[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-85021-en":3,"doc-seo-85021-105":29,"detail-sidebar-cat-0-en-105":90},{"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":4,"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},85021,1374391975076,"Riley","https://ap-avatar.wpscdn.com/avatar/14000253ca4ec9f6853?x-image-process=image/resize,m_fixed,w_180,h_180&k=1783305029341752051",8,"Research & Report","A Sparse and Truncated State Vector Simulator for Peaked Circuits","A Sparse and Truncated State Vector Simulator for Peaked Circuits focuses on approximating the output of peaked quantum circuits by predicting the most probable output bit string. The approach replaces dense state-vector simulation with a sparse representation storing only nonzero amplitudes and introduces truncation based on a term limit or a probability-mass threshold. Efficiency is achieved through aggressive vectorization of state-vector operations and optional hardware acceleration (e.g., GPU). Performance and limitations are evaluated in an open-source implementation.","A Sparse and Truncated State Vector Simulator for  \nPeaked Circuits  \nDiogo R. Ferreira  \nInstituto Superior Te´cnico (IST)  \nUniversidade de Lisboa  \nLisbon, Portugal  \n[diogo.ferreira@tecnico.ulisboa.pt](diogo.ferreira@tecnico.ulisboa.pt)  \narXiv :2607 .07816v1 [ quant-ph] 8 Jul 2026  \nAbstract—In a class of quantum circuits known as peaked circuits, the goal is to predict the most probable bit string atthe output of the circuit. Since these circuits are designed to have a sharp peak in their output distribution, in principle it should be possible to simulate them using a truncated state vector with a limited number of terms, or a fraction of the total probability mass. This approximate simulation can be carried out on a classical computer with a sparse representation that stores only the nonzero amplitudes of the state vector, in contrast to the dense representations that are common in most quantum simulators. For efficiency, all operations on the state vector should be vectorized to the furthest possible extent and, if available, hardware acceleration can also be used. This work describes how these requirements were met in an open-source implementation, and discusses its performance and limitations.  \nIndex Terms—Quantum circuits, state-vector evolution, array vectorization, GPU computing.  \nI. INTRODUCTION  \nTo set up the context and terminology, we recall some basic concepts about state vectors. For a single-qubit system, the state vector can be written as:  \n|ψ⟩ = α0 |0⟩ + α 1 |1⟩ , with |α0 | 2 + |α1 | 2 = 1 (1)  \nwhere |0⟩ and |1⟩ are the basis states, and α0 and α 1 are complex amplitudes, whose squared magnitudes (interpreted as probabilities) sum to 1 .  \nA unitary operation on a single qubit can be written as:  \nU = 􀀔u00u10 u01u11􀀕 , with U†U = I (2)  \nwhere U† denotes the adjoint (conjugate transpose) of U.  \nApplying this unitary U on a single qubit yields a new state vector |ψ′ ⟩ = α′0 |0⟩ + α′1 |1⟩, with amplitudes:  \nα′0 = u00 α0 + u01 α 1  \n(3)  \nα′1 = u 10 α0 + u 11 α 1  \nIn a multiple-qubit system,  \n2n −1 2n −1  \n|ψ⟩ = X αi |i⟩ , with X |αi | 2 = 1 (4)  \ni=0 i=0  \nwhere |i⟩ is a basis state of the n-qubit system (usually expressed in binary form, in the range |00...00⟩ to |11... 11⟩), and αi is the corresponding amplitude.  \nIn an n-qubit system, applying U on a single qubit involves applying the expressions in Eq. (3) to every pair of amplitudes (αi ,αj) whose basis states differ only in that qubit (i.e. |i⟩ has 0 at the corresponding bit position, and |j⟩ has 1) while having the same bit pattern in all other qubits.  \nTherefore, in an n-qubit system, an operation on a single qubit might require updating all amplitudes across an entire state vector with 2n terms. The same applies to operationson two, three or more qubits, where the unitary matrix has dimensions 4×4, 8×8 or in general 2m ×2m for m≤n qubits.  \nIn practice, it often happens that many amplitudes are zero, which avoids the need to handle a full-blown state vector with 2n terms. For example, in most quantum circuits, the convention is that the circuit begins in the ground state (where α0 =1, and αi =0 for i>0), so the initial state of the system can be represented with a single term. However, the number of nonzero terms can grow rapidly:  \n• A unitary operation that creates superposition (e.g. the Hadamard gate) may double the number of nonzero terms in the state vector, because a qubit that was in the ground state (with an amplitude for 0) now has an amplitude for 0 and an amplitude for 1. All fixed patterns for other qubits will have to be duplicated and their amplitudes updated.  \n• For an operation that creates entanglement between two qubits, the number of nonzero terms in the state vector could quadruple (although the CNOT gate, for example, operates on two qubits and may create entanglement, but it just permutes the amplitudes without increasing the number of nonzero terms) .  \n• Operations on three and four qubits could potentially mul","cbCaibof8cE4j8xg","https://ap.wps.com/l/cbCaibof8cE4j8xg","pdf",248527,1,6,"English","en",105,"# Introduction\n## State-vector basics and unitary evolution\n## Sparse vs dense representations\n## Truncation for approximate simulation\n## Peaked circuits and the paper’s objective\n# Proposed simulator approach\n# Truncation methods\n# Vectorized implementation and acceleration\n# Performance and limitations","[{\"question\":\"What problem does the simulator address for peaked circuits?\",\"answer\":\"It targets predicting the most probable output bit string produced by peaked quantum circuits, whose output distributions have a sharp peak at a specific string.\"},{\"question\":\"How does the method reduce classical simulation cost?\",\"answer\":\"It uses a sparse state-vector representation that stores only nonzero amplitudes, and it applies truncation to keep only the most significant terms rather than the full exponentially growing vector.\"},{\"question\":\"On what criteria can truncation be performed?\",\"answer\":\"Truncation can be based on a hard maximum number of stored terms (memory limit) or on a threshold that keeps only a minimum fraction of the total probability 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problem does the simulator address for peaked circuits?","Question",{"text":74,"@type":75},"It targets predicting the most probable output bit string produced by peaked quantum circuits, whose output distributions have a sharp peak at a specific string.","Answer",{"name":77,"@type":72,"acceptedAnswer":78},"How does the method reduce classical simulation cost?",{"text":79,"@type":75},"It uses a sparse state-vector representation that stores only nonzero amplitudes, and it applies truncation to keep only the most significant terms rather than the full exponentially growing vector.",{"name":81,"@type":72,"acceptedAnswer":82},"On what criteria can truncation be performed?",{"text":83,"@type":75},"Truncation can be based on a hard maximum number of stored terms (memory limit) or on a threshold that keeps only a minimum fraction of the total probability 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