One hurdle to performing reliable quantum computations is overcoming noise. One possibility is to reduce the number of particles needing to be protected from noise and instead use systems with more states, so-called qudit quantum computers. In this paper we show that codes for these systems can be derived from already known codes, and, in particular, that degenerate stabilizer codes can have their distance also promised upon sufficiently large local dimension as well as an alternative bound on the local dimension required to preserve the distance of local-dimension-invariant codes, which is a result which could prove to be useful for error-corrected qudit quantum computers.
One hurdle to performing reliable quantum computations is overcoming noise. One possibility is to reduce the number of particles needing to be protected from noise and instead use systems with more states, so-called qudit quantum computers. In this paper we show that codes for these systems can be derived from already known codes, and, in particular, that degenerate stabilizer codes can have their distance also promised upon sufficiently large local dimension as well as an alternative bound on the local dimension required to preserve the distance of local-dimension-invariant codes, which is a result which could prove to be useful for error-corrected qudit quantum computers.
The protection of quantum information in systems leveraging more than two basis states (qudits) is an ongoing challenge. A tool for transforming the plethora of qubit stabilizer codes into qudit stabilizer codes is the local-dimension-invariant (LDI) extension to the stabilizer formalism. A drawback of the LDI method is that generally in order to ensure the distance of the resultant code is at least preserved the newly selected local-dimension will need to be any prime bigger than a large cutoff value. In this work we consider the repercussions of this large cutoff value, then by focusing on quantum low-density parity-check (qLDPC) codes we are able to bring down this cutoff value. Notably, we are able to reduce it in some cases to the initial local-dimension and for arbitrary CSS qubit-derived codes to singly exponential in d with no dependence on the number of logicals k. This reduces the size of the caveat in LDI forms in their distance preservation condition.
Protection of quantum information from noise is a massive challenge. One avenue people have begun to explore is reducing the number of particles needing to be protected from noise and instead use systems with more states, so-called qudit quantum computers. These systems will require codes which utilize the full computational space. Many prior qudit codes are very restrictive on relations between the parameters of number of qudits, number of logical qudits, distance of the code, and number of bases. In this paper we show that codes for these systems can be derived from already known codes, often relaxing the constraints somewhat, a result which could prove to be very useful for fault-tolerant qudit quantum computers.
Protecting information in systems that have more than two basis states (qudits) not only offers a promising route for reducing the number of individual quantum locations that must be protected, while more accurately reflecting the structure of realistic quantum hardware, but also has some possibly enticing foundational strengths. While work in the past has largely focused on protecting information in quantum devices with locations that are some consistent local structure, this work considers coding-theoretic constraints on devices constructed from locations which may vary in their local structures -- these are mixed-register quantum devices. In this work we provide some general results for mixed-register Pauli operators, then identify some stabilizer encoded information forms that are forbidden. Building on these insights, we construct coding-theoretically optimal mixed-register stabilizer codes from sets of codes defined on coprime local-dimensions. The construction of such codes results in codes with logical subspaces that do not directly correspond to any of the constituent local-dimensions.
Having protected quantum information is essential to perform quantum computations. One possibility is to reduce the number of particles needing to be protected from noise and instead use systems with more states, so called qudit quantum computers. In this paper we show that codes for these systems can be derived from already known codes, and in particular this procedure removes the need for shared entanglement in entanglement-assisted quantum error-correcting codes, which is a result which could prove to be useful for fault-tolerant qudit, and even qubit, quantum computers.
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.
We investigate Permutation-Invariant (PI) quantum error-correcting codes encoding a logical qudit of dimension dL in PI states using physical qudits of dimension dP. We extend the Knill--Laflamme (KL) conditions for d−1 deletion errors from qubits to qudits and investigate numerically both qubit (dL=dP=2) and qudit (dL>2 or dP>2) PI codes. We analyze the scaling of the block length n in terms of the code distance d, and compare to existing families of PI codes due to Ouyang, Aydin--Alekseyev--Barg (AAB) and Pollatsek--Ruskai (PR). Our three main findings are: (i) We conjecture that qubit PI codes correcting up to d−1 deletion errors have block length n(d)≥(3d2+1)/4, which implies an upper bound d≤12n−3/3 on their code distance, and that PR codes can saturate this bound. (ii) For qudit PI codes encoding a single qudit we numerically observe that increasing dP results in n monotonically decreasing and approaching the quantum Singleton bound n(d)≥2d−1. (iii) We propose a semi-analytic extension of the qubit AAB construction to qudits that finds explicit solutions by solving a linear program. Our results therefore provide key insights into lower bounds on the block length scaling of both qubit and qudit PI codes, and demonstrate the benefit of increased physical local dimension in the context of PI codes.