Programmability of covariant quantum channels

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.

Covariant catalysis requires correlations and good quantum reference frames degrade little

“…Note added. Results closely related to Theorem 5, also using the quantitative information-disturbance tradeoff as a proof-technique, have been previously derived in the context of so-called "no-programming theorems" for quantum computers [70,71] and for estimating the fundamental energy costs of implementing unitary gates on a quantum computer [72,73] (where the energy-source corresponds to the quantum reference frame in our formulation). We thank Yuxiang Yang for making us aware of the close resemblance between these results and ours after the first version of this manuscript appeared as a preprint.…”

Infinitesimal reference frames suffice to determine the asymmetry properties of a quantum system

“…Given this, such concepts find application in quantum metrology [21], symmetry-constrained dynamics [17,22,23], quantum reference frames [19,[24][25][26][27][28], thermodynamics [29][30][31], measurement theory [32][33][34][35][36], macroscopic coherence [37], and quantum speed-limits [38]. More recent work has seen a renewed interest in quantum reference frames in a relativistic setting and the problem of time in quantum physics [39][40][41][42][43][44][45][46][47][48][49][50], as well as applications in quantum computing and covariant quantum error-correcting codes [51][52][53][54][55][56][57], where the Eastin-Knill theorem provides an obstacle to transversal gate-sets forming a continuous unitary sub-group [51].…”

Infinitesimal reference frames suffice to determine the asymmetry properties of a quantum system

“…Given this, such concepts find application in quantum metrology [21], symmetry-constrained dynamics [17,22,23], quantum reference frames [19,[24][25][26][27][28], thermodynamics [29][30][31], measurement theory [32][33][34][35][36], macroscopic coherence [37], and quantum speed-limits [38]. More recent work has seen a renewed interest in quantum reference frames in a relativistic setting and the problem of time in quantum physics [39][40][41][42][43][44][45][46][47][48][49][50], as well as applications in quantum computing and covariant quantum error-correcting codes [51][52][53][54][55][56][57] where the Eastin-Knill theorem provides an obstacle to transversal gate-sets forming a continuous unitary subgroup.…”