Quantum error mitigation by hierarchy-informed sampling: chiral dynamics in the Schwinger model

Abstract

Quantum simulations on current NISQ hardware are limited by its noisy nature, making efficient quantum error mitigation methods highly demanded. In this paper we introduce a novel mitigation scheme, applicable to arbitrary quantum simulations of time-dependent Hamiltonian dynamics on NISQ devices. The scheme uses a polynomial subset of extended qubit Bogoliubov-Born-Green-Kirkwood-Yvon (BBGKY) hierarchy equations as a sampling criterion of possible mitigated candidates for the quantum observables. We show that for favorable Hamiltonians the polynomial subset of BBGKY hierarchy equations leads to a polynomial overhead in both classical and quantum resources. We employ the method to mitigate simulations of the chiral magnetic effect (CME), a chiral feature of the Schwinger model. We empirically show the effectiveness of our scheme at recovering the real-time dynamics of the CME from noisy quantum simulations of the Schwinger model, for a range of different parameter values of the model. We numerically demonstrate a systematic reduction of quantum noise, together with an increasing noise reduction capability as the amount of BBGKY constraints grows.

Figures (8)

• Figure 1: Example of a BBGKY hierarchy and its structure. Each node represents a Pauli string of length $n$, while each edge represents an immediate connection. The full hierarchy can split into multiple disconnected subhierarchies, in this example $\mathcal{Q}_R$ and $\mathcal{Q}'_{R'}$. The sequence of $Q_r$ subsets, from $Q_0$ to the full subhierarchy $Q_R$, showcases the growth of the radius $r \in \qty{0,\dots,R}$. For instance, Pauli strings $\sigma_0$ and $\sigma_1$ are connected, while Pauli strings $\sigma_1$ and $\sigma_2$ are immediately connected.
• Figure 2: Schematic representation of a configuration $\va{x}$ (blue) over all time points. The $\va{x}'$ (red) configuration approximates the real-time dynamics of $\ev{\sigma_q}$ (gray).
• Figure 3: Mitigation results of the electric current evolution in time, with $r = 1$ ($R=3$) and $m = 0.1$. Each color corresponds to a different choice of $\mu_5$.
• Figure 4: Mitigation results of the electric current evolution in time, with $r = 2$ ($R=3$) and $m = 0.5$. Again each color corresponds to a different choice of $\mu_5$.
• Figure 5: Top panel: dependence of the errors with respect to the radius $r$ in the $m = 0.1$ realization. The two colors represent the two different choices of $\mu_5 \neq 0$, for which the corresponding two $L_\text{Trotter}$ overlap. Bottom panel: dependence of the two definitions of $z$ in $r$. The subhierarchy radius is $R=3$.