Quantum speeding-up of computation demonstrated in a superconducting two-qubit processor

We operate a superconducting quantum processor consisting of two tunable transmon qubits coupled by a swapping interaction, and equipped with nondestructive single-shot readout of the two qubits. With this processor, we run the Grover search algorithm among four objects and find that the correct answer is retrieved after a single run with a success probability between $0.52$ and $0.67$, which is significantly larger than the $0.25$ achieved with a classical algorithm. This constitutes a proof of concept for the quantum speed-up of electrical quantum processors.

Figure 1

Schematic blueprint of a quantum processor based on quantum gates, represented here in the two-qubit case relevant for our experiment. A quantum processor consists of a qubit register that can perform any unitary evolution needed by an algorithm under the effect of a universal set of quantum gates (single-qubit gate $U_1$, two-qubit gate $U_2$). Ideally, all the qubits may be read projectively, and may be reset.

Figure 2

Electrical scheme of the two-qubit circuit operated and typical sequence during processor operation. (a) Two capacitively coupled transmon qubits have tunable frequencies controlled by the flux induced in their SQUID loop by a local current line. The coupling capacitance (center) yields a swapping evolution between the qubits when on resonance. Each transmon is embedded in a nonlinear resonator used for single-shot readout. Each reflected readout pulse is routed to a cryogenic amplifier through circulators, homodyned at room temperature, and acquired digitally, which yields a two-bit outcome. (b) Typical operation of the processor showing the resonant microwave pulses $a\left(t\right)$ applied to the qubits and to the readouts, on top of the dc pulses (polylines) that vary the transition frequencies of qubit I (solid) and II (dashed). With the qubits tuned at a first working point for single-qubit gates, resonant pulses are applied for performing $X$ and $Y$ rotations, as well as small flux pulses for $Z$ rotations; qubits are then moved to the interaction point for two-qubit gate operations. Such sequences can be combined as needed by the algorithm. Qubits are then moved to their initial working points for applying tomography pulses as well as a $\left|1\right.〉\to \left|2\right.〉$ pulse $X_{12}\left(\pi \right)$ to increase the fidelity of the forthcoming readout. Finally, they are moved to better readout points and read.

Figure 3

(a) Experimental sequence used for implementing the Grover search algorithm on four objects. First, $Y(\pi /2)$ rotations are applied to produce the superposition $\left|\phi \right.〉=(1/2){\sum }_{u,v}\left|uv\right.〉$ of all basis states; then one of the four possible oracles (corresponding to the four sign combinations of the $Z$ rotations) is applied. The tagged state is then decoded in all cases using an $\text{i}swap$ operation followed by $X(\pi /2)$ rotations. (b) State tomography at two steps of the algorithm ($\rho$ matrices) and success probability after a single run (histograms). The bright dots on the left-hand side mark the basis state tagged by each oracle operator used. The amplitude of each matrix element is represented by a disk [black for the ideal density matrix, red (gray) for the measured one] and its phase by an arrow (as well as a filling color for the ideal matrix). After applying the oracle, the information on the tagged state is encoded in the phase of six particular elements of $\rho$. After decoding, the tagged state should be the only matrix element present in $\rho$. The fidelity $F_{\mathrm{final}}$ actually obtained is indicated in this element. The probability distribution of the single-run readout outcomes is shown on the right-hand side (bright box for the correct answer, solid dark boxes for the wrong ones).