{"arxiv_id":"quant-ph/0010117","incoming":{"papers":91,"sentences":149,"groups":[{"kind":"method_or_tool","label":"Uses its method or tool","papers":13,"sentences":20,"quotes":[{"arxiv_id":"1509.07112","title":"Quantum Walk on the Line through Potential Barriers","url":"/paper/1509.07112","section":"Analysis with Potential Barriers","sentence":"As described in [ref] , this can be done using the method of steepest descent, where we examine the time-oscillatory portion of [ref] and [ref], which is [equation] where we have defined [equation] From the analysis of [ref] , the part we care about is the movement of the peak in probability (e.g., the leading right peak in Figure [ref]).","intent_p":1.0,"contests_p":null},{"arxiv_id":"1009.0200","title":"Interference Phenomena in Quantum Information","url":"/paper/1009.0200","section":"Meeting problem for distinguishable particles","sentence":"Concerning the quantum walk, in Appendix [ref] we approximate the single particle probability distribution according to [ref] and replace the sum in ([ref]) by integral.","intent_p":0.95,"contests_p":null},{"arxiv_id":"1301.2905","title":"Quantum walks with memory on cycles","url":"/paper/1301.2905","section":"Fourier analysis","sentence":"In order to obtain the expression for the amplitudes of the quantum walk with memory on cycle we use the method introduced in [ref] and represent time evolution of the walk using the Fourier transform [equation] From the above we get a recursive relation for the time evolution in the Fourier basis [equation] where [equation]","intent_p":0.96,"contests_p":null},{"arxiv_id":"1909.09214","title":"Asymptotic reduced density matrix of discrete-time quantum walks","url":"/paper/1909.09214","section":"formalism","sentence":"([ref]) as discussed in detail in Ref. [ref] .","intent_p":0.81,"contests_p":null},{"arxiv_id":"1409.8134","title":"Limit theorems of a two-phase quantum walk with one defect","url":"/paper/1409.8134","section":"Introduction","sentence":"The Fourier analysis [ref] and the stationary phase method [ref] are useful to study position-independent QWs.","intent_p":0.83,"contests_p":null}]},{"kind":"comparison","label":"Compares with it","papers":3,"sentences":4,"quotes":[{"arxiv_id":"1504.07418","title":"Asymptotic properties of the Dirac quantum cellular automaton","url":"/paper/1504.07418","section":"Stationary phase method","sentence":"Eq (8) in [ref] ), where one obtains a common factor $1+(-1)^t+tα$, reflecting the parity properties of the QW: for even (odd) $t$, the probability distribution vanishes at odd (even) sites.","intent_p":0.99,"contests_p":null},{"arxiv_id":"1109.1866","title":"Quantum Walks on the Line with Phase Parameters","url":"/paper/1109.1866","section":"Contributions","sentence":"This method is in fact stronger than the stationary point method from [ref] and [ref] , where the authors use it to study the asymptotics of the resulting probability distribution from coin operators with real eigenvalues.","intent_p":0.96,"contests_p":null},{"arxiv_id":"2603.16837","title":"Ergodicity in discrete-time quantum walks","url":"/paper/2603.16837","section":"Coins with nonzero entries","sentence":"The paper [ref] asked the same question but had a different approach.","intent_p":0.95,"contests_p":null}]},{"kind":"background","label":"Cites it as background","papers":51,"sentences":56,"quotes":[{"arxiv_id":"1611.00634","title":"Field Induced Oscillation of Two Majorana Modes for a finite Quantum Wire","url":"/paper/1611.00634","section":"Introduction","sentence":"In the literature of discrete quantum walk [ref] , the Kitaev model [ref] for quantum wire is one of the simplest systems that supports these topological phases, which are identified as Majorana boundary modes [ref] .","intent_p":0.95,"contests_p":null},{"arxiv_id":"2010.07127","title":"Entanglement transfer, accumulation and retrieval via quantum-walk-based qubit-qudit dynamics","url":"/paper/2010.07127","section":"Background","sentence":"Background on QWs Discrete-time QWs embody a widely studied type of interaction between a two-dimensional “coin” degree of freedom, and a high-dimensional “walker” one [ref] .","intent_p":1.0,"contests_p":null},{"arxiv_id":"1309.5253","title":"Quantum walks on embedded hypercubes","url":"/paper/1309.5253","section":"Introduction","sentence":"Both continuous [ref] and discrete [ref] models of QW have been formulated, which exhibit properties that are qualitatively different from the classical analog [ref] .","intent_p":0.88,"contests_p":null},{"arxiv_id":"1202.4208","title":"Symmetry and localization of quantum walk induced by extra link in cycles","url":"/paper/1202.4208","section":"Introduction","sentence":"Most of previous studies have studied quantum walks on some simple graphs, such as the line [ref] , cycle [ref] , hypercube [ref] , trees [ref] , dendrimers [ref] , ultrametric spaces [ref] , threshold network [ref] , and other regular networks with simple topology [ref] .","intent_p":0.89,"contests_p":null},{"arxiv_id":"1502.06293","title":"Moments of Coinless Quantum Walks on Lattices","url":"/paper/1502.06293","section":"Introduction","sentence":"Moments and the mean square displacement (or variance) of the coined quantum walk on one-dimensional lattices were analyzed in Refs. [ref] and on two-dimensional lattices in Refs. [ref] .","intent_p":0.92,"contests_p":null}]}],"unlabelled_sentences":69},"outgoing":{"papers":0,"sentences":0,"groups":[],"unlabelled_sentences":0},"claims":[],"thresholds":{"intent_p":0.8,"supports_p":0.95,"contests_p":0.7},"method":"Citing sentences come from each paper's LaTeX source; every \\cite marks the sentence and the works it cites. Pith's classifier typed each sentence. A label is shown at intent probability 0.8 or more (0.95 for support), a contest at 0.7 or more. Measured accuracy against two blind model families is at https://pith.science/accuracy. The quoted sentences are the citing authors' own.","held_up":{"arxiv_id":"quant-ph/0010117","kind":"model","paper_version":1,"prompt_version":"held-up-v3.5","generated_at":"2026-10-27T18:11:43+00:00","loaded_at":"2026-10-28T01:03:06.257802+00:00","evidence":"substantial","account":[],"groups":[{"kind":"method_or_tool","label":"Uses its method or tool","papers":6,"entries":[{"p":0.95,"quote":null,"citing":[{"self":false,"year":2010,"title":"Interference Phenomena in Quantum Information","arxiv_id":"1009.0200"}],"verified":true,"paraphrase":"Approximated the single particle probability distribution according to this paper and replaced the sum by an integral."},{"p":0.96,"quote":null,"citing":[{"self":false,"year":2013,"title":"Quantum walks with memory on cycles","arxiv_id":"1301.2905"}],"verified":true,"paraphrase":"Used the method introduced in this paper to obtain the amplitudes of the quantum walk with memory on a cycle and represented its time evolution using the Fourier transform."},{"p":0.81,"quote":null,"citing":[{"self":false,"year":2015,"title":"An Effective Hamiltonian Approach to Quantum Random Walk","arxiv_id":"1505.01435"}],"verified":true,"paraphrase":"Following another work and this paper, considered the basis state of the particle as a product state of the chirality and position basis states."},{"p":0.97,"quote":null,"citing":[{"self":false,"year":2012,"title":"Controlling and reversing the transition from classical diffusive to quantum ballistic transport in a quantum walk by driving the coin","arxiv_id":"1209.3376"}],"verified":true,"paraphrase":"Used path integration from this paper to obtain the solution for the quantum part of the walk."},{"p":0.99,"quote":null,"citing":[{"self":false,"year":2023,"title":"Closed-form expressions for the probability distribution of quantum walk on a line","arxiv_id":"2308.05213"}],"verified":true,"paraphrase":"Following the analysis given in another work, transformed the state vector in this paper to momentum space."},{"p":0.82,"quote":null,"citing":[{"self":false,"year":2021,"title":"Quantum Corralling","arxiv_id":"2104.03141"}],"verified":true,"paraphrase":"Used this paper for the inverse Fourier transform."}],"papers_all":14,"self_papers":0,"self_entries":[]}],"counts":{"by_kind":{"comparison":3,"method_or_tool":14},"dropped":{"relied_on":0,"unknown_ids":0,"account_lines":0,"contest_vetoed":0,"below_threshold":0,"contest_drawback":0,"contest_without_basis":0,"contest_not_one_sentence":0,"contest_on_group_citation":0},"papers_read":21,"self_papers":0,"citing_papers":90,"sentences_read":29,"citing_sentences":149,"unambiguous_read":29,"typed_independent":16,"independent_papers":90,"by_kind_independent":{"comparison":3,"method_or_tool":14}},"input_sha256":"c6f9870c2a22a1c119fe865f501bfd30d6141f06aee9db3768713aa0c3020483","method":"For a paper with five or more citing arXiv papers, Pith selected up to 40 citing sentences its citation classifier typed, read those whose citation markers point at this paper alone, and wrote one paraphrase for each later paper in each group, each resting on one sentence. A later paper is listed under a label only when the classifier gives one of its sentences that label at intent probability 0.8 or more (0.95 for support), Pith's reading of the sentence finds that the citing paper itself did it, and a second check of the paraphrase against the sentence passes. The section lists uses only: building on the paper, using its method, tool, data or values, comparing with it and citing it as support. It does not list criticism or disagreement, so their absence says nothing about how the paper has been received. Code counted the papers, wrote the summary from the listed entries, set the evidence level and linked each entry to its citing paper. Quotes are copied from the citing sentence and shown only when the citing paper's licence is CC BY, CC BY-SA or CC0; otherwise the paraphrase is shown. Papers that share an author with this one are listed apart. A paper with fewer citing papers gets a line counted by code, which names uses only: a later paper whose only label is a contest is counted as not described.","line":null,"summary":"6 later papers by other authors are listed below: 6 used its method or tool."},"_links":{"html":"https://pith.science/paper/quant-ph/0010117"}}