Loops and Strings in a Superconducting Lattice Gauge Simulator

week ending PRL 117, 240504 (2016) PHYSICAL REVIEW LETTERS 9 DECEMBER 2016

G. K. Brennen,1 G. Pupillo,2 E. Rico,3,4 T. M. Stace,5 and D. Vodola2 1Centre for Engineered Quantum Systems, Department of Physics and Astronomy, Macquarie University, Sydney, NSW 2109, Australia 2icFRC, IPCMS (UMR 7504) and ISIS (UMR 7006), Universite de Strasbourg and CNRS,67000 Strasbourg, France 3Department of Physical Chemistry, University of the Basque Country UPV/EHU, Apartado 644, E-48080 Bilbao, Spain 4IKERBASQUE, Basque Foundation for Science, Maria Diaz de Haro 3, E-48013 Bilbao, Spain 5Center for Engineered Quantum Systems, School of Mathematics and Physics, The University of Queensland, St Lucia, Queensland 4072, Australia (Received 12 January 2016; published 7 December 2016) We propose an architecture for an analog quantum simulator of electromagnetism in 2 þ 1 dimensions, based on an array of superconducting fluxonium devices. The encoding is in the integer (spin-1) representation of the quantum link model formulation of compact Uð1Þ lattice gauge theory. We show how to engineer Gauss’ law via an ancilla mediated gadget construction, and how to tune between the strongly coupled and intermediately coupled regimes. The witnesses to the existence of the predicted confining phase of the model are provided by nonlocal order parameters from Wilson loops and disorder parameters from ’t Hooft strings. We show how to construct such operators in this model and how to measure them nondestructively via dispersive coupling of the fluxonium islands to a microwave cavity mode. Numerical evidence is found for the existence of the confined phase in the ground state of the simulation Hamiltonian on a ladder geometry.

DOI: 10.1103/PhysRevLett.117.240504

Gauge theories play a fundamental role in modern to replicate the equilibrium and dynamical properties of a physics, including quantum electrodynamics and quantum system of interest. Indeed, this is one of the motivations for chromodynamics. The discretized version of gauge theory, quantum technologies based on atomic [24–40] and super- lattice gauge theory (LGT), is key to understanding physics conducting platforms [41–44]. A way to measure space-time ranging from quantum spin liquids to quark-gluon plasmas Wilson loops in atomic lattice gauge simulators (assuming [1–3]. A fundamental phenomenon in gauge theories is the localized excitations) was given in Ref. [32] but a critical notion of confinement, which manifests in the absence of outstanding problem has been the reliable measurement of isolated, color-charged particles in nature; i.e., the only nonlocal, space-like Wilson loops and ’t Hooft strings. “physical” states are those that transform “trivially” under a Here, we propose an analog simulator of a pure compact gauge transformation. Yet, quantum phases of gauge field Uð1Þ QLM in 2 þ 1 dimensions [45], based on super- theories cannot be characterized by local order parameters. conducting fluxonium [46] devices placed on a square Instead, nonlocal order parameters such as Wilson loops [1] lattice. The devices operate in a finite-dimensional mani- and ’t Hooft strings [4] have been introduced to indicate the fold of low-lying eigenstates, to represent “discrete” presence or absence of a confined phase. electric fluxes on the lattice. By engineering local couplings Quantum link models (QLMs) provide a formulation of between devices, we show how to replicate the local LGTs, in which finite-dimensional subsystems associated interactions and constraints of the QLM. The couplings with edges of the lattice encode the gauge field [5–7]. can be tuned to access different phases of the quantum Related Uð1Þ gauge models are important for understand- phase diagram of the model. We demonstrate how to ing various condensed matter systems, including quantum measure nonlocal, space-like Wilson loops and ’t Hooft spin ice models or quantum dimer models, which may strings in the proposed architecture. Moreover, we report exhibit deconfined critical points at T ¼ 0 [8]. In principle, density-matrix renormalization group (DMRG) calcula- QLMs break Lorentz invariance while relativistic Uð1Þ tions of a ’t Hooft disorder parameter in a QLM, and gauge theories in 2 þ 1 dimensions are always in a show that the QLM indeed captures confinement physics. confinement phase at T ¼ 0 but may undergo a phase Quantum link model.—In the pure gauge Uð1Þ QLM, transition at T > 0 to a deconfined phase [9]. In either ˆ c electric fluxes Eα;β are defined on the links hα; βi of a square case, confinement physics is a key to understanding the 1 latticewith local link state space CNþ [circles in Fig. 1(a)]. In phenomenology. the electric basis, the Hilbert space is labeled by the electric Numerical simulation of LGTs can be computationally ˆ costly due to the size of the Hilbert space or the sign problem fluxes on the links, Eα;βjEα;βi¼Eα;βjEα;βi.Foracompact 1 with quantum Monte Carlo techniques [10] (for recent Uð Þ gauge group, fluxes take integer or half integer values, − 2 ≤ ≤ 2 ∈ Zþ proposals using tensor networks see Refs. [11–23]). An ðN= Þ Eα;β ðN= Þ, N . The local link electric- ˆ alternative approach is to build analog quantum simulators displacement operator Uα;β satisfies the commutation

smallest Wilson loop operator is defined on a plaquette, ˆ ˆ ˆ ˆ W ¼ Uα;βUβ;δUδ;γUγ;α. This is a discrete approximation to ei⨖A·dl where A is the magnetic vector potential. Over a longer closed path C, a Wilson loop operator WC is the ˆ path-ordered multiplication of Uα;β along links in C. In the −areaðCÞ confined phase, WC satisfies an area law hWCi ∼ e . −perimeterðCÞ In the deconfined phase, it satisfies hWCi ∼ e . ’ A t Hooft string operator is definedQ as a directed product ϒˆ φ φ ˆ of electric link operators ð Þ¼ n exp ði Enax;naxþay Þ; in Ref. [47] (Sec. IB) we show that in the QLM it acts a disorder parameter. This operator changes the value of the magnetic flux by an amount φ on the plaquettes where it starts and ends, introducing a pair of magnetic vortices. In ˆ the confining phase it is ordered, i.e., hϒðφÞi ≠ 0 for φ ≠ 0 2 1 FIG. 1. Uð1Þ quantum link model engineered in a fluxonium in þ dimensions. The fact that a nonzero expectation ˆ ˆ value of the disorder parameter characterizes a confinement array. (a) “Electric” Eα;β, and “magnetic” Uα;β, degrees of freedom are associated with links hα; βi of a square lattice. The link degrees phase in a gauge model may simplify the signal-to-noise of freedom (red circles) are encoded in eigenstates of the fluxonia. problem in an actual quantum simulation. ϒˆ ˆ The ancillae (blue diamonds) on vertices are inductively coupled In Fig. 2(a) we show the disorder parameter for HQLM to neighboring link islands to mediate the Gauss constraint and on the quasi-2D ladder lattice, shown in Fig. 1(c), calcu- plaquette interactions are obtained via link nearest neighbor lated using DMRG calculation. The ladder is the minimal capacitive coupling. (b) Superconducting circuit elements used lattice exemplifying a 2 þ 1 dimensional system. Clearly, to build and couple components of the simulation. The link devices ϒˆ 2 2 ≫ 1 ϕˆ is nonzero in the strong coupling regime g gmag , have local phase link and capacitive, inductive, and flux-biased elec Josephson energies EC, EL, and EJ, respectively, and similarly for indicative of a confining phase. Thus, even in this limited the ancilla devices. The capacitive and inductive coupling energies geometry, the QLM captures confinement physics. In what c c “ ” are EC and EL. (c) A minimal quasi-1D ladder implementation follows, we propose an analog QLM simulator to study embedded in a microwave cavity (black box), in which a ’t Hooft ground state and dynamical phenomena on computation- string of link fluxonia (green circles) can be measured via an ally challenging 2D lattices. ancilla coupled to the cavity (green triangle).

ˆ ˆ ˆ relation ½Eα;β; Uα;β¼−Uα;β [for a detailed description see the Supplemental Material [47] (Sec. I)]. In the charge-free sector, the net electric flux at a given vertex is zero; hence, ˆ ˆ ˆ ˆ there is a conserved quantity Gα ¼ Eμ;α þ Eν;α − Eα;β− ˆ Eα;γ. The phase of the operators can be changed locally ˆ with the Uð1Þ gauge transformation eiθαGα and the dynamics remain invariant. The gauge invariant subspace satisfies ˆ Gαjphysi¼0, which is the discretized Gauss law ∇~ ~ 0 · Ejphys ¼ . In a pure gauge model, there are two com- ˆ peting terms in the Hamiltonian: the electric term penalizes FIG. 2. Expectation value of the ’t Hooft string ϒðφÞ electric flux on each link hα; βi and the magnetic term which inserts a flux φ at the plaquette in the middle of a ladder penalizes magnetic flux on each plaquette □, [see Fig. 1(c)]. Left panel: value in the ground state of the pure ˆ gauge model HQLM as a function of the electric coupling 2 X 1 X gelec with a perturbative value of the magnetic coupling ˆ 2 ˆ 2 − ˆ ˆ ˆ ˆ 1 2 ∼ 2 2 → 2 75 HQLM ¼ gelec Eα;β 2 ðUα;βUβ;δUδ;γUγ;α þH:c:Þ; ð =gmagÞ ð J =UÞ ð = Þ. Right panel: value in the ground α β gmag □ ˆ h ; i state of the two-body Hamiltonian Himp as a function of the on- 1 75 ð1Þ site energy V, with J ¼ and U ¼ . Numerics were performed for a size L ¼ 29 rung ladder with 85 spins using DMRG calculation with 300 states and a truncation error estimated at < 10−12. 2 2 ˆ where gelec and gmag are the coupling constants for the electric From the plots, hϒi ≥ 0.5 is indicative of a confining phase. The term and magnetic term, respectively. equivalence between the implemented model (3) and the gauge We characterize the confinement of electric charges, invariant model (4) in the strongly coupled and intermediately which locally violate Gauss’ law, using Wilson loops. The coupled regimes is evident.

240504-2 week ending PRL 117, 240504 (2016) PHYSICAL REVIEW LETTERS 9 DECEMBER 2016

Implementation with superconducting devices.—To sim- The Gauss law constraint is enforced by ancillary fluxo- 1 a a a ulate a Uð Þ QLM, there are three elements: (i) the local nium devices at lattice vertices, with parameters EJ;EC;EL ϕa π Hilbert space, labeled by the electric flux on the lattice and off ¼ . The lowest energy states jgi and jei are shown links; here, the Hilbert space is spanned by a discrete set of in Fig. 3(b). Each ancillaP is inductivelyP coupled to its states of a fluxonium device, (ii) Gauss’ law on the lattice ˆ c 2 4 ϕˆ − ϕˆ 2 neighbors, Hind ¼ðEL= Þ v k¼1ð k aÞ , where vertices; here, this is imposed by strong interactions c ℏ 2 2 1 EL ¼ð = eÞ ð =LcÞ, and Lc is the coupling inductance. between devices, mediated by tunable inductive couplings, Ancillae are initialized in the long-lived excited state jei, for (iii) the gauge invariant dynamics; here, this emerges at which T1 > 1 ms [53]. second order of perturbation with capacitive couplings At first order in Ec =Δ, the interaction Hˆ is zero since between neighboring devices. L ind jei is antisymmetric. At second order we obtain an effec- We propose a lattice of fluxonium devices [51], which are tive Hamiltonian acting on the links around each vertex. inductively shunted superconducting Josephson junctions We only include the terms diagonal in the jmki basis since with demonstrated relaxation times on the order of 1 ms 2 2 ∼ −π =σ [52,53], located on the edges and vertices of the square the off diagonal terms are much smaller by a factor e . lattice, as shown in Figs. 1(a) and 1(b). The Hamiltonian for The total Hamiltonian, local plusP inductive interactionP for a ˆ ˆ z2 device k is spin-P SPrepresentation is then kHPk þ Hind ¼ V kSk þ ˆ z 2 ˆ z S U ð ∈N S Þ , where S ¼ − mjmihmj, U ¼ ˆ 4 ˆ 2 ˆ ϕˆ v k ðvÞ k m¼ S Hk ¼ ECnk þ Pð kÞ; ð2Þ c 2 ϕˆ 2 ϕˆ 2 Δ 0 N EL jhgj ajeij jh im¼1j = > , ðvÞ is the neighborhood Δ − 0 ˆ ϕˆ − ϕˆ ϕ ϕˆ 2 2 of a vertex v (see Ref. [47], Sec. IIB), ¼ Ee Eg > is where Pð kÞ¼ EJ cosð k þ offÞþEL k= is the local 2 the ancilla qubit energy splitting including local contri- potential, EC ¼ e =ð2CÞ is the charging energy of the island with total capacitance C, E ℏ=2e 2 1=L is the butions from the inductive coupling, calculated using J ¼ð Þ ð JÞ a → a 4 c Eq. (2) with the replacement EL EL þ EL, and V Josephson energy with LJ the effective inductance of the 2 Josephson junction, E ℏ=2e 2 1=L is the shunt induc- (which generates the QLM electric coupling gelec) is the L ¼ð Þ ð Þ − tive energy, and E ≥ E >E . qutrit energy splitting E1 E0, computed using Eq. (2) J C L → 2 c ˆ with EL EL þ EL. The phase ϕk is proportional to the (physical) flux in the 2 The QLM magnetic coupling gmag is generated by a capa- device. It is not compact, so the conjugate charge nˆ k ¼ P − ∂ ∂ϕ citive coupling between link devices, Hˆ ¼8Ec nˆ nˆ , ið = kÞ takes continuous values. The offset phase pffiffiffiffiffiffiffiffiffiffiffiffiffiffi cap C hk;ji k j ϕ 2πΦ Φ Φ c −2 −1 2 −2 off ¼ ext= 0, where ext is a tunable flux [53] and where E =E ≃f 8þξ K0ðξ Þ=4ξK½−16ξ =ð8þξ Þg, Φ 2 pffiffiffiffiffiffiffiffiffiffiffiC C 0 ¼ h=ð eÞ is the flux quantum. The potential terms can ξ ≡ C =C and C is the capacitance between nearest be tuned to support integer representations of the electric c c ϕ 0 neighbors, K0ðxÞ is a modified Bessel function, and KðxÞ is flux by setting off ¼ , shown in Fig. 3(a), or half-integer ˆ ϕ π an elliptic integral (see Ref. [47], Sec. IIC). The operators n representations with off ¼ , Fig. 3(b). In the limit ∼ ≫ generate displacements in phase, so the interaction drives EJ EC EL, the lowest energy states are the first band fluctuations in the electric flux m . Longer range couplings ϕˆ 2π k Wannier functions with mean (physical) flux h ki¼ mk, decay exponentially in island separation, with a correlation σ 8 1=4 and zero-point phase fluctuations ϕ ¼ð EC=EJÞ . length ξ. The total two body Hamiltonian is (a) X X X 2 ˆ z2 ˆ z Himp ¼ V Sk þ U Sk k v ∈N X k ðvÞ ˆ þ ˆ − ˆ − ˆ þ þ J ðSj Sk þ Sj Sk Þð3Þ (b) hj;ki −8 c 1 ˆ 0 2 ≫ with J ¼ ECjh jnj ij . In the limit U jJj, the second term projects the ground states into the gauge invariant subspace and the effective Hamiltonian is X ˆ 2 2 ˆ z2 Heff ¼ðV þ J =UÞ Sj FIG. 3. (a) Link fluxonium devices operate as qutrits with X j ϕ 0 ϕ off ¼ . The potential Pð linkÞ (red) and the eigenfunctions are − 2 2 ˆ þ ˆ − ˆ þ ˆ − 0 1 ð J =UÞ ðS S S S þ H:c:Þ plotted for qutrit states j i; j i, which represent electric flux, □ and the next energy state jsi. A cavity field couples states j0i ↔ X 2 4 ˆ z ˆ z 1 − ˆ z ˆ z jsi with Rabi frequency Ω and detuning δ to tune the electric þðJ = UÞ SjSkð SjSkÞ: ð4Þ coupling in the QLM. (b) Ancilla fluxonium potential PðϕaÞ hj;ki (blue) with ϕoff ¼ π, and eigenfunctions jgi and jei. Ancillae are initialized in state jei and generate the Gauss constraint on link This is the first key element of our proposal: defining 2 2 2 1 2 2 2 devices through an effective interaction. gelec ¼ V þ J =U and =gmag ¼ J =U, the first two

240504-3 week ending PRL 117, 240504 (2016) PHYSICAL REVIEW LETTERS 9 DECEMBER 2016 ˆ δ 0 terms of Heff are equivalent to the gauge Hamiltonian can be reduced by choosing < . There are energy shifts Hˆ , Eq. (1), once we identify Eˆ ↦ Sˆ z and Uˆ ↦ Sˆ −; from off-resonant coupling to multiple excited states by all QLM ω notice the latter is nonunitary unlike the continuum case. three qutrit states. Optimizing F and g to minimize V ω 1 588 2 0 2 2 2 ≃ 1 The third term respects all the symmetries of the gauge gives F ¼ . EJ and jgj ¼ . , so that gelecgmag . invariant model, and renormalizes the electric field and U. Decoherence.—Spin decoherence limits the ultimate Comparison of DMRG calculations of Hˆ [Fig. 2(a)] size of the simulator. WeQ envision starting in the gapped QLM 0 0 and Hˆ [Fig. 2(b)] on a ladder geometry shows that Hˆ product state jGð Þi ¼ linksj i by tuning parameters to eff eff the extremely strongly coupled regime (with Ω ¼ 0), and replicates the confinement physics (ϒˆ ≠ 0) of the original adiabatically evolving the ground state to an intermediate QLM. We note that higher order corrections to the gauge coupling regime. The adiabatic evolution could be done by invariant Hamiltonian in Eq. (4) may lead to an effective slowly increasing the driving field Rabi frequency over a coupling to matter field. In this case, the behavior of the time T and, as described below, nonlocal order param- Wilson loops and ’t Hooft strings in 2 1 D is uncertain, sim ð þ Þ eters can be measured as a function of final coupling and left open to further study [54]. strength [see Fig. 2(b)]. As shown in Ref. [47] (Sec. IIIB), We now discuss different limits of the model based on the ground state jGðtÞi is gapped throughout with energy the three level (spin-1) Uð1Þ QLM shown in Fig. 3(a). The ΔE t ∼ 4g2 t Hˆ interaction energies are determined by diagonalizing the gapð Þ elecð Þ, and from the effective model eff is 0 Δ min ∼ 8 2 local fluxonium Hamiltonian and computing wave- minimal at V ¼ where Egap J =U. The decoherence function overlaps (see Ref. [47], Sec. IID). We show times for fluxonium tuned to the qutrit point have been ∼ ∼ 50 μ 0 032 how to prepare the strongly coupled limit, which is close reported at T1 T2 s [53]. Consider U ¼ . EJ ⊗ 0 40 to the product state links j ij, and then reduce the coupling as above and choose EJ ¼ GHz and Tsim ¼ 2 Δ min 0 135 μ to the intermediate limit. = Egap ¼ . s. The inverted ancilla qubit lifetime — ˆ is Ta ∼ 1 ms [53], giving an error rate per ancilla of Strong coupling, Because Hind contributes both to U 1 −T =Ta −4 and V in Hˆ with comparable magnitudes and jJj ≪ U, 1 − e sim 1 ∼ 10 , allowing reliable simulations on a imp ∼1000 in order to satisfy the Gauss constraint, the system will lattice with link spins. — be in the strong coupling regime. To enforce this, the Nonlocal measurement. The second key element of our a proposal is the measurement of spin-1 Wilson loop oper- Josephson energy on the ancilla islands EJ is the biggest ˆ þ ˆ − ˆ þ ˆ − energy scale of the model, which determines the cascade of ators WC ¼ S ⊗ S ⊗ ⊗ S ⊗ S on C, and ’t Hooft a a a c a ˆ − φˆ z φˆ z φˆ z 0 2 0 06 0 04 ϒ φ i S0 ⊗ i S1 ⊗ ⊗ i Sn−1 energies: EJ ¼ EC ¼ . EJ, EC ¼ . EJ, EC ¼ . EJ, disorder operators ð Þ¼e e e a 0 01 a 0 003 a c 0 0002 a “ ” EL ¼ . EJ, EL ¼ . EJ, and EL ¼ . EJ. The on a line extending from a spin 0 on the boundary. link states j1i are then nearly degenerate with splitting Importantly, the measurement does not alter the observable 0 0006 a c Δ 0 023 . EJ and EL= ¼ . ensuring that the perturbation being measured, and repeated measurements give the same a 0 006 theory on the ancilla is valid. We find U=EJ ¼ . , result; i.e., it is nondemolition. The idea is to prepare the a 0 06 −0 04 V=EJ ¼ . , and J=U ¼ . , which by Eq. (1) gives ground state of the spin-1 lattice Hamiltonian, turn off 2 2 ∼ 3000 210 ˆ W ϒˆ φ gelecgmagn . Josephson energies EJ ¼ GHz [55], Himp, and then measure C or ð Þ. Thus, the measure- capacitive energies EC ¼ 14.2 GHz [56], and inductive ment can “quench” the system, in order to study the energies EL ¼ 0.52 GHz [57] have been reported, sug- ensuing dynamics and multitime correlations when the gesting the simulation coupling strengths here are within Hamiltonian is turned on again [58]. reach of experimentally demonstrated values. To measure nonlocal operators, a subset of spins in Intermediate coupling.—To reduce the electric field term the array are coupled to a single microwave cavity mode, we shift the energy of state j0i by off-resonant coupling to Fig. 1(c). Ultimately, only a single qubit degree of freedom an excited state jsi, above the qutrit subspace. Consider need be measured, which is advantageous if the measure- a driving field that couples to the fluxonium at frequency ment error is significant. By contrast, if spins were ω 0 → F, which is detuned from the j i jsi transition by measured independently the fidelity would decrease expo- δ ω − − ¼ F ðEs E0Þ. Because of the anharmonic energy nentially with operator size. ω R spacing of the fluxonium the frequency F can be chosen We require aP dispersive coupling of spins in a region , to be very far off resonant for other possible transitions. ˆ −χ ˆ † ˆ 0 0 Hint ¼ a a j∈R j ijh j, and a coupling between an Inductive coupling via a quarter wavelength transmission ancilla A (such as one of the ancilla qubits) and the bosonic line gives a time dependent fluxonium-field interac- ˆ A −χ ˆ † ˆ ˆ † ˆ field, H ¼ Aa ajeiAhej, where a and a are bosonic ˆ −iωFt int tion HFF ¼ðΩ=2Þjsih0je þ H:c:, where the Rabi fre- ˆ creation and annihilation operators. Selectively addressing quency is Ω ¼ −ghsjϕj0i and g depends on physical cavity coupling within the region R or at the ancilla properties of the transmission line and fluxonium [57]. location can be done by coherently mapping noninteracting ˆ Assuming other excited states are far detuned, Himp in the spins to noninteracting local states, which are far detuned qutrit subspace is modified by V → V þjΩj2=ð4δÞ,soV from the cavity coupling. In Ref. [47] (Sec. III) we describe

240504-4 week ending PRL 117, 240504 (2016) PHYSICAL REVIEW LETTERS 9 DECEMBER 2016 ˆ in detail two methods to measure WC or ϒðφÞ. In brief, one proposals. Moreover, we provide a physical encoding of the method uses a geometric phase gate, requiring only the states for the QLM, where local electric field terms are ability to prepare the vacuum state of the cavity and a nontrivial. The protocol is rather robust to inhomogeneities, sequence of displacement operators and evolution gener- allowing for implementations in superconducting arrays, ˆ and we have presented numerical evidence that lattice QED ated by Hint. An alternative method can be done in a single step but requires the preparation of a superposition of in “quasi-2” þ 1 dimensions exhibits confinement. Beyond vacuum and a single photon state of the cavity. ground state characterization, the simulator can be used to Fidelity.—To estimate process fidelity, we assume the probe dynamics and measure the evolution of nonlocal cavity has a decay rate κ, and system and ancilla spins order or disorder parameters. γ depolarize independently with an error rate .Onn spins, This work was partially supported by the ARC Centre the geometric phase-gate measurement process fidelity is of Excellence for Engineered Quantum Systems EQUS (Grant No. CE110001013). We also acknowledge financial FðgpÞðθ; ΩÞ pro support from Basque Government Grants IT472-10, > ηA½1 − nð4π þ 6θÞγ=jχj Spanish MINECO FIS2012-36673-C03-02, UPV/EHU −3θκ χ −θκ χ Project No. EHUA15/17, UPV/EHU UFI 11/55 and the × ½1 − πΩκðe =j j þ e =j jÞð1 þ πκ=2jχjÞ=jχj; SCALEQIT EU project. G. P. and D. V. acknowledge ð5Þ support by the ERC-St Grant ColdSIM (No. 307688), EOARD, UdS via Labex NIE and IdEX, RYSQ. where η < 1 describes the finite detection fidelity of the A pffiffiffi ancilla spin. For measuring Wilson loops Ω ¼ π= 3, θ ¼ 2π=3, while for measuring ’t Hooft strings Ω ¼ π=2, θ ¼ π=2. For the single photon implementation, the process [1] K. G. Wilson, Confinement of quarks, Phys. Rev. D 10, ðspÞ −γt¯ðθÞ fidelity is Fpro ðθÞ > ηp½1 − nð1 − e Þ, where ηp ≤ 1 2445 (1974). describes finite single photon detection fidelity, and the [2] H. J. Rothe, Lattice Gauge Theories (World Scientific, ¯ θ 1 2θκ=jχj 2θ 2κ χ 2 Singapore, 2012). mean gate time is tð Þ¼½ð þ e Þð Þ =j j = [3] X. G. Wen, Quantum Field Theory of Many-Body Systems: 2θκ=jχj ð2θκ=jχjþe − 1Þ. In the presence of inhomogeneities From the Origin of Sound to an Origin of Light and in the dispersive coupling strength χ, the error E for the Electrons, Oxford Graduate Texts (Oxford University Press, global gates with angle θ is E ≈ θ2jRjðjRj − 1Þϵ2=2, where New York, 2004). ϵ is the fractional cavity mode field variation across the [4] G. ’t Hooft, On the phase transition towards permanent lattice (Ref. [47], Sec. IIIA). quark confinement, Nucl. Phys. B138, 1 (1978). Using transmons coupled to a 3D microwave cavity [60] [5] D. Horn, Finite matrix models with continuous local gauge the following values were reported for one island: γ ¼ invariance, Phys. Lett. 100B, 149 (1981). [6] P. Orland and D. Rohrlich, Lattice gauge magnets: Local 66.7 kHz, jχj=2π ¼ 99.8 MHz, κ ¼ 22.2 kHz. Single-shot isospin from spin, Nucl. Phys. B338, 647 (1990). transmon qubit measurements have also been reported with [7] S. Chandrasekharan and U-J. Wiese, Quantum link models: η 0 919 A ¼ . [61]. With efficient single microwave photon A discrete approach to gauge theories, Nucl. Phys. B492, detectors, the single photon protocol allows for a meas- 455 (1997). urement of a Wilson loop on n − 1 spins with fidelity [8] D. S. Rokhsar and S. A. Kivelson, Superconductivity and ðspÞ −3 the quantum hard-core dimer gas, Phys. Rev. Lett. 61, 2376 Fpro > ηpð1 − 2.5 × 10 nÞ. Microwave photon number resolution can be achieved with η ≃ 0.90 [62,63]. (1988). p [9] A. M. Polyakov, Gauge Fields and Strings (Harwood Assuming similar parameters for fluxonium and local Academic Publishers, London, 1987). addressability, using either the geometric phase gate or [10] Lattice QCD for Nuclear Physics, edited by H.-W. Lin and the single photon gate, a Wilson loop of length 8 or a H. B. Meyer (Springer, Heidelberg, 2015). ’t Hooft string of size 10 could be measured with ∼90% [11] T. Byrnes, P. Sriganesh, R. J. Bursill, and C. J. Hamer, fidelity. By the nondestructive nature of the measurement, Density matrix renormalisation group approach to the the imperfect detection efficiency can be improved by massive Schwinger model, Phys. Rev. D 66, 013002 (2002). repeating the measurement until the presence or absence of [12] M. C. Banuls, K. Cichy, K. Jansen, and J. I. Cirac, The mass a photon is known with high confidence, enabling meas- spectrum of the Schwinger model with matrix product urement of much larger loops. states, J. High Energy Phys. 11 (2013) 158. [13] M. C. Banuls, K. Cichy, J. I. Cirac, K. Jansen, and H. In summary, we provide a proposal for an analog 2 þ 1D Saito, Matrix product states for lattice field theories, PoS, QLM simulator using a 2D array of superconducting LATTICE 2013, 332 (2013). devices. The simulator can be tuned between intermediate [14] E. Rico, T. Pichler, M. Dalmonte, P. Zoller, and S. and strong coupling regimes, and allows nondestructive Montangero, Tensor Networks for Lattice Gauge Theories measurement of nonlocal, spacelike QLM order and dis- and Atomic Quantum Simulation, Phys. Rev. Lett. 112, order parameters, resolving an outstanding gap in other 201601 (2014).

240504-5 week ending PRL 117, 240504 (2016) PHYSICAL REVIEW LETTERS 9 DECEMBER 2016

[15] B. Buyens, J. Haegeman, K. Van Acoleyen, H. Verschelde, [33] P. Hauke, D. Marcos, M. Dalmonte, and P. Zoller, Quantum and F. Verstraete, Matrix product states for gauge field simulation of a lattice Schwinger model in a chain of theories, Phys. Rev. Lett. 113, 091601 (2014). trapped ions, Phys. Rev. X 3, 041018 (2013). [16] P. Silvi, E. Rico, T. Calarco, and S. Montangero, Lattice [34] E. Zohar, J. I. Cirac, and B. Reznik, Quantum simulations of gauge tensor networks, New J. Phys. 16, 103015 (2014). gauge theories with ultra-cold atoms: Local gauge invari- [17] L. Tagliacozzo, A. Celi, and M. Lewenstein, Tensor net- ance from angular-momentum conservation, Phys. Rev. A works for lattice gauge theories with continuous groups, 88, 023617 (2013). Phys. Rev. X 4, 041024 (2014). [35] K. Stannigel, P. Hauke, D. Marcos, M. Hafezi, S. Diehl, M. [18] T. Pichler, M. Dalmonte, E. Rico, P. Zoller, and S. Dalmonte, and P. Zoller, Constrained dynamics via the Zeno Montangero, Real-time dynamics in U(1) lattice gauge effect in quantum simulation: Implementing non-Abelian theories with tensor networks, Phys. Rev. X 6, 011023 lattice gauge theories with cold atoms, Phys. Rev. Lett. 112, (2016). 120406 (2014). [19] B. Buyens, J. Haegeman, H. Verschelde, and F. Verstraete, [36] S. Notarnicola, E. Ercolessi, P. Facchi, G. Marmo, S. and K. Van Acoleyen, Confinement and string breaking for Pascazio, and F. V. Pepe, Discrete Abelian gauge theories QED2 in the Hamiltonian picture, arXiv:1509.00246. for quantum simulations of QED, J. Phys. A: Math. Theor. [20] J. Haegeman, K. Van Acoleyen, N. Schuch, J. I. Cirac, and 48, 30FT01 (2015). F. Verstraete, Gauging quantum states: From global to local [37] A. Bazavov, Y. Meurice, S.-W. Tsai, J. Unmuth-Yockey, and symmetries in many-body systems, Phys. Rev. X 5, 011024 J. Zhang, Gauge-invariant implementation of the Abelian (2015). Higgs model on optical lattices, Phys. Rev. D 92, 076003 [21] S. Kuhn, E. Zohar, J. I. Cirac, and M. C. Banuls, Non- (2015). Abelian string breaking phenomena with matrix product [38] A. Bermudez and D. Porras, Interaction-dependent photon- states, J. High Energy Phys. 07 (2015) 130. assisted tunneling in optical lattices: A quantum simulator [22] E. Zohar, M. Burrello, T. B. Wahl, and J. I. Cirac, Fermionic of strongly correlated electrons and dynamical gauge fields, projected entangled pair states and local U(1) gauge New J. Phys. 17, 103021 (2015). theories, Ann. Phys. (Amsterdam) 363, 385 (2015). [39] U.-J. Wiese, Towards quantum simulating QCD, Nucl. [23] E. Zohar and M. Burrello, Building projected entangled Phys. A931, 246 (2014). pair states with a local gauge symmetry, New J. Phys. 18, [40] E. Zohar, J. I. Cirac, and B. Reznik, Quantum simulations 043008 (2016). of lattice gauge theories using ultra-cold atoms in optical [24] H. Weimer, M. Muller, I. Lesanovsky, P. Zoller, and H. P. lattices, Rep. Prog. Phys. 79, 014401 (2016). Büchler, A Rydberg quantum simulator, Nat. Phys. 6, 382 [41] D. Marcos, P. Rabl, E. Rico, and P. Zoller, Superconducting (2010). circuits for quantum simulation of dynamical gauge fields, [25] L. Tagliacozzo, A. Celi, A. Zamora, and M. Lewenstein, Phys. Rev. Lett. 111, 110504 (2013). Optical Abelian lattice gauge theories, Ann. Phys. [42] D. Marcos, P. Widmer, E. Rico, M. Hafezi, P. Rabl, U.-J. (Amsterdam) 330, 160 (2013). Wiese, and P. Zoller, Two-dimensional lattice gauge theories [26] A. W. Glaetzle, M. Dalmonte, R. Nath, I. Rousochatzakis, with superconducting quantum circuits, Ann. Phys. R. Moessner, and P. Zoller, Quantum spin-ice and dimer (Amsterdam) 351, 634 (2014). models with Rydberg atoms, Phys. Rev. X 4, 041037 [43] L. Garcia-Alvarez, J. Casanova, A. Mezzacapo, I. L. (2014). Egusquiza, L. Lamata, G. Romero, and E. Solano, [27] E. Kapit and E. Mueller, Optical-lattice Hamiltonians for Fermion-fermion scattering in quantum field theory with relativistic quantum electrodynamics, Phys. Rev. A 83, superconducting circuits, Phys. Rev. Lett. 114, 070502 033625 (2011). (2015). [28] E. Zohar and B. Reznik, Confinement and lattice quantum- [44] A. Mezzacapo, E. Rico, C. Sabin, I. L. Egusquiza, L. electrodynamic electric flux tubes simulated with ultra-cold Lamata, and E. Solano, Non-Abelian lattice gauge theories atoms, Phys. Rev. Lett. 107, 275301 (2011). in superconducting circuits, Phys. Rev. Lett. 115, 240502 [29] J. Casanova, L. Lamata, I. L. Egusquiza, R. Gerritsma, C. F. (2015). Roos, J. J. Garcia-Ripoll, and E. Solano, Quantum simu- [45] J. Kogut and L. Susskind, Hamiltonian formulation of lation of quantum field theories in trapped ions, Phys. Rev. Wilsons lattice gauge theories, Phys. Rev. D 11, 395 (1975). Lett. 107, 260501 (2011). [46] V. E. Manucharyan, J. Koch, L. I. Glazman, and M. H. [30] E. Zohar, J. I. Cirac, and B. Reznik, Simulating compact Devoret, Fluxonium: Single cooper-pair circuit free of quantum electrodynamics with ultra-cold atoms: Probing charge offsets, Science 326, 113 (2009). confinement and non-perturbative effects, Phys. Rev. Lett. [47] See Supplemental Material at http://link.aps.org/ 109, 125302 (2012). supplemental/10.1103/PhysRevLett.117.240504 for analy- [31] D. Banerjee, M. Dalmonte, M. Muller, E. Rico, P. Stebler, sis on: large-N representation and duality transformation, U.-J. Wiese, and P. Zoller, Atomic quantum simulation of the implementation with fluxonia, measurement of non- dynamical gauge fields coupled to fermionic matter: From local order parameters, and gauge invariance and “dressed” string breaking to evolution after a quench, Phys. Rev. Lett. quantum states, which includes Ref. [48–50]. 109, 175302 (2012). [48] A. M. Rey, G. Pupillo, C. W. Clark, and C. J. Williams, [32] E. Zohar, J. I. Cirac, and B. Reznik, Simulating (2 þ 1)- Ultra-cold atoms confined in an optical lattice plus parabolic dimensional lattice QED with dynamical matter using potential: A closed-form approach, Phys. Rev. A 72, 033616 ultracold atoms, Phys. Rev. Lett. 110, 055302 (2013). (2005).

240504-6 week ending PRL 117, 240504 (2016) PHYSICAL REVIEW LETTERS 9 DECEMBER 2016

[49] L. Jiang, G. K. Brennen, A. V. Gorshkov, K. Hammerer, quantum state of an electrical circuit, Science 296, 886 M. Hafezi, E. Demler, M. D. Lukin, and P. Zoller, Anyonic (2002). interferometry and protected memories in atomic spin [57] N. A. Masluk, Ph.D. thesis, Yale University, 2012. lattices, Nat. Phys. 4, 482 (2008). [58] It was proven in Ref. [59] that nondemolition measurements [50] G. K. Brennen, K. Hammerer, L. Jiang, M. D. Lukin, and P. of spacelike non-Abelian Wilson loops are not physical Zoller, Global operations for protected quantum memories because they would allow for faster than light signaling. in atomic spin lattices, arXiv:0901.3920. However, Abelian Wilson loops such as considered here [51] S. M. Girvin, Circuit QED: superconducting qubits coupled have no such obstruction provided the spins on the loop to microwave photons, in Quantum Machines: Measure- have access to a shared entanglement resource, which here is ment and Control of Engineered Quantum Systems, Lecture the common cavity mode. Notes of the Les Houches Summer School Vol. 96, edited by [59] D. Beckman, D. Gottesman, A. Kitaev, and J. Preskill, M. Devoret, B. Huard, R. Schoelkopf, and L. F. Cugliandolo Measurability of Wilson loop operators, Phys. Rev. D 65, (Oxford University Press, Oxford, 2011). 065022 (2002). [52] U. Vool, I. M. Pop, K. Sliwa, B. Abdo, C. Wang, T. Brecht, [60] H. Paik, D. I. Schuster, L. S. Bishop, G. Kirchmair, G. Y. Y. Gao, S. Shankar, M. Hatridge, G. Catelani, M. Catelani, A. P. Sears, B. R. Johnson, M. J. Reagor, Mirrahimi, L. Frunzio, R. J. Schoelkopf, L. I. Glazman, L. Frunzio, L. I. Glazman, S. M. Girvin, M. H. Devoret, and M. H. Devoret, Non-Poissonian quantum jumps of a and R. J. Schoelkopf, Observation of high coherence in Fluxonium qubit due to quasiparticle excitations, Phys. Rev. Josephson junction qubits measured in a three-dimensional Lett. 113, 247001 (2014). circuit QED architecture, Phys.Rev.Lett.107, 240501 (2011). [53] I. M. Pop, K. Geerlings, G. Catelani, R. J. Schoelkopf, L. I. [61] Y. Liu, S. Srinivasan, D. Hover, S. Zhu, R. McDermott, and Glazman, and M. H. Devoret, Coherent suppression of A. A. Houck, High fidelity single-shot readout of a trans- electromagnetic dissipation due to superconducting quasi- mon qubit using a SLUG micro-wave amplifier, New J. particles, Nature (London) 508, 369 (2014). Phys. 16, 113008 (2014). [54] E. Fradkin, Field Theories of Condensed Matter Physics, [62] B. R. Johnson, M. D. Reed, A. A. Houck, D. I. Schuster, 2nd ed. (Cambridge University Press, Cambridge, England, L. S. Bishop, E. Ginossar, J. M. Gambetta, L. DiCarlo, L. 2013). Frunzio, S. M. Girvin, and R. J. Schoelkopf, Quantum [55] J. Bylander, S. Gustavsson, F. Yan, F. Yoshihara, K. Harrabi, nondemolition detection of single microwave photons in G. Fitch, D. G. Cory, Y. Nakamura, J.-S. Tsai, and W. D. a circuit, Nat. Phys. 6, 663 (2010). Oliver, Noise spectroscopy through dynamical decoupling [63] S. R. Sathyamoorthy, L. Tornberg, A. F. Kockum, B. Q. with a superconducting flux qubit, Nat. Phys. 7, 565 (2011). Baragiola, J. Combes, C. M. Wilson, T. M. Stace, and G. [56] D. Vion, A. Aassime, A. Cottet, P. Joyez, H. Pothier, C. Johansson, Quantum non-demolition detection of a propagat- Urbina, D. Esteve, and M. H. Devoret, Manipulating the ing microwave photon, Phys. Rev. Lett. 112, 093601 (2014).

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