There is, I think, something ethereal about i —the square root of minus one. I remember first hearing about it at school. It seemed an odd beast at that time—an intruder hovering on the edge of reality.

Usually familiarity dulls this sense of the bizarre, but in the case of i it was the reverse: over the years the sense of its surreal nature intensified. It seemed that it was impossible to write mathematics that described the real world in …

I and i

Sphere Packings, Lattices and Groups

Fault tolerance is the study of reliable computation using unreliable components. With a given noise model, can one still reliably compute? For example, one can run many copies of a classical calculation in parallel, periodically using majority gates to catch and correct faults. Von Neumann showed in 1956 that if each gate fails independently with probability p, flipping its output bit 0 $ 1, then such a fault tolerance scheme still allows for arbitrarily reliable computation provided that p is below some constant threshold (whose value depends on the model details) [10]. In a quantum computer, the basic gates are much more vulnerable to noise than classical transistors – after all, depending on the implementation, they are manipulating single electron spins, photon polarizations, and similarly fragile subatomic particles. It might not be possible to engineer systems with noise rates less than 10 , or perhaps 10 , per gate. Additionally, the phenomenon of entanglement makes quantum systems inherently fragile. For example, in Schrödinger’s cat state – an equal superposition between a living cat and a dead cat, often idealized as 1= p 2.j0ni C j1ni/ – an interaction with just one quantum bit (“qubit”) can collapse, or decohere, the entire system. Fault tolerance techniques will therefore be essential for achieving the considerable potential of quantum computers. Practical fault tolerance techniques will need to control high noise rates and do so with low overhead, since qubits are expensive. Quantum systems are continuous, not discrete, so there are many possible noise models. However, the essential features of quantum noise for fault tolerance results can be captured by a simple discrete model similar to the one Von Neumann used. The main difference is that, in addition to bit-flip X errors which swap 0 and 1, there can also be phase-flip Z errors which swap jCi 1=p2.j0i C j1i/ and j i 1=p2.j0i j1i/ (Fig. 1). A noisy gate is modeled as a perfect gate followed by independent introduction of X, Z, or Y (which is both X and Z) errors with respective probabilities pX; pZ; pY. One popular model is independent depolarizing noise .pX D pZ D pY p=3/; a depolarized qubit is completely randomized. Faulty measurements and preparations of single-qubit states must additionally be modeled, and there can be memory noise on resting qubits. It is often assumed that measurement results can

Fault-Tolerant Quantum Computation.

Quantum information processing exploits the quantum nature of information. It offers fundamentally new solutions in the field of computer science and extends the possibilities to a level that cannot be imagined in classical communication systems. For quantum communication channels, many new capacity definitions were developed in comparison to classical counterparts. A quantum channel can be used to realize classical information transmission or to deliver quantum information, such as quantum entanglement. Here we review the properties of the quantum communication channel, the various capacity measures and the fundamental differences between the classical and quantum channels.

/pdf/a-survey-on-quantum-channel-capacities-4y952c0kbw.pdf

A Survey on Quantum Channel Capacities

We propose a new class of post-quantum digital signature schemes that: (a) derive their security entirely from the security of symmetric-key primitives, believed to be quantum-secure, and (b) have extremely small keypairs, and, (c) are highly parameterizable. In our signature constructions, the public key is an image y=f(x) of a one-way function f and secret key x. A signature is a non-interactive zero-knowledge proof of x, that incorporates a message to be signed. For this proof, we leverage recent progress of Giacomelli et al. (USENIX'16) in constructing an efficient Σ-protocol for statements over general circuits. We improve this Σ-protocol to reduce proof sizes by a factor of two, at no additional computational cost. While this is of independent interest as it yields more compact proofs for any circuit, it also decreases our signature sizes. We consider two possibilities to make the proof non-interactive: the Fiat-Shamir transform and Unruh's transform (EUROCRYPT'12, '15,'16). The former has smaller signatures, while the latter has a security analysis in the quantum-accessible random oracle model. By customizing Unruh's transform to our application, the overhead is reduced to 1.6x when compared to the Fiat-Shamir transform, which does not have a rigorous post-quantum security analysis. We implement and benchmark both approaches and explore the possible choice of f, taking advantage of the recent trend to strive for practical symmetric ciphers with a particularly low number of multiplications and end up using Low MC (EUROCRYPT'15).

/pdf/post-quantum-zero-knowledge-and-signatures-from-symmetric-3odro7h178.pdf

Post-Quantum Zero-Knowledge and Signatures from Symmetric-Key Primitives

We present a general construction of asymmetric quantum codes based on additive codes under the trace Hermitian inner product. Various families of additive codes over F4 are used in the construction of many asymmetric quantum codes over F4.

https://dr.ntu.edu.sg/bitstream/10220/7503/1/additive_asymmetric%20Quantum%20Codes.pdf

Additive Asymmetric Quantum Codes

Asymmetric quantum error-correcting codes (AQCs) may offer some advantage over their symmetric counterparts by providing better error-correction for the more frequent error types. The well-known CSS construction of q-ary AQCs is extended by removing the Fq-linearity requirement as well as the limitation on the type of inner product used. The proposed constructions are called CSS-like constructions and utilize pairs of nested subfield linear codes under one of the Euclidean, trace Euclidean, Hermitian, and trace Hermitian inner products. After establishing some theoretical foundations, best-performing CSS-like AQCs are constructed. Combining some constructions of nested pairs of classical codes and linear programming, many optimal and good pure q-ary CSS-like codes for q ∈ {2,3,4,5,7,8,9} up to reasonable lengths are found. In many instances, removing the Fq-linearity and using alternative inner products give us pure AQCs with improved parameters than relying solely on the standard CSS construction.

/pdf/css-like-constructions-of-asymmetric-quantum-codes-4fn8q5w6xp.pdf

CSS-Like Constructions of Asymmetric Quantum Codes

The weights in maximum distance separable (MDS) codes of length n and dimension k over the finite field GF(q) are studied. Up to some explicit exceptional cases, the MDS codes with parameters given by the MDS conjecture are shown to contain all k weights in the range n - k + 1 to n. The proof uses the covering radius of the dual code.

The Weights in MDS Codes

We solve an open question in code-based cryptography by introducing the first provably secure group signature scheme from code-based assumptions. Specifically, the scheme satisfies the CPA-anonymity and traceability requirements in the random oracle model, assuming the hardness of the McEliece problem, the Learning Parity with Noise problem, and a variant of the Syndrome Decoding problem. Our construction produces smaller key and signature sizes than the existing post-quantum group signature schemes from lattices, as long as the cardinality of the underlying group does not exceed the population of the Netherlands $${\approx }2^{24}$$ users. The feasibility of the scheme is supported by implementation results. Additionally, the techniques introduced in this work might be of independent interest: a new verifiable encryption protocol for the randomized McEliece encryption and a new approach to design formal security reductions from the Syndrome Decoding problem.

/pdf/a-provably-secure-group-signature-scheme-from-code-based-4ynhj1al8p.pdf

A Provably Secure Group Signature Scheme from Code-Based Assumptions

Using the Calderbank–Shor–Steane (CSS) construction, pure q-ary asymmetric quantum error-correcting codes attaining the quantum Singleton bound are constructed. Such codes are called pure CSS asymmetric quantum maximum distance separable (AQMDS) codes. Assuming the validity of the classical maximum distance separable (MDS) Conjecture, pure CSS AQMDS codes of all possible parameters are accounted for.

Martianus Frederic Ezerman

Papers

Additive Asymmetric Quantum Codes

CSS-Like Constructions of Asymmetric Quantum Codes

The Weights in MDS Codes

A Provably Secure Group Signature Scheme from Code-Based Assumptions

Pure asymmetric quantum MDS codes from CSS construction : a complete characterization