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arXiv:2610.14192

Lifting an $[n,k,d]$ Binary Linear Code to an $[[n,k,d]]$ Quantum Stabilizer Code

Thamadathil Aswanth, Navin Kashyap

quant-phcs.ITSubmitted Oct 9, 2026• Announced Oct 13, 2026• 0 citations
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Abstract

The codeword stabilized (CWS) framework provides a systematic method for constructing quantum error correcting codes from a simple undirected graph and a classical code. When the underlying classical code is linear, the resulting quantum code is a stabilizer code. In this work, we investigate the conditions under which an arbitrary $[n,k,d]$ classical binary linear code can be lifted to an $[[n,k,d]]$ quantum stabilizer code using the CWS framework. We analyze the random CWS code constructed from a given $[n,k,d]$ binary linear code $\mathcal{C}$ and an Erdős-Rényi random graph $\mathrm{G}(n, \frac12)$. We establish a lower bound on the probability that the resulting quantum code preserves the minimum distance $d$ of the classical code. This probabilistic analysis yields a Gilbert-Varshamov-type condition on the $n,k,d$ parameters of the classical code that guarantees the existence of a graph resulting in a quantum code with the same $n,k,d$ parameters.

Comments
6 pages, Submitted to ITW 2027

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Lifting an [n,k,d][n,k,d][n,k,d] Binary Linear Code to an [[n,k,d]][[n,k,d]][[n,k,d]] Quantum Stabilizer Code
The codeword stabilized (CWS) framework provides a systematic method for constructing quantum error correcting codes from a simple undirected graph and a classical code. When the underlying classical code is linear, the resulting quantum code is a stabilizer code. In this work, we investigate the conditions under which an arbitrary [n,k,d][n,k,d][n,k,d] classical binary linear code can be lifted to an [[n,k,d]][[n,k,d]][[n,k,d]] quantum stabilizer code using the CWS framework. We analyze the random CWS code constructed from a given [n,k,d][n,k,d][n,k,d] binary linear code C\mathcal{C}C and an Erdős-Rényi random graph G(n,12)\mathrm{G}(n, \frac12)G(n,21​). We establish a lower bound on the probability that the resulting quantum code preserves the minimum distance ddd of the classical code. This probabilistic analysis yields a Gilbert-Varshamov-type condition on the n,k,dn,k,dn,k,d parameters of the classical code that guarantees the existence of a graph resulting in a quantum code with the same n,k,dn,k,dn,k,d parameters.
Abstract We propose a systematic procedure for the construction of graphs associated with binary quantum stabilizer codes. The procedure is characterized by means of the following three step process. First, the stabilizer code is realized as a codeword-stabilized (CWS) quantum code. Second, the canonical form of the CWS code is determined and third, the input vertices are attached to the graphs. In order to verify the effectiveness of the procedure, we implement the Gottesman stabilizer code characterized by multi-qubit encoding operators for the resource-efficient error correction of arbitrary single-qubit errors. Finally, the error-correcting capabilities of the Gottesman eight-qubit quantum stabilizer code is verified in graph-theoretic terms as originally advocated by Schlingemann and Werner.
Codeword-stabilized quantum codes give a unified graph-state description of stabilizer and nonadditive quantum error-correcting codes. Although each individual CWS word state is stabilizer, coherent superpositions of different word states can be nonstabilizer. We develop a CWS-adapted magic-witness framework that isolates this codeword coherence and converts it into certified lower bounds on robustness of magic. The main result is an exact reduction of the stabilizer threshold of a natural CWS coherence witness to a finite-geometric problem over the classical CWS word set. For general weighted superpositions, the threshold is computed by enumerating affine intersections and affine-quadratic phases. For equal-weight superpositions, the phase optimization collapses, and the threshold is determined entirely by how many CWS words can lie in affine flats of each dimension. Thus a quantum optimization over stabilizer states becomes a classical affine-incidence problem. This reduction yields a fixed-parameter algorithm, an analytic lower bound for an infinite union-stabilizer family, and exact rational certificates for several standard nonadditive CWS examples. The framework also clarifies why exact enumeration fails for large structured families and identifies the remaining task as an affine-intersection problem. The result provides a geometric mechanism by which nonlinear CWS word sets generate certifiable magic.
An m-uniform quantum state on n qubits is an entangled state in which every m-qubit subsystem is maximally mixed. Starting with an m-uniform state realized as the graph state associated with an m-regular graph, and a classical n,k,d≥m+1 binary linear code with certain additional properties, we show that pure [n,k,m+1]]2 quantum error-correcting codes (QECCs) can be constructed within the codeword stabilized (CWS) code framework. As illustrations, we construct pure [[22r-1,22r-2r-3,3]]2 and [[(24r-1)2,(24r-1)2-32r-7,5]]2 QECCs. We also give measurement-based protocols for encoding into code states and for recovery of logical qubits from code states.
nnn-qubit,mmm-uniform states are[[n, 0, m+1]][[n,\ 0,\ m+1]][[n, 0, m+1]]nonde-generate quantum error correcting codes (QECC). The codeword stabilized (CWS) codes use [n,k,d≥m+1n, k, d\geq m+1n,k,d≥m+1] classical codes andmmm-uniform graph states to construct[[n, k, m+1]][[n,\ k,\ m+1]][[n, k, m+1]]QECCs. We provide a set of conditions on the classical codes to obtain CWS codes, which can be encoded in a proposed measurement-based approach. We arrive at an explicit construction of a CWS code when the startingmmm-uniform graph state is a cluster state of qubits on a one-dimensional lattice. The proposed encoding protocol admits sequential encoding of logical qubits, making it suitable for quantum memory applications.
We show that for any R,ε>0R,ε>0R,ε>0 and any prime p≥2p\geq 2p≥2, there exists an infinite family of ppp-ary quantum low-density parity-check (QLDPC) codes, rate RRR, checks of weight Oε(1)O_ε(1)Oε​(1), and normalized distance at least δGV(p,R)−εδ_{\mathrm{GV}}(p,R)-εδGV​(p,R)−ε. Here, δGV(p,R)δ_{\mathrm{GV}}(p,R)δGV​(p,R) denotes the quantum Gilbert-Varshamov (GV) bound for ppp-ary stabilizer codes. In fact, we construct an ensemble of such QLDPC codes, such that a random code from this ensemble is close to the GV bound with high probability. Moreover, this ensemble matches the performance of random stabilizer codes on several quantum channels. Specifically, it approaches the quantum capacity of the erasure channel and the hashing bound for memoryless Pauli channels, including the depolarizing channel. A significant challenge in working with QLDPC codes is that they are necessarily \emph{degenerate}, i.e., contain many low-weight stabilizers. A key contribution of our work is a construction of QLDPC codes with quantitative control on their degeneracy. These codes are obtained by combining known constructions of asymptotically good QLDPC codes with the expander-based distance amplification procedure of Alon, Edmonds, and Luby [FOCS'95]. Our random ensemble is constructed by starting with these low-degeneracy QLDPC codes near the quantum Singleton bound and concatenating each coordinate with a random inner code. This can be viewed as a quantum analogue of Thommesen's construction, and as an LDPC version of a result of Ouyang, with an appropriately designed outer code.
Universal quantum computation requires non-Clifford gates, which stabilizer codes can provide transversally only up to a finite level of the Clifford hierarchy. Non-stabilizer codes are not subject to this bound and are the natural candidates to fill the gap. However, it is difficult to efficiently evaluate the nonstabilizerness (or magic) of non-stabilizer codes. Here, we resolve this problem for codeword-stabilized (CWS) codes by showing that the nonstabilizerness of a CWS code is determined by the difference structure of its codewords. Whenever a Sidon set of the required size exists, the most magical codes are related to the Sidon sets, whose pairwise differences are all distinct. Structural analysis demonstrates that nonstabilizerness is invariant under coset closure, giving rise to non-stabilizer codes with arbitrarily many logical qubits and constant nonstabilizerness. Nonstabilizerness of whole code families becomes exactly computable, and we obtain closed-form values for the Kerdock codes. Moreover, we reveal a tradeoff between nonstabilizerness and transversal gates. For a code with kkk logical qubits, a diagonal transversal gate that is non-Clifford on ttt coordinates forces the nonstabilizerness benchmark to be at most 2(k−t)2(k-t)2(k−t). These results reduce the search for codes with high nonstabilizerness and for transversal non-Clifford gates to classical counting problems, to which the standard tools of additive combinatorics apply.