Superconducting quantum simulator for topological order and the toric code

Topological order is now being established as a central criterion for characterizing and classifying ground states of condensed matter systems and complements categorizations based on symmetries. Fractional quantum Hall systems and quantum spin liquids are receiving substantial interest because of their intriguing quantum correlations, their exotic excitations, and prospects for protecting stored quantum information against errors. Here, we show that the Hamiltonian of the central model of this class of systems, the toric code, can be directly implemented as an analog quantum simulator in lattices of superconducting circuits. The four-body interactions, which lie at its heart, are in our concept realized via superconducting quantum interference devices (SQUIDs) that are driven by a suitably oscillating flux bias. All physical qubits and coupling SQUIDs can be individually controlled with high precision. Topologically ordered states can be prepared via an adiabatic ramp of the stabilizer interactions. Strings of qubit operators, including the stabilizers and correlations along noncontractible loops, can be read out via a capacitive coupling to read-out resonators. Moreover, the available single-qubit operations allow to create and propagate elementary excitations of the toric code and to verify their fractional statistics. The architecture we propose allows to implement a large variety of many-body interactions and thus provides a versatile analog quantum simulator for topological order and lattice gauge theories.

Figure 1

Toric code lattice and its implementation. (a) Lattice for the toric code. Small rectangles on the lattice edges indicate spin-$\frac{1}{2}$ systems (qubits). Star interactions $A_s$ are marked with orange diamonds and plaquette interactions $B_p$ with green diamonds. The transition frequencies of the qubits form a periodic pattern in both lattice directions such that no nearest neighbors and no next-nearest neighbors of spins have the same transition frequency and, therefore, all four spins at the corners of a physical cell have different transition frequencies. (b) Circuit of a physical cell. The rectangles at the corners represent qubits with different transition frequencies (see inset for the circuit of each qubit). Qubits $(n,m)$ and $(n,m+1) \left[\right(n+1,m)$ and $(n+1,m+1)]$ are connected via the inductances $L$ to the node $(a;n,m) \left[\right(b;n,m\left)\right]$. The nodes $(a;n,m)$ and $(b;n,m)$ in turn are connected via a dc-SQUID with an external magnetic flux ${\mathrm{\Phi }}_{\mathrm{ext}}$ threaded through its loop. The Josephson junctions in the dc-SQUID have Josephson energies $E_J$ and capacitances $C_J$, which include shunt capacitances.

Figure 2

Macroscopic analogy providing an intuitive explanation of our approach to generate the stabilizer interactions. The qubits are nonlinear oscillators so that one can think of them as masses (corresponding to the qubits' capacitances) attached to anharmonic springs (corresponding to the qubits Josephson energy). We here sketch four nonlinear oscillators representing four qubits as green spheres connected by black (anharmonic) springs to the support frame. Note that all four qubits have different transition frequencies so that all four spheres representing them here would have different masses and their springs different spring constants. For vanishing critical current, the relevant SQUID modes ${\phi }_{-;j}$ behave as harmonic oscillators and in our macroscopic analogy we can represent them by a mass (here sketched as a red sphere, corresponding to the capacitance $2C_J$) that is attached via four springs (sketched in orange, corresponding to the inductors $L$) to the four masses representing the neighboring qubits. For vanishing critical current, the only potential energy of the modes ${\phi }_{-;j}$ comes from their inductive coupling to the neighboring qubits, and hence the mass representing the SQUID mode (red sphere) has no spring connecting it to a support frame. We now imagine that we shake the mass representing the SQUID mode with a driving term that is proportional to the fourth power of the mass' position variable ${\phi }_{-;j}^4$. This corresponds to the oscillating flux bias $\propto {\phi }_{\text{ac}}F\left(t\right)$. If we shake this mass at a frequency ${\omega }_d$ that is distinct from its eigenfrequency ${\omega }_{-;j}$ but equals the sum of the oscillation frequencies of all four neighboring masses (neighboring qubits), we drive a term proportional to $q_1^{†}{q}_2^{†}{q}_3^{†}{q}_4^{†}$, where $q_j^{†}$ creates an excitation in qubit $j$. If, on the other hand, we shake the SQUID mass at a frequency ${\omega }_d={\omega }_1+{\omega }_2-{\omega }_3-{\omega }_4$, we drive a term proportional to $q_1^{†}{q}_2^{†}{q}_3{q}_4$. The selective driving of specific four-body terms, as we consider it here, of course relies on choosing the transition frequencies of all four qubits to be different. Our approach considers a superposition of various drive frequencies such that the sum of the triggered four-body terms adds up to the desired stabilizer interaction (see text).

Figure 3

Lattice for minimal realization on a torus with eight physical qubits including four star interactions (orange diamonds) and four plaquette interactions (green diamonds). The qubits are indicated by small colored squares, where the coloring encodes their transition frequencies and the numbering indicates the periodic boundary conditions. Here, eight different transition frequencies are needed.

Figure 4

Pattern of transition frequencies for an implementation of a large lattice. The qubits are indicated by small colored squares, where the coloring encodes their transition frequencies. Here, four principal transition frequencies are needed. These differ by a few GHz and are indicated by the filling colors brown, blue, red, and purple. To suppress two-body interactions between next-nearest-neighbor qubits, each pair of these needs to be detuned from each other by $\sim 100\text{MHz}$. The resulting frequency differences are indicated by the colored frames around the respective squares. The gray diamond highlights the frequency set which is periodically repeated in both lattice directions.

Figure 5

The energy eigenvalues of $H\left(\lambda \right)$ as a function of $\lambda$ for the eight-qubit lattice with $\mathrm{\Delta }=2\pi \times 5\mathrm{MHz}$ and $J_s=J_p=2\pi \times 2\mathrm{MHz}$. The initial ground state is the unique vacuum $|{\psi }_0\rangle ={\prod }_{j=1}^8|0_j\rangle$. At $\lambda \sim 0.6$ the ground state starts to become degenerate but the degenerate ground-state manifold is always separated by a gap $\sim J_s\sim J_p$ from higher excited states.

Figure 6

Adiabatic preparation of an eight-qubit lattice in the protected subspace. (a) Fidelity $F$ for being in the protected subspace and (b) purity $P$ of the state. $J_s=J_p=h\times 2\mathrm{MHz}$ and coherence times $T_1=T_2=50\mu \mathrm{s}$ (see Sec. 2d).