[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-84734-en":3,"doc-seo-84734-105":29,"detail-sidebar-cat-0-en-105":91},{"code":4,"msg":5,"data":6},0,"success",{"doc_id":7,"user_id":8,"nickname":9,"user_avatar":10,"doc_module":4,"category_id":11,"category_name":12,"doc_title":13,"doc_description":14,"doc_content":15,"file_id":16,"file_url":17,"file_type":18,"file_size":19,"view_count":20,"is_deleted":4,"is_public":20,"is_downloadable":20,"audit_status":20,"page_count":21,"language":22,"language_code":23,"site_id":24,"html_lang":23,"table_of_contents":25,"faqs":26,"seo_title":13,"seo_description":14,"update_tm":27,"read_time":28},84734,137441390410,"Hazel","https://ap-avatar.wpscdn.com/avatar/2000252f4ab5702993?_k=1776741390130283984",8,"Research & Report","Noise-Aware Synthesis of Quantum LDPC Encoder Circuits via Two-Sided Hamming Descent","Quantum low-density parity-check (LDPC) codes support fault-tolerant quantum computation but require efficient preparation of encoded quantum states. The work casts encoder construction as a circuit-resynthesis task: starting from the linear-reversible CNOT block implementing the encoder matrix, it searches for a lower-cost equivalent circuit suitable for routing and explicitly mitigating noise. A two-sided Hamming descent optimization and a noise-aware pipeline reduce gate counts and two-qubit depth across CSS families, improving fidelity after routing. Scheduling further lowers routed preparation failure without increasing two-qubit gates for Bivariate Bicycle encoders.","Noise-Aware Synthesis of Quantum LDPC Encoder Circuits via Two-Sided Hamming Descent  \nAditya Sodhani∗ and Keshab K. Parhi†  \narXiv :2607 .04462v1 [ quant-ph] 5 Jul 2026  \nDepartment of Electrical and Computer Engineering,  \nUniversity of Minnesota, Minneapolis, Minnesota, USA  \nAbstract  \nQuantum low-density parity-check (LDPC) codes are a promising route to fault-tolerant quantum computation, but their use requires efficient preparation of encoded states. Standard encoder constructions generate circuits through fixed algebraic procedures, yet the resulting circuit can contain substantial redundancy. We formulate LDPC encoder preparation as a circuit-resynthesis problem: given the linear-reversible matrix implemented by the encoder’s CNOT block, we seek a lower-cost equivalent circuit that can be routed efficiently on the target hardware and which mitigates noise. We propose a novel optimization approach referred as two-sided Hamming descent and a noise-aware optimization pipeline for this task.  \nAcross several families of Calderbank-Shor-Steane (CSS) LDPC encoders, including Bivariate Bicycle, hypergraph-product, and entanglement-assisted codes, the proposed pipeline produces substantially smaller and shallower encoder circuits than the standard constructions and the synthesis baselines considered, cutting gate counts by 53 . 8% in aggregate across the benchmark and by up to 68% on the Bivariate Bicycle family. The gains remain visible after routing, where the two-qubit depth is reduced by up to 71% and translate into higher-fidelity state preparation under circuit-level noise. On the Bivariate Bicycle family, live-range scheduling further reduces routed preparation failure by up to 13.7% without adding two-qubit gates to the selected circuit. These results indicate that encoder-matrix resynthesis, combined with hardware-calibrated selection and scheduling, is an effective compiler-level tool for preparing quantum LDPC code states.  \nKeywords: quantum LDPC codes, encoder synthesis, linear reversible circuits, CNOT optimization, twosided Hamming descent, noise-aware optimization, circuit depth, commutation-aware scheduling, Bivariate Bicycle codes, fault-tolerant quantum computing.  \n1 Introduction  \nQuantum low-density parity-check (LDPC) codes are central to practical fault-tolerant quantum computation [1, 2, 3] . Bivariate Bicycle (BB) codes have recently been proposed for superconducting hardware and shown to achieve a high pseudo-threshold under circuit-level noise [4] . Hypergraph-product (HGP) codes [5] provide the first constant-rate family with polynomial distance and underlie subsequent asymptotically good constructions [6, 7, 8] . Entanglement-assisted quasi-cyclic LDPC (EA QC-LDPC) codes extend these constructions to settings with pre-shared entanglement [9, 10, 11] . Realizing any of these codes begins with  \n∗ [sodha005@umn.edu](sodha005@umn.edu)[ ](sodha005@umn.edu)†Corresponding author: [parhi@umn.edu](parhi@umn.edu)  \nthe same operational step. An encoder circuit maps unencoded qubits into a valid encoded state, the first step of the fault-tolerant stack. A noisier encoder injects more error into this initial logical state before the error-correction cycle begins, so high-fidelity preparation is a prerequisite for fault-tolerant performance [4] .  \nThe BB, HGP, and EA QC-LDPC families are all Calderbank-Shor-Steane (CSS) [12, 13] constructions, and their encoder circuits consist of Hadamard gates on a subset of qubits followed by a CNOT subsequence that entangles the qubits into the stabilizer state [14, 15, 16, 17] . The standard constructions for this task are the Cleve-Gottesman (CG) reduction [14] for standard CSS codes and the Sharma-Kumar-Garani (SKG) construction [18] for EA codes. Both construct the encoder circuit from the code’s parity-check matrix along a fixed algebraic path (reduced row echelon form with deterministic column ordering) . Their fixed elimination paths guarantee a valid encoder thr","cbCaivivKRyrZkW0","https://ap.wps.com/l/cbCaivivKRyrZkW0","pdf",820836,1,33,"English","en",105,"# Introduction\n## Fault-tolerant encoding and noise sensitivity\n## CSS encoder circuit structure\n## Existing encoder construction methods\n## Quantum circuit optimization categories","[{\"question\":\"What problem does the paper address in quantum LDPC encoder preparation?\",\"answer\":\"It targets inefficient and noise-exposing encoder circuits by reducing redundancy in the encoder’s CNOT-block implementation while producing routable, lower-cost equivalents for hardware execution.\"},{\"question\":\"What is the main idea behind “two-sided Hamming descent”?\",\"answer\":\"It reformulates encoder preparation as circuit resynthesis and applies an optimization approach that finds a lower-cost equivalent circuit from the linear-reversible matrix implemented by the encoder’s CNOT block, while being noise-aware.\"},{\"question\":\"How do the reported improvements behave after routing?\",\"answer\":\"The reduction in circuit metrics remains visible after routing: two-qubit depth decreases up to 71%, yielding higher-fidelity state preparation under circuit-level noise, and live-range scheduling further reduces preparation failure for the Bivariate Bicycle 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problem does the paper address in quantum LDPC encoder preparation?","Question",{"text":75,"@type":76},"It targets inefficient and noise-exposing encoder circuits by reducing redundancy in the encoder’s CNOT-block implementation while producing routable, lower-cost equivalents for hardware execution.","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"What is the main idea behind “two-sided Hamming descent”?",{"text":80,"@type":76},"It reformulates encoder preparation as circuit resynthesis and applies an optimization approach that finds a lower-cost equivalent circuit from the linear-reversible matrix implemented by the encoder’s CNOT block, while being noise-aware.",{"name":82,"@type":73,"acceptedAnswer":83},"How do the reported improvements behave after routing?",{"text":84,"@type":76},"The reduction in circuit metrics remains visible after routing: two-qubit depth decreases up to 71%, yielding higher-fidelity state preparation under circuit-level noise, and live-range scheduling 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