Abstract
Quantum compiling, where a parameterized quantum circuit is trained to learn a target unitary, is an important primitive for quantum computing that can be used as a subroutine to obtain optimal circuits or as a tomographic tool to study the dynamics of an experimental system. While much attention has been paid to quantum compiling on discrete variable hardware, less has been paid to compiling in the continuous variable paradigm. Here we motivate several, closely related, short depth continuous variable algorithms for quantum compilation. We analyse the trainability of our proposed cost functions and numerically demonstrate our algorithms by learning arbitrary Gaussian operations and Kerr non-linearities. We further make connections between this framework and quantum learning theory in the continuous variable setting by deriving No-Free-Lunch theorems. These generalization bounds demonstrate a linear resource reduction for learning Gaussian unitaries using entangled coherent-Fock states and an exponential resource reduction for learning arbitrary unitaries using Two-Mode-Squeezed states.
Highlights
Progress in experimental implementations of quantum optical neural networks [1,2,3] and extensions of quantum machine learning frameworks to the continuous-variable (CV) setting [4,5,6] indicate that quantum photonics is a viable platform for near-term quantum algorithms
We focus on the Loschmidt echo two-mode-squeezed states (TMSSs) cost, but analogous arguments follow for the ricocheted TMSS cost and the averaged coherent state cost
We note that, for the example of compiling the identity operation considered earlier, the averaged coherent state cost function CACSE in Eq (16), and its approximation in Eq (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
Summary
Progress in experimental implementations of quantum optical neural networks [1,2,3] and extensions of quantum machine learning frameworks to the continuous-variable (CV) setting [4,5,6] indicate that quantum photonics is a viable platform for near-term quantum algorithms. Our CV compiling algorithms make use of Gaussian measurements and CV resources such as intensity and quadrature squeezing, and so do not require preparation of exotic optimal probe states as in an optimal quantum sensing protocol, nor a large number of measured observables as in process tomography In this sense, CV compiling provides a new tool for experimental physics. The bounds highlight how utilizing entangled training states can reduce the amount of training data required to learn an unknown unitary and entanglement could be seen to provide a “free lunch.” We further use these results as an alternative motivation for the cost functions we propose for quantum compiling.
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