Quantum Fourier analysis for multivariate functions and applications to a class of Schrödinger-type partial differential equations

In this work we develop a highly efficient representation of functions and differential operators based on Fourier analysis. Using this representation, we create a variational hybrid quantum algorithm to solve static, Schrödinger-type, Hamiltonian partial differential equations (PDEs), using space-efficient variational circuits, including the symmetries of the problem, and global and gradient-based optimizers. We use this algorithm to benchmark the performance of the representation techniques by means of the computation of the ground state in three PDEs, i.e., the one-dimensional quantum harmonic oscillator and the transmon and flux qubits, studying how they would perform in ideal and near-term quantum computers. With the Fourier methods developed here, we obtain low infidelities of order ${10}^{-4}–{10}^{-5}$ using only three to four qubits, demonstrating the high compression of information in a quantum computer. Practical fidelities are limited by the noise and the errors of the evaluation of the cost function in real computers, but they can also be improved through error mitigation techniques.

Figure 1

Quantum interpolation algorithms. (a) Algorithm to recreate a finer interpolation in position space $|f^{(n+m)}\rangle$ adding $m$ qubits to a previous discretization $|f^{\left(n\right)}\rangle$. (b) Algorithm to recreate a momentum space discretization with $n+m$ qubits from $|f^{\left(n\right)}\rangle$.

Figure 2

Variational representation of one-dimensional smooth functions. (a) $R_Y$ Ansatz of depth one, with full entanglement over three qubits. (b) ZGR Ansatz to represent a function $f\left(x\right)$ with three qubits. (c) Ansatz to represent an (anti)symmetric function $g\left(x\right)={\left(\text{sgn}x\right)}^{\text{sym}}f\left(\right|x\left|\right)$, where $U_f$ encodes $f(x>0)$.

Figure 3

Variational quantum PDE solver. We use a quantum computer to initialize two quantum circuits in the states $|f_{{\mathit{\theta }}_i}^{\left(n\right)}\rangle$ and $|{\tilde{f}}_{{\mathit{\theta }}_i}^{\left(n\right)}\rangle$. We estimate the expectation values of the $V\left(\widehat{\mathbf{x}}\right)$ and $D\left(\widehat{\mathbf{p}}\right)$ operators from measurements in the quantum computer, thereby approximating the energy functional $E\left[\mathit{\theta }\right]$. A classical computer uses these estimates to iteratively update the parameters of the variational quantum circuit, until convergence.

Figure 4

Comparison of the infidelity figures of merit (the median and the standard deviation around the mean of 100 repetitions) for the ZGR Ansatz for the harmonic oscillator for the numerical limit. The theoretical infidelity $1-F^t$ is the infidelity of the theoretical $2^n$-point wave function interpolated up to $2^{12}$ points and the theoretical $2^{12}$-point wave function. The $n$-qubit infidelity $1-F^{\left(n\right)}$ is the infidelity of the $n$-qubit function obtained from the optimization and the $2^n$-point theoretical function. The continuous infidelity $1-F^{\infty }$ is the infidelity of the $n$-qubit function interpolated up to $2^{12}$ points and the $2^{12}$-point theoretical function.

Figure 5

Results of the simulations for two, three, four, five, and six qubits with 8192 evaluations for the harmonic oscillator using the ZGR and the $R_Y$ Ansätze with depths 1 (RY1) and 2 (RY2) and the COBYLA, SPSA, and Adam optimizers. (a) Continuous infidelity with $n+m=12$. (b) Rescaled energy $\overline{\varepsilon }$ [Eq. (40)].

Figure 6

Lowest infidelity results of the simulations for two, three, four, five, and six qubits with 8192 evaluations for the harmonic oscillator using the ZGR and the $R_Y$ Ansätze with depths 1 (RY1) and 2 (RY2). (a) Continuous infidelity with $n+m=12$. (b) Rescaled energy $\overline{\varepsilon }$ [Eq. (40)].

Figure 7

Result of the optimization for the Adam optimizer and the ZGR Ansatz for three qubits for the harmonic oscillator. (a) Absolute value of the theoretical and optimization wave functions ($\beta x$ is a dimensionless coordinate where $\beta =\sqrt{m\omega /\hslash }$). (b) Value of the energy for each iteration.

Figure 8

Result of the optimization for the Adam optimizer and the ZGR Ansatz for 8192 and $32768$ evaluations for the harmonic oscillator. (a) Continuous infidelity with $n+m=12$. (b) Rescaled energy $\overline{\varepsilon }$ [Eq. (40)].

Figure 9

Results of the simulations for two, three, four, five, and six qubits with 8192 evaluations for the transmon qubit using the ZGR and the $R_Y$ Ansätze with depths 1 (RY1) and 2 (RY2) and the COBYLA, SPSA, and Adam optimizers. (a) Continuous infidelity with $n+m=12$. (b) Rescaled energy $\overline{\varepsilon }$ [Eq. (40)].

Figure 10

Results of the simulations for two, three, four, five, and six qubits with 8192 evaluations for the flux qubit using the ZGR and the $R_Y$ Ansätze with depths 1 (RY1) and 2 (RY2) and the COBYLA, SPSA, and Adam optimizers. (a) Continuous infidelity with $n+m=12$. (b) Rescaled energy $\overline{\varepsilon }$ [Eq. (40)].

Figure 11

Absolute value of the ground state of the flux qubit. We show the theoretical eigenstate as well as the discretized and interpolated solution of our algorithm for the four-qubit RY1 Ansatz using the L-BFGS-B optimizer.

Figure 12

Results for the ideal (noiseless) and noisy (ibmq_santiago noise model) optimization for the SPSA optimizer and the RY1 Ansatz with 8192 evaluations for the harmonic oscillator. (a) Continuous infidelity with $n+m=12$. (b) Rescaled energy $\overline{\varepsilon }$ [Eq. (40)].

Figure 13

Circuit infidelity for the position $1-F_x$ and momentum $1-F_p$ circuits for the harmonic oscillator for the RY1 Ansatz for 100 repetitions with 8192 evaluations each and using the ibmq_santiago noise model.

Figure 14

Zero-noise extrapolation results (RY1 Ansatz with thermal relaxation and 100 repetitions with 8192 evaluations for each simulation). (a) Harmonic oscillator (three qubits). (b) Flux qubit (four qubits).