Additive approximation of Turaev-Viro 3-manifold invariants
Given a compact, orientable three-manifold presented by a Heegaard splitting, compute a certain additive approximation to its Turaev-Viro invariant, the scalar topological invariant that takes the same value on homeomorphic manifolds.
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Given a compact, orientable three-manifold presented by a Heegaard splitting, compute a certain additive approximation to its Turaev-Viro invariant, the scalar topological invariant that takes the same value on homeomorphic manifolds. The Turaev-Viro invariants are scalar topological invariants of compact, orientable 3-manifolds, and Alagic, Jordan, Koenig and Reichardt give a quantum algorithm that additively approximates them for a manifold presented by a Heegaard splitting. The paper describes the algorithm as motivated by the relationship between topological quantum computers and (2+1)-D topological quantum field theories. The abstract states that its accuracy is shown to be nontrivial in the following sense: the same algorithm, after efficient classical preprocessing, can solve any problem efficiently decidable by a quantum computer, so approximating certain Turaev-Viro invariants of manifolds presented by Heegaard splittings is a universal problem for quantum computation, which the Zoo records by saying that this approximation is BQP-complete. The Zoo sets the result beside an earlier polynomial-time quantum algorithm that additively approximates the Witten-Reshitikhin-Turaev (WRT) invariant of a manifold given by a surgery presentation, notes that squaring the WRT invariant yields the Turaev-Viro invariant, and states that whether the earlier approximation is BQP-complete is unknown.
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How it works
The Turaev-Viro invariants are scalar topological invariants of compact, orientable 3-manifolds, and Alagic, Jordan, Koenig and Reichardt give a quantum algorithm that additively approximates them for a manifold presented by a Heegaard splitting. The paper describes the algorithm as motivated by the relationship between topological quantum computers and (2+1)-D topological quantum field theories. The abstract states that its accuracy is shown to be nontrivial in the following sense: the same algorithm, after efficient classical preprocessing, can solve any problem efficiently decidable by a quantum computer, so approximating certain Turaev-Viro invariants of manifolds presented by Heegaard splittings is a universal problem for quantum computation, which the Zoo records by saying that this approximation is BQP-complete. The Zoo sets the result beside an earlier polynomial-time quantum algorithm that additively approximates the Witten-Reshitikhin-Turaev (WRT) invariant of a manifold given by a surgery presentation, notes that squaring the WRT invariant yields the Turaev-Viro invariant, and states that whether the earlier approximation is BQP-complete is unknown. This record's speedup class, "Superpolynomial", is a secondary source's classification of the approximation and simulation algorithms it files this under — not a claim its primary paper makes. Not checked against the primary source yet. The sources read state no complexity bound for this record (Two sources were read and neither states a cost for this algorithm. The abstract of arXiv:1003.0923 quotes no running time, query count, gate count or qubit count: it says only "We give a quantum algorithm for additively approximating Turaev-Viro invariants of a manifold presented by a Heegaard splitting" and that "the same algorithm, after efficient classical preprocessing, can solve any problem efficiently decidable by a quantum computer". The Quantum Algorithm Zoo entry "Three-manifold Invariants" is likewise unquantified for this result: "a quantum computer can efficiently find a certain additive approximation to its Turaev-Viro invariant, and this approximation is BQP-complete [ 129 ]." The only clause in that entry naming a running time belongs to a different algorithm in a different reference — "Earlier, in [ 114 ], a polynomial-time quantum algorithm was given to additively approximate the Witten-Reshitikhin-Turaev (WRT) invariant of a manifold given by a surgery presentation" — and that paper is not part of this record, so its bound is not carried over here. BQP-completeness is a statement of hardness rather than a cost, so the field is left empty rather than filled from elsewhere.).
Implementation
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turaev-viro-invariants.txt
ALGORITHM: Additive approximation of Turaev-Viro 3-manifold invariants
PROBLEM: Given a compact, orientable three-manifold presented by a Heegaard splitting, compute a certain additive approximation to its Turaev-Viro invariant, the scalar topological invariant that takes the same value on homeomorphic manifolds.
IDEA: The Turaev-Viro invariants are scalar topological invariants of compact, orientable 3-manifolds, and Alagic, Jordan, Koenig and Reichardt give a quantum algorithm that additively approximates them for a manifold presented by a Heegaard splitting. The paper describes the algorithm as motivated by the relationship between topological quantum computers and (2+1)-D topological quantum field theories. The abstract states that its accuracy is shown to be nontrivial in the following sense: the same algorithm, after efficient classical preprocessing, can solve any problem efficiently decidable by a quantum computer, so approximating certain Turaev-Viro invariants of manifolds presented by Heegaard splittings is a universal problem for quantum computation, which the Zoo records by saying that this approximation is BQP-complete. The Zoo sets the result beside an earlier polynomial-time quantum algorithm that additively approximates the Witten-Reshitikhin-Turaev (WRT) invariant of a manifold given by a surgery presentation, notes that squaring the WRT invariant yields the Turaev-Viro invariant, and states that whether the earlier approximation is BQP-complete is unknown.
REPORTED COST: Not stated by the sources read
BASIS: Two sources were read and neither states a cost for this algorithm. The abstract of arXiv:1003.0923 quotes no running time, query count, gate count or qubit count: it says only "We give a quantum algorithm for additively approximating Turaev-Viro invariants of a manifold presented by a Heegaard splitting" and that "the same algorithm, after efficient classical preprocessing, can solve any problem efficiently decidable by a quantum computer". The Quantum Algorithm Zoo entry "Three-manifold Invariants" is likewise unquantified for this result: "a quantum computer can efficiently find a certain additive approximation to its Turaev-Viro invariant, and this approximation is BQP-complete [ 129 ]." The only clause in that entry naming a running time belongs to a different algorithm in a different reference — "Earlier, in [ 114 ], a polynomial-time quantum algorithm was given to additively approximate the Witten-Reshitikhin-Turaev (WRT) invariant of a manifold given by a surgery presentation" — and that paper is not part of this record, so its bound is not carried over here. BQP-completeness is a statement of hardness rather than a cost, so the field is left empty rather than filled from elsewhere.
PRIMARY SOURCE: Gorjan Alagic, Stephen P. Jordan, Robert Koenig, Ben W. Reichardt (2010), Approximating Turaev-Viro 3-manifold invariants is universal for quantum computation — https://arxiv.org/abs/1003.0923
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Quantum vs classical
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Literature & references
Approximating Turaev-Viro 3-manifold invariants is universal for quantum computation2010 · Gorjan Alagic, Stephen P. Jordan, Robert Koenig, Ben W. Reichardt
Primary source: it gives the quantum algorithm that additively approximates Turaev-Viro invariants of a manifold presented by a Heegaard splitting, motivates the construction by the relationship between topological quantum computers and (2+1)-D topological quantum field theories, and establishes that approximating certain such invariants is a universal problem for quantum computation, which it presents as a novel relation between distinguishing non-homeomorphic 3-manifolds and the power of a general quantum computer. Consult it for the accuracy the approximation achieves and for the classical preprocessing the universality claim assumes, since the abstract quantifies neither and states no running time.