We give a general procedure for weight reducing quantum codes. This corrects a previous work\cite{owr}, and introduces a new technique that we call "coning" to effectively induce high weight stabilizers in an LDPC code. As one application, any LDPC code (with arbitrary $O(1)$ stabilizer weights) may be turned into a code where all stabilizers have weight at most $5$ at the cost of at most a constant factor increase in number of physical qubits and constant factor reduction in distance. Also, by applying this technique to a quantum code whose $X$-stabilizers are derived from a classical log-weight random code and whose $Z$-stabilizers have linear weight, we construct an LDPC quantum code with distance $\tilde Ω(N^{2/3})$ and $\tildeΩ(N^{2/3})$ logical qubits.
We give a general procedure for weight reducing quantum codes. This corrects a previous work\cite{owr}, and introduces a new technique that we call "coning" to effectively induce high weight stabilizers in an LDPC code. As one application, any LDPC code (with arbitrary O(1) stabilizer weights) may be turned into a code where all stabilizers have weight at most 5 at the cost of at most a constant factor increase in number of physical qubits and constant factor reduction in distance. Also, by applying this technique to a quantum code whose X-stabilizers are derived from a classical log-weight random code and whose Z-stabilizers have linear weight, we construct an LDPC quantum code with distance Ω~(N2/3) and Ω~(N2/3) logical qubits.
Quantum weight reduction is the task of transforming a quantum code with large check weight into one with small check weight. This problem is important in practice, since low-weight measurements are necessary for reliable implementations of quantum error correction on physical hardware. It is also important theoretically, as it can be used to construct constant-locality versions of quantum locally testable codes, which may be relevant to the quantum PCP conjecture. We give a streamlined geometric procedure for quantum weight reduction based entirely on coning and treating X and Z checks symmetrically, which simplifies Hastings' previous approach. In particular, given an arbitrary [[n,k,d]] quantum code with weight w, our method produces a code with parameters [[O(nw2logw),k,Ω(dw)]], check weight 5, and qubit weight 6; these bounds are optimal or close to optimal within the current geometric framework. Applied to random dense CSS codes, our procedure yields explicit quantum codes surpassing the square-root distance barrier, with parameters [[n,Ω~(n1/3),Ω~(n2/3)]]. These codes also admit a three-dimensional embedding that saturates the Bravyi-Poulin-Terhal (BPT) bound, recovering the layer-code result. Our weight reduction technique also improves fault-tolerant logical-operator measurements by reducing the required number of ancilla qubits.
We present an algorithm that takes a CSS stabilizer code as input, and outputs another CSS stabilizer code such that the stabilizer generators all have weights O(1) and such that O(1) generators act on any given qubit. The number of logical qubits is unchanged by the procedure, while we give bounds on the increase in number of physical qubits and in the effect on distance and other code parameters, such as soundness (as a locally testable code) and “cosoundness” (defined later). Applications are discussed, including to codes from high-dimensional manifolds which have logarithmic weight stabilizers. Assuming a conjecture in geometry[11], this allows the construction of CSS stabilizer codes with generator weight O(1) and almost linear distance. Another application of the construction is to increasing the distance to X or Z errors, whichever is smaller, so that the two distances are equal.
We describe a general method for turning quantum circuits into sparse quantum subsystem codes. The idea is to turn each circuit element into a set of low-weight gauge generators that enforce the input-output relations of that circuit element. Using this prescription, we can map an arbitrary stabilizer code into a new subsystem code with the same distance and number of encoded qubits but where all the generators have constant weight, at the cost of adding some ancilla qubits. With an additional overhead of ancilla qubits, the new code can also be made spatially local. Applying our construction to certain concatenated stabilizer codes yields families of subsystem codes with constant weight generators and with minimum distance d = n1-∈, where E = O(1/√log n). For spatially local codes in D dimensions, we nearly saturate a bound due to Bravyi and Terhal and achieve d = n1-∈-1/D. Previously the best code distance achievable with constant-weight generators in any dimension, due to Freedman, Meyer, and Luo, was O(√n log n) for a stabilizer code.
Stabilizer codes are the most widely studied class of quantum error-correcting codes and form the basis of most proposals for a fault-tolerant quantum computer. A stabilizer code is defined by a set of parity-check operators, which are measured in order to infer information about errors that may have occurred. In typical settings, measuring these operators is itself a noisy process and the noise strength scales with the number of qubits involved in a given parity check, or its weight. Hastings has proposed a method for reducing the weights of the parity checks of a stabilizer code, though it has previously only been studied in the asymptotic regime. Here, we instead focus on the regime of small to medium-size codes suitable for quantum computing hardware. We both provide a fully explicit description of Hastings’s method and propose a substantially simplified weight-reduction method that is applicable to the class of quantum product codes. Our simplified method allows us to reduce the check weights of hypergraph- and lifted-product codes to at most six, while preserving the number of logical qubits and at least retaining (in fact, often increasing) the code distance. The price we pay is an increase in the number of physical qubits by a constant factor but we find that our method is much more efficient than Hastings’s method in this regard. We benchmark the performance of our codes on a simulated photonic quantum computing architecture based on Gottesman-Kitaev-Preskill qubits and passive linear optics, finding that our weight-reduction method substantially improves code performance. Published by the American Physical Society 2024
Quantum codes with low-weight stabilizers known as LDPC codes have been actively studied recently due to their simple syndrome readout circuits and potential applications in fault-tolerant quantum computing. However, all families of quantum LDPC codes known to this date suffer from a poor distance scaling limited by the square-root of the code length. This is in a sharp contrast with the classical case where good families of LDPC codes are known that combine constant encoding rate and linear distance. Here we propose the first family of good quantum codes with low-weight stabilizers. The new codes have a constant encoding rate, linear distance, and stabilizers acting on at most n qubits, where n is the code length. For comparison, all previously known families of good quantum codes have stabilizers of linear weight. Our proof combines two techniques: randomized constructions of good quantum codes and the homological product operation from algebraic topology. We conjecture that similar methods can produce good stabilizer codes with stabilizer weight na for any a>0. Finally, we apply the homological product to construct new small codes with low-weight stabilizers.
Abstract Quantum stabilizer codes are essential for protecting quantum information, but systematic searches for high-distance codes become computationally prohibitive as code length increases. Existing code constructions exhibit structural regularities, yet no general framework quantifies how such constraints reduce the stabilizer-code search space or incorporates them into code searches. This project asks whether structural properties of known quantum codes can be formulated as mathematical constraints that quantitatively reduce search spaces while remaining useful for discovering high-distance codes. We develop a unified framework based on q-Galois matrices and define the search-space retention factor as the asymptotic fraction of logarithmic search space retained after imposing a constraint. We derive retention factors for weight-spectrum, qudit-participation, overlap, support, Pauli-distribution, local non-commutation, graph-theoretic, and symmetry constraints, including combinations. We introduce finite-size correction factors C and C′ and find empirically that they decay approximately as logn/n and 1/n, respectively. The framework is evaluated using 58,902 quantum-code parameter sets, including 34,775 codes with usable matrix data across q=2,3,4,5,7,8; 1,909 semi-perfect quantum codes are analyzed. We develop a general method to empirically estimate the weight-spectrum and average-weight for unresolved semi-perfect quantum codes, achieving runtime speedups ranging from 108.3608 to 10193.9298 across 4 distinct unresolved codes. Runtime analysis validates that these reductions translate into substantial computational savings. Applying the framework to a (14,3) qubit search produced a code detecting 91,765 of 91,770 errors of weight at most 4 and achieving 99.9999933% and 99.9733% of the theoretically optimal performance of [[14,3,5]]2 code at p=0.01. Thus, the framework converts structural properties into explicit candidate-space reductions that can be imposed during code enumeration, providing a quantitative basis for reducing the computational cost of high-distance stabilizer-code searches.