Victor Albert

We study relationships between permutation-invariant, bosonic Fock-state, and spin codes, which arise in different physical systems, but exhibit close mathematical affinity. We show that, starting with classical ell-1 codes, it is possible to construct qudit permutationally invariant (PI) codes of arbitrary dimension, spin codes, and Fock state codes, called collectively SU(q) codes. To maintain control of the code parameters in this transition, we rely on a classic result from convex geometry known as Tverberg's theorem. Constructing ell-1 codes based on combinatorial patterns called Sidon sets and utilizing their Tverberg partitions, we obtain new families of SU(q) codes with distance that scales almost linearly with the code length N. This improves upon the existing designs for all the three code families and yields a conceptually new framework for constructing spin codes. We further present explicit constructions of SU(2) codes with shorter length or lower total spin/excitation than the known codes with similar parameters, new bosonic codes with exotic Gaussian gates, as well as examples of some short codes with distance larger than the known constructions.

I'll introduce a framework for constructing quantum codes defined on spheres, taking inspiration from classical spherical codes. The framework is applied to bosonic coding, obtaining multimode extensions of the cat codes that can outperform previous constructions while requiring a similar type of overhead. Our polytope-based cat codes consist of sets of points with large separation that at the same time form averaging sets known as spherical designs, a property sufficient to guarantee protection against photon losses.

We develop quantum information processing primitives for the planar rotor, the state space of a particle on a circle. The n-rotor Clifford group, U(1)n(n+1)/2 ⋊ GLn(Z), is represented by continuous U(1) gates generated by polynomials quadratic in angular momenta, as well as discrete GLn(Z) gates generated by momentum sign-flip and sum gates. Understandings in rotor Clifford group allow us to establish connections between homological rotor error-correcting codes and oscillator quantum codes, including Gottesman-Kitaev-Preskill codes and rotation-symmetric bosonic codes. Inspired by homological rotor codes, we provide a systematic construction of multi-mode rotation-symmetric bosonic codes by analoging Fock states to rotor states with non-negative angular momentum. This new family of multi-mode bosonic codes protect against dephasing and changes in occupation numbers, which we call homological number-phase codes. Their encoding and decoding circuits are readily derived from the corresponding rotor Clifford operations. In particular, we show how to non-destructively measure the oscillator phase using conditional occupation-number addition and post-selection. We also outline several rotor and oscillator varieties of the GKP-stabilizer codes. References: arXiv:2311.07679, homological rotor error-correcting codes (arXiv:2303.13723), GKP-stabilizer codes (arXiv:1903.12615)

We develop an extension of a recently introduced subspace coset state monogamy-of-entanglement game [Coladangelo, Liu, Liu, and Zhandry; Crypto'21] to general group coset states, which are uniform superpositions over elements of a subgroup to which has been applied a group-theoretic generalization of the quantum one-time pad. We give a general bound on the winning probability of a monogamy game constructed from subgroup coset states that applies to a wide range of finite and infinite groups. To study the infinite-group case, we use and further develop a measure-theoretic formalism that allows us to express continuous-variable measurements as operator-valued generalizations of probability measures. We apply the monogamy game bound to various physically relevant groups, yielding realizations of the game in continuous-variable modes as well as in rotational states of a polyatomic molecule. We obtain explicit strong bounds in the case of specific group-space and subgroup combinations. As an application, we provide the first proof of one sided-device independent security of a squeezed-state continuous-variable quantum key distribution protocol against general coherent attacks.

Abstract Noise in quantum metrology reduces the sensitivity to which one can determine an unknown parameter in the evolution of a quantum state, such as time. Here, we consider a probe system prepared in a pure state that evolves according to a given Hamiltonian. We study the resulting local sensitivity of the probe to time after the application of a given noise channel. We show that the decrease in sensitivity due to the noise is equal to the sensitivity that the environment gains with respect to the energy of the probe. We obtain necessary and sufficient conditions for when the probe does not suffer any sensitivity loss; these conditions are analogous to, but weaker than, the Knill-Laflamme quantum error correction conditions. New upper bounds on the sensitivity of the noisy probe are obtained via our uncertainty relation, by applying known sensitivity lower bounds on the environments system. Our time-energy uncertainty relation also generalizes to any two arbitrary parameters whose evolutions are generated by Hermitian operators. This uncertainty relation asserts a general trade-off between the sensitivities that two parties can achieve for any two respective parameters of a single quantum system, in terms of the commutator of the associated generators. We consider applications to strongly interacting many-body probes. We find probe states for general interaction graphs of Ising and Heisenberg interactions that are robust to any single located error. For a 1D spin chain with nearest-neighbor interactions subject to amplitude damping noise on each site, we verify numerically that our probe state does not lose any sensitivity to first order in the noise parameter.