Approximating a given unitary operator into a discrete set of gates is of great importance in quantum computation. This problem can be seen as a source quantization problem where the unitary operator is considered as the source. Elements of the Clifford hierarchy can be implemented fault-tolerantly using quantum gate teleportation. This motivates the development of algorithms that quantize unitary operators into Clifford hierarchy elements. In this paper, we present a fast algorithm for approximating arbitrary n-qubit unitaries with Clifford matrices. The algorithm relies on the Clifford transvection decomposition of the Clifford group. The algorithm greedily applies Clifford transvections taking the input unitary closer to the identity matrix. To make the algorithm fast, we take advantage of the Pauli basis representation for unitary matrices to parallelize matrix distance computations. The complexity of the proposed algorithm is O(n 4n). Based on simulation results, the proposed algorithm outperforms the state of the art method.

We consider a global phase-invariant distance in the projective unitary group ${\mathcal{P}}{{\mathcal{U}}_n},$ relevant for universal quantum computing. We obtain the volume and measure of small metric ball in ${\mathcal{P}}{{\mathcal{U}}_n}$ and derive the Gilbert-Varshamov and Hamming bounds in ${\mathcal{P}}{{\mathcal{U}}_n}$. In addition, we provide upper and lower bounds for the kissing radius of the codebooks in ${\mathcal{P}}{{\mathcal{U}}_n}$ as a function of the minimum distance. Using the lower bound of the kissing radius, we find a tight Hamming bound. Also, we establish bounds on the distortion-rate function for quantizing a source uniformly distributed over ${\mathcal{P}}{{\mathcal{U}}_n}$. As example codebooks in ${\mathcal{P}}{{\mathcal{U}}_n}$ we consider the projective, Pauli and Clifford groups, as well as the projective group of diagonal gates in the Clifford hierarchy, and find their minimum distances. Finally, we verify the analytical results by simulation.

We consider massive grant-free random access in a massive multiple-input multiple-output (mMIMO) system. Users transmit a preamble selected at random from a pool, together with data in a grant-free manner. While superposed users with different preambles can be separated and their data decoded, colliding users selecting the same preamble may not be separated. We introduce a preamble collision resolution mechanism, where the multiuser channel of colliding users is resolved with a combination of preamble-based and data-aided channel estimation. For this, we use the sample covariance matrix of the data to estimate the superposition channel from the preamble. For the most likely case of two colliding users, based on the knowledge of the modulation alphabet is used to resolve a phase ambiguity in the multiuser channel. Through numerical evaluations, we validate the efficacy of the proposed method in effectively resolving preamble collisions.

We investigate an unsourced coded slotted random access (RA) protocol which combines physical and MAC layer approaches. In contrast to the literature, we consider MAC-layer slotting which carries information. We distinguish three categories w.r.t. the amount of information carried by the slotting principle; non/partial/full information carrying setups. We model physical layer performance in a slot with a pilot sequence transmission selected based on information, combined with a finite block length code. For MAC-layer non/partial/full information carrying slotting, we use interference-free (IF) constant weight codes with different numbers of packet repetitions, and compare averaged per user block error rate performance with grant-free 5G NR 2-step RA. We observe that the full information carrying setup provides the best results for all slot repetition schemes, and in addition such schemes have the potential to significantly outperform 5G-NR RA. Also, partial information carrying schemes can provide a large portion of the gain over non-information carrying schemes for the same receiver complexity.

Binary chirps (BCs) are exponentiated 2nd-order Reed-Muller codes, which have interesting geometric and algebraic features, one of which is the close connection to the diagonal part of the Clifford group, which is the 2nd level of the Clifford hierarchy. We develop a novel transvection based method to analyze the diagonal Clifford hierarchy. Using this, we identify a connection of recently proposed generalized BCs with the 3rd level of Clifford hierarchy. Then, we propose two systematic extensions of the BC codebook to an arbitrary Clifford hierarchy level and find their minimum distances. In these extensions, the number of codewords grows exponentially with the hierarchy level. For decoding, we design a low-complexity decoding approach for the extended BCs, using the Howard algorithm for BC decoding as a component. Through simulations, we show that the performance of the proposed low-complexity decoder can achieve performance very close to the exhaustive search with significantly reduced complexity.

For achieving long-distance quantum communication, Quantum Repeaters (QRs) have to be used, with commu-nication range and reliability increased by using intermediate stations. The physical requirements of second generation QRs may be achievable in the near future. They use Quantum Error Correction Codes (QECC) to protect logical qubits against environmental interaction using physical redundancy. In this work, we study the types of errors that can corrupt quantum codewords in intermediate stations. Our studies show that the errors are distance-dependent, and also consist of correlated errors and biased errors. To mitigate this error model, we use non-symmetric CSS codes as well as mirrored structure coding. We show that using non-symmetric CSS codes results in better performance. Also, we prove the logical CZ gate transversality of the mirrored structure coding. The effectiveness of the proposed methods is verified by numerical simulations.

Binary Chirp (BC) codebooks consist of ${N^{\left( {{{\log }_2}N + 3} \right)/2}}$ lines in ${\mathbb{C}^N}$, equivalent up to overall phase rotations. Exploiting the underlying algebraic structure, the BCs allow suboptimal decoders with complexity N(logN)2, based on autocorrelations between the received signal and its permuted versions. We analyze the performance of these decoders in additive white Gaussian noise channels, providing lower bounds of decoding error probability, which are tight in the limits of low and high signal-to-noise ratio. Due to the autocorrelation nature of the receiver, the error probability becomes a function of order statistics of χ2-distributed random variables. Our results can be used when dimensioning communication systems where BCs are used as component codes.