Quantum Simulation Architecture for Lattice Bosons in Arbitrary, Tunable External Gauge Fields
QQuantum Simulation Architecture for Lattice Bosons in Arbitrary, Tunable ExternalGauge Fields
Eliot Kapit
Rudolf Peierls Center for Theoretical Physics, Oxford University,1 Keble Road, Oxford OX1 3NP, United Kingdom ∗ I describe a lattice of asymmetrical qubit pairs in arbitrary dimension, with couplings arranged sothat the motion of single-qubit excited states mimics the behavior of charged lattice bosons hoppingin a magnetic field. I show in particular that one can choose the parameters of the many-bodycircuit to reach a regime where the complex hopping phase between any two elements can be tunedto any value by simply adjusting the relative phases of two applied oscillating voltage signals. Ialso propose specific realizations of our model using coupled three junction flux qubits or transmonqubits, in which one can reach the strongly interacting bosonic quantum Hall limit where one willfind anyonic excitations. This model could also be studied in trapped ions, and the superconductingcircuits could be used for topological quantum computation.
PACS numbers: 03.67.Lx,03.67.-a,73.43.-f,75.10.Kt
I. INTRODUCTION
Fractional quantum Hall effects [1–3] are among themost profound collections of phenomena to emerge in in-teracting quantum many-body systems. The elementaryexcitations in these systems do not act like bosons orfermions; rather, they are anyons , which in some casescan be used for a robust form of quantum computing[4, 5]. All physical examples of fractional quantum Halleffects are in two dimensional (2D) electron gases. Herewe propose a method for linking standard qubit designswhich will realize a bosonic fractional quantum Hall ef-fect. The rich theoretical literature on bosonic fractionalquantum Hall effects suggests that there will be a largenumber of interesting states [6–13] that could be exploredin our system. These include ‘Pfaffians’ and their gener-alizations. Furthermore, one could anticipate that someimportant experiments (such as directly braiding quasi-particles) may be simpler in a qubit array than in a GaAslayer surrounded by AlGaAs.There are several competing approaches to engineer-ing bosonic fractional quantum Hall effects. One pro-posal uses Raman lasers to simulate the magnetic vec-tor potential in neutral cold atoms [14, 15]. The tech-nical challenges are, however, quite daunting: new cool-ing methods need to be designed to offset heating fromthe Raman lasers, and the most natural probes are in-direct. Another scheme is to use lattices of tiny su-perconducting grains (charge qubits, [16–21]) connectedthrough Josephson junctions. Suitably low temperaturescan be reached in a dilution refrigerator, and the systemis readily studied using transport measurements. Unfor-tunately, random charge noise, which scales linearly withthe interaction strength, would prevent the quantum Hallregime from being reached without significant local tun- ∗ Electronic address: [email protected] ing of the potentials on hundreds or thousands of latticesites. Other proposals include superconducting Jaynes-Cummings lattices [22] and “photon lattices” of coupledoptical waveguides [23–25], each of which have their ownadvantages and shortcomings.I here propose a new and promising approach. Con-sider a circuit of qubits, with a geometry which naturallymaps onto a system of charged bosons hopping in a mag-netic field. In order to produce complex hopping matrixelements I study a lattice of coupled asymmetrical pairsof qubits, which I label as A or B . I choose device pa-rameters so that excitation energy ω A of the A qubits issignificantly smaller than the excitation energy of the B qubits, and place a B qubit on each link between neigh-boring A qubits. Further, I couple them to each otherthrough alternating hopping ( σ + A σ − B + H.C. , henceforthreferred to as a “ ± ” coupling) and potential ( σ zA σ zB , a“ zz ” coupling) terms. Coupling qubits through higherenergy auxilliary qubits has been considered previouslyboth theoretically [26] and experimentally [27, 28]. I alsoapply an external oscillating electromagnetic field of fre-quency ω to each qubit, with the relative phase of thesignal applied to the B qubits shifted relative to thatof the A qubits by a locally tunable ϕ s . Since the B qubits are higher energy than the A qubits, they can beintegrated out, leading to complex tunneling matrix el-ements (the amplitude of a process where the states ofneighboring qubit pairs are exchanged) between A qubitswith phases that can be tuned to any value by adjusting ϕ s .As I will describe below, a particularly attractive re-alization of this architecture would be to use three junc-tion “flux qubits” (FQs) [29–38]. The flux qubits aremesoscopic superconducting rings interrupted by threeJosephson junctions, placed in a magnetic field which istuned so that nearly 1/2 of a magnetic flux quantum pen-etrates the ring. The energies of the flux qubits can betuned by adjusting this magnetic field, or by varying theareas of the Josephson junctions, so that the B qubits arehigher energy than the A qubits as outlined above. We a r X i v : . [ c ond - m a t . s t r- e l ] J un then capacitively couple all the flux qubits to an external,oscillating voltage V sin ωt , and arrange the couplings sothat the phases of the voltage applied to the B qubitsare shifted relative to the A qubits. The subtle interplayof the oscillating applied voltages with the mix of charge(capacitive) and phase (Josephson) couplings introducesphase shifts which make these hopping matrix elementscomplex, mimicking the Peierls phases found for chargedparticles in magnetic fields.All of the flux qubits in my design are operated inthe regime where the Josephson energy E J is large com-pared to the charging energy E C , so charge noise effectsare exponentially suppressed. The system is therefore al-most completely insensitive to stray low-frequency elec-tric fields. The many-body excitation gap, a key featureof anyon states, can be measured through the single-qubitresponse to applied oscillating voltages. The large non-linearities of the flux qubit devices imply that the firstexcited states experience an effectively infinite on-siterepulsion. I note also that our scheme is not intendedto function as a dynamical circuit QED architecture orJaynes-Cummings model (in contrast to the recent workof Koch et al [39, 40] and others); the device parame-ters should be chosen so that the external voltages canbe treated as purely classical sources, with no dynamicalphotons present in our system. Further, though I will notdiscuss them in any detail here, my proposal could alsobe studied in trapped ions, perhaps using the methodsof Korenbilt et al [41] to engineer the anisotropic spininteractions, or through the digital simulation methoddetailed in Lanyon et al [42]. Finally, since originallysubmitting this article, I have become aware of a similarmethod in cold atoms [43] for engineering artificial gaugefields through time-dependent drive fields that are out ofphase from one lattice site to the next.The remainder of this paper is organized as follows. Insection II, I write down the basic coupled qubit Hamil-tonian, and outline the conditions under which arbitraryexternal gauge fields can be simulated. In section III, Idescribe three junction flux qubits, and how they can becoupled to obtain the arbitrary complex hopping phasesderived in section II. Following this, I provide a briefdiscussion of how these phases might be implemented intransmon qubits [44–47] as well. I also describe a differ-ent regime of operation in the superconducting qubits,where different excited states are mixed by the drivefield, which has some advantages over the formulation de-scribed in the main body of the text. Having derived thecomplex hopping phases, in section IV I show how the cir-cuits of the two previous sections can be used as buildingblocks for exotic boson fractional quantum Hall states.Finally, in section V, I show how a simple arrangement offour qubits could experimentally demonstrate a nonzeroeffective gauge field, and offer concluding remarks. AA BB b V sin ωt b V sin ωt b V sin ( ωt + ϕ s ) b V sin ( ωt + ϕ s ) D z D z D ± FIG. 1: (Color online) Basic coupling structure for the A and B qubits. Each site in our many-body lattice wouldcorrespond to a single A qubit, which couples to its neigh-bors through one B qubit per link, joined through alternat-ing hopping ( ± ) and potential ( zz ) couplings as described insection II. Though drawn in one dimension in the figure, weultimately intend to construct 2d lattices in this manner, andgeneralizations to even higher dimensions are also possible. II. GENERAL FORMALISMA. Berry’s Phase of a Rotating Spin
Before outlining the physics of the larger qubit array,I would first like to discuss a simple example to morestraightforwardly elucidate the origin of the complex hop-ping phases. Specifically, I will consider a pair of spinsand examine the Berry’s phase effects generated duringa process where an excitation is transfered from one spinto its neighbor (whose eigenstates lie on a different axisfrom the first spin) by rotating both spins about z andthen transferred back by rotating both spins about y .Let us consider two initially uncoupled spin- degrees offreedom, with the Hamiltonian and eigenstates, H = σ xA + cos θσ zB + sin θ (cos ϕσ xB + sin ϕσ yB ) , (1) | A (cid:105) = 1 √ , − , | A (cid:105) = 1 √ , , | B (cid:105) = 1 √ − θ (cid:0) e iϕ ( − θ ) , sin θ (cid:1) , | B (cid:105) = 1 √ θ (cid:0) θ, e − iϕ sin θ (cid:1) . Let us assume that initially spin A is excited and spin B is in its ground state. We first act with the operator σ zA σ zB to transfer the excitation from A to B , and as-sume energy is conserved in this process so that the finalstate after acting with σ zA σ zB is | A B (cid:105) ; since the B spinis quantized along a different direction from A , the exci-tation must rotate to be transferred to the B spin. Wethen act with σ yA σ yB to transfer the excitation back; theresulting matrix element M for the entire process is M = (cid:104) A B | σ yA σ yB | A B (cid:105) (cid:104) A B | σ zA σ zB | A B (cid:105) (2)= sin θ (cos ϕ + i sin ϕ cos θ ) . For θ (cid:54) = π/ ϕ (cid:54) = 0 , π , M is complex, and the re-sulting phase can be understood as a consequence of theBerry’s phase acquired by a rotating spin, though we noteof course that the Berry’s phase discussed here is only ananalogy, since we are considering the action of pairs ofoperators and not continuous, adiabatic changes to thesystem’s wavefunction. When a spin m is rotated alonga closed path, the resulting phase is equal to m timesthe area subtended by the path on the unit sphere. Inthis case, we have two spins which rotate, but both endin the same states in which they started, so we obtaina gauge-invariant phase equal to the sum of the phasespicked up by both spins. The area subtended by A isjust π , but the area subtended by B depends on the pro-jection of σ y and σ z onto its quantization axis, and thusdepends on ϕ s , yielding the result above. Note that ifwe’d acted with σ zA σ zB or σ yA σ yB twice instead of using acombination of the two, the outcome would necessarilybe real, since M would be the product of a matrix ele-ment and its Hermitian conjugate. In the Berry’s phasepicture, the phase is zero simply because the path of eachspin’s rotation would be a 1d line, and thus each area iszero. Both inequivalent eigenstates and anisotropic op-erations are necessary for spin transfer matrix element tobe complex.It is precisely this effect–the phase picked up by a spinwhich rotates as it propagates in space–which I will use toengineer artificial hopping phases in our lattice. Specif-ically, imagine the case in which we had two (identical) A spins with a B spin in between them, and after actingwith σ zA σ zB to pass an excitation from A B spin,we then act with σ yA σ yB to transfer the excitation to thesecond A qubit instead of sending it back to the first.Since the A spins are identical, the matrix element M should be the same as the one derived above, and there-fore by letting B spins mediate a hopping coupling, wecan introduce tunable phases in a lattice of A spins.Engineering this structure in a real spin (or qubit) lat-tice is by no means trivial. For real spins, one could in-troduce a spatially varying magnetic field to generate theinequivalent local eigenstates, but adding the anisotropicspin-spin interactions ( σ zA σ zB or σ yA σ yB instead of S A · S B )is very difficult. Conversely, for a more general lat-tice of qubits, generating passive anisotropic couplingsis often straightforward, but generating inequivalent lo-cal eigenstates is not. I here demonstrate that couplingthe qubits to a continuously oscillating monochromaticexternal field can introduce the required rotations, pro-vided that the phases of the signals applied to the B qubits are different from those applied to the A qubits.By adjusting these phases at a local level, we can inde-pendently tune the tunneling phase between any linkedsites on the lattice, and can thus simulate any desiredexternal gauge field, at least in principle. B. Qubit Coupling Hamiltonian
We will consider a lattice of qubits, arranged such thatthere is a higher energy B qubit between each pair of linked A qubits. We shall assume throughout that thefollowing conditions hold:(1) The nonlinearities of each physical system which weuse as a qubit are large enough that we can consider themto be purely two-level systems, and ignore all eigenstatesbesides | (cid:105) and | (cid:105) . This requirement ultimately con-strains the magnitudes of the couplings between qubits,which must be small compared to the physical devices’absolute nonlinearities. I describe an alternate regime,where states | (cid:105) and higher are considered and mixed bythe drive fields, later in this work.(2) The qubits can be coupled to an external electro-magnetic field. We shall further require that the elec-tromagnetic field operator (cid:98) V (which could represent thecoupling to magnetic fields as well) has no expectationvalue in either state, so (cid:104) | (cid:98) V | (cid:105) = (cid:104) | (cid:98) V | (cid:105) = 0. Thesefields will always be present in the qubit array Hamil-tonian, and we will treat them in the standard rotatingwave approximation.(3) We must be able to introduce two types of couplingbetween the qubits, so that the qubit-qubit Hamiltoniantakes the form H int = D ± (cid:0) σ + A σ − B + σ − A σ + B (cid:1) + D z σ zA σ zB . (3)We must have independent control over both D ± and D z for our method to succeed. Note that any physicalcoupling between the qubits will typically include termswhich violate number conservation. However, when wetransform to the rotating frame when the external oscil-lating voltage is applied, the terms in H int are unchangedbut anomalous terms such as σ − A σ zB or σ + A σ + B will be-come rapidly oscillating and can be dropped from thelow-energy Hamiltonian.(4) We must be able to tune the relative phase ϕ s of theexternal electromagnetic field applied to B qubits relativeto the A qubits, as shown in fig. 1. If ϕ s (cid:54) = 0 , π then timereversal symmetry is broken, since we cannot chose a zeropoint for the time t so that both V A ( t ) = V A ( − t ) and V B ( t ) = V B ( − t ). Breaking time reversal symmetry is abasic requirement for obtaining nontrivial effective gaugefields.These requirements could be fulfilled by a large numberof physical systems, including spin qubits, trapped ions,and superconducting devices, which will be the focus ofthis work. Let us now consider the Hamiltonian of agiven qubit pair, H AB . Before turning on the oscillatingfields, our qubit Hamiltonian is H AB = ω A σ zA + ω B σ zB (4)+ (cid:8) D ± (cid:0) σ + A σ − B + σ − A σ + B (cid:1) or D z σ zA σ zB (cid:9) . We now turn on the oscillating fields. When acting on A or B , we have: (cid:98) V = Ω A/B σ yA/B , (5)with Ω A/B = (cid:10) A/B (cid:12)(cid:12) (cid:98) V (cid:12)(cid:12) A/B (cid:11) , which we choose to bereal. We now examine (cid:98) V sin ωt = Ω A/B (cid:16) e iωt σ − A/B + e − iωt σ + A/B (cid:17) (6)+ Ω
A/B (cid:16) e − iωt σ − A/B + e iωt σ + A/B (cid:17) . We now transform to the rotating frame by applying theunitary transformation | ψ (cid:105) → exp − i ω ( σ zA + σ zB ) t | ψ (cid:105) .The time dependence of terms on the first line of (6) iscancelled out, leaving us with Ω A/B σ xA/B / ω .We now make the rotating wave approximation (RWA)to neglect these terms. After transforming to the rotatingframe and invoking the RWA, H AB is: H AB = ( ω A − ω )2 σ zA + ( ω B − ω )2 σ zB (7)+ Ω A σ xA + Ω B ϕ s σ xB + sin ϕ s σ yB )+ (cid:8) D ± (cid:0) σ + A σ − B + σ − A σ + B (cid:1) or D z σ zA σ zB (cid:9) . From now on we will assume ω is tuned to resonancewith the A qubits, so that ω = ω A and the single-siteHamiltonian for the A qubits is just Ω A σ xA / A qubits consists of a zz coupling toa B qubit followed by a ± coupling to the other A qubit. For simplicity, we will ignore cases where A qubitsare coupled directly; such couplings will produce eitherneighbor-neighbor potential interactions or real-valuedhopping matrix elements, depending on their structure.We assume that the energy difference E B − E A = (cid:113) ( ω B − ω A ) + Ω B − Ω A ≡ δE is large compared to D ± and D z , so that we can treat the A − B coupling pertur-batively. We now eliminate the B qubits using secondorder perturbation theory; noting that all A qubits areidentical, the resulting Hamiltonian, to order D /δE , isgiven by: H = (cid:88) ij (cid:16) J ij a † i a j + H.C. (cid:17) + ˜Ω A (cid:88) i a † i a i , (8) J ij = − D zij D ± ij δE sin θ (cid:0) cos ϕ s ( ij ) + i cos θ sin ϕ s ( ij ) (cid:1) , cos θ = ω B − ω A (cid:113) ( ω B − ω A ) + Ω B . Here a † i /a i creates/annihilates an excitation in the A qubit at site i , and ˜Ω A is equal to Ω A plus O ( J ) shiftswhich depend on the coordination number of the lat-tice and magnitudes of the couplings. Since the qubitsare spin- , we have an effective hard-core constraint, so a † i | i (cid:105) = 0. If we now identifyarg J ij ≡ q (cid:90) r j r i A · d r , (9) we see that the complex phases of J are identical to thePeierls phases of a charged particle moving on a lattice inan external gauge field A . Further, if we choose param-eters so that the B qubits are far off-resonance, θ will besmall and J ij → − D zij D ± ij δE θe iϕ s ( ij ) + O (cid:0) θ (cid:1) . (10)In this regime, we can freely adjust the phase of J withoutsignificantly altering its magnitude, and can thus simu-late any time-dependent external gauge field configura-tion we desire, simply by adjusting the B qubit phaseshifts ϕ s ( ij ) at each link.Before continuing, it is worth keeping in mind that therotating wave approximation is simply the zeroth-orderterm in a power series in Ω A/B /ω , and is therefore notan exact description of the system’s dynamics. Correc-tions to the RWA have been treated in many ways, butfor our purposes the treatment of Thimmel et al [48] isthe most useful, since it details the effect of terms beyondthe RWA on the time-independent rotating frame effec-tive Hamiltonian. Generalizing their result, we obtain aneffective correction term at first order in Ω A/B /ωδH = 3Ω A ω σ zA − Ω A ( ω A − ω )4 ω σ xA (11)+ 3Ω B ω σ zB − Ω B ( ω B − ω )4 ω (cos ϕ s σ xB + sin ϕ s σ yB ) . These small shifts can be eliminated by further tuningof the device parameters and applied frequencies, andshould not change the basic physics of the system or itsartificial gauge field. In particular, in the flux qubitsI will describe below, ω is 15-30 times larger than theRabi frequencies Ω A/B , so these corrections are stronglysuppressed.In addition to these corrections, in a physical qubitsystem the applied voltage ˆ V will include matrix ele-ments that mix the qubit’s basis states with higher ex-cited modes [72]. These transitions can be treated inperturbation theory by integrating out the higher modes,and produce σ z terms which scale as Ω / ( ω − ω ij ), where ω ij is the energy difference between the coupled states.These corrections can become significant if the Rabi fre-quencies Ω approach the absolute nonlinearities of thequbit spectra, but for flux qubits these nonlinearities arelarge and mixing with higher excited states can be ig-nored. Alternatively, the higher excited states can beused to our advantage by driving the system near a dif-ferent resonance to leave the ground state unchanged; Iwill describe this approach in section III C.The ability to engineer artificial gauge fields of anydesired configuration has tremendous potential to un-lock new physics, and I will discuss the most natural ap-plication, simulating a uniform magnetic field to realizestrongly interacting bosons in the quantum Hall regime,later in the work. Before doing so, however, I will firstdescribe a possible implementation of this architecture in (1) (1)(1) (2) (2)(2) αE J , αC, fαE J , αC, f E J , C E J , C E J , C E J , CE J , C E J , C ( L ) ( R ) κE J , κCκE J , κC βE J , βC, f gCV sin ( ωt + ϕ s ) V sin ωtV sin ωt ηCηC ηCηCηCηC f ′ f ′′ FIG. 2: (Color online) Basic circuit architecture. The regions enclosed in dashed boxes are three-junction flux qubits, whichare connected to a physical ground. The blue ( A , left and right) and red ( B , center) qubits differ from each other by a rescalingof the area of the central Josephson junction, which is tuned so that the B qubits have higher energy excitations. A magneticfield penetrates the plane so that f flux quanta are enclosed by each ring. An oscillating voltage V E ( t ) is applied near resonanttransitions to both qubits, mixing their ground and first excited states. Excitations in the A flux qubits can tunnel throughthe B qubits to each other; the oscillating voltage will make this transition matrix element complex. The qubit properties andthe couplings between them are discussed in section III. superconducting flux qubits. While flux qubits are cer-tainly not the only– or even necessarily the best– qubitsto use for this purpose, our proposal will demonstratethat a fairly robust implementation of my architecture can be realized using device parameters from previousexperiments. Thus, small lattices should be within reachof current technology. III. QUBIT IMPLEMENTATIONSA. Flux Qubits
The three-junction flux qubit consists of a supercon-ducting ring interrupted by three Josephson junctions asshown in fig. 2, with one junction whose area is rescaledby α relative to the other two. A constant, tunable mag-netic flux bias of f (cid:54) = 1 / φ = 0, then the two remaining degrees of freedomof the flux qubit are the phases φ and φ of the othertwo superconducting regions. The derivation of the fluxqubit Hamiltonian is descrbed in detail in Orlando et al [30]; in terms of the phases φ and φ , the flux qubit Hamiltonian H F Q is H F Q = (1 + α + η ) (cid:0) Q + Q (cid:1) + 2 αQ Q (1 + η ) (1 + 2 α + η ) C (12) − E J [cos φ + cos φ + α cos (2 πf + φ − φ )]+ 2 η ( αQ + (1 + α + η ) Q ) V sin ωt (1 + η ) (1 + 2 α + η ) . Here, Q j = − ei∂/∂φ j , E J is the Josephson energy ofthe Josephson junctions and f is the total magnetic fluxthrough the loop in units of the magnetic flux quantumΦ . The terms on the third line of (12) represent thecoupling of the flux qubit to the applied voltage V sin ωt .For the moment, we will consider this Hamiltonian with V = 0.We let φ ± = ( φ ± φ ) /
2. For f (cid:54) = 0, the symme-try between φ and φ is broken, and for f close to1/2, the ground and first excited states are distinguishedby their behavior along the φ − direction, as excitationsalong φ + are significantly more expensive. The typicalexcitation energy for 0 . < α < . . < f < . ω F Q / π = 12 − E J /h ∼ E C = e / C = E J /
40, and the nonlinearities of the spec-trum are all reasonably large. In this work we will onlyconsider flux qubits operated at the symmetry point of f = 1 /
2, in which case the ground and first excited statesare both even along φ + and even or odd, respectively,along φ − . From this, we can readily translate operatorsin the phase basis to Pauli matrices acting in the qubitbasis. We will define the following compact notation formatrix elements: M ij (cid:98) O,s ≡ (cid:104) i s | (cid:98) O | j s (cid:105) e . g . M , A = (cid:104) A | Q | A (cid:105) . (13)In this notation, we have: Q j → e ( − j M ∂ φ − σ y , (14)sin φ j → ( − j M φ σ x , cos φ j → M φ + M φ + M φ − M φ σ z . For consistency, all matrix elements M are calculatedbetween the V = 0 (non-rotating) eigenstates of the fluxqubit Hamiltonians.Let us now turn to the coupling Hamiltonian betweenthe qubits shown in fig. 2. Direct and indirect coupling offlux qubits, including through intermediary qubits [27],has been considered theoretically and demonstrated ex-perimentally [26, 28, 33, 34, 49–52]. We label the two A qubits by L and R . The coupling of the B qubit to theright qubit is a simple capacitive coupling, and so is givenby a constant times σ yB σ yR , which becomes a ± couplingin the rotating frame: H BR = 8 E C M ∂ φ ,B M ∂ φ ,R (1 + 2 α + η ) (1 + 2 β + η ) (cid:0) σ + B σ − R + σ − B σ + R (cid:1) . (15)It is important to note that both σ x σ x and σ y σ y become ± couplings in the rotating frame, as the components ofthem which lead to net creation or destruction of excita-tions are rapidly oscillating and should be dropped. Thecoupling between the left qubit and the B qubit consistsof two Josephson junctions; since these junctions defineclosed loops through ground, they pick up flux biases f (cid:48) and f (cid:48)(cid:48) from the external magnetic field. For simplicity,we choose the wiring geometry so that these biases areboth zero mod 2 π . When we write the coupling between L and B as a set of Pauli matrices, the ± terms vanishdue to the sign flips in (14), but the zz term survives: H LB = − κE J (cid:0) M φ ,L − M φ ,L (cid:1) (16) × (cid:0) M φ ,B − M φ ,B (cid:1) σ zL σ zB . Alternately, one could obtain a pure zz coupling by sim-ply placing a single Josephson junction between a pairof regions, and choosing the wiring geometry so that theflux bias f (cid:48) is nonzero, leading to an interaction term of the form − κE J cos ( φ L − φ B + 2 πf (cid:48) ) plus a capac-itive term with the same structure as (15). One couldthen tune f (cid:48) so that the ± components of the xx and yy terms from these couplings interfere with each other,leaving only the zz part of the coupling.We are now in a position to plug in numbers and eval-uate J for this architecture. Consider flux qubits wiredas in fig. 2. If we choose the realistic device parameterslisted below, taking into account the single-qubit energyshifts from the D z coupling gives us: I c = 400nA , C = 3 . , α = 0 . , f = 0 . , (17) η = 0 . , β = 0 . , κ = 0 . , g = 0 . E J /h = 200GHz = 33 E C /h, ω A = 2 π × ,ω B = 2 π × , Ω A V (cid:39) Ω B V = 2 π × . ,D z = 2 π × . ,D ± = 2 π × . . (18)A plot of J for V = 0 . , . V , | J | is almostcompletely independent of ϕ s , but for larger V the mag-nitude fluctuations become significant. | J | can be furtherincreased by up to an order of magnitude by choosingdevice parameters to work in the regime where f > / B. Transmon Qubits
I will now briefly outline the implementation of mymodel in transmon qubits. The basic circuit of a trans-mon qubit reduces to that of a single Josephson junction(with large E J /E C ) connecting a small superconductingisland to ground. The superconducting phase φ of theisland is the qubit’s sole quantum degree of freedom, andpartly as a result of this, these qubits are extremely sta-ble, achieving decay and dephasing times over an order ofmagnitude greater than flux qubits [53, 54]. Further, re-quiring only a single Josephson junction, they are simplerto fabricate than flux qubits. This stability and simplic-ity comes at a cost, however, in that the natural non-linearities of transmon qubits are only a few percent ofthe excitation energy ω , which strongly constrains themagnitudes of the coupling terms.The Transmon qubit Hamiltonian is a quantum anhar- ϕ s /π | J | / h ( G H z ) , a r g J / π FIG. 3: (Color online) Magnitude and phase of J for thedevice parameters given in eq. (17) at the resonance point ω = ω A . The blue and purple curves are | J | and arg J/π ,respectively, for V = 0 . V = 0 . J can be made sig-nificantly larger by increasing α and working away from the f = 1 / E J , CE J , C E ′′ J , C ′′ , fE ′ J , C ′ gCV sin ( ωt + ϕ s ) V sin ωtV sin ωt ηCηCηC FIG. 4: (Color online) Implementation of the ± − B − zz linkwith transmon qubits. The effective circuit of a transmonqubit is a single Josephson junction connecting a small su-perconducting island to ground; the superconducting phase φ of the island is the qubit’s single quantum degree of freedom.The ± coupling can be implemented by a capacitive couplingbetween two qubits ( gC in the figure). The zz coupling issomewhat more challenging, but can be realized by connect-ing two qubits with a Josephson junction ( E (cid:48)(cid:48) J and C (cid:48)(cid:48) ) witha flux bias f . The Josephson junction naturally produces xx , yy and zz couplings, and since both xx and yy couplings be-come ± couplings in the rotating frame, by tuning the fluxbias f and appropriately choosing E (cid:48)(cid:48) J and C (cid:48)(cid:48) , we can getthem to cancel each other out, leaving only the zz term. monic oscillator, H T = − E C ∂ ∂φ − E J cos φ. (19)The charge and phase operators map to σ matrices in thequbit basis as they did in flux qubits: Q → − e M ∂ φ σ y , sin φ → M φ σ x , (20)cos φ → M φ + M φ + M φ − M φ σ z . To construct the alternating ± and zz couplings requiredto obtain tunable phases, we introduce alternating capac-itive and flux-biased Josephson junction couplings. Thecapacitive part is simply a σ y σ y term: H C = + 4 e gC (cid:48)(cid:48) CC (cid:48) M ∂ φ,A M ∂ φ,B σ yA σ yB . (21)On the other hand, the Josephson junction with a fluxbias (fig. 4) contains xx , yy and zz terms by default;dropping xz terms that will vanish in the rotating frame, H JJ = + 4 e C (cid:48)(cid:48) CC (cid:48) M ∂ φ,A M ∂ φ,B σ yA σ yB (22) − (cid:0) cos πf − sin πf (cid:1) E (cid:48)(cid:48) J × (cid:2) M φ,A M φ,B σ xA σ xB +2 (cid:0) M φ,A − M φ,A (cid:1) × (cid:0) M φ,B − M φ ,B (cid:1) σ zA σ zB (cid:3) . Upon transitioning to the rotating frame, both xx and yy become ± couplings, so by tuning f , E (cid:48)(cid:48) J and C (cid:48)(cid:48) ,we can cause the ± components to exactly cancel eachother, leaving a pure zz coupling. For appropriate E (cid:48)(cid:48) J and C (cid:48)(cid:48) , the bias field can be set to zero. The low-energymany-body Hamiltonian (8) will be the same whether thecircuit is comprised of flux qubits, transmons, or a mix ofthe two, though the magnitudes of J and Ω will of coursevary from one implementation to the next. C. An Alternative Formulation: Near-ResonantlyDriven Transitions Between Excited States
An alternate, and potentially more attractive, formu-lation of this architecture in physical qubits is to ex-ploit the nonlinearities of the physical device spectra todrive a transition between different excited states ratherthan between | (cid:105) and | (cid:105) . Depending on the structure ofthese nonlinearities, such a drive signal may not be use-ful or possible, but for systems whose nonlinearities arevery large and positive (flux qubits) or small and neg-ative (transmons), driving the system at the | (cid:105) ↔ | (cid:105) or | (cid:105) ↔ | (cid:105) transition can have a number of practicaladvantages over the | (cid:105) ↔ | (cid:105) formalism described in theprevious section. In this subsection, I will demonstratehow artificial gauge fields can be generated in a systemdriven at ω instead of ω , and work out the detailsof implementing this architecture in transmons or fluxqubits.Consider a superconducting qubit device whose low-lying states have energies E = 0 , E = ω and E =2 ω + δ , and let us assume that the energies higher ex-cited states are all far detuned in energy from ω or ω + δ , so we can restrict ourselves to the basis of | (cid:105) , | (cid:105) and | (cid:105) . We now consider the familiar set of charge andphase operators Q , cos φ and sin φ , and make the crucialassumption that the wavefunction describing state | (cid:105) inthe phase basis has opposite parity in φ compared to | (cid:105) and | (cid:105) . This property holds for both transmons and fluxqubits operated at the f = 1 / | (cid:105) , | (cid:105) and | (cid:105) , we can write the charge andphase operators as: Q = iq − iq iq − iq , (23)sin φ = s s s s , cos φ = c c c c c . Here, q ij = (cid:104) i | Q | j (cid:105) and the other coefficients aredefined analogously. Let us now consider the caseof a single transmon qubit, where the nonlinear-ity δ is small and negative, driven by the voltage V sin (( ω + δ + (cid:15) ) t + ϕ s ). In the rotating frame, theHamiltonian for this qubit is: H B ( T ) = − δ − (cid:15) Ω e iϕ s e − iϕ s − δ − (cid:15) Ω e iϕ s e − iϕ s (24)We now choose parameters so that | δ + (cid:15) | (cid:29) Ω . In thislimit, the rotating frame eigenstates and energies of (24)are: cos θ = (cid:15) (cid:112) (cid:15) + Ω ; | B (cid:105) = { , , } , (25) | B (cid:105) = (cid:8) e iϕ s (cos θ − , e iϕ s sin θ, (cid:9) √ − θ | B (cid:105) = (cid:8) e iϕ s (cos θ + 1) , e iϕ s sin θ, (cid:9) √ θE = 0 , E / = − δ − (cid:15) ( − / +) (cid:113) (cid:15) + Ω . Since δ is negative, | (cid:105) is the ground state, and at reso-nance, θ = π/ A and B , where as before ϕ s = 0 for qubit A . As in the previoussection, we can evaluate the hopping matrix elementsfrom the three possible couplings Q A Q B , sin φ A sin φ B and cos φ A cos φ B to be (cid:104) A B | Q A Q B | A B (cid:105) = q sin θ √ − θ e − iϕ s , (26) (cid:104) A B | sin φ A sin φ B | A B (cid:105) = s sin θ √ − θ e − iϕ s , (cid:104) A B | cos φ A cos φ B | A B (cid:105) = − c (cid:114) − cos θ e − iϕ s . We readily see from these equations that a chain of “ ± ”and “ zz ” couplings will again produce hopping matrixelements with arbitrarily tunable complex phases. Thephysical origin for these phases is as follows. In the rotat-ing frame, the phase offset ϕ s causes the phase of physicalqubit wavefunctions to advance by ± ϕ s as they absorbor emit photons into the drive field, which can only in-duce transitions between | (cid:105) ↔ | (cid:105) and | (cid:105) ↔ | (cid:105) . Sincethe first excited state in the rotating frame is a super-position of states | (cid:105) and | (cid:105) in the rest frame, a tran-sition between it and the ground state driven by Q orsin φ will have a phase shift of ± ϕ s , since this processchanges the state just as it would be changed by a singlephoton. The cos φ operator, however, can mix states | (cid:105) and | (cid:105) in the rest frame, and thus acts as an effectivetwo-photon process in a single step, advancing the phaseby ± ϕ s . Consequently, the Q/ sin φ ( ± ) and cos φ ( zz )operators see the phase shifts differently, and the phaseaccumulated around a loop which chains together ± and zz operators can be nonzero, indicating the presence ofan artificial gauge field.This method can be implemented identically in fluxqubits, where δ is positive, and is between 6 and 24 times ω for E J = 40 E C and 0 . < α < .
85. In this case,since the energy scales are so widely separated we cansimply assume that a | (cid:105) ↔ | (cid:105) transition from the drivefield is forbidden, giving us the rotating frame Hamilto-nian: H B ( F Q ) = ω − (cid:15) Ω e iϕ s e − iϕ s ω
00 0 0 . (27)Aside from additional sign flips in the Q and sin φ oper-ators which depend on which region of the qubit is beingcoupled to, the calculation proceeds identically as it didin transmons. In both cases, the physical wiring depictedin figs. 2 and 4 can be left unchanged.Driving the qubits at this transition instead of | (cid:105) ↔| (cid:105) has a number of advantages. First, the energies ofthe rotating frame excited states are larger, avoiding theneed for strong Rabi frequencies to get high energy exci-tations. Second, the hopping phase between two qubitscan be tuned arbitrarily without changing the magnitudeeven if both qubits are on resonance; in the | (cid:105) ↔ | (cid:105) caseone of the qubits had to be significantly detuned to ob-tain these phases. Note, however, that if one wishes totune the phase between any two sites without changingthe phases between any other sites, one must still include“auxilliary” qubits to mediate the tunneling. Third andmost importantly, decays in the rest frame (such as anenergy loss process which sends | (cid:105) → | (cid:105) ) cannot sponta-neously create rotating frame excitations from the groundstate when the qubits are driven near ω , since the ro-tating and rest frame ground states are the same and donot mix with state | (cid:105) . This means that the qubit arraywill be empty of excitations unless the system is popu-lated by a second pulse near the rotating frame energy,making the population easier to control and eliminatinga significant heating source in the many-body system.We see from these calculations that both flux qubitsand transmons could be used to simulate artifcial gaugefields in exotic many-body systems. I will now describeboth the simplest and most interesting of these systems:strongly interacting bosons in a uniform magnetic field,which would realize a bosonic fractional quantum Halleffect. IV. MANY-BODY STATES AND THE LOWESTLANDAU LEVEL
By considering a 2d lattice of qubits we arrive atthe final hopping Hamiltonian (8). Previous studies[7, 8, 11, 55–63] have shown that the square lattice ver-sion of this Hamiltonian is analogous to the 2d lowestLandau level problem of strongly interacting bosons, andrealizes abelian and non-abelian fractional ground statesat the appropriate fixed densities. I expect that small ar-rays should be sufficient to observe quantum Hall physics,since the magnetic length l B = 1 / √ π Ψ (where Ψ is thegauge-invariant phase accumulated when a particle cir-culates around a plaquette) can be less than a latticespacing [73]. Connections between flux qubits beyondnearest neighbors can reproduce the exact lowest Landaulevel of the continuum [61, 63] and lead to more robustfractional quantum Hall states, but they may not be nec-essary to observe the Laughlin state at ν = 1 / ν as the ratio of particle to flux density. A wide range ofother possible quantum spin-1/2 models with 2-body in-teractions, both with complex phases and without, couldbe studied in this device architecture; I find quantumHall systems to be the most intriguing, due to the ex-istence of abelian anyons at ν = 1 / ν = 1 and 3/2[5, 63], along with other exotic states at different fillingfractions. The boson density could be controlled by usinga second external field at frequency ω (cid:48) near the rotatingframe energy E A to populate the lattice; the ω (cid:48) depen-dence of the system’s response to this field could be usedto measure the gaps of the many-body states.The incoherent particle loss rates in my array fromsingle qubit decay and dephasing effects should not be asignificant obstacle to studying strongly correlated many-body states. Using values from the previous section andfrom the superconducting qubit literature [21, 64], a typi-cal hopping parameter would be J/ (cid:126) = 1GHz. The decay rate would be roughly given by the decay/dephasing rateof the qubits, which for flux qubits is of order 1MHz.With a Landau band spacing of ω LLL (cid:39) J in a squarelattice at Ψ = 1 / A qubit), random fluctuations in the mag-nitude of J , and random fluctuations in the phase of J between neighboring sites. In a real system, these noisesources would be correlated, but as the details of thosecorrelations would depend in part on the physical imple-mentation of the qubits, I have assumed that each typeof quenched disorder is applied randomly to every sitewith no dependence on the other types or on the disor-der at nearby sites. To determine the broadening fromeach noise source, I numerically diagonalized the single-particle hopping matrix on 8 × ×
12 lattices withperiodic boundary conditions, given by the Hamiltonian: H LLL = (cid:88) ij ( F ij ) J ij (cid:16) e i ( φ ij + πδφ ij ) + H.C. (cid:17) (28)+ (cid:88) i J NN δU i n i . Here, the hopping matrix elements are restricted to near-est and next nearest neighbors with relative magnitudeschosen as in [61], δU i and δφ ij are dimensionless param-eters which are Gaussian distributed about 0, J NN is theaverage nearest neighbor hopping energy, and F ij is adimensionless parameter which is Gaussian distributedabout 1. I diagonalized (28) for 25 random distributionsof noise for each data point (in steps of 0.02 for each σ ), and from the spectrum I extracted the lowest Lan-dau level broadening ∆, which is the ratio of the energysplitting between the lowest and highest LLL states tothe splitting between the highest LLL state and the bot-tom of the first excited band. I then fit ∆ (cid:0) σ U/J/ Ψ (cid:1) asa function of the standard deviation of each noise sourcewith the other two sources set to zero; this relationship0 Flux Density ∆ C U C J C Ψ + C U σ U + C J σ J + C Ψ σ Ψ , wherethe σ ’s are the standard deviation of each noise source (localpotential, hopping magnitude, and hopping phase) which isapplied randomly to every site ( σ U ) and link between sites( σ J and σ Ψ ). As seen in the Hamiltonian (28), the potentialfluctuations are in units of J NN and the phase fluctuationsare in units of π . Above the flux density Ψ = 1 /
3, trunca-tion to nearest and next nearest neighbor hopping introducessignificant broadening even in the clean system, so flux den-sities of 1/3 or less should be the focus of experiments on ourdesign. was linear in each case for small fluctuations. The re-sults of our simulations are shown in table I; note that∆ is nonzero even without defects, as a consequence oftruncating the Hamiltonian in [61] to nearest and nextnearest neighbor hopping.It is important to note that this calculation only cap-tures distortions to the single particle spectrum and thatthe many-body response to noise of this type is a subtleproblem beyond the scope of this work. However, oneshould qualitatively expect that the topological statesshould be disrupted when the normalized Landau levelsplitting ∆ approaches the dimensionless quasiparticleexcitation gap ∆ qp /J NN . In numerical studies of thissystem in the clean limit with hard-core 2-body repul-sion (largely unpublished), ∆ qp /J NN typically rangedbetween 0 . V. A SIMPLE EXPERIMENT TODEMONSTRATE THE GAUGE FIELD
While the ultimate purpose of this proposal is to studyexotic many-body states in an array of hundreds or thou-sands of flux qubits, the existence of a nontrivial gaugefield can be demonstrated by studying an arrangement offour flux qubits, connected in a loop. Consider a squareloop of four flux qubits labeled (1-4), where qubit 1 sitsat the top left corner and qubit 4 at the bottom right,as shown in Fig. 5. For this choice, any hop through a D z coupling will accumulate a phase ψ , giving a total ofΨ = 2 ψ for a complete circuit of the loop. Conversely,if the phases of the voltages applied to the B qubits areshifted by π from one qubit pair to the next, the mag-nitude of the hopping matrix element will be unchanged but there will be no complex phase accumulation. In thiscase, the B qubits have identical rotating frame energiesto the A qubits, and differ from them through the rela-tive phases ϕ si of the applied voltages. We will assumefor simplicity that the magnitudes of the hopping matrixelements from the D z and D ± couplings are both equalto J .To demonstrate that the alternating voltages generatea nonzero effective flux through the four-qubit loop, wefirst initialize the array by letting all four qubits relax totheir ground states. At time t = 0, we apply a microwavepulse to qubit 1 to excite it into the rotating frame excitedstate | A (cid:105) , and then at time t we measure the state ofqubit 4. The probability of qubit 4 being excited is givenby P ( t ) = (cid:12)(cid:12)(cid:12) (cid:104) | e iHt/ (cid:126) | (cid:105) (cid:12)(cid:12)(cid:12) (29)= 14 (cid:18) cos (cid:18) tJ (cid:126) cos Ψ4 (cid:19) − cos (cid:18) tJ (cid:126) sin Ψ4 (cid:19)(cid:19) . This interference pattern is particularly striking when Ψis nearly equal to π . If we let Ψ = π + (cid:15) , the probabilitydistribution becomes P ( t ) = (cid:32) sin √ Jt (cid:126) (cid:33) (cid:18) sin Jt(cid:15) √ (cid:126) (cid:19) . (30)In the limit of (cid:15) →
0, the probability of qubit 4 beingoccupied becomes zero at all times, due to the perfectinterference of the two paths. This is a dramatic effect,and while field fluctuations and fabrication defects wouldprevent perfect interference in a real device, the strongslowing of the occupation periodicity of qubit 4 as Ψ ap-proaches π would be readily observable. Such interfer-ence is only possible if there is a gauge-invariant phasedifference between the two paths, and would thereforedemonstrate that nontrivial effective gauge fields are re-alized in my architecture.An alternative experiment would be to connect a mi-crowave source to qubit 1 and a microwave drain at qubit2, and measure the transmission coefficient as a functionof Ψ for photons near the rotating frame excitation en-ergy E A . At Ψ = 0 the transmission coefficient shouldbe maximal, and at Ψ = π it should be zero (or nearlyzero when defects are taken into account), owing to thedestructive interference of the two paths which is the keysignature of a gauge field. I note also that a similar ar-rangement of three qubits with two ± links and one zz link could potentially engineer a charge noise free variantof the circulator design in [39]; as a microwave circulatorrequires time reversal symmetry breaking to function, itcould also demonstrate the existence of a nontrivial gaugefield.1 VI. CONCLUSION
I have demonstrated a method for realizing a quan-tum Hall state of bosons using asymmetric qubit pairs,driven by applied oscillating electric fields. I also demon-strated that my model could be implemented in latticesof flux or transmon qubits. With appropriate protocolsfor stabilizing the average particle density and measuringthe conductivity, I expect that conductivity quantizationcould be observed on small arrays, though I note that thedetails of how to measure the conductivity are beyondthe scope of this article. The statistics of anyonic collec-tive modes could be determined through similar methods[65–68].Further, the dynamical tunability of my model couldbe exploited to realize exotic combinations of states thatwould be difficult or impossible to study in cold atomor solid state systems. One could locally adjust the ap-plied voltage V sin ( ωt + ϕ s ) and flux bias f to changethe gauge field density and effective chemical potential ina given region, creating islands of arbitrary shape whichcould be at a different filling fraction than the surround-ing lattice and thus have different anyonic modes. Alter-nately, by reversing the signs of all the phase shifts ϕ s in a region, one can crate a sharp boundary between re-gions with effective gauge fields of equal magnitude butopposite sign. In both cases, we expect physics along the boundaries to be rich.Finally, by locally tuning V , ϕ s and f to manipulatevortices in the qubit lattice, arrays of ordinary qubitscould be used to construct a topological non-abeliananyon qubit [4, 5, 69], trading information density fortopological protection against decoherence. Though fardown the road, in that sense my proposal could be similarto the surface code and cluster state [70, 71] ideas devel-oped in recent years, and could provide a new potentialmechanism for reducing decoherence in superconductingquantum information devices. VII. ACKNOWLEDGMENTS
I would like to thank John Chalker, Greg Fuchs, ChrisHenley, Matteo Mariantoni, John Martinis and DavidPappas for useful discussions related to this project. I amindebted to Paul Ginsparg for his critical comments andadvice, and to Steve Simon for his assistance in the finalpreparation of this manuscript. Most of all, I would liketo thank Erich Mueller for his guidance at many stages ofthis project. This material is based on work supportedin part by the National Science Foundation GraduateProgram, EPSRC Grant No. EP/I032487/1, and OxfordUniversity. [1] R. B. Laughlin, Phys. Rev. Lett. , 1395 (1983).[2] R. E. Prange and S. M. Girvin, The Quantum Hall Effect (Springer, 1986).[3] D. Yoshoika,