Entanglement in a Quantum Annealing Processor

Entanglement lies at the core of quantum algorithms designed to solve problems that are intractable by classical approaches. One such algorithm, quantum annealing (QA), provides a promising path to a practical quantum processor. We have built a series of architecturally scalable QA processors consisting of networks of manufactured interacting spins (qubits). Here, we use qubit tunneling spectroscopy to measure the energy eigenspectrum of two- and eight-qubit systems within one such processor, demonstrating quantum coherence in these systems. We present experimental evidence that, during a critical portion of QA, the qubits become entangled and entanglement persists even as these systems reach equilibrium with a thermal environment. Our results provide an encouraging sign that QA is a viable technology for large-scale quantum computing.

Popular Summary

Quantum algorithms hold the promise of helping to solve a broad range of problems that are simply intractable with classical algorithms. The advantage of quantum calculations stems from exploiting the strange and nonintuitive properties of quantum systems: tunneling, superposition, quantum coherence, and entanglement. Building a general-purpose quantum computer, however, is extremely challenging; a more scalable and feasible approach may involve implementing a single, simpler quantum algorithm, such as quantum annealing. It is critical to demonstrate that such a scalable processor has access to quantum-mechanical resources such as coherence and entanglement. We build a processor based on quantum annealing and verify that specific two- and eight-qubit systems become entangled, a necessary and significant step in developing quantum annealing into a viable quantum-computing technology.

We run quantum annealing on a processor chip composed of magnetically coupled superconducting flux qubits. The chip is mounted on the mixing chamber of a dilution refrigerator held at 12.5 mK. We use qubit tunneling spectroscopy to infer nonclassical correlations in two- and eight-qubit systems based on eigenspectra and level occupations, effects that persist even at thermal equilibrium. Our measurements of spectral lines are dominated by the noise of the qubit tunneling spectroscopy probe, however, and we expect that follow-up experiments with improved probes will enable larger systems of qubits to be studied.

Our work provides promise that quantum annealing is a viable approach to realizing quantum-computing technologies. Moreover, our technique represents an effective way of studying quantum behavior in a practical processor, helping us to further understand the capabilities of quantum algorithms.

Figure 1

(a) Photograph of the QA processor used in this study. We report measurements performed on the eight-qubit unit cell indicated. The bodies of the qubits are extended loops of Nb wiring (highlighted with red rectangles). Interqubit couplers are located at the intersections of the qubit bodies. (b) Electron micrograph showing the cross section of a typical portion of the processor circuitry (described in more detail in Appendix pp1). (c) Schematic diagram of a pair of coupled superconducting flux qubits with external control biases ${\mathrm{\Phi }}_{qi}^x$ and ${\mathrm{\Phi }}_{\mathrm{ccjj}}^x$ and with flux through the body of the $i$th qubit denoted as ${\mathrm{\Phi }}_{qi}$. An inductive coupling between the qubits is tuned with the bias ${\mathrm{\Phi }}_{\mathrm{co},ij}^x$. (d) Energy scales $\mathrm{\Delta }(s)$ and $\mathcal{E}(s)$ in Hamiltonian (1) calculated from a rf SQUID model based on the median of independently measured device parameters for these eight qubits. See Appendix pp1 for more details. (e),(f) The two- and eight-qubit systems studied were programmed to have the topologies shown. Qubits are represented as gold spheres, and interqubit couplers, set to $J=-2.5$, are represented as silver lines.

Figure 2

An illustration of entanglement between two qubits during QA with $h_i=0$ and $J<0$. We plot calculations of the two-qubit ground-state wave-function modulus squared in the basis of ${\mathrm{\Phi }}_{q1}$ and ${\mathrm{\Phi }}_{q2}$, the flux through the bodies of $q_1$ and $q_2$, respectively. The color scale encodes the probability density with red corresponding to high probability density and blue corresponding to low probability density. We use Hamiltonian (1) and the energies in Fig. 1 for the calculation. The four quadrants represent the four possible states of the two-qubit system in the computation basis. We also plot the single-qubit potential energy ($U$ versus ${\mathrm{\Phi }}_{q1}$) calculated from measured device parameters. (a) At $s=0$ ($\mathrm{\Delta }\gg 2|J|\mathcal{E}\sim 0$), the qubits weakly interact and are each in their ground state $(1/\sqrt{2})(|\uparrow ⟩+|\downarrow ⟩)$, which is delocalized in the computation basis. The wave function shows no correlation between $q_1$ and $q_2$ and, therefore, their wave functions are separable. (b) At intermediate $s$ ($\mathrm{\Delta }\sim 2|J|\mathcal{E}$), the qubits are entangled. The state of one qubit is not separable from the state of the other, as the ground state of the system is approximately $|+⟩\equiv (|\uparrow \uparrow ⟩+|\downarrow \downarrow ⟩)/\sqrt{2}$. A clear correlation is seen between $q_1$ and $q_2$. (c) As $s\to 1$, $\mathrm{\Delta }\ll 2|J|\mathcal{E}$, and the ground state of the system approaches $|+⟩$. However, the energy gap $g$ between the ground state ($|+⟩$) and the first excited state ($|-⟩$) is closing. When the qubits are coupled to a bath with temperature $T$ and $g<k_{B}T$, the system is in a mixed state of $|+⟩$ and $|-⟩$ and entanglement is extinguished.

Figure 3

Spectroscopic data for two- and eight-qubit systems plotted in false color (color indicates normalized qubit tunnel spectroscopy rates). A nonzero measurement (false color) indicates the presence of an eigenstate of the probed system at a given energy (ordinate) and $s$ (abscissa). Panel (a) shows the measured eigenspectrum for the two-qubit system as a function of $s$. Panel (b) shows a similar set of measurements for the eight-qubit system. The ground-state energy $E_1$ has been subtracted from the data to aid in visualization. The solid curves indicate the theoretical expectations for the energy eigenvalues using independently calibrated qubit parameters and Hamiltonian (1). We emphasize that the solid curves are not a fit, but rather a prediction based on Hamiltonian (1) and measurements of $\mathrm{\Delta }$ and $\mathcal{E}$. The slight differences between the high-energy spectrum prediction and measurements are due to the additional states in the rf SQUID flux qubits. A full rf SQUID model that is in agreement with the measured high-energy spectrum is explored in the Supplemental Material [34]. Panels (c) and (d) show measured eigenspectra of the two-qubit system versus $h_1=h_2\equiv h_i$ for two values of annealing parameter $s$, $s=0.339$ and $s=0.351$ from left to right, respectively. Note the avoided crossing at $h_i=0$. Panels (e) and (f) show analogous measured eigenspectra for the eight-qubit system (with $h_1=\cdots =h_8\equiv h_i$). Because the eight-qubit gap closes earlier in QA for this system, we show measurements for smaller $s$, $s=0.271$ and $s=0.284$ from left to right, respectively.

Figure 4

(a),(b) Measurements of the occupation fraction, or population, of the ground state ($P_1$) and first excited state ($P_2$) of the two-qubit and eight-qubit system, respectively, versus $s$. Early in the annealing trajectory, $g\gg k_{B}T$, and the system is in the ground state with $P_1\lesssim 1$. The solid curves show the equilibrium Boltzmann predictions for $T=12.5\mathrm{mK}$. (c) Concurrence $\mathcal{C}$, negativity $\mathcal{N}$, and witness ${\mathcal{W}}_{\chi }$ versus $s$ for the two-qubit system. Early in QA, the qubits are weakly interacting, thus resulting in limited entanglement. Entanglement peaks near $s=0.37$. For larger $s$, the gap between the ground and first excited state shrinks and thermal occupation of the first excited state rises, thus extinguishing entanglement. Solid curves indicate the expected theoretical values of each witness or measure using Hamiltonian (1) and Boltzmann statistics. (d) Negativity $\mathcal{N}$ and witness ${\mathcal{W}}_{\chi }$ versus $s$ for the eight-qubit system. For all $s$ shown, the nonzero negativity $\mathcal{N}$ and nonzero witness ${\mathcal{W}}_{\chi }$ report entanglement. For $s>0.39$ and $s>0.312$ for the two-qubit and eight-qubit systems, respectively, the shaded gray region denotes the regime in which the ground and first excited states cannot be resolved via our spectroscopic method. Solid curves indicate the expected theoretical values of each witness or measure using Hamiltonian (1) and Boltzmann statistics.

Figure 5

Upper limit of the quantity $\mathrm{Tr}[{\mathcal{W}}_{AB}\rho ]$ versus $s$ for several bipartitions $A-B$ of the eight-qubit system. When this quantity is less than 0, the system is entangled with respect to this bipartition. The solid dots show the upper limit on $\mathrm{Tr}[{\mathcal{W}}_{AB}\rho ]$ for the median bipartition. The open dots above and below these are derived from the two bipartitions that give the highest and lowest upper limits on $\mathrm{Tr}[{\mathcal{W}}_{AB}\rho ]$, respectively. For the points at $s>0.3$, the measurements of $P_1$ and $P_2$ do not constrain $\rho$ enough to certify entanglement.

Figure 6

(a) $\mathrm{\Delta }(s)$ versus $s$. We show measurements for all eight qubits studied in this work. We use a single-qubit Landau-Zener experiment to measure $\mathrm{\Delta }/h<100\mathrm{MHz}$ [39]. We use QTS to measure $\mathrm{\Delta }/h>1\mathrm{GHz}$ [33]. The red line shows the theoretical prediction for a rf SQUID model employing the median qubit parameters of the eight devices. The vertical black line separates coherent (left) and incoherent (right) evolution as estimated by analysis of single-qubit spectral line shapes. (b) $|I_q^p|(s)$ versus $s$. We show measurements for all eight qubits studied in this work. We use a two-qubit coupled flux measurement with the interqubit coupling element set to 1.37 pH [8]. The red line shows the theoretical prediction for a rf SQUID model employing the median qubit parameters of the eight devices.

Figure 7

Typical waveforms during QTS. We prepare the initial state by annealing probe and system qubits from $s=0$ to $s=1$ in the presence of a large polarization bias ${\epsilon }_{\text{pol}}$. We then bias the system qubit $q_1$ (to which the probe is attached) to a bias ${\epsilon }_1$ and the probe qubit to a bias ${\epsilon }_P$. With these biases asserted, we then adjust the system qubits’ annealing parameter to an intermediate point $s^{*}$ and the probe qubit to a point $s_P$ and dwell for a time $\tau$. Finally, we complete the anneal $s\to 1$ and readout the state of the qubits.

Figure 8

Spectroscopy data for two FM-coupled qubits at ${\tilde{J}}_P<0$. (a) Measurements of tunneling rate $\mathrm{\Gamma }$ for three values of $h_1=h_2\equiv h_i$. These data were taken at $s=0.339$. Peaks in $\mathrm{\Gamma }$ reveal the energy eigenstates of the two-qubit system. (b) Multiple scans of $\mathrm{\Gamma }$ for different values of $h_i$ assembled into a two-dimensional color plot. For better interpretability, we have subtracted off a baseline energy with respect to (a) such that the ground and first excited levels are symmetric about zero. Note the avoided crossing at $h_i=0$. The peak tunneling rate $\mathrm{\Gamma }\sim |{\mathrm{\Delta }}_P⟨{\psi }_0^L|n⟩{|}^2$ [33]. The solid black and white curves plot the theoretical expectations for the energy eigenvalues using independent measurements shown in Fig. 6 and Hamiltonian (1).