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arXiv:1911.00099 · Physical Review X

Robust Encoding of a Qubit in a Molecule

Victor V. Albert, Jacob P. Covey, John P. Preskill

Published Sep 1, 2020• 153 citations
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Abstract

We construct quantum error-correcting codes that embed a finite-dimensional code space in the infinitedimensional Hilbert 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 non-Abelian groups and develop orthogonality relations for coset spaces, laying the groundwork for quantum information processing with exotic configuration spaces.

DOI
10.1103/physrevx.10.031050

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We construct quantum error-correcting codes that embed a finite-dimensional code space in the infinitedimensional Hilbert 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 non-Abelian groups and develop orthogonality relations for coset spaces, laying the groundwork for quantum information processing with exotic configuration spaces.
Diatomic molecular codes [V. V. Albert, J. P. Covey, and J. Preskill, Robust encoding of a qubit in a molecule, Phys. Rev. X 10, 031050 (2020).PRXHAE2160-330810.1103/PhysRevX.10.031050] are designed to encode quantum information in the orientation of a diatomic molecule, allowing error correction from small torques and changes in angular momentum. Here, we directly study noise native to atomic and molecular platforms-spontaneous emission, stray electromagnetic fields, and Raman scattering-and show that diatomic molecular codes fail against this noise. We derive simple conditions that are sufficient for codes to protect against such noise. We also identify existing and develop new absorption-emission (Æ) codes that are more practical than molecular codes, require lower average momentum, can directly protect against photonic processes up to arbitrary order, and are applicable to a broader set of atomic and molecular systems.
The rotation of trapped molecules offers a promising platform for quantum technologies and quantum information processing. In parallel, quantum error correction codes that can protect quantum information encoded in rotational states of a single molecule have been developed. These codes are currently an abstract concept, as no implementation strategy is yet known. Here, we present a step towards experimental implementation of one family of such codes, namely absorption-emission codes. We first construct architecture-agnostic check and correction operators. These operators are then decomposed into elements of the quantum logic spectroscopy toolbox that is available for molecular ions. We then describe and analyze a measurement-based sequential as well as an autonomous implementation strategy in the presence of thermal background radiation, a major noise source for rotation in polar molecules. The presented strategies and methods might enable robust sensing or even fault-tolerant quantum computing using the rotation of individual molecules.
Quantum error-correcting codes are constructed that embed a finite-dimensional code space in the infinite-dimensional Hilbert space of a system described by continuous quantum variables. These codes exploit the noncommutative geometry of phase space to protect against errors that shift the values of the canonical variables q and p. In the setting of quantum optics, fault-tolerant universal quantum computation can be executed on the protected code subspace using linear optical operations, squeezing, homodyne detection, and photon counting; however, nonlinear mode coupling is required for the preparation of the encoded states. Finite-dimensional versions of these codes can be constructed that protect encoded quantum information against shifts in the amplitude or phase of a d-state system. Continuous-variable codes can be invoked to establish lower bounds on the quantum capacity of Gaussian quantum channels.
We formulate a quantum phase space for rotational and nuclear-spin states of rigid molecules. For each nuclear spin isomer, we re-derive the isomer's admissible angular momentum states from molecular geometry and nuclear-spin data, introduce its angular position states using quantization theory, and develop a generalized Fourier transform converting between the two. We classify molecules into three types -- asymmetric, rotationally symmetric, and perrotationally symmetric -- with the last type having no macroscopic analogue due to nuclear-spin statistics constraints. We discuss two general features in perrotationally symmetric state spaces that are Hamiltonian-independent and induced solely by symmetry and spin statistics. First, we quantify when and how an isomer's state space is completely rotation-spin entangled, meaning that it does not admit any separable states. Second, we identify isomers whose position states house an internal pseudo-spin or "fiber" degree of freedom, and the fiber's Berry phase or matrix after adiabatic changes in position yields naturally robust operations, akin to braiding anyonic quasiparticles or realizing fault-tolerant quantum gates. We outline how the fiber can be used as a quantum error-correcting code and discuss scenarios where these features can be experimentally probed
Owing to their rich internal structure and significant long-range interactions, ultracold molecules have been widely explored as carriers of quantum information. Several different schemes for encoding qubits into molecular states, both bare and field-dressed, have been proposed. At the same time, the rich internal structure of molecules leaves many unexplored possibilities for qubit encodings. We show that all molecular qubit encodings can be classified into four classes by the type of the effective interaction between the qubits. In the case of polar molecules, the four classes are determined by the relative magnitudes of matrix elements of the dipole moment operator in the single-molecule basis. We exemplify our classification scheme by considering the encoding of the effective spin-1/2 system into nonadjacent rotational states (e.g., N = 0 and 2) of polar and nonpolar molecules with the same nuclear spin projection. Our classification scheme is designed to inform the optimal choice of molecular qubit encoding for quantum information storage and processing applications, as well as for dynamical generation of many-body entangled states and for quantum annealing.