Abrupt transitions in variational quantum circuit training

Variational quantum algorithms dominate gate-based applications of modern quantum processors. The so-called layerwise trainability conjecture appears in various works throughout the variational quantum computing literature. The conjecture asserts that a quantum circuit can be trained piecewise, e.g., that a few layers can be trained in sequence to minimize an objective function. Here, we prove this conjecture false. Counterexamples are found by considering objective functions that are exponentially close (in the number of qubits) to the identity matrix. In the finite setting, we found abrupt transitions in the ability of quantum circuits to be trained to minimize these objective functions. Specifically, we found that below a critical (target-gate-dependent) threshold, circuit training terminates close to the identity and remains near to the identity for subsequently added blocks trained piecewise. A critical layer depth will abruptly train arbitrarily close to the target, thereby minimizing the objective function. These findings shed light on the divide-and-conquer trainability of variational quantum circuits and apply to a wide collection of contemporary literature.

Figure 1

One layer of the hardware efficient Ansatz.

Figure 2

Top: Two layers of the checkerboard Ansatz. The dashed region encloses one Ansatz layer. Bottom: The entangling gate used herein, where $R_{ZZ}\left(\omega \right)=e^{-i\omega Z\otimes Z}$.

Figure 3

$k$-Toffoli gate trainability for multiple numbers of layers per stack of the HEA. For example, the green plot (corresponding to the 4-Toffoli) illustrates untrainability for up to eight layers per stack.

Figure 4

Trainability transition with the normalized value of the cost function for stacks of the HEA. We observe a waterfall effect on the critical number of layers per stack that increases with $k=n-1$.

Figure 5

Contour of 10 million points representing unitaries made by a single layer of the HEA with random parameters, and their distance to the identity and the Toffoli gate illustrating $e^{i\alpha \left({\mathit{\theta }}_{\mathbb{1}}\right)}\mathbb{1}$ is the global minimum.

Figure 6

Trainability convergence for one-, two-, and three-layer stacks of the checkerboard Ansatz approximating a 2-Toffoli gate. One- and two-layer stacks stop training after one stack trains, but the three-layer stack improves the approximation with every added stack.

Figure 7

Single layer of the HEA. Inside the dashed area $R\left({\mathit{\theta }}_{\text{R}}\right)$, outside $D\left({\mathit{\theta }}_{\text{D}}\right)$.

Figure 8

Our implementation of a single layer of the checkerboard Ansatz for three qubits, $R_{ZZ}\left(\omega \right)=e^{-i\omega Z\otimes Z}$.