In this paper, we prove how to extend a subset of quantum stabilizer codes into a qudit hybrid code storing $\log_2 p$ classical bits over a qudit space with dimension $p$, with $p$ prime. Our proof also gives an explicit procedure for finding the entire collection of stabilizer algebras for all of the subcodes of the hybrid code. This allows extra classical information to be transmitted without having to arduously search for additional codes and their associated codewords, and also provides a first lower bound to the amount of classical information able to be transmitted in a qudit hybrid code, but unfortunately only allows for $\log_2 p$ classical bits to be decoded by a receiver.
In this paper, we prove how to extend a subset of quantum stabilizer codes into a qudit hybrid code storing log2p classical bits over a qudit space with dimension p, with p prime. Our proof also gives an explicit procedure for finding the entire collection of stabilizer algebras for all of the subcodes of the hybrid code. This allows extra classical information to be transmitted without having to arduously search for additional codes and their associated codewords, and also provides a first lower bound to the amount of classical information able to be transmitted in a qudit hybrid code, but unfortunately only allows for log2p classical bits to be decoded by a receiver.
Quantum computing’s seemingly perpetual promise of nearness still has a few hurdles \nto surmount before it can become a reality. Among these hurdles is that of protection of \ninformation from random errors. A potential solution for this challenge is stabilizer codes, \nwhich are the analog of classical linear error-correcting codes, however, with an additional \naxis of error possibility. By and large, quantum computing is discussed in the standard \nqubit, or two-level, language, however, it is worth considering the case of qudits, or more \nthan two-level systems. Often times results follow simply from some form of algebraic \nextension: typically group theory or linear algebra. In this work we consider some features \nthat are not immediately apparent from that approach, more often appealing to physical \nintuition to guide our mathematical ideas, then proving these ideas using the language of \nqudit operators. \n \nHere we consider two particular previously unexplored ideas. The first idea is that of \nembedding and inscribing of codes into spaces of different sizes than the stabilizer code \nwas originally designed for. Here, we show that all codes can be embedded, and that for \ninfinitely many primes we can in fact guarantee that the distance is at least preserved–a \nsomewhat surprising result. The second idea is, in a way an application of those presented \nin the first idea, that of turning many stabilizer codes into hybrid codes by taking advantage \nof relabelling of syndromes and the rapid increase in syndrome space upon using codes in \nlarge spaces. Both of these are somewhat useful in their current forms, however, with some \nadditional mathematical work could be turned into potentially very powerful tools for the \nprotection of quantum information. We finish off by, and along the way, mentioning various \nfuture directions to carry work on this.
Classical coding theory contains several techniques to obtain new codes from other codes, including puncturing and shortening. Both of these techniques have been generalized to quantum codes. Restricting to stabilizer codes, this paper introduces more freedom in the choice of the encoded states after puncturing. Furthermore, we also give an explicit description of the stabilizers for the punctured code. The additional freedom in the procedure also opens up for new ways to construct new codes from old, and we present several ways to utilize this in the search for codes with good or even optimal parameters. In particular, we use the construction to obtain codes whose parameters exceed the best previously known and which are better than what general puncturing guarantees. Lastly, the freedom in our puncture procedure allowed us to generalize the proof of the Griesmer bound from the classical setting to stabilizer codes for qudits of prime dimension since the proof relies heavily on the puncturing technique.
We consider the characterization as well as the construction of quantum codes that allow to transmit both quantum and classical information, which we refer to as `hybrid codes'. We construct hybrid codes [![n,k:m,d]!]q with length n and distance d, that simultaneously transmit k qudits and m symbols from a classical alphabet of size q. Many good codes such as [![7,1:1,3]!]2, [![9,2:2,3]!]2, [![10,3:2,3]!]2, [![11,4:2,3]!]2, [![11,1:2,4]!]2, [![13,1:4,4]!]2, [![13,1:1,5]!]2, [![14,1:2,5]!]2, [![15,1:3,5]!]2, [![19,9:1,4]!]2, [![20,9:2,4]!]2, [![21,9:3,4]!]2, [![22,9:4,4]!]2 have been found. All these codes have better parameters than hybrid codes obtained from the best known stabilizer quantum codes.
We consider the characterization as well as the construction of quantum codes that allow to transmit both quantum and classical information, which we refer to as `hybrid codes'. We construct hybrid codes [n, k:m, d]qwith length n and distance d, that simultaneously transmit k qudits and m symbols from a classical alphabet of size q. Many good codes such as [7,1:1, 3]2, [9,2:2,3]2, [10, 3:2, 3]2, [11,4:2, 3]2, [11,1:2,4]2, [13,1:4,4]2, [13,1:1, 5]2, [14,1:2, 5]2, [15,1:3, 5]2, [19, 9:1,4]2, [20, 9:2,4]2, [21, 9:3, 4]2, [22, 9:4,4]2have been found. All these codes have better parameters than hybrid codes obtained from the best known stabilizer quantum codes.
Encoding of Quantum Stabilizer Codes Over Qudits with d=pk
Stabilizer codes over qudits have been widely studied but their encoding procedures have not been investigated in detail by many. The encoding procedure proposed previously by Grassl et al. generates the stabilizer codes over qudits for prime d, while it may only generate a subspace of the codespace over qudits where d is a power of a prime. In this paper, we propose an encoding procedure to generate the stabilizer codes over qudits for d being a power of a prime. The procedure involves reducing the problem of encoding over qudits with d = pkto a problem similar to encoding over qudits with d = p.
We develop a theory of stabilizer codes whose stabilizer groups consist of commuting qudit Paulis, oscillator displacements, and/or planar-rotor displacements. We consider codes whose stabilizer groups form generalized lattices in quantum phase space, a condition that guarantees a finite logical dimension. We construct oscillator-rotor, rotor-qudit, and oscillator-rotor-qubit codes that cannot be decomposed into separate subsystems by generalized Clifford transformations. We also introduce an oscillator-qubit code arising from a partially lifted Golay code, where Construction A is applied to half of the Golay code coordinates. We derive analytical forms of logical operators from the symplectic dual lattice, determine Clifford-Gaussian gates from lattice symmetries, show that the code dimension equals the covolume of the stabilizer lattice, and organize the syndrome subspaces into a vector bundle. Our technique builds on earlier non-commutative geometric results by Rieffel, but it can often be interpreted simply as concatenating a given stabilizer code with a Gottesman-Kitaev-Preskill (GKP) code, applying conventional lattice-theoretic results, and unconcatenating. We formulate distances, optimal decoding, and quantum weight enumerators, recovering the Pauli, qudit, and GKP versions together with known MacWilliams identities. We construct finite-energy codewords whose encoding is an isometry up to an error exponentially small in the inverse damping parameter.