Impact and mitigation of Hamiltonian characterization errors in digital-analog quantum computation

TL;DR

A protocol for mitigating calibration errors which resembles dynamical-decoupling techniques is proposed and opens the possibility of scaling digital-analog to intermediate and large scale systems while having an estimation on the errors committed.

Abstract

Digital-analog is a universal quantum computing paradigm which employs the natural entangling Hamiltonian of the system and single-qubit gates as resources. Here, we study the stability of these protocols against Hamiltonian characterization errors. For this, we bound the maximum separation between the target and the implemented Hamiltonians. Additionally, we obtain an upper bound for the deviation in the expected value of an observable. We further propose a protocol for mitigating calibration errors which resembles dynamical-decoupling techniques. These results open the possibility of scaling digital-analog to intermediate and large scale systems while having an estimation on the errors committed.

Figures (5)

• Figure 1: Representation of a generic stepwise-DAQC circuit. These circuits are composed of single qubit gates (blue) that act in between the analog blocks (yellow), which correspond to the free evolution of the system under the natural interaction Hamiltonian for a set time $t_k$. The sandwiching of SQGs changes the effective sign of the analog Hamiltonian for each block, $e^{-it_kH_\text{S}^{(k)}}$. Note that in stepwise-DAQC, the interaction Hamiltonian is turned off during the application of the SQGs. In contrast, in banged-DAQC the interaction Hamiltonian is always on while the gates are applied.
• Figure 2: Operator norm of the error Hamiltonian for different system sizes and topologies: nearest-neighbours (red), random connected graphs (yellow) and all-to-all (ATA). Here $\mathcal{S}=\mathcal{P}$. The solid line represents the mean over 10000 different problems employing the usual DAQC protocol and the colored area the interquartile range. The dashed line shows the mean value of the bound given in Eq. \ref{['eq:errorHamiltonian']}. More details about the simulations are given in the supplementary material.
• Figure 3: Representation of a hardware with a square connectivity $\mathcal{S}$ (blue) and all-to-all characterization errors $\mathcal{D}$ (red). Taking this as the unit cell of a square lattice, we can represent the second example given in Sec. \ref{['sec:stability']}. For this example, we can pictorially check that $\mathcal{S}\subset\mathcal{D}$ and $\text{deg}(\mathcal{S})<\text{deg}(\mathcal{D})$.
• Figure 4: Repetition of the simulations for Fig. \ref{['fig:bound']} but for the characterization error mitigated DAQC protocol. As the second term in Eq. \ref{['eq:errorHamiltonian']} is taken out, the bounds are tighter compared to the simulation results.
• Figure 5: Total analog time $t_\text{A}$ for a Hamiltonian simulation task with the 3 different problem topologies. We show the times for both the usual DAQC protocols and the one with the error mitigation technique. The results for the ATA and the ATA with error mitigation overlap, as both protocols are identical for this case.

Theorems & Definitions (2)

• Definition 1: Stability of a quantum simulation
• Definition 2: Stable expectation value adrian2023