Results for “Low Density Parity Check (LDPC) codes”

DOE OSTI2025

Hierarchical memories: Simulating quantum LDPC codes with local gates

Constant-rate low-density parity-check (LDPC) codes are promising candidates for constructing efficient fault-tolerant quantum memories. However, if physical gates are subject to geometric-locality constraints, it becomes challenging to realize these codes. In this paper, we construct a new family of [[N,K,D]] codes, referred to as hierarchical codes, that encode a number of logical qubits K=Ω(N/log(N) 2 ). The N th element of this code family is obtained by concatenating a constant-rate quantum LDPC code with a surface code; nearest-neighbor gates in two dimensions are sufficient to implement the corresponding syndrome-extraction circuit and achieve a threshold. Below threshold the logical failure rate vanishes superpolynomially as a function of the distance D(N). We present a bilayer architecture for implementing the syndrome-extraction circuit, and estimate the logical failure rate for this architecture. Under conservative assumptions, we find that the hierarchical code outperforms the basic encoding where all logical qubits are encoded in the surface code.

Pattison, Christopher A. [California Institute of ↗

DOE OSTI2025

Power and Limitations of Linear Programming Decoder for Quantum LDPC Codes

Decoding quantum error-correcting codes is a key challenge in enabling fault-tolerant quantum computation. In the classical setting, linear programming (LP) decoders offer provable performance guarantees and can leverage fast practical optimization algorithms. Although LP decoders have been proposed for quantum codes, their performance and limitations remain relatively underexplored. In this work, we uncover a key limitation of LP decoding for quantum low-density parity-check (LDPC) codes: certain constant-weight error patterns lead to ambiguous fractional solutions that cannot be resolved through independent rounding. To address this issue, we incorporate a post-processing technique known as ordered statistics decoding (OSD), which significantly enhances LP decoding performance in practice. Our results show that LP decoding, when augmented with OSD, can outperform belief propagation with the same post-processing for intermediate code sizes of up to hundreds of qubits. These findings suggest that LP-based decoders, equipped with effective post-processing, offer a promising approach for decoding near-term quantum LDPC codes.

Gu, Shouzhen [Yale U.]↗

DOE OSTI2025

Power and Limitations of Linear Programming Decoder for Quantum LDPC Codes

Decoding quantum error-correcting codes is a key challenge in enabling fault-tolerant quantum computation. In the classical setting, linear programming (LP) decoders offer provable performance guarantees and can leverage fast practical optimization algorithms. Although LP decoders have been proposed for quantum codes, their performance and limitations remain relatively underexplored. In this work, we uncover a key limitation of LP decoding for quantum low-density parity-check (LDPC) codes: certain constant-weight error patterns lead to ambiguous fractional solutions that cannot be resolved through independent rounding. To address this issue, we incorporate a post-processing technique known as ordered statistics decoding (OSD), which significantly enhances LP decoding performance in practice. Our results show that LP decoding, when augmented with OSD, can outperform belief propagation with the same post-processing for intermediate code sizes of up to hundreds of qubits. These findings suggest that LP-based decoders, equipped with effective post-processing, offer a promising approach for decoding near-term quantum LDPC codes.

Gu, Shouzhen [Yale U.]↗

DOE OSTI2025

Power and Limitations of Linear Programming Decoder for Quantum LDPC Codes

Decoding quantum error-correcting codes is a key challenge in enabling fault-tolerant quantum computation. In the classical setting, linear programming (LP) decoders offer provable performance guarantees and can leverage fast practical optimization algorithms. Although LP decoders have been proposed for quantum codes, their performance and limitations remain relatively underexplored. In this work, we uncover a key limitation of LP decoding for quantum low-density parity-check (LDPC) codes: certain constant-weight error patterns lead to ambiguous fractional solutions that cannot be resolved through independent rounding. To address this issue, we incorporate a post-processing technique known as ordered statistics decoding (OSD), which significantly enhances LP decoding performance in practice. Our results show that LP decoding, when augmented with OSD, can outperform belief propagation with the same post-processing for intermediate code sizes of up to hundreds of qubits. These findings suggest that LP-based decoders, equipped with effective post-processing, offer a promising approach for decoding near-term quantum LDPC codes.

Gu, Shouzhen [Yale U.]↗

DOE OSTI2025

Low-Density Parity-Check Stabilizer Codes as Gapped Quantum Phases: Stability under Graph-Local Perturbations

We generalize the proof of stability of topological order, due to Bravyi, Hastings, and Michalakis, to stabilizer Hamiltonians corresponding to low-density parity-check (LDPC) codes without the restriction of geometric locality in Euclidean space. We consider Hamiltonians 𝐻 0 defined by ⟦𝑁,𝐾,𝑑⟧ LDPC codes, which obey certain topological quantum order conditions: (i) code distance 𝑑 ≥ 𝑐⁢log (𝑁), implying local indistinguishability of ground states, and (ii) a mild condition on local and global compatibility of ground states—these include good quantum LDPC codes and the toric code on a hyperbolic lattice, among others. We consider stability under weak perturbations that are quasilocal on the interaction graph defined by 𝐻 0 and that can be represented as sums of bounded-norm terms. As long as the local perturbation strength is smaller than a finite constant, we show that the perturbed Hamiltonian has well-defined spectral bands originating from the 𝑂⁡(1) smallest eigenvalues of 𝐻 0 . The band originating from the smallest eigenvalue has 2 𝐾 states, is separated from the rest of the spectrum by a finite energy gap, and has exponentially narrow bandwidth 𝛿 =𝐶⁢𝑁⁢𝑒 −Θ⁡(𝑑) , which is tighter than the best-known bounds even in the Euclidean case. We also obtain that the new ground-state subspace is related to the initial-code subspace by a quasilocal unitary, allowing one to relate their physical properties. Our proof uses an iterative procedure that performs successive rotations to eliminate non-frustration-free terms in the Hamiltonian. Our results extend to quantum Hamiltonians built from classical LDPC codes, which give rise to stable symmetry-breaking phases. These results show that LDPC codes very generally define stable gapped quantum phases, even in the non-Euclidean setting, initiating a systematic study of such phases of matter.

mathematical physics↗

DOE OSTI2025

Low-Density Parity-Check Codes as Stable Phases of Quantum Matter

Phases of matter with robust ground-state degeneracy, such as the quantum toric code, are known to be capable of robust quantum information storage. Here, we address the converse question: given a quantum error-correcting code, when does it define a stable gapped quantum phase of matter, whose ground-state degeneracy is robust against perturbations in the thermodynamic limit? We prove that a low-density parity-check (LDPC) code defines such a phase, robust against all few-body perturbations, if its code distance grows at least logarithmically in the number of degrees of freedom, and it exhibits “check soundness.” Many constant-rate quantum LDPC expander codes have such properties, and define stable phases of matter with a constant zero-temperature entropy density, violating the third law of thermodynamics. Our results also show that quantum toric-code phases are robust to spatially nonlocal few-body perturbations. Similarly, phases of matter defined by classical codes are stable against symmetric perturbations. In the classical setting, we present improved locality bounds on the quasiadiabatic evolution operator between two nearby states in the same code phase.

quantum error correction↗

DOE OSTI2026

Efficient Routing of Quantum LDPC Codes on Programmable 2D Toric Architectures

Quantum low-density parity-check codes are promising candidates towards scalable fault-tolerant quantum computation. Among these, bivariate bicycle (BB) codes offer superior encoding rates and large code distance compared to surface codes. However, their requirement on long-range stabilizer measurements poses significant challenges for implementation on realistic hardware with limited connectivity, such as superconducting circuit platforms. In this work, we introduce a novel hardware-software co-design that leverages a programmable communication network architecture to address these limitations. Our approach utilizes a 2D toric network of oscillators as a flexible communication fabric linking qubits at each site. Such architecture significantly reduces the number of long-range couplers required from O ( n ) to O (√ n ). Dual-rail qubits, along with native gates including Swap-Wait-Swap gates and beamsplitter SWAPs, ensure that long-range two-qubit gates can be executed with high fidelity and low latency. To further enhance performance, our qubit layout and routing algorithm utilize symmetries of the codes and enable maximum parallelism for long-range two-qubit gates, maintaining a low syndrome extraction cycle duration and scalability over the code length. We perform circuit-level simulation with realistic noise modeling based on experimental hardware parameters, observing an logical error rate per logical qubit per cycle of 3.06% for [[18,4,4]] BB code, 2.6× less than the existing experimental result. These findings provide a practical roadmap and identify key technological advancements needed to achieve low-overhead fault-tolerant quantum computing at scale.

Liu, Kun [Yale Univ., New Haven, CT (United States↗