps. Slower ramping in parameter space (longer T f ) would result is sharper transitions. In order tosuccessfully use slower ramps longer coherence times are required, which based on our current understanding of decoherencemechanisms gathered from this first generation of gmon devices, is achievable and will be implemented in the next generationof this experiment. ∗ These authors contributed equally to this work. † [email protected][1] M. Geller, Y. Donate, E. Chen, C. Neill, P. Roushan, and J. Martinis, “Qubit architecture with high coherence and fasttunable coupling,” under preperation (2014).[2] Y. Chen, C. Neill, P. Roushan, N. Leung, M. Fang, R. Barends, J. Kelly, B. Campbell, Z. Chen, B. Chiaro, A. Dunsworth,E. Jeffrey, A. Megrant, J. Mutus, P. O’Malley, C. Quintana, D Sank, J. Vainsencher, A. Wenner, M. White, T. Geller,A. Cleland, and J. Martinis, “Qubit architecture with high coherence and fast tunable coupling,” submitted (2014).[3] R. Barends, “Logic gates at the surface code threshold: Superconducting qubits poised for fault-tolerant quantum com-puting,” .[4] R. Barends, J. Kelly, A. Megrant, D. Sank, E. Jeffrey, Y. Chen, Y. Yin, B. Chiaro, J. Mutus, C. Neill, P. O’Malley,P. Roushan, J. Wenner, T. C. White, A. N. Cleland, and John M. Martinis, “Coherent josephson qubit suitable forscalable quantum integrated circuits,” Phys. Rev. Lett. , 080502 (2013).[5] F. D. M. Haldane, “Model for a quantum hall effect without landau levels: Condensed-matter realization of the "parityanomaly",” Phys. Rev. Lett. , 2015–2018 (1988).[6] P. J. Leek, J.M Fink, A. Blais, R. Bianchetti, M. Göppl, J. M. Gambetta, D. I. Schuster, L. Frunzio, R. J. Schoelkopf,and A. Wallraff, “Observation of berry’s phase in a solid-state qubit,” Science , 1889–1892 (2007).[7] M. Berry, “Quantal phase factors accompanying adiabatic changes,” Proc. R. Soc. Lond. A , 45–57 (1984).[8] D. Griffiths, Introduction to Quantum Mechanics (Prentice Hall, 1994).[9] In the language of geometry, A µ intuitively gives (up to a factor of i ) the component of | ψ i that one must subtractfrom ∂ µ | ψ i in order to turn the derivative ∂ µ | ψ i into a “covariant” derivative D µ | ψ i with the property that | D µ ψ i isorthogonal to | ψ i , which defines what it means to keep a quantum state “parallel” as R is moved from R to R + d R .[10] B. A. Bernevig and T. L. Hughes, Topological Insulators and Topological Superconductors (Princeton University Press,2013).[11] D. J. Thouless, M. Kohmoto, M. P. Nightingale, and M. den Nijs, “Quantised singularities in the electromagnetic field.”Proc. R. Soc. Lond. A , 60–72 (1931).[12] V. Gritsev and A. Polkovnikov, “Dynamical quantum hall effect in the parameter space,” PNAS , 6457–6462 (2012).[13] This is an instance of a general result [17] stating that unless there is some special kind of symmetry in the Hamiltonian,three parameters must be tuned in order to reach a degeneracy.[14] J. Sakurai,
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