Mitigating the measurement overhead of ADAPT-VQE with optimised informationally complete generalised measurements

TL;DR

The paper tackles the measurement bottleneck of ADAPT-VQE, which arises from estimating gradients for many potential pool operators. It introduces AIM-ADAPT-VQE, combining Adaptive Informationally Complete (AIM) measurements with ADAPT-VQE so that energy estimation using IC POVMs provides data that can be reused to compute all commutators needed to select the next operator, eliminating extra quantum shots. Across H$_4$ and N$_2$ test Hamiltonians and multiple fermion-to-qubit mappings and pools, the approach converges to chemical precision with CNOT counts matching or improving upon the standard ADAPT-VQE, and remains effective even with scarce measurement data, albeit sometimes at the cost of deeper circuits. The results demonstrate a practical pathway to reduce measurement overhead, enabling larger operator pools and more scalable quantum simulations of chemical systems on near-term hardware.

Abstract

ADAPT-VQE stands out as a robust algorithm for constructing compact ansätze for molecular simulation. It enables to significantly reduce the circuit depth with respect to other methods, such as UCCSD, while achieving higher accuracy and not suffering from so-called barren plateaus that hinder the variational optimisation of many hardware-efficient ansätze. In its standard implementation, however, it introduces a considerable measurement overhead in the form of gradient evaluations trough estimations of many commutator operators. In this work, we mitigate this measurement overhead by exploiting a recently introduced method for energy evaluation relying on Adaptive Informationally complete generalised Measurements (AIM). Besides offering an efficient way to measure the energy itself, Informationally Complete (IC) measurement data can be reused to estimate all the commutators of the operators in the operator pool of ADAPT-VQE, using only classically efficient post-processing. We present the AIM-ADAPT-VQE scheme in detail, and investigate its performance with several H4 Hamiltonians and operator pools. Our numerical simulations indicate that the measurement data obtained to evaluate the energy can be reused to implement ADAPT-VQE with no additional measurement overhead for the systems considered here. In addition, we show that, if the energy is measured within chemical precision, the CNOT count in the resulting circuits is close to the ideal one. With scarce measurement data, AIM-ADAPT-VQE still converges to the ground state with high probability, albeit with an increased circuit depth in some cases.

Figures (5)

• Figure 1: Flowchart of the AIM-ADAPT-VQE algorithm. At each step of the ADAPT algorithm, the state is prepared on the quantum computer and measured with an IC POVM. A first batch of shots is collected and it is used to estimate the energy of the state, as well as the gradients of the operators in the pool. The POVM is optimised, a new batch of shots is collected and the process is repeated until the target precision is reached. The operator with the largest gradient is added to the ansatz, with an initial parameter chosen with Rotosolve. All the parameters in the ansatz are then optimised using a classical optimiser. Notice that this step requires additional measurements. However, in this work, the latter task is performed using exact simulations (that is, without shot noise), as the focus is in the estimation of the gradients of the operators in the ADAPT pool.
• Figure 2: AIM-ADAPT-VQE simulations for different target measurement precisions for a hydrogen chain $\text{H}_4$ with interatomic distance $1.5\angstrom$, using different fermion-to-qubit mappings and pools: a), d), g) JW mapping and QEB pool; b), e), h) BK mapping and spin-dependent fermionic pool; c), f), i) JKMN mapping and spin-dependent fermionic pool. All the results are obtained from 10 realisations of each experiment. The top panels, a), b), c), show the average error as a function of the ADAPT iteration. The dashed black line shows an exact statevector simulation and the horizontal dotted line the chemical precision ($1.6m\hartree$). The error bars show one standard error distance from the average. The opacity of the lines is proportional to the number of runs that have not met the stopping criteria up until that point. The second row of panels, d), e), f) show the average error as a function of the cumulative number of shots used by AIM-ADAPT-VQE. The dotted line corresponds to the chemical precision $1.6m\hartree$. The bottom panels, g), h), i), are box-and-whisker plots of the CNOT counts of the constructed circuits at the end of the AIM-ADAPT-VQE simulations. The boxes represent the distribution of CNOT counts over the realisations of each experiment. The dashed lines show the number of CNOTs obtained with a statevector simulation of ADAPT-VQE.
• Figure 3: AIM-ADAPT-VQE simulations for $\text{H}_4$ with interatomic distance $1.5\angstrom$, using the qubit-ADAPT pool and different fermion-to-qubit mappings: a), d), g) JW mapping; b), e), h) BK mapping; c), f), i) JKMN mapping. The same quantities as in Fig. \ref{['fig:fermionics']} are shown. Top panels: average error vs number of iterations; mid panels: average error vs cumulative number of measurement shots; bottom panels: distribution of the CNOT counts of the transpiled final circuit.
• Figure 4: Optimisation of the IC POVM for different AIM-ADAPT-VQE iterations. The plots show the estimated standard error in Ha as a function of the total number of shots used during steps 3-7 of the algorithm, for one run of the experiment. The panels correspond to those of Fig. \ref{['fig:fermionics']}, for the strategy $T_E = 1.6m\hartree$. The horizontal dotted line represents the chemical precision and the dashed line the shot-noise $1 / \sqrt{\text{shots}}$ scaling. At each iteration, the optimal POVM of the previous iteration is used as the initial condition. In the first iterations the lines decrease faster than the shot noise, meaning that the initial POVM is not optimal for the current ansatz and can be further refined by the algorithm. In later iterations the lines follow the shot-noise line, i.e., the POVM of the previous iteration is already quite close to optimal.
• Figure 5: AIM-ADAPT-VQE simulations for $\text{N}_2$ with interatomic distance $1.098\angstrom$, using the fermionic pool and Jordan-Wigner fermion-to-qubit mapping. The same quantities as in Fig. \ref{['fig:fermionics']} are shown: a) average error vs number of iterations; b) average error vs cumulative number of measurement shots; c) distribution of the CNOT counts of the transpiled final circuit.