Demonstration of a Nonstoquastic Hamiltonian in Coupled Superconducting Flux Qubits

Hamiltonian-based quantum computation is a class of quantum algorithms in which the problem is encoded in a Hamiltonian and the evolution is performed by a continuous transformation of the Hamiltonian. Universal adiabatic quantum computing, quantum simulation, and quantum annealing are examples of such algorithms. Up to now, all implementations of this approach have been limited to qubits coupled via a single degree of freedom. This gives rise to a stoquastic Hamiltonian that has no sign problem in quantum Monte Carlo simulations. In this paper, we report implementation and measurements of two superconducting flux qubits coupled via two canonically conjugate degrees of freedom—charge and flux—to achieve a nonstoquastic Hamiltonian. We perform microwave spectroscopy to extract circuit parameters and show that the charge coupling manifests itself as a ${\sigma }^y{\sigma }^y$ interaction in the computational basis. We observe destructive interference in quantum coherent oscillations between the computational-basis states of the two-qubit system. Finally, we show that the extracted Hamiltonian is nonstoquastic over a wide range of parameters.

Figure 1

Schematic and potential energy of coupled rf SQUIDs. (a) Schematic circuit of two rf SQUIDs coupled inductively via a tunable coupler and capacitively via a fixed capacitor. (b) Effective potential energy of the circuit shown in (a). Arrows show tunneling paths between $|\uparrow \uparrow ⟩$ and $|\downarrow \downarrow ⟩$ states within the two-state approximation for each rf SQUID. The solid arrows indicate tunneling due to two consecutive single-qubit flips facilitated by ${\sigma }_i^x$. The dotted arrow represents direct two-qubit cotunneling due to ${\sigma }_1^x{\sigma }_2^x$ and ${\sigma }_1^y{\sigma }_2^y$. The tunneling amplitudes may have opposite signs, thus leading to destructive interference.

Figure 2

Microwave spectroscopy of the coupled two-qubit system at $M_{12}=0$ and ${\mathrm{\Phi }}_{q,1}^x={\mathrm{\Phi }}_{q,2}^x=0$. (a) Pulse sequences for two-qubit microwave spectroscopy. At ${\mathrm{\Phi }}_{\mathrm{CJJ,monostable}}$ the tunneling barrier is at its lowest, resulting in a monostable potential. At ${\mathrm{\Phi }}_{\mathrm{CJJ,latched}}^x$ the tunneling barrier is high and the tunneling amplitude is negligible. The effective single-qubit spectroscopy has the same pulse sequence except for ${\mathrm{\Phi }}_{\mathrm{CJJ},2}^x=0.5{\mathrm{\Phi }}_0$. (b)–(d) Two-qubit-spectroscopy energy versus ${\mathrm{\Phi }}_{\mathrm{CJJ},1}^x$, which controls the qubit-1 barrier height. Qubit 2 is kept at ${\mathrm{\Phi }}_{\mathrm{CJJ},2}^x$, corresponding to effective single-qubit tunneling amplitudes (b) ${\mathrm{\Delta }}_2=1.5\mathrm{GHz}$, (c) ${\mathrm{\Delta }}_2=2.0\mathrm{GHz}$, and (d) ${\mathrm{\Delta }}_2=3.0\mathrm{GHz}$. Energies are measured relative to the ground state. The solid lines are obtained by our fitting the rf-SQUID model, Eq. (2), to the experimental data. Dashed lines represent the excited-state energies of uncoupled qubits ($M_{12}=0,C_{12}=0$) given the other fitting parameters.

Figure 3

Two-qubit spectra at nonzero energy bias $h_i\ne 0$. In all cases, qubit 2 is biased away from degeneracy with ${\mathrm{\Phi }}_{q,2}^x=0.1\mathrm{m}{\mathrm{\Phi }}_0$ and ${\mathrm{\Phi }}_{\mathrm{CJJ},2}^x$ is set such that the effective single-qubit tunneling ${\mathrm{\Delta }}_2=1.5\mathrm{GHz}$. Energy is measured relative to the ground state. Solid white lines correspond to numerical simulations obtained with the two-qubit Hamiltonian (3) with no fitting parameters. (a) Energy as a function of ${\mathrm{\Phi }}_{\mathrm{CJJ},1}^x$ at fixed ${\mathrm{\Phi }}_{q,i}^x=0.1\mathrm{m}{\mathrm{\Phi }}_0$ and $M_{12}=0$. (b) Energy as a function of ${\mathrm{\Phi }}_{q,1}^x$ at $M_{12}=0.55\mathrm{pH}$ and effective single-qubit ${\mathrm{\Delta }}_1=1.5\mathrm{GHz}$. The vertical dotted line goes through ${\mathrm{\Phi }}_{q,1}^x=0$ to highlight the asymmetry in the spectrum due to fixed ${\mathrm{\Phi }}_{q,2}^x=0.1\mathrm{m}{\mathrm{\Phi }}_0$. (c) Energy as a function of $M_{12}$ at ${\mathrm{\Phi }}_{q,i}^x=0.1\mathrm{m}{\mathrm{\Phi }}_0$ and ${\mathrm{\Delta }}_i=1.5\mathrm{GHz}$. The dashed lines correspond to numerical simulations at zero energy biases. The observed avoided crossing at $M_{12}=0.55\mathrm{pH}$ is due to nonzero biases. The vertical dotted line separates ferromagnetic (FM) and antiferromagnetic (AFM) regions. The asymmetry in the data with respect to this line is due to capacitive coupling. (d) Extracted interaction parameters in Hamiltonian (3) that provide the theoretical (solid) lines in (c). The Hamiltonian is nonstoquastic in the white unshaded area according to Ref. [13].

Figure 4

Two-qubit coherent oscillations as a function of mutual inductance. (a) Population of state $|\downarrow \uparrow ⟩$ when the system is initialized in $|\uparrow \downarrow ⟩$. (b),(c) Population of state $|\downarrow \downarrow ⟩$ when the system is initialized in $|\uparrow \uparrow ⟩$. The final flux bias on qubits for (c) is set to $0.1\mathrm{m}{\mathrm{\Phi }}_0$, corresponding to Figs. 3 and 3. For (a),(b) the final flux bias is zero. (d)–(f) Numerical simulations corresponding to (a)–(c), obtained with the reduced four-level model given in Eq. (3), respectively.

Figure 5

(a) Two-qubit-Hamiltonian parameters as functions of ${\mathrm{\Phi }}_{\mathrm{CJJ},i}^x$, which controls the barrier height, at zero flux bias and zero inductive coupling. (b) $J_{xx}$ and $J_{yy}$ as a function of ${\mathrm{\Delta }}_i$ for the same setting as in (a). (c) The nonstoquastic region as a function of mutual inductance and ${\mathrm{\Phi }}_{\mathrm{CJJ},i}^x$ for nonzero flux bias. The Hamiltonian is nonstoquastic in the white area.

Figure 6

Measured coupler $M_{12}$ versus coupler flux bias ${\mathrm{\Phi }}_{\mathrm{co}}^x$ and a fit to the classical model of the coupler.

Figure 7

Measured persistent current $I_p$ versus ${\mathrm{\Phi }}_{\mathrm{CJJ}}^x$, which controls the barrier height of the double-well potential, and a fit to the classical model of qubit 1.

Figure 8

(a) Pulse sequence and the effective qubit potential at each pulse segment for the single-qubit-spectroscopy experiments. (b) Color plot of the microwave-spectroscopy data versus the flux bias ${\mathrm{\Phi }}_{\mathrm{CJJ}}^x$, which controls the barrier height for qubit 1. (c) Extracted spectroscopy peak positions that represent the qubit frequency versus ${\mathrm{\Phi }}_{\mathrm{CJJ}}^x$ and a fit to the numerical rf-SQUID model.

Figure 9

Two-qubit-spectroscopy energy versus ${\mathrm{\Phi }}_{\mathrm{CJJ},1}^x$ at (a) ${\mathrm{\Delta }}_2/h\approx 1.5\mathrm{GHz}$ and $M_{12}=-2.0\mathrm{pH}$, (b) ${\mathrm{\Delta }}_2/h\approx 2.0\mathrm{GHz}$ and $M_{12}=-2.0\mathrm{pH}$, (c) ${\mathrm{\Delta }}_2/h\approx 3.0\mathrm{GHz}$ and $M_{12}=-2.0\mathrm{pH}$, (d) ${\mathrm{\Delta }}_2/h\approx 1.5\mathrm{GHz}$ and $M_{12}=-0.0\mathrm{pH}$, (e) ${\mathrm{\Delta }}_2/h\approx 2\mathrm{GHz}$ and $M_{12}=0.0\mathrm{pH}$, (f) ${\mathrm{\Delta }}_2/h\approx 3.0\mathrm{GHz}$ and $M_{12}=0.0\mathrm{pH}$, (g) ${\mathrm{\Delta }}_2/h\approx 1.5\mathrm{GHz}$ and $M_{12}=2.0\mathrm{pH}$, (h) ${\mathrm{\Delta }}_2/h\approx 2.0\mathrm{GHz}$ and $M_{12}=2.0\mathrm{pH}$, and (i) ${\mathrm{\Delta }}_2/h\approx 3.0\mathrm{GHz}$ and $M_{12}=2.0\mathrm{pH}$. The solid and dashed lines represent numerical calculations of energy differences. The solid lines correspond to excitations from the ground state and the dashed lines correspond to excitations from the first excited state.

Figure 10

Characterization and compensation of pulse distortion. (a) The coherent-oscillation data, (b) measured tunneling amplitude ${\mathrm{\Delta }}_m$, and (c) applied barrier pulses without (blue) and with (green) pulse-distortion corrections.

Figure 11

(a) The effective qubit potential during each segment of the coherent-oscillation pulse shown in (b). The potential landscape represents a cut across the landscape depicted in Fig. 1. Its direction depends on the initial tilt pulse applied on the two qubits. Two-qubit coherent oscillation between the computational-basis states with strong (c) ferromagnetic coupling at $M_{12}=-2\mathrm{pH}$ and strong (d) antiferromagnetic coupling at $M_{12}=2\mathrm{pH}$ versus time. Color plots of the two-qubit coherent oscillation versus the magnetic coupling strength $M_{12}$ with the system initially prepared at (e) $|\downarrow \downarrow ⟩$ and (f) $|\downarrow \uparrow ⟩$.

Figure 12

$T_1$ and $T_2$ measurements on qubit 1 at $\mathrm{\Delta }=2\mathrm{GHz}$ and at zero bias, $h_1=0$. (a) By fitting the decay envelope of the coherent oscillation to an exponential function, we obtain $T_2=16.2\mathrm{ns}$. (b) By fitting the energy relaxation of the first excited state, we obtain $T_1=17.5\mathrm{ns}$.