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arXiv:2511.14797

An Error Correctable Implication Algebra for a System of Qubits

Morrison Turnansky

quant-phmath-phSubmitted Nov 16, 2025• Announced Nov 18, 2025• 0 citations
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Abstract

We present the Lukasiewicz logic as a viable system for an implication algebra on a system of qubits. Our results show that the three valued Lukasiewicz logic can be embedded in the stabilized space of an arbitrary quantum error correcting stabilizer code. We then fully characterize the non trivial errors that may occur up to group isomorphism. Lastly, we demonstrate by explicit algorithmic example, how any algorithm consistent with the Lukasiewicz logic can immediately run on a quantum system and utilize the indeterminate state.

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We present the Lukasiewicz logic as a viable system for an implication algebra on a system of qubits. Our results show that the three valued Lukasiewicz logic can be embedded in the stabilized space of an arbitrary quantum error correcting stabilizer code. We then fully characterize the non trivial errors that may occur up to group isomorphism. Lastly, we demonstrate by explicit algorithmic example, how any algorithm consistent with the Lukasiewicz logic can immediately run on a quantum system and utilize the indeterminate state.
Ternary quantum systems are being studied because they provide more computational state space per unit of information, known as qutrit. A qutrit has three basis states, thus a qubit may be considered as a special case of a qutrit where the coefficient of one of the basis states is zero. Hence both $(2\ifmmode\times\else\texttimes\fi{}2)$-dimensional and $(3\ifmmode\times\else\texttimes\fi{}3)$-dimensional Pauli errors can occur on qutrits. In this paper, we (i) explore the possible $(2\ifmmode\times\else\texttimes\fi{}2)$-dimensional as well as $(3\ifmmode\times\else\texttimes\fi{}3)$-dimensional Pauli errors in qutrits and show that any pairwise bit swap error can be expressed as a linear combination of shift errors and phase errors, (ii) propose a special type of error called a quantum superposition error and show its equivalence to arbitrary rotation, (iii) formulate a nine-qutrit code which can correct a single error in a qutrit, and (iv) provide its stabilizer and circuit realization.
Leakage beyond the computational subspace is a major source of error in multilevel quantum hardware. We show that any [[n,k,d]] [[n,k,d]] [[n,k,d]] stabilizer code can be embedded isometrically into a single DDD-dimensional system while preserving its complete error-correcting structure. We further derive a necessary and sufficient condition for exact leakage correction, proving that leakage is correctable precisely when it does not distinguish between logical states. These results establish a unified framework for quantum error correction in multilevel quantum systems.
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.
Quantum decoherence and errors represent some of the major challenges arising in quantum computations. Quantum error correcting codes protect quantum states against these errors and make quantum computing more reliable. One of the main problems of quantum error correction is the design of feasible decoding algorithms that can simplify error-correction for general quantum codes. The dissertation addresses decoding of general Codeword Stabilized (CWS) codes. This class of quantum codes also includes some other important classes such as additive Stabilizer codes and non-additive Union Stabilizer (USt) codes.We first design a generic error-correcting algorithm for CWS codes and analyze the number of decoding measurements and quantum gates. This algorithm performs exhaustive screening of different error patterns, similar to decoding of classical non-linear codes. For an n-qubit quantum code correcting up to t erroneous qubits, this brute-force approach consecutively tests all correctable error patterns and employs a separate n-qubit measurement in each test.The main result is a new error grouping technique that enables simultaneous testing of large groups of errors in a single measurement. To achieve this reduction, we first proceed with a new error-correction algorithm for the USt codes. Secondly, we design an algorithm that converts generic non-linear CWS codes into the simpler quasi-linear USt codes. Each decoding measurement can then either locate the actual error in a given group of errors or entirely eliminate this group. This technique yields a much simpler algorithm that exponentially reduces the number of measurements about 3^t times in any t-error-correcting CWS code.
By defining projective error models we study the mathematical structure of Clifford codes and stabilizer codes using tools from projective representation theory. Furthermore, we introduce a new class of codes which we have called weak stabilizer codes and we determine some relationships between these three classes of codes. We show that the obstruction for a stabilizer code to be non-trivial is given by a class in group cohomology, and we are able to determine similar obstructions for weak stabilizer codes to be non-trivial. In the case where the projective error model corresponds to a nice error basis we give a complete characterization of when a Clifford code is a weak stabilizer code in terms of the size of the group of logical operators and the size of the group of stabilizers of the code. Lastly, we produce two infinite families of Clifford codes that are not stabilizer codes, as well as a method of combining these examples into more examples of non-stabilizer Clifford codes.