Quantum energy landscape and circuit optimization

We study the effects of entanglement and control parameters on the symmetry of the energy landscape and optimization performance of the variational quantum circuit. Through a systematic analysis of the Hessian spectrum, we characterize the local geometry of the energy landscape at a random point and along an optimization trajectory. We argue that decreasing the entangling capability and increasing the number of circuit parameters have the same qualitative effect on the Hessian eigenspectrum. Both the low-entangling capability and the abundance of control parameters increase the curvature of nonflat directions, contributing to the efficient search of area-law entangled ground states as to the optimization accuracy and the convergence speed.

Figure 1

Variational quantum circuit used in this paper: (a) circuit architecture and (b) structure of the two-qubit gate.

Figure 2

Plot of (a) ${\text{Var}}_{\mathit{\theta }}\left[{\mathbf{\nabla }}_a\mathcal{L}\right]$ and (b) ${\mathbb{E}}_{\mathit{\theta }}\left[{\rho }_{\alpha \beta }\left(\mathit{\theta }\right){\rho }_{\alpha \beta }\left(\mathit{\theta }\right)\right]$ with 56 circuit layers, estimated from 1500 samples over different probability $p\in \{0.0,0.1,...,0.9\}$ that removes the two-qubit entanglers.

Figure 3

Collections of 50 VQA instances to approximate the $n=12$ Ising ground state with the circuits with $L=56$ layers, for each probability $p$ of omitting the cz gates: (a) energy difference $\mathrm{\Delta }E$, (b) Rényi-2 entropy ${\mathcal{R}}^{\left(2\right)}$, (c) percent that reaches $\mathrm{\Delta }E<0.1$, (d) number of updates $\tau$ to reach $\mathrm{\Delta }E<0.1$. The VQA optimizations are successful for $p\in [0.2,0.8]$.

Figure 4

Collection of 500 sample Hessian eigenspectra at $L=56$ with different probabilities $p$ of omitting cz gates, after $\tau$ parameter updates: (a) $\tau =0$ and (b) $\tau =5\times {10}^3$.

Figure 5

Characteristic plots for the Hessian eigenspectrum with $n=12$ qubits and $L=56$ layers, based on 50 instances for each probability $p$ to omit the cz gates, at randomly initialized circuit parameters: (a) top and bottom eigenvalues, (b) percent of large eigenvalues satisfying $\left|\lambda \right|>5$, (c) percent of small eigenvalues satisfying $\left|\lambda \right|<0.2$, and (d) gradient overlap with ${\mathcal{P}}_{\text{small}}$.

Figure 6

Characteristic plots for the Hessian eigenspectrum with $n=12$ qubits and $L=56$ layers, based on 50 VQA instances for each probability $p$ to omit the cz gates, after 5000 steps of the parameter update: (a) top and bottom eigenvalues, (b) percent of large eigenvalues satisfying $\left|\lambda \right|>25$, (c) percent of small eigenvalues satisfying $\left|\lambda \right|<0.2$, and (d) gradient overlap with ${\mathcal{P}}_{\text{small}}$.

Figure 7

Evolution of the Hessian eigenvalues, the overlap of the energy gradient with the small or large curvature subspaces, the energy gap from the ground state, and the Rényi-2 entropy: (a) top eigenvalue ${\lambda }_{\text{max}}$, (b) bottom eigenvalue ${\lambda }_{\text{min}}$, (c) percent of large eigenvalues ($\left|\lambda \right|>25$), (d) percent of small eigenvalues ($\left|\lambda \right|<0.2$), (e) gradient overlap with ${\mathcal{P}}_{\text{small}}$, (f) gradient overlap with ${\mathcal{P}}_{\text{large}}$, (g) energy difference $\mathrm{\Delta }E$, and (h) Rényi-2 entropy ${\mathcal{R}}^{\left(2\right)}$. Their estimation is based on a collection of 26 Hessian samples for every $0\le p<1$ and $\tau \in \{a\times {10}^b:a\times {10}^b\le 5000\text{and}0\le a,b\le 9\}$.

Figure 8

Collection of 50 VQA instances to approximate the $n=12$ Ising ground state for each circuit depth $L$ that counts both 56 entangling and $L-56$ rotation layers: (a) energy difference $\mathrm{\Delta }E$, (b) Rényi-2 entropy ${\mathcal{R}}^{\left(2\right)}$, (c) percent that reaches $\mathrm{\Delta }E<0.1$, and (d) number of parameter updates $\tau$ to reach $\mathrm{\Delta }E<0.1$. Adding control parameters with a fixed amount of the entanglement capability improves the optimization performance.

Figure 9

Visualization of 286 sample Hessian eigenspectra, after $\tau$ gradient-descent updates, for each circuit depth $L$ that counts 56 entangling and $L-56$ rotation layers: (a) $\tau =0$ and (b) $\tau =5\times {10}^3$. It looks qualitatively similar to Fig. 4.

Figure 10

Characteristic plots for the Hessian eigenspectrum with 56 entangling and $L-56$ rotation layers, based on 26 instances for each $L\in \{56,64,...,112,120,144,168\}$, at randomly initialized circuit parameters: (a) top and bottom eigenvalues, (b) percent of large eigenvalues satisfying $\left|\lambda \right|>5$, (c) percent of small eigenvalues satisfying $\left|\lambda \right|<0.2$, and (d) gradient overlap with ${\mathcal{P}}_{\text{small}}$. Adding control parameters with a fixed amount of the entanglement capability has qualitatively the same effect as fixing the number of control parameters and reducing the entanglement capability. See Fig, 5, which looks qualitatively analogous.

Figure 11

Characteristic plots for the Hessian eigenspectrum with 56 entangling and $L-56$ rotation layers, based on 26 VQA instances for each $L\in \{56,64,...,112,120,144,168\}$, after 5000 steps of the parameter update: (a) top and bottom eigenvalues, (b) percent of large eigenvalues satisfying $\left|\lambda \right|>25$, (c) percent of small eigenvalues satisfying $\left|\lambda \right|<0.2$, and (d) gradient overlap with ${\mathcal{P}}_{\text{small}}$. They are qualitatively similar to Fig. 6.