Coherent Coupled Qubits for Quantum Annealing

Quantum annealing is an optimization technique which potentially leverages quantum tunneling to enhance computational performance. Existing quantum annealers use superconducting flux qubits with short coherence times limited primarily by the use of large persistent currents $I_{\mathit{p}}$. Here, we examine an alternative approach using qubits with smaller $I_{\mathit{p}}$ and longer coherence times. We demonstrate tunable coupling, a basic building block for quantum annealing, between two flux qubits with small (approximately 50-nA) persistent currents. Furthermore, we characterize qubit coherence as a function of coupler setting and investigate the effect of flux noise in the coupler loop on qubit coherence. Our results provide insight into the available design space for next-generation quantum annealers with improved coherence.

Figure 1

Coupled-qubit geometry. (a) Device schematic. Qubit $A$ (left loop) and qubit $B$ (right loop) are capacitively shunted three-junction flux qubits coupled through a shared inductance with a rf-SQUID coupler (center loop). On-chip bias currents $I_1$, $I_2$, and $I_3$ control the external fluxes ${\mathrm{\Phi }}_{\mathit{A}}$, ${\mathrm{\Phi }}_{\mathit{C}}$, and ${\mathrm{\Phi }}_{\mathit{B}}$ through the qubit and coupler loops. (b) Optical micrograph of the aluminum (light gray) device on a silicon (dark gray) substrate. (c) Optical image showing the qubits, coupler, and flux bias lines. (d) SEM image of the galvanic connection between qubit $B$ (lower right) and the coupler (upper left).

Figure 2

Qubit spectroscopy. Dashed black traces: Semiclassical model. Solid green traces: Simulations of the full-circuit Hamiltonian. (a) Spectroscopy of qubit $A$ vs $f_A$, with $f_B=f_C=0$. (b) Spectroscopy of qubit $B$ vs $f_{\mathit{B}}$ with $f_{\mathit{A}}=f_{\mathit{C}}=0$. The yellow dashed line represents the starting point and range of qubit frequencies in the following panel. (c) Spectroscopy of qubit $B$ vs $f_{\mathit{C}}$ for $f_{\mathit{A}}=0$ and $f_{\mathit{B}}=0.516$. The regions of antiferromagnetic (AFM), ferromagnetic (FM), and zero coupling are indicated. The inset shows detailed data for the FM region.

Figure 3

Qubit-qubit coupling. Dashed black traces: Semiclassical model. Solid green traces: Simulations of the full-circuit Hamiltonian. (a)–(f) Spectroscopy of qubit-level crossings for different coupling strengths. Panels (a)–(c) and (d)–(f) show measurements using resonator $A$ and resonator $B$, respectively. In each panel, we scan $f_A$ while holding $f_B$ at a fixed bias point approximately $10\mathrm{m}{\mathrm{\Phi }}_0$ away from degeneracy. The left, middle, and right panels correspond to zero ($f_{\mathit{C}}=0.402$), intermediate ($f_{\mathit{C}}=0.48$), and maximum ($f_{\mathit{C}}=0.5$) coupling, as indicated by the insets. (g) Avoided level crossings as qubit $A$ (red) is tuned across qubit $B$ (blue) with $f_{\mathit{C}}=0.5$. (h) $J$ vs coupler bias. Error bars are derived from the error of fitting the qubit spectroscopy to a double Gaussian function.

Figure 4

Qubit properties vs coupler bias. Results for qubit $B$, with $f_{\mathit{A}}=0$. Left column: Full range of coupler biases. Right column: Enlargement of the region near coupler degeneracy. (a),(b) ${\mathrm{\Delta }}_{\mathit{B}}$ vs coupler bias. Dashed black traces: Semiclassical model. Solid green traces: Simulations of the full-circuit Hamiltonian. (c),(d) Qubit energy-relaxation time vs coupler bias. The red circles and the magenta triangles correspond to measurements taken at different times. The gray band indicates the typical range of $T_1$ variations when the coupler is biased away from degeneracy. The solid line represents an upper bound on qubit $T_1$ due to flux noise in the coupler loop with an exponent $\alpha =0.91$ and an amplitude of $15\mu {\mathrm{\Phi }}_0/\sqrt{\mathrm{Hz}}$ combined in parallel with a coupler-independent relaxation time of $3.5\mu \mathrm{s}$. (e),(f) Ramsey (left axis) and echo (right axis) $1/e$ decay times vs coupler bias. Solid lines show the expected dependence due to $1/f^{\alpha }$-flux noise in the coupler loop with the same amplitude and exponent as above.

Figure 5

Schematic diagram of the full galvanic circuit. The nodes of the circuit labeled 1–9 are used to define its canonical flux and charge variables [39].

Figure 6

Direct coupling. (a) Circuit schematic for two loops of inductance $L_{A,B}$ coupled through a mutual inductance $M$. (b) Circuit schematic for two loops which are galvanically coupled through a shared inductance $M$. (c) Circuit schematic for two flux qubits with persistent currents $I_{\mathit{p}}^{A,B}$ coupled through a mutual inductance $\tilde{M}$. (d) Illustration of the energies of the ground (blue) and first excited (red) states of a flux qubit as a function of the external flux ${\mathrm{\Phi }}_{\mathrm{ext}}$ through the qubit loop. At the degeneracy point (${\mathrm{\Phi }}_{\mathrm{ext}}={\mathrm{\Phi }}_0/2$), the ground and excited states are separated in energy by $\hslash \mathrm{\Delta }$. When biased away from degeneracy, the qubit states are approximately persistent current states $|\pm z⟩$.

Figure 7

Mediated coupling. (a) Circuit schematic for two flux qubits with persistent currents $I_{\mathit{p}}^{A,B}$ which each couple through a mutual inductance $\tilde{M}$ to an intermediate loop of inductance $L$. (b) Circuit schematic for a similar configuration but with the intermediate loop replaced with a rf-SQUID coupler. (c) Illustration of the energies of the ground (blue) and first excited (gray) states of a rf-SQUID coupler as a function of the external flux ${\mathrm{\Phi }}_{\mathit{c}}$ through the coupler loop. (d) Left axis: Comparison of the coupler circulating current calculated using the slope of the ground-state energy (red) and using the current operator (green). Right axis: Effective inductance of the coupler vs ${\mathrm{\Phi }}_{\mathit{c}}$.

Figure 8

Inductive loading model. Circuit schematic used to model the loading of the qubit inductance due to the coupler $L_{\mathrm{eff}}$.

Figure 9

Repeated $T_1$ measurements. (a) Repeated measurements of $T_1$ of qubit $B$ as a function of time. Here, $f_{\mathit{A}}=0$, $f_{\mathit{B}}=0.5$, and $f_{\mathit{C}}=0$. (b) Histogram of measured $T_1$ values.

Figure 10

Flux-noise analysis. (a) $\sqrt{{\eta }_{0,1}}$ vs $\gamma$ determined through numerical integration. When calculating ${\eta }_0$, we assume ${\omega }_{\mathrm{low}}/2\pi =3\mathrm{mHz}$ and $t=200\mathrm{ns}$. (b) Estimated coupler flux-noise-amplitude-based measured Ramsey, echo, and $T_1$ times as a function of $\gamma$.

Figure 11

Zero coupling. Detailed data for the qubit-level crossing with the coupler nominally biased for zero coupling ($f_{\mathit{C}}=0.5$). Red hourglasses: Qubit $A$. Blue circles: Qubit $B$.