Abstract
We construct quantum error-correcting codes that embed a finite-dimensional code space in the infinite-dimensional Hilbert state space of rotational states of a rigid body. These codes, which protect against both drift in the body's orientation and small changes in its angular momentum, may be well suited for robust storage and coherent processing of quantum information using rotational states of a polyatomic molecule. Extensions of such codes to rigid bodies with a symmetry axis are compatible with rotational states of diatomic molecules, as well as nuclear states of molecules and atoms. We also describe codes associated with general nonabelian compact Lie groups and develop orthogonality relations for coset spaces, laying the groundwork for quantum information processing with exotic configuration spaces.
Highlights
Quantum systems described by continuous variables arise in many laboratory settings
We develop the theory of molecular codes and generalizations thereof. Laboratory realizations of these coding schemes that improve the coherence times of molecular qubits may still be far off, but we propose laying the foundations for molecular quantum error correction as a challenging goal for the physicists and chemists of the noisy intermediate-scale quantum (NISQ) era [36]
Though our work is partially motivated by advances in molecular physics, the coding methods we use are best explained in an abstract group-theoretic framework, which we summarize
Summary
A microwave resonator in a superconducting circuit or the motional degree of freedom of a trapped ion can be viewed as a harmonic oscillator with an infinite-dimensional Hilbert space. Such continuous-variable systems have potential applications to quantum information processing. A GKP code is a quantum error-correcting code designed to protect against noise that slightly shifts the position or momentum of an oscillator. Couplings between a molecule’s internal degrees of freedom can be readily engineered and utilized These features beg the question of whether it is possible to utilize the rich yet spatially compact molecular Hilbert space for quantum error correction; our work shows that such is the case.
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