Efficient trainability of linear optical modules in quantum optical neural networks

Abstract

The existence of "barren plateau landscapes" for generic discrete variable quantum neural networks, which obstructs efficient gradient-based optimization of cost functions defined by global measurements, would be surprising in the case of generic linear optical quantum neural networks with coherent state inputs due to the tunability of the intensity of coherent states and the relevant unitary group having exponentially smaller dimension. We demonstrate that coherent light in m modes can be generically compiled efficiently if the total intensity scales linearly with m below a critical rate, and extend this result to cost functions based on homodyne, heterodyne, or photon detection measurement statistics. We further demonstrate efficient trainability of m mode linear optical quantum circuits for variational mean field energy estimation of positive quadratic Hamiltonians for input states that do not have energy exponentially vanishing with m.

Universal compiling and (No-)Free-Lunch theorems for continuous variable quantum learning

“…Finally, we note that for the example of compiling the identity operation considered earlier, the averaged coherent state cost function C ACS E in (16), and its approximation in (17), do not exhibit barren plateaus if the energy bound E is taken to depend on the mode number m in such a way that the maximal energy per mode E(m)/m grows sublinearly as a function of m (see Section 2 of Ref. [48]). However, we expect that more general compiling problems, such as Gaussian compiling or compiling of Kerr non-linearities, will exhibit barren plateaus.…”

“…that the probability that the cost gradient deviates from zero vanishes exponentially. Thus the landscape exhibits a barren plateau [48]. This barren plateau phenomenon can be circumvented by using a local variant of our proposed costs.…”

“…Recently, it has been shown that discrete variable VQAs can exhibit so called 'barren plateaus', where under certain conditions the gradient of the cost function vanishes exponentially with the size of the system [35][36][37][38][39][40][41][42][43][44][45][46][47]. Preliminary results further indicate continuous variable systems [48] may exhibit an analogous barren plateau phenomenon where the cost gradients vanish exponentially with the number of system modes. On such barren plateau landscapes (potentially untenably) precise measurements are required to determine the direction of steepest descent and navigate to the minimum.…”

“…A continuous variable NFL theorem cannot be derived that is strictly analogous to (48) in the discrete variable setting because there is no Haar measure over the unitary group in B(H) for infinite dimensional H. On the other hand, one is often only interested in the action of the target unitary U on Fock states only up to a finite cutoff. For example, recent proposals for efficient updates and derivatives of Gaussian gates in parameterized CV circuits utilize cutoff recursion relations for the Fock matrix elements of the gates [58].…”

“…Eq. (48) implies that a single full rank state, i.e. a state with rank d, can be used to fully learn a unitary of rank d. Thus, the truncated TMSS states defined in Eq.…”