Layered KIK quantum error mitigation for dynamic circuits

TL;DR

This work proposes a layer-based noise amplification approach that overcomes challenges without incurring additional overhead or experimental complexity and allows error correction to address dominant noise mechanisms, while the Layered KIK suppresses residual errors arising from leakage and correlated noise sources.

Abstract

Quantum Error Mitigation is essential for enhancing the reliability of quantum computing experiments. The adaptive KIK error mitigation method has demonstrated significant advantages, including resilience to temporal noise drifts, applicability to non-Clifford gates, and guaranteed performance bounds. However, its reliance on global noise amplification introduces limitations, such as incompatibility with mid-circuit measurements and dynamic circuits, as well as small residual errors due to unaccounted high-order Magnus noise terms. In this work, we propose a layer-based noise amplification approach that overcomes these challenges without incurring additional overhead or experimental complexity. Since the Layered KIK method is inherently compatible with mid-circuit measurements, it enables seamless integration with quantum error correction codes. This synergy allows error correction to address dominant noise mechanisms, while the Layered KIK suppresses residual errors arising from leakage and correlated noise sources. Similarly, for reducing sampling costs, Layered KIK can be combined with complementary mitigation methods for providing drift resilience and broadening the range of addressable errors.

Figures (7)

• Figure 1: A four-qubit simulation demonstrating the advantage of using the layered-based KIK (LKIK) amplification over the global KIK amplification (GKIK - single layer). In (a) and (b) the y axis is the difference $\Delta \langle A\rangle$ between the mitigated expectation value (see main text) and the ideal value, and the $x$ axis is the mitigation order $M$ with $M=0$ indicating no mitigation. The ideal expectation value is $\langle A\rangle \simeq 0.025$. In (a) the noise strength parameter is $\xi = 0.02$ and in (b) it is $\xi=0.2$. Figure (a) shows that LKIK is essential for achieving high accuracy even when the noise is weak, while Fig. (b) shows that LKIK is important when the noise is strong even when the requested target accuracy is modest. The dashed lines show the prediction of the Layered KIK formula \ref{['eq: Layered KIK formula']}. (c) An illustration of a three-layer circuit (top) being noise amplified with GKIK (middle) and LKIK (bottom). The amplification factor is three for both cases. Squares below the black horizontal line represent pulse-inverse operation.
• Figure 1: Experimental results from the AQT trapped ion quantum computer IBEX. The plots show mitigation of time-dependent error using (a) drift-resilient execution order and (b) non drift-resilient execution order. The time-dependent error is injected by using randomize compiling to convert a controllable time-dependent coherent error into a controllable time-dependent incoherent error. See text for the description of the coherent error time-dependence. In (a), a different level of noise amplification is executed every twenty shots, so that a full cycle of the KIK algorithm is executed before the noise changes. In (b), 4k shots are taken sequentially for each of the amplification factors. The results show that protocol (b) leads to unphysical results (survival probability beyond one) while protocol (a) converges to the correct result (dashed line). The blue curves exploit the Taylor coefficients while the green curve shows mitigation using the adaptive coefficient described in Ref. npjqiKIK.
• Figure 2: Local contributions to the global $\Omega_2$. (a, b) The performance of the global KIK introduced in ref. npjqiKIK is limited by the second-order Magnus term of the entire circuit, $\Omega_{2}^{G}$. $\Omega_{2}^{G}$ is calculated using a double integral whose integration domain is depicted by the purple triangle in (a). $\tau$ is the time duration of the unmitigated circuit. The same circuit can be described as a sequence of $L$ consecutive layers ($L = 4$ in (a)). As a result, ${\Omega }_{2}^{G}$ can be divided into two different types of contributions: i) the blue triangles that arise from the $\Omega_2$ of each layer, and ii) the orange squares which originate from the $\Omega_1$ commutator of different layers. Crucially, we show that in the Layered KIK protocol, the contribution of the squares is eliminated, leaving only the blue triangles contribution. Furthermore, as the layers get thinner (b), the contribution of the blue triangles becomes negligible. From this argument one can derive an upper bound on the LKIK mitigation error that scales as $1/L$. Interestingly, when the layers are sufficiently thin, we find a tighter error bound that scales as $1/L^2$. As such, LKIK is bias-free in practice, since it is always possible to choose a sufficiently large $L$ that guarantees that the residual mitigation error remains below the target experimental accuracy.
• Figure 2: An experimental comparison of the fidelity of the mitigated quantum state, using gate insertion (red) and pulse-inverse KIK (blue) on the IBM processor ibm_jakarta. The circuit comprises of two qubits that undergo ten swap operations.
• Figure 3: For the same example as in Fig. \ref{['fig:A-four-qubit-simulation']}a, the error with respect to the ideal expectation value (red dots) is plotted for order seven as a function of number of layers. The good fit to the $1/L^2$ curve confirms the prediction of Eq. \ref{['eq:34']}.