Abstract
A programmable quantum processor uses the states of a program register to specify one element of a set of quantum channels which is applied to an input register. It is well-known that such a device is impossible with a finite-dimensional program register for any set that contains infinitely many unitary quantum channels (Nielsen and Chuang's No-Programming Theorem), meaning that a universal programmable quantum processor does not exist. The situation changes if the system has symmetries. Indeed, here we consider group-covariant channels. If the group acts irreducibly on the channel input, these channels can be implemented exactly by a programmable quantum processor with finite program dimension (via teleportation simulation, which uses the Choi-Jamiolkowski state of the channel as a program). Moreover, by leveraging the representation theory of the symmetry group action, we show how to remove redundancy in the program and prove that the resulting program register has minimum Hilbert space dimension. Furthermore, we provide upper and lower bounds on the program register dimension of a processor implementing all group-covariant channels approximately.
Highlights
Programmable quantum processors are devices which can apply desired quantum operations, specified by the user via program states, to arbitrary input states
A programmable quantum processor uses the states of a program register to specify one element of a set of quantum channels which is applied to an input register
It is well-known that such a device is impossible with a finite-dimensional program register for any set that contains infinitely many unitary quantum channels (Nielsen and Chuang’s No-Programming Theorem), meaning that a universal programmable quantum processor does not exist
Summary
Programmable quantum processors are devices which can apply desired quantum operations, specified by the user via program states, to arbitrary input states. We give a different construction of covariant programmable quantum processors based on teleportation in Subsection 3.3 This construction is subsequently concatenated with a compression map which allows us to utilize the special structure of the Choi-Jamiołkowski states corresponding to the covariant channels. They are in general worse than the exact bounds in Theorem 25, but they apply more generally This result is the only one in which we consider arbitrary representations U instead of irreducible ones. In Theorem 31, we provide lower bounds on the program dimension of approximate covariant quantum processors This shows that the construction in Theorem 25 is optimal for the exact case
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