# Programmability of covariant quantum channels

Martina Gschwendtner1,2, Andreas Bluhm3, and Andreas Winter4,5

1Munich Center for Quantum Science and Technology (MCQST), 80799 München, Germany
2Zentrum Mathematik, Technical University of Munich, 85748 Garching, Germany
3QMATH, Department of Mathematical Sciences, University of Copenhagen, 2100 Copenhagen, Denmark
4Institució Catalana de Recerca i Estudis Avançats (ICREA), Pg. Lluis Companys, 23, 08001 Barcelona, Spain
5Grup d'Informació Quàntica, Departament de Física, Universitat Autònoma de Barcelona, 08193 Bellaterra (Barcelona), Spain

### 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 $\textit{No-Programming Theorem}$), 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 $\textit{teleportation simulation}$, 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.

The idea of programmable quantum processors resembles very much its classical equivalent to program a machine such that it implements several operations. The quantum version is able to apply quantum channels to an input state specified by the user via a program state from a program register containing all relevant information. Nielsen and Chuang show in their No-Programming Theorem that devices implementing any infinite set of unitaries exactly with a finite-dimensional program register do not exist. This result kicked off the search for approximate processors, optimizing the program size for any given precision. However, we can bypass this no-go theorem by implementing particular families of quantum channels, which is the scope of this article.

Since symmetries are of fundamental importance in physics, we consider covariant quantum channels. To those, the No-Programming Theorem is not necessarily applicable. We show that an exact implementation is possible if the group describing the symmetry acts irreducibly on the channel input. Using the special block-diagonal structure of the Choi-Jamiolkowski states corresponding to the covariant quantum channels allows us to show that the program dimension is at most the sum of the dimensions of the blocks occurring in this structure. Addressing the interest in approximate versions of programmable quantum processors, we first provide upper bounds on the program dimension, which we show for arbitrary representations instead of irreducible ones. These are finally complemented by lower bounds. The approximate bounds are in general worse than the exact ones, but they apply in a more general setting. Moreover, they show that our upper bounds in the exact case are tight.

This article opens up possibilities for future work on the exact implementation in the broader setting with reducible representations.

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