Layering and subpool exploration for adaptive variational quantum eigensolvers: Reducing circuit depth, runtime, and susceptibility to noise

Adaptive variational quantum eigensolvers (ADAPT-VQEs) are promising candidates for simulations of strongly correlated systems on near-term quantum hardware. To further improve the noise resilience of these algorithms, recent efforts have been directed towards compactifying, or layering, their Ansatz circuits. Here, we broaden the understanding of the algorithmic layering process in three ways. First, we investigate the noncommutation relations between the different elements that are used to build ADAPT-VQE Ansätze. In doing so, we develop a framework for studying and developing layering algorithms, which produce shallower circuits. Second, based on this framework, we develop a new subroutine that can reduce the number of quantum-processor calls by optimizing the selection procedure with which a variational quantum algorithm appends Ansatz elements. Third, we provide a thorough numerical investigation of the noise-resilience improvement available via layering the circuits of ADAPT-VQE algorithms. We find that layering leads to an improved noise resilience with respect to amplitude-damping and dephasing noise, which, in general, affect idling and nonidling qubits alike. With respect to depolarizing noise, which tends to affect only actively manipulated qubits, we observe no advantage of layering.

Figure 1

Layering: A sparse Ansatz circuit (left), as produced by standard ADAPT-VQEs, can be compressed to a dense structure (right) by layering. Boxes denote Ansatz elements. Each line represents a single qubit. Note that Ansatz circuit elements entangle two or four qubits.

Figure 2

A diagram comparing support and operator commutativity. (a) The elements both support- and operator-commute. Note that $R_y\left(\theta \right)$ only has qubit support on the top qubit line. (b) The elements operator-commute but do not support-commute. (c) The elements neither support- nor operator-commute.

Figure 3

A diagram of the distance $d$ from an element $A\in \mathcal{P}$ under the pool metric, Definition 6. The Ansatz element $A$ (black dot) is surrounded by Ansatz elements of the noncommuting set ${\mathcal{N}}_{\text{G}}(\mathcal{P},A)$ of distance 1 (white circle). All other elements $\mathcal{P}\setminus {\mathcal{N}}_{\text{G}}(\mathcal{P},A)$ have distance 2 (gray shaded region).

Figure 4

Visualization of our strategy for subpool exploration. The pool $\mathcal{P}$ is successively explored in subpools ${\mathcal{S}}_m$, with Ansatz elements $A_m$ of minimal loss generating future subpools through their generalized noncommuting set.

Figure 5

A toy example of subpool exploration. (a) The Ansatz element pool $\mathcal{P}$ comprises of six two-qubit operators acting on four qubits. Each Ansatz element is labeled 1,$...$,6. (b) Graphs visualize Ansatz elements as vertices. A vertex pair is connected by an edge if the corresponding Ansatz elements do not support-commute. The dynamics of subpool exploration are visualized as follows. Left: The initial pool is ${\mathcal{S}}_0≔\left\{3\right\}$, and $A_0≔3$ is found to be the Ansatz element of minimal loss. Center: Next, we search the noncommuting set of $A_0≔3$ given by ${\mathcal{S}}_1≔\{1,2,4,5\}$. Within ${\mathcal{S}}_1$ we determine the loss of all elements. Assume the element of minimal loss is found to be element $A_1≔1$. Right: Next, we search the noncommuting set of $A_1≔1$ given by $\{3,4,5,6\}$. We have already evaluated the losses of $\{3,4,5\}$, so we set the subpool to ${\mathcal{S}}_2≔\left\{6\right\}$ and determine its minimum loss. Assuming the loss of $A_2=6$ is larger than the loss of $A_1=1$, we return $A_{\mathcal{S}}^{*}=1$ as the local minimum. In this toy example, the whole pool is searched, and the local minimum is the global minimum.

Figure 6

Visualization of layer construction and successive pool reduction. Gray areas indicate the removal of generalized noncommuting sets corresponding to Ansatz elements $A_n$ added to the layer $\mathcal{A}$. Parameters can either be optimized once the whole layer is fixed (static layering) or after adding each Ansatz element (dynamic layering).

Figure 7

Histograms of the relative frequencies of the number of subpools searched for identifying a suitable Ansatz element $m_s$ with Explore-ADAPT-VQE. The mean and uncertainty in the mean are indicated by solid and dashed lines, respectively.

Figure 8

The energy accuracy is plotted as a function of Ansatz-circuit depth (left) and the number of Ansatz-circuit parameters (Ansatz elements, right), for QEB standard, Explore-, Static-(TETRIS)-, and Dynamic-ADAPT-VQE. These simulations use support commutation. Each row shows data for a specific molecule. The region of chemical accuracy is shaded in cream.

Figure 9

Energy accuracy against the number of loss function calls. Using the QEB pool and support commutation, we compare standard-, Explore-, Static-(TETRIS)-, and Dynamic-ADAPT-VQE. Each row shows data for a specific molecule, with the number of orbitals increasing up the page. Energy accuracies better than chemical accuracy are shaded in cream.

Figure 10

Energy accuracy against the number of times the Ansatz is optimized (left column), and the number of expectation values calculated during optimizer calls (right column). Using the QEB pool and support commutation, we compare standard-, Explore-, Static-(TETRIS)-, and Dynamic-ADAPT-VQE. Each row shows data for a specific molecule, with the number of orbitals increasing up the page. Energy accuracies better than chemical accuracy are shaded in cream.

Figure 11

Circuit diagram visualizing the layer-by-layer noise model: on the left, a noiseless Ansatz-element layer with two support-commuting Ansatz elements is decomposed into columns of native gates. On the right, noise is added to the Ansatz-element layer. For global amplitude damping (or dephasing) noise, the channel $\mathcal{F}$ (or $\mathcal{C}$) is applied to each qubit. For local depolarizing noise, the channels $\mathcal{D}$ are applied to the target of the noisy (two-qubit, cnot) gate.

Figure 12

Energy error ${\mathrm{\Delta }}_t\left(\alpha \right)$ for an Ansatz circuit ${\mathrm{\Lambda }}_t$ of ${\mathrm{H}}_4$ as a function of noise strength $\alpha ={\omega }_1,{\omega }_z,p$ going from the left to right, respectively. Connected dots and crosses are calculated using full density-matrix simulations. Dashed lines show the corresponding extrapolation using noise susceptibility.

Figure 13

Amplitude-damping noise susceptibility as a function of (left) energy accuracy and (right) the number of parameters in Ansatz circuits ${\mathrm{\Lambda }}_t$. QEB-ADAPT is compared to support-based Dynamic-ADAPT-VQE. Each row shows data for a specific molecule, with the number of orbitals increasing up the page. Energy accuracies better than chemical accuracy are shaded in cream.

Figure 14

Same as Fig. 13, but for dephasing noise.

Figure 15

Same as Fig. 13, but for depolarizing noise. In addition, black crosses correspond to density matrix simulations corroborating noise susceptibility via finite differences. The black crosses are discussed further in Appendix pp4.

Figure 16

The ratio of the minimal $T_1$ and $T_2$ times and maximal depolarizing probability $p$ for Dynamic- and Static-ADAPT-VQE to standard ADAPT-VQE required to reach an accuracy of $\mathrm{\Delta }={10}^{-7} \mathrm{mHa}$ for each molecule.

Figure 17

Noise susceptibility for (a) amplitude-damping, (b) dephasing, and (c) depolarizing noise as a function of (a),(b) Ansatz-circuit depths $d$ and (c) the number of noisy cnot-gates $N_{II}$. The top panels show noise susceptibility as a function of $d$ (a),(b) and $N_{II}$ (c). The bottom panels of (a) and (b) show noise susceptibility divided by $d$, as a function $d$. The bottom panel of (c) shows noise susceptibility divided by $N_{II}$ as a function of $N_{II}$. The solid lines correspond to the simulation of Dynamic-ADAPT-VQE using support commutation for a range of molecules. The dashed lines represent simulations based on QEB-ADAPT-VQE.

Figure 18

Energy accuracy against Ansatz-circuit depths (left) and the number of Ansatz-circuit parameters (Ansatz elements, right), for qubit-Dynamic-ADAPT-VQE using support and operator commutation—dashed and solid lines, respectively. Each row shows data for a specific molecule, with the number of orbitals increasing up the page. Energy accuracies better than chemical accuracy are shaded in cream.

Figure 19

Energy accuracy against Ansatz-circuit depths (left), the number of Ansatz-circuit parameters (Ansatz elements, center), and the number of cnot gates in the Ansatz circuit (right), for qubit-Dynamic-ADAPT-VQE using support commutation as well as QEB standard and Dynamic-ADAPT-VQE using support commutation with both steepest gradient and largest energy reduction decision rules, are compared. Each row shows data for a specific molecule, with the number of orbitals increasing up the page. Energy accuracies better than chemical accuracy are shaded in cream.

Figure 20

Energy accuracy against the number of loss function calls (left), the number of times the Ansatz is optimized (center), and the number of expectation values calculated during optimizer calls (right), for qubit-Dynamic-ADAPT-VQE using support and operator commutation, as well as QEB standard and Dynamic-ADAPT-VQE using support commutation with both the steepest gradient and largest energy reduction decision rules, are compared. Each row shows data for a specific molecule, with the number of orbitals increasing up the page. Energy accuracies better than chemical accuracy are shaded in cream.