Emulating the one-dimensional Fermi-Hubbard model by a double chain of qubits

The Jordan-Wigner transformation maps a one-dimensional (1D) spin-$1/2$ system onto a fermionic model without spin degree of freedom. A double chain of quantum bits with $XX$ and $ZZ$ couplings of neighboring qubits along and between the chains, respectively, can be mapped on a spin-full 1D Fermi-Hubbard model. The qubit system can thus be used to emulate the quantum properties of this model. We analyze physical implementations of such analog quantum simulators, including one based on transmon qubits, where the $ZZ$ interaction arises due to an inductive coupling and the $XX$ interaction due to a capacitive interaction. We propose protocols to gain confidence in the results of the simulation through measurements of local operators.

Figure 1

Layout of a qubit system consisting of two chains (labeled $\uparrow ,\downarrow$) of $n$ qubits, where the neighboring qubits of each chain are coupled via a SWAP interaction of the form ${\sigma }_{j,s}^{+}{\sigma }_{j+1,s}^{-}+\text{H.c.}$ The two chains from a ladder, with qubits belonging to the same rung coupled through terms involving ${\sigma }_{j,\uparrow }^z{\sigma }_{j,\downarrow }^z$. With the resulting Hamiltonian $H_{\mathrm{QS}}$ (3) the qubit system can serve as an analog quantum simulator (AQS) of the 1D Fermi-Hubbard model $H_{\mathrm{FH}}$ (7).

Figure 2

Circuit diagram for a physical realization of the quantum simulator of Fig. 1 by a ladder array of charge qubits (gray box). Superconducting islands (gray circles) within the same chain are coupled via Josephson junctions $J$, neighboring islands belonging to the two different chains are coupled via capacitances $C^z$. Each island is connected via a capacitance $C$ to a control voltage $V_{j,s}$. Two charge states of the islands form the basis of the qubit. The circuit provides the desired couplings of Fig. 1 and allows tuning the chemical potential without further restriction, thus allowing any occupation of the fermionic states.

Figure 3

Superconducting circuit based on tunable transmon qubits (gray box). Each transmon consists of two Josephson junctions $J$ forming a loop with geometric inductance $L$; the junctions are shunted by capacitances $C$. Each individual transmon is tunable via applied magnetic fluxes ${\mathrm{\Phi }}_{j,s}$ threading the loop. Additional capacitances $C^x$ create an $XX$ coupling between two qubits. Coupling the inductances of two qubits via the mutual inductance $M$ produces a $ZZ$ interaction. The circuit is therefore a realization of the qubit system of Fig. 1.

Figure 4

For a quantum simulator realized by transmon qubits, the parameter $U/t$ of the Fermi-Hubbard model (7) to be emulated is plotted against the external phase ${\varphi }_{\mathrm{e}}$ (assuming all external fluxes to be equal). The shaded area indicates the accessible range for ${\varphi }_{\mathrm{e}}$, and $\gamma$ is a scaling factor (see main text).

Figure 5

Circuit diagram of a tunable transmon with Josephson junctions J, the capacitances C in parallel, and the intrinsic inductance L of the loop. The phase differences across the junctions (and capacitances) are labeled ${\varphi }_{\mathrm{l}}$ and ${\varphi }_{\mathrm{r}}, {\mathrm{\Phi }}_{\mathrm{e}}$ denotes an external flux through the loop.

Figure 6

Circuit diagram of two inductively coupled transmons, where their inductances $L$ form a mutual inductance $M=k_{M}L$ with $k_M\in (0,1)$. The transmons should be built identically with Josephson junctions $J$ and shunt capacities $C$, but with individually controlled external fluxes ${\mathrm{\Phi }}_{1\mathrm{e}}$ and ${\mathrm{\Phi }}_{2\mathrm{e}}$ for tuning. ${\varphi }_{(1,2)\mathrm{l}}$ and ${\varphi }_{(1,2)\mathrm{r}}$ denote the phase differences across the junctions (and capacitances).

Figure 7

The transmon's level splitting $\epsilon$ depends on the phase ${\varphi }_{\mathrm{e}}$ related to the external flux ${\mathrm{\Phi }}_{\mathrm{e}}=\frac{\hslash }{2e}{\varphi }_{\mathrm{e}}$. Fluctuations $\mathrm{\Delta }{\varphi }_{-}$ of the phase across the inductance of the transmon vary the energy splitting by an amount which depends on the slope of $\epsilon$.

Figure 8

Circuit diagram of a chain of coupled identical transmons, with Josephson junctions $J$ (with critical current $I_{\mathrm{c}}$), shunt capacitances $C$, and loop inductance $L$. They are individually tunable though external fluxes ${\mathrm{\Phi }}_{j\mathrm{e}}$, and the phase differences across the junctions (and capacitances) of the $j\mathrm{th}$ qubit are denoted ${\varphi }_{j\mathrm{l}}$ and ${\varphi }_{j\mathrm{r}}$. The transmons are coupled by capacitances $C^x$ between neighboring transmons.

Figure 9

(a) Schematic of the concentric transmon qubit: It consists of a large central disk electrode ($230\mu \mathrm{m}$ diameter) surrounded by a ring electrode, constituting the shunt capacitance for the two Josephson junctions interconnecting the electrodes. (b) Possible implementation of the simulator scheme depicted in Fig. 1 with concentric transmon qubits. In the vertical direction, the magnetic $ZZ$ coupling, mediated by a nonvanishing mutual inductance $M$, is relevant. The $XX$ coupling can effectively be suppressed by detuning vertically adjacent qubits. Transverse coupling is dominant in horizontal direction, where the $ZZ$ coupling is suppressed by exploiting the asymmetric gradiometry of the qubits.