Yingkai Ouyang

In the quest to unlock the maximum potential of quantum sensors, it is of paramount importance to have practical measurement strategies that can estimate incompatible parameters with best precisions possible. However, it is still not known how to find practical measurements with optimal precisions, even for uncorrelated measurements over probe states. Here, we give a concrete way to find uncorrelated measurement strategies with optimal precisions. We solve this fundamental problem by introducing a framework of conic programming that unifies the theory of precision bounds for multiparameter estimates for uncorrelated and correlated measurement strategies under a common umbrella. Namely, we give precision bounds that arise from linear programs on various cones defined on a tensor product space of matrices, including a particular cone of separable matrices. Subsequently, our theory allows us to develop an efficient algorithm that calculates both upper and lower bounds for the ultimate precision bound for uncorrelated measurement strategies, where these bounds can be tight. In particular, the uncorrelated measurement strategy that arises from our theory saturates the upper bound to the ultimate precision bound. Also, we show numerically that there is a strict gap between the previous efficiently computable bounds and the ultimate precision bound.

Abstract We introduce a framework for constructing a quantum error correcting code from {\it any} classical error correcting code. This includes CSS codes and goes beyond the stabilizer formalism to allow quantum codes to be constructed from classical codes that are not necessarily linear or self-orthogonal. We give an algorithm that explicitly constructs quantum codes with linear distance and constant rate from classical codes with a linear distance and rate. As illustrations for small size codes, we obtain Steane's $7-$qubit code uniquely from Hamming's [7,4,3] code, and obtain other error detecting quantum codes from other explicit classical codes of length 4 and 6. Motivated by quantum LDPC codes and the use of physics to protect quantum information, we introduce a new 2-local frustration free quantum spin chain Hamiltonian whose ground space we analytically characterize completely. By mapping classical codewords to basis states of the ground space, we utilize our framework to demonstrate that the ground space contains explicit quantum codes with linear distance. This side-steps the Bravyi-Terhal no-go theorem because our work allows for more general quantum codes beyond the stabilizer and/or linear codes. This model may be called an example of {\it subspace} quantum LDPC codes with linear distance.

The gold-standard for security in quantum cryptographic protocols is information-theoretic security, because such a form of security that makes no assumptions on the hardness of any computational problems and relies only on the fundamental laws of quantum mechanics, will be surely future-proof. Here, we revisit a permutational-key quantum homomorphic encryption protocol with information-theoretic security. We explain how this protocol can be integrated with quantum error correction that allows the error correction encoding to be a homomorphism. This feature enables both client and server to apply the encoding and decoding step for the quantum error correction, without use of the encrypting permutation-key.