Variational quantum algorithms for nonlinear problems

We show that nonlinear problems including nonlinear partial differential equations can be efficiently solved by variational quantum computing. We achieve this by utilizing multiple copies of variational quantum states to treat nonlinearities efficiently and by introducing tensor networks as a programming paradigm. The key concepts of the algorithm are demonstrated for the nonlinear Schrödinger equation as a canonical example. We numerically show that the variational quantum ansatz can be exponentially more efficient than matrix product states and present experimental proof-of-principle results obtained on an IBM Q device.

Figure 1

(a) Quantum network for summing a nonlinear function of the form $F={f^{\left(1\right)}}^{*}{\prod }_{j=1}^r\left(O_j{f}^{\left(j\right)}\right)$. The ancilla qubit on the top line undergoes Hadamard gates $\widehat{H}$ and controls operations of the rest of the network via the control port (CP) and is measured in the computational basis. Starting from product states $|\mathit{0}\rangle$ of $n$ qubits shown as thick lines, variational states ${|\psi \left(\mathit{\lambda }\right)}_j\rangle ={\widehat{U}}_j\left(\mathit{\lambda }\right)|\mathit{0}\rangle$ representing the functions $f^{\left(j\right)}$ are created and fed to the QNPU through input ports (IPs). The QNPU contains problem-specific quantum networks defining the linear operators $O_j$ and has the output ports (OPs). (b) Network $\widehat{U}\left(\mathit{\lambda }\right)$ of depth $d=5$ with $n=6$. The values $\mathit{\lambda }=\{{\mathit{\lambda }}_1,{\mathit{\lambda }}_2,...\}$ determine the form of the two-qubit gates. (c) Cost function $\mathcal{C}\left(\lambda \right)$ for the nonlinear Schrödinger equation (1) for a harmonic potential $V$, a single variational parameter $\lambda$, and $g={10}^4$ (solid line) and $g=10$ (dashed line). The arrows indicate optimal values $\lambda ={\lambda }_{\min }$. (d) Solutions ${\left|f\left(x\right)\right|}^2$ for $\lambda ={\lambda }_{\min }$ with $N=4$ grid points and periodic boundary conditions $f\left(b\right)=f\left(a\right)$ for $g={10}^4$ (solid line) and $g=10$ (dashed line). The circles and squares are numerically exact results. Experimental data in (c) and (d) were obtained, from a modified circuit, on IBM Q devices (see [35] for details).

Figure 2

(a) QNPU circuit calculating the nonlinear term ${\left|\psi \right|}^4$. The networks are shown for $n=3$ and all input ports are fed the same variational quantum states created by $\widehat{U}\left(\mathit{\lambda }\right)$. IP2 is fed $\widehat{U}\left(\mathit{\lambda }\right)$ and IP3 is fed ${\widehat{U}}^{*}\left(\mathit{\lambda }\right)$. (b) QNPU circuit for working out the potential energy term $\tilde{V}{\left|\psi \right|}^2$.

Figure 3

(a) Numerically exact solution ${\left|f\left(x\right)\right|}^2$ of Eq. (1) on a logarithmic scale, $V\left(x\right)$ in Eq. (6) with $s_1=2\times {10}^4$ and ${\kappa }_1=2\pi \times 32$. The green thin line (blue thick line) is for $s_1/s_2=200$ ($s_1/s_2=2$). (b) A log-log plot of the IPR (top panel) and lin-log plot of $S_{\text{max}}$ of the exact solution $|{\psi }^{\text{exact}}\rangle$ (bottom panel) as a function of ${\kappa }_1$. Green crosses (blue circles) correspond to $s_1/s_2=200$ ($s_1/s_2=2$). (c) Representation error ${\epsilon }_{\text{R}}$ of the exact solution for the quantum ansatz (thick lines) and the MPS ansatz (thin lines) as a function of $\mathcal{N}/N$. All curves are for $s_1/s_2=2$ and correspond to $s_1=2\times {10}^4$ and ${\kappa }_1=2\pi \times 32$ (green dash-dotted line), $s_1=8\times {10}^4$ and ${\kappa }_1=2\pi \times 64$ (blue dashed line), and $s_1=3.2\times {10}^5$ and ${\kappa }_1=2\pi \times 128$ (purple solid line). The inset shows $\mathcal{N}$ as a function of ${\kappa }_1$ for ${\epsilon }_{\text{R}}=0.05$ and $s_1/s_2=2$. Blue squares (green triangles) correspond to the quantum ansatz (MPS ansatz). All curves in (a)–(c) are for $N=2^{13}=8192$ grid points and $g=50$.