Section: New Results

Quantum Information

Participants : Kaushik Chakraborty, André Chailloux, Anthony Leverrier, Jean-Pierre Tillich.

Quantum codes

Protecting quantum information from external noise is an issue of paramount importance for building a quantum computer. It also worthwhile to notice that all quantum error-correcting code schemes proposed up to now suffer from the very same problem that the first (classical) error-correcting codes had: there are constructions of good quantum codes, but for the best of them it is not known how to decode them in polynomial time.

Recent results:

• A. Leverrier and JP. Tillich, together with G. Zémor, proposed a new class of quantum LDPC codes, “Quantum expander codes”, which feature a simple and very efficient decoding algorithm which can correct arbitrary patterns of errors of size scaling as the square-root of the length of the code. These are the first codes with constant rate for which such an efficient decoding algorithm is known (see Section  5.1.3 ) [55] , [35] , [73] .

• Error analysis for Boson Sampling, a simplified model for quantum computation [21]

Quantum cryptography

A recent approach to cryptography takes into account that all interactions occur in a physical world described by the laws of quantum physics. These laws put severe constraints on what an adversary can achieve, and allow for instance to design provably secure key distribution protocols. We study such protocols as well as more general cryptographic primitives such as coin flipping with security properties based on quantum theory.

Recent results:

• A. Leverrier gave the first composable security proof for a continuous-variable quantum key distribution protocol with coherent states [22] . This essentially completes the security analysis of continuous-variable protocols with coherent states, which are by far the most practical protocols relying on continuous variables.

• A. Leverrier and E. Diamanti reviewed the state-of-the-art concerning quantum key distribution with continuous variables [18] .

• A. Leverrier and M. Tomamichel gave the most complete security proof of the BB84 protocol to date, including all finite-size effects and a full description of the protocol [89] .

• K. Chakraborty and A. Leverrier studied a general family of quantum protocols for position verification and present a new class of attacks based on the Clifford hierarchy that outperform previously known attacks [17] .

Quantum correlations and nonlocality

Since the seminal work from Bell in the 60's, it has been known that classical correlations obtained via shared randomness cannot reproduce all the correlations obtained by measuring entangled quantum systems. This impossibility is for instance witnessed by the violation of a Bell inequality and is known under the name of “Quantum Nonlocality”. In addition to its numerous applications for quantum cryptography, the study of quantum nonlocality and quantum games has become a central topic in quantum information theory, with the hope of bringing new insights to our understanding of quantum theory.

Recent results:

• Development of a general framework for the study of quantum correlations with combinatorial tools [14]

Relativistic cryptography

(see Section  5.1.2 ).

Quantum cryptanalysis of symmetric primitives

Symmetric cryptography seems at first sight much less affected in the post-quantum world than asymmetric cryptography: its main known threat is Grover's algorithm, which allows for an exhaustive key search in the square root of the normal complexity. For this reason, it is usually believed that doubling key lengths suffices to maintain an equivalent security in the post-quantum world. However, a lot of work is certainly required in the field of symmetric cryptography in order to “quantize” the classical families of attacks in an optimized way. G. Leurent, A. Leverrier and M. Naya Plasencia have recently started working in this area in collaboration with M. Kaplan, especially on differential cryptanalysis. Some preliminary results show that counter-intuitive and surprising cases appear: in general, it is not sufficient to consider the best classical attacks and try to “quantize” them if one wants to find the best post-quantum attack [34] , [85] .