{"arxiv_id":"quant-ph/0511096","incoming":{"papers":39,"sentences":82,"groups":[{"kind":"extends","label":"Builds on it","papers":2,"sentences":2,"quotes":[{"arxiv_id":"0809.2307","title":"Quantum Computation Beyond the Circuit Model","url":"/paper/0809.2307","section":"Approximation and BQP-complete Problems","sentence":"This result was reformulated and extended to $e^i 2 π/k$ for arbitrary $k$ by Aharonov et al. 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Qubit","url":"/paper/0807.4688","section":"HOMFLY polynomials","sentence":"Our presentation closely[However, for consistency with [ref] , we use $k$ and $r$ to represent the parameters called $l$ and $k$, respectively, in [ref] .] follows that of [ref] .","intent_p":0.83,"contests_p":null},{"arxiv_id":"2505.00457","title":"On estimating Schatten norm and power distances between quantum states","url":"/paper/2505.00457","section":"Proof techniques: containment for constantly above","sentence":"Our algorithm is sketched as follows: * Find a good approximation polynomial for $x ·x^α- 1$. * Implement a unitary block-encoding $U$ of $ν_- ·ν_-^α-1$ using Quantum Singular Value Transformation (QSVT) [ref] and Linear Combinations of Unitaries (LCU) [ref] , given the state-preparation circuits of $ρ_0$ and $ρ_1$. * Perform the Hadamard test [ref] on $U$ and $ρ_j$ with outcome $b_j ∈ 0,1$ for each $j ∈ 0,1$. * Estimate $(ρ_0,ρ_1)$ by computing the expected value of $b_0-b_1$.","intent_p":1.0,"contests_p":null},{"arxiv_id":"1003.0923","title":"Approximating Turaev-Viro 3-manifold invariants is universal for quantum computation","url":"/paper/1003.0923","section":"Introduction","sentence":"An estimate to within $ϵ$ of the desired matrix element $⟨v_, g| ρ_, g(x) |v_, g⟩$ can be given, except with probability $δ$, using $O(log(1/δ) /ϵ^2)$ Hadamard tests, as in [ref] .","intent_p":0.99,"contests_p":null},{"arxiv_id":"2410.13559","title":"On estimating the trace of quantum state powers","url":"/paper/2410.13559","section":"Proof techniques: containment for q constantly larger than 1","sentence":"Our approach to estimating $ρ^q$ is via one-bit precision phase estimation [ref] , also known as the Hadamard test [ref] , equipped with the quantum singular value transformation (QSVT) [ref] .","intent_p":1.0,"contests_p":null}]},{"kind":"comparison","label":"Compares with it","papers":4,"sentences":4,"quotes":[{"arxiv_id":"0707.2831","title":"Estimating Jones polynomials is a complete problem for one clean qubit","url":"/paper/0707.2831","section":"Fibonacci and Path Model Representations","sentence":"This sign difference is a minor detail which arises only because [ref] chooses a different fourth root of $t$ for $A$ than we do.","intent_p":0.99,"contests_p":null},{"arxiv_id":"1105.5100","title":"Approximating the Turaev-Viro Invariant of Mapping Tori is Complete for One Clean Qubit","url":"/paper/1105.5100","section":"Analogy with Jones polynomials","sentence":"Similarly, estimating the Jones polynomial of the trace closure of a braid is DQC1-complete [ref] , while estimating the Jones polynomial of the plat closure of a braid is BQP-complete [ref] .","intent_p":0.98,"contests_p":null},{"arxiv_id":"0809.2307","title":"Quantum Computation Beyond the Circuit Model","url":"/paper/0809.2307","section":"Fibonacci and Path Model Representations","sentence":"This sign difference is a minor detail which arises only because [ref] chooses a 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closure of braid $b$. $w(b^tr)$ is denotes the writhe of the link $b^tr$.","intent_p":0.86,"contests_p":null},{"arxiv_id":"0809.2307","title":"Quantum Computation Beyond the Circuit Model","url":"/paper/0809.2307","section":"Introduction","sentence":"That is, this problem exactly captures the power of the quantum circuit model [ref] .","intent_p":0.82,"contests_p":null},{"arxiv_id":"1105.5100","title":"Approximating the Turaev-Viro Invariant of Mapping Tori is Complete for One Clean Qubit","url":"/paper/1105.5100","section":"Introduction","sentence":"In particular, approximating the Jones polynomial, a famous invariant of links, is a BQP-complete problem [ref] .","intent_p":0.86,"contests_p":null},{"arxiv_id":"0805.0007","title":"Superpolynomial speedups based on almost any quantum circuit","url":"/paper/0805.0007","section":"Background","sentence":"Apart from the BQP-complete problems [ref] , the main tool for solving most of these problems is a quantum Fourier transform (QFT) over some group.","intent_p":0.91,"contests_p":null},{"arxiv_id":"1009.0114","title":"On the degeneracy of $SU(3)_k$ topological phases","url":"/paper/1009.0114","section":"Introduction","sentence":"The representation of the braid group corresponding to $SU(N)_k$ also produces a topological invariant of knots and links called the single-variable HOMFLY polynomial at the $(N+k)^th$ root of unity. $N=2$ yields the Jones polynomial as a special case (see [ref] ).","intent_p":0.86,"contests_p":null}]}],"unlabelled_sentences":50},"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. 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