Solving Systems of Linear Equations with a Superconducting Quantum Processor

Superconducting quantum circuits are a promising candidate for building scalable quantum computers. Here, we use a four-qubit superconducting quantum processor to solve a two-dimensional system of linear equations based on a quantum algorithm proposed by Harrow, Hassidim, and Lloyd [Phys. Rev. Lett. 103, 150502 (2009)], which promises an exponential speedup over classical algorithms under certain circumstances. We benchmark the solver with quantum inputs and outputs, and characterize it by nontrace-preserving quantum process tomography, which yields a process fidelity of $0.837\pm 0.006$. Our results highlight the potential of superconducting quantum circuits for applications in solving large-scale linear systems, a ubiquitous task in science and engineering.

Figure 1

(a) False color photomicrograph and (b) simplified circuit schematic of the superconducting quantum circuit for solving $2\times 2$ linear equations. Shown are the four Xmon qubits, marked from $Q_1$ to $Q_4$, and their corresponding readout resonators, marked from $R_1$ to $R_4$.

Figure 2

(a) Compiled quantum circuits for solving $2\times 2$ linear equations with four qubits. There are three subroutines and more than 15 gates as indicated. (b) The Bloch-sphere illustration of the controlled rotation subroutine for $C=1$, where the two rotation angles of $\pi$ and $\pi /3$ are achieved for eigenvalues ${\lambda }_1=1$ ($|01⟩$) and ${\lambda }_2=2$ ($|10⟩$), respectively. (c) The $Q_3–Q_4$ cz gate. Top: qubit energy level arrangement showing that the $|0⟩\leftrightarrow |1⟩$ transition of the target $Q_4$ is on resonance with the $|1⟩\leftrightarrow |2⟩$ transition of the control $Q_3$. Only when $Q_3$ is in $|1⟩$, the state in $Q_4$ will make a full cycle and gain an additional phase of $\pi$. Middle: the quantum circuits for calibrating the cz gate sandwiched in between two $\pi /2$ rotations, where the second $\pi /2$ rotation axis has an angle $\theta$ from the $X$ axis in the $XY$ plane of the Bloch sphere. Bottom: the calibrated Ramsey interference curves of $Q_4$ when $Q_3$ is in $|0⟩$ (blue) and $|1⟩$ (red), which differ by a phase of $\pi$. No single-qubit phase gates were used during this measurement to cancel the dynamical phase due to the change of qubit frequency.

Figure 3

(a) 18 input quantum states indexed by the number $j$ on the Bloch sphere that are used to test the quantum linear solver. (b) Expectation values of the three operators, where $\{X,Y,Z\}$ are the Pauli operators $\{{\sigma }_x,{\sigma }_y,{\sigma }_z\}$ for the output state $|x⟩$ when the input state $|b_1⟩=|0⟩$. (c) Fidelity values of the output states corresponding to the 18 input states $|b_j⟩$ that are labeled on the Bloch sphere in (a). Statistical errors are shown by error bars, defined as $\pm 1$ s.d., using the repeated sets of the QST measurement.

Figure 4

Nontrace-preserving QPT characterizing the quantum linear solver. Shown are the real parts of the experimental ${\chi }_{\exp }$ matrix (bars with color) and the ideal ${\chi }_{\mathrm{id}}$ matrix (black frames), where $I$ is the identity and $\{X,Y,Z\}$ are the Pauli operators $\{{\sigma }_x,{\sigma }_y,{\sigma }_z\}$. All imaginary components (data not shown) of ${\chi }_{\exp }$ are measured to be no higher than 0.015 in magnitude.