Robustness of quantum algorithms against coherent control errors

Coherent control errors, for which ideal Hamiltonians are perturbed by unknown multiplicative noise terms, are a major obstacle for reliable quantum computing. In this paper we present a framework for analyzing the robustness of quantum algorithms against coherent control errors using Lipschitz bounds. We derive worst-case fidelity bounds which show that the resilience against coherent control errors is mainly influenced by the norms of the Hamiltonians generating the individual gates. These bounds are explicitly computable even for large circuits and they can be used to guarantee fault tolerance via threshold theorems. Moreover, we apply our theoretical framework to derive a guideline for robust quantum algorithm design and transpilation, which amounts to reducing the norms of the Hamiltonians. Using the three-qubit quantum Fourier transform as an example application, we demonstrate that this guideline targets robustness more effectively than existing ones based on circuit depth or gate count. Furthermore, we apply our framework to study the effect of parameter regularization in variational quantum algorithms. The practicality of the theoretical results is demonstrated via implementations in simulation and on a quantum computer.

Figure 1

Evolution of the quantum state resulting from applying $U_{\mathrm{ideal}}=R_{\mathrm{z}}\left(\frac{\pi }{4}\right)R_{\mathrm{y}}\left(\frac{\pi }{2}\right)$ (blue dotted line) and $U_{\mathrm{ideal}}^{\prime }=R_{\mathrm{z}}(-\frac{3\pi }{4})R_{\mathrm{y}}(-\frac{\pi }{2})$ (red dashed line) to $|0\rangle$. Additionally, the final states for random realizations of coherent control errors are shown for both circuits (in blue and red, respectively).

Figure 2

Ideal and noisy quantum circuits.

Figure 3

Fidelities, Lipschitz bounds, gate counts, and circuit depths for five elementary gate set implementations of the three-qubit QFT, which are affected by coherent control errors. For each gate set, the figure shows the average fidelity including the standard deviation (left bar, orange), the worst-case fidelity (middle bar, blue), the Lipschitz bound (right bar, gray), the gate count (left number in parentheses), and the circuit depth (right number in parentheses).

Figure 4

Ideal circuits ${\widehat{U}}_A$ and ${\widehat{U}}_B$, and their noisy versions $U_A\left({\varepsilon }_A\right)$ and $U_B\left({\varepsilon }_B\right)$.

Figure 5

Fidelities for circuits $U_A$ (orange lines) and $U_B$ (blue lines) depending on the noise level: average (solid lines), standard deviation (shaded area), and worst case (dashed lines) on ibm_nairobi over 80 noise samples for each noise level $\overline{\varepsilon }$, and worst-case bound from Theorem 2 (dotted lines).

Figure 6

Average and standard deviation of the Lipschitz bounds for the trained quantum circuits with varying regularization parameter $\lambda$.

Figure 7

Untranspiled quantum circuit for the three-qubit QFT, which is taken from qiskit [72]. In total, the circuit consists of three Hadamard gates, three controlled-phase gates, and one swap gate.

Figure 8

Transpiled quantum circuit for the three-qubit QFT with gate set A. In total, the circuit consists of $18R_{\mathrm{Z}}, 30\sqrt{X}$, and 6 cx gates and has a Lipschitz bound of $L_{\text{QFT}}^{\text{A}}=117.95$.

Figure 9

Transpiled quantum circuit for the three-qubit QFT with gate set B. In total, the circuit consists of $20R_{\mathrm{Z}}, 30\sqrt{X}$, and 6 cz gates and has a Lipschitz bound of $L_{\text{QFT}}^{\text{B}}=106.79$.

Figure 10

Transpiled quantum circuit for the three-qubit QFT with gate set C. In total, the circuit consists of $11U_3$ and 6 cx gates and has a Lipschitz bound of $L_{\text{QFT}}^{\text{C}}=45.26$.

Figure 11

Transpiled quantum circuit for the three-qubit QFT with gate set D. In total, the circuit consists of 11 phased-xz gates, four cz gates, and one FSIM gate and has a Lipschitz bound of $L_{\text{QFT}}^{\text{D}}=36.95$.

Figure 12

Transpiled quantum circuit for the three-qubit QFT with gate set E. In total, the circuit consists of $16R_{xy}$ and $6U_{zz}$ gates and has a Lipschitz bound of $L_{\text{QFT}}^{\text{E}}=29.42$.