Variational quantum algorithms with invariant probabilistic error cancellation on noisy quantum processors

TL;DR

This work tackles the challenge of running variational quantum algorithms on noisy quantum processors by integrating probabilistic error cancellation (PEC) into iterative VQAs via invariant-PEC (IPEC) and an adaptive extension (APPEC). IPEC fixes the PEC sampling circuits across iterations to stabilize variance, while APPEC gradually increases mitigation to further reduce sampling costs without sacrificing convergence. The authors demonstrate that PEC can be made practical for QAOA through these schemes, achieving large gains in convergence reliability and substantial sampling-cost reductions (up to ≈90–95% in experiments and simulations) and enabling escape from local minima in larger networks. The results suggest a promising route toward scalable, near-term quantum optimization on noisy devices, with implications for MaxCut problems and broader VQA applications.

Abstract

In the noisy intermediate-scale quantum era, emerging classical-quantum hybrid optimization algorithms, such as variational quantum algorithms (VQAs), can leverage the unique characteristics of quantum devices to accelerate computations tailored to specific problems with shallow circuits. However, these algorithms encounter biases and iteration difficulties due to significant noise in quantum processors. These difficulties can only be partially addressed without error correction by optimizing hardware, reducing circuit complexity, or fitting and extrapolation. A compelling solution is applying probabilistic error cancellation (PEC), a quantum error mitigation technique that enables unbiased results without full error correction. Traditional PEC is challenging to apply in VQAs due to its variance amplification, contradicting iterative process assumptions. This paper proposes a novel noise-adaptable strategy that combines PEC with the quantum approximate optimization algorithm (QAOA). It is implemented through invariant sampling circuits (invariant-PEC, or IPEC) and substantially reduces iteration variance. This strategy marks the first successful integration of PEC and QAOA, resulting in efficient convergence. Moreover, we introduce adaptive partial PEC (APPEC), which modulates the error cancellation proportion of IPEC during iteration. We experimentally validated this technique on a superconducting quantum processor, cutting sampling cost by 90.1\%. Notably, we find that dynamic adjustments of error levels via APPEC can enhance escape from local minima and reduce sampling costs. These results open promising avenues for executing VQAs with large-scale, low-noise quantum circuits, paving the way for practical quantum computing advancements.

Figures (16)

• Figure 1: Schematic of applying IPEC to VQA. (a) The flowchart of the application process. (1) Ideal VQA process. (2) Noise impact on VQA. (3) Challenges in direct PEC application. (4) Addressing challenges with IPEC. (5) Further reducing sampling costs with APPEC. (6) Getting better outcomes with IPEC. (b) Iteration dynamics of IPEC. The iteration procedure initiates with predefined parameters $\bm{[\beta,\gamma]}_{init}$. The right pathway, highlighted in red, depicts APPEC's gradual increase in the error mitigation proportion of IPEC during VQA. The parameters converged in the previous step are used as the initial parameters for the subsequent step, resulting in the final parameters $\bm{[\beta,\gamma]}_N$. On the left, depicted in blue, full IPEC mitigates 100% error throughout the iteration, yielding similar parameters $\bm{[\beta,\gamma]}\approx\bm{[\beta,\gamma]}_N$. In each iteration, labelled with $s, s_0, \cdots, s_N$, represent the iteration step, while $\Gamma, \Gamma_0, \cdots, \Gamma_N$, indicate the corresponding sampling cost.
• Figure 2: QAOA circuit design, graph representation, and error assumption. (a) Quantum circuit diagram for an n-sided 2-regular QAOA, where the circuit within the curly braces will be repeated $p$ times. (b) Graphs corresponding to the MaxCut problem in Fig.(a). The top graph, the square, corresponds to the case of $n=4$; the bottom graph corresponds to the $n$-sided 2-regular graph for any integer value of $n$. (c) Error model, denoted by $\Lambda$, targeting the 2-qubit gates in circuits, specifically the CNOT gate. $\Lambda^{-m}$ represents the corresponding (partial or full) error mitigation with PEC where $0 \leq m \leq 1$.
• Figure 3: Challenges and solutions in applying PEC to QAOA. (a) Iteration trajectories in QAOA under various conditions. The red and green lines represent the trajectory of the QAOA in the noisy and ideal noiseless cases, respectively. The purple line represents the direct use of PEC (the original PEC) in each iteration of a 4-qubit QAOA with $p=2$. To avoid obscuration, its values are scaled to 1/25. The blue lines show the trajectories after using IPEC with 10 independent tests. The orange lines represent the ZNE using linear fitting with 10 different noise scaling factors: 1 and $m$ ($m=1.2, 1.4, \cdots, 3.0$). All trajectories are normalised to show the reduction values during iteration. (b) Comparison of energy landscapes projected based on the ideal convergence parameters obtained in (a) under the constraint of $\bm{\beta} + \bm{\gamma} = \text{constant}$. (c) The probability expectations of the quantum states with and without IPEC/ZNE for distribution using parameters obtained from IPEC 4 (Tab.\ref{['table:results']}). The IPEC for distribution was conducted on 10000 samples each. The x-axis represents the measurement results corresponding to the binary representation. (d) Performance of the solutions, including IPEC and APPEC under a noise parameter $\epsilon=0.05$ and a QAOA depth $p=2$. The red line, which represents the noisy scenario, overlaps with the blue APPEC line, indicating similar performance at the start of the iterations. (e) Analysis of convergence performance at various noise levels. The blue dots indicate the distance $D$ at noise levels $\epsilon$ from 0.1 to 0.01, which displays a linear relationship as indicated by the fitted blue line. Correspondingly, the red dots and lines represent convergence positions under the same noise levels, obtained using APPEC to mitigate partial errors. (f) Assessing convergence in the context of APPEC steps. The cost function $f_{A,B}(N)$ in Eq.\ref{['eq:fab']}, directly quantifying the sampling costs in QAOA, is computed for each value of $N$. This data is fitted, depicted by the red line and dots, obtaining the constants $A=30$ and $B=12$. Initial parameters $[\bm{\beta}_{init},\bm{\gamma}_{init}]$ are specified as $[0.1, 0.5, 0.7, 0.9]$. The "Nelder-Mead" optimiser is used for all iterations. We perform a fixed 100-step iteration for (a). For (d), the optimiser stops automatically with an acceptable absolute error of $10^{-2}$ between iterations.
• Figure 4: Experimental verification of APPEC in QAOA. (a) Superconducting quantum devices used in the experiment consisted of four qubits interconnected by four couplers. (b) Evaluation of the noise characteristics by employing cycle benchmarking, which measures the occupation of the two-qubit state $|00\rangle$ across various Pauli bases to refine the quantum error model. (c) Comparison of quantum state distributions with three sets of data derived from the last few iteration points of the three curves represented in (d). The distributions of states $|101\rangle$ and $|010\rangle$ demonstrate the significant outcomes for MaxCut in QAOA. The blue and cyan bars represent experimental results with and without APPEC, respectively, with the former aligning closely with ideal results (black dashed bars) and showing an accuracy improvement from 0.887 to 0.979. The negative values in the data originate from the data post-processing in the IPEC error mitigation strategy. They can be understood as an over-correction during the process of adjusting the data. To enhance the persuasiveness of the data, this very small negative value is set to zero during the calculation of accuracy. The negative values in the data originate from the data post-processing in the IPEC error mitigation strategy. They can be understood as overcorrection during the process of rectifying data. In the calculation of fidelity, to enhance the persuasiveness of the data, this very small negative value is set to 0. (d) Iterative process diagram of QAOA. The red line represents the experimental data of QAOA using APPEC, and the black dashed lines indicate the four points where each IPEC method was applied, showing a clear downward trend compared to the orange line representing noise-containing experiments. Light and dark blue lines represent the ideal and noisy simulations, respectively, based on the error model identified by experimental measurements. We used the "Nelder-Mead" optimiser with fixed 42-step iterations. The initial parameters were [0.1, 0.7].
• Figure 5: APPEC with 6 qubits. (a) Performance of the APPEC with 6 qubits under local depolarising noise $\epsilon=0.02$. Results compare convergence in four scenarios: ideal, noisy, IPEC, and APPEC with $N=4$. The red star highlights the point where APPEC successfully avoids local minima and achieves convergence. Here, initial parameters $[\bm{\beta}_{init},\bm{\gamma}_{init}]$ are set as $[0.1, 0.5, 0.7, 0.7, 0.9, 0.95]$. We used the "Nelder-Mead" optimiser, which stops automatically with an acceptable absolute error of $10^{-2}$ between iterations. (b) Comparative analysis of the energy landscapes, projected onto the line connecting the convergence positions in ideal $(x = 0)$ and noisy $(x = 1)$ cases. Dashed lines illustrate the relative depths between two local minima in the noisy and 2/4 IPEC cases.