Experimental investigation of an eight-qubit unit cell in a superconducting optimization processor

A superconducting chip containing a regular array of flux qubits, tunable interqubit inductive couplers, an XY-addressable readout system, on-chip programmable magnetic memory, and a sparse network of analog control lines has been studied. The architecture of the chip and the infrastructure used to control it were designed to facilitate the implementation of an adiabatic quantum optimization algorithm. The performance of an eight-qubit unit cell on this chip has been characterized by measuring its success in solving a large set of random Ising spin-glass problem instances as a function of temperature. The experimental data are consistent with the predictions of a quantum mechanical model of an eight-qubit system coupled to a thermal environment. These results highlight many of the key practical challenges that we have overcome and those that lie ahead in the quest to realize a functional large-scale adiabatic quantum information processor.

Figure 1

(Color online) Mapping of ISG problems, specified by a set of values denoted as $h_i$ and $K_{ij}$, onto superconducting hardware. Two qubits, two $\left|I_q^p\right|$ compensators, and one interqubit coupler are shown. A global current bias $I_{ccjj}\left(t\right)$ provides the fluxes ${\Phi }_{ccjj}^x\left(t\right)$ that drive the annealing process to multiple CCJJ rf-SQUID flux qubits. Interqubit coupling is mediated by tunable mutual inductances $M_{ij}\propto K_{ij}$ that are controlled by static fluxes ${\Phi }_{\text{co},ij}^x$. Qubit bodies are subjected to the sum of static flux biases ${\Phi }_i^0$ and time-dependent flux biases driven by a global current bias $I_g\left(t\right)$. The latter signals are mediated to each qubit via tunable mutual inductances $M_i\propto h_i$ that are controlled by static fluxes ${\Phi }_{I_p,i}^x$.

Figure 2

(Color online) Device schematic. (a) A single CCJJ rf SQUID with the two time-dependent biases relevant for this study. The flux bias ${\Phi }_{ccjj}^x\left(t\right)$ drives the annealing process. The global $\left|I_q^p\right|$-compensation bias is provided by $I_g\left(t\right)$. (b) Schematic layout of the eight-qubit unit cell. Qubits are shown in gray. Example readout, CCJJ, $L$ tuner (LT), $\left|I_q^p\right|$ compensator (IPC), internal coupler (ICO), and portions of external couplers (XCO) have been noted. (c) Graph representation of the hardware connectivity. Qubits are represented by vertices (solid dots) and couplings by edges (solid lines). Dark (shaded) edges correspond to ICO (XCO) couplings.

Figure 3

Scanning electron micrograph of fabrication cross section. Superconducting Nb layers identified as BASE, WIRA, WIRB, and WIRC. Resistive TiPt layer identified as RESI. Planarized dielectric denoted as ${\text{SiO}}_2$. An example Josephson-junction trilayer and a via between WIRA and WIRB have been denoted as TRI and VIA, respectively.

Figure 4

(Color online) (a) Optical image of an eight-qubit unit cell completed through the processing of WIRB. Qubits, labeled as $q_0\to q_7$, reside within the trenches formed by ground plane in WIRA and extended vias in WIRB. PMM elements are visible as spirals and washers between qubits. (b) Optical image of a $4\times 4$ array of eight-qubit unit cells completed through the processing of WIRB. Unit cell shown in (a) is enclosed within the dashed box.

Figure 5

(Color online) Measured and predicted qubit parameters as a function of ${\Phi }_{ccjj}^x$. (a) Magnitude of the persistent current $\left|I_q^p\right|$. (b) Tunneling energy ${\Delta }_q$. Solid curves are predictions from the CCJJ rf-SQUID Hamiltonian using independently calibrated device parameters.

Figure 6

(Color online) (a) Mutual inductance $M_{ij}$ versus coupler flux bias ${\Phi }_{\text{co},ij}^x$ for all 16 interqubit couplers in the unit cell. Solid curves are the theoretical $M_{ij}$ obtained using the mean of the individual coupler fit parameters. (b) Mutual inductance $M_i$ versus $\left|I_q^p\right|$-compensator flux bias ${\Phi }_{I_p,i}^x$. Solid curve is the theoretical $M_i$ obtained using the mean of the individual $\left|I_q^p\right|$-compensator fit parameters.

Figure 7

(Color online) Verification of readout fidelity for the eight-qubit unit cell. (a) Example results from preparing the eight qubits into each of the $2^8=256$ possible final spin configurations and recording the number of correct reads for 1000 repetitions. (b) Probability of observing a given number of errors within 1000 repetitions, as taken from the data in (a). Data have been fit to a Poisson distribution with mean number of errors $\lambda =0.36$.

Figure 8

(Color online) Depiction of the measurement wave form sequence. Only time-dependent biases have been shown. All qubits are subjected to common ${\Phi }_{ccjj}^x\left(t\right)$ and $I_g\left(t\right)$. All QFPs are subjected to a common ${\Phi }_{\text{latch}}^x\left(t\right)$. Example dc-SQUID flux ${\Phi }_{\text{ro}}^x$ and current $i_{\text{ro}}\left(t\right)$ wave forms shown only for device connected to $q_0$.

Figure 9

(Color online) Histograms of the experimentally observed and the simulated probability of finding the system in the ground state $P_{\text{gs}}$ from a test involving 6400 unique random ISG instances at three different temperatures: (a) $T=20\text{mK}$, (b) $T=35\text{mK}$, and (c) $T=50\text{mK}$.

Figure 10

(Color online) Experimental (symbols) and simulated (curves) probabilities of finding the system in low-energy states. (a) Example A. (b) Example B. (c) Example C. Solid (dashed) curves have been color coded to match filled (hollow) symbols.

Figure 11

(Color online) Evaluation of the dimensionless objective function $E\left(\vec{s}\right)$ [Eq. (1)] for examples A, B, and C. Instance settings are given in the Appendix. Results plotted relative to the lowest value $E\left({\vec{s}}_{\text{gs}}\right)$. Spin configurations $|\vec{s}⟩$ are identified to the right of each level.

Figure 12

(Color online) Calculated low-energy eigenspectra versus $-{\Phi }_{ccjj}^x$. (a) Example A. (b) Example B. (c) Example C. Energies have been plotted relative to the ground-state energy $E_0$ and truncated above that of the second excited state of the CCJJ rf SQUID, $E_{\text{rfs}}-E_0$. Initial states are labeled as $|{\Pi }_0⟩$, $|{\Pi }_{1,m}⟩$, and $|{\Pi }_{2,m,n}⟩$ on the left. Final states are labeled as $|\vec{s}⟩$ on the right. Vertical dashed lines roughly delineate the value of $-{\Phi }_{ccjj}^x$ beyond which the population statistics appear to freeze during annealing.