Research map / Learning theory and algorithms / Sample complexity
Quantum Speedup for Hypergraph Sparsification
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Proposes the first quantum algorithm for hypergraph spectral sparsification, producing an ε-spectral sparsifier with Õ(n/ε²) hyperedges in Õ(r√(mn)/ε + r√(mnr)) time, by quantizing hyperedge leverage score overestimates and quantum graph sparsification.
Problem
Hypergraph spectral sparsification is costly classically (state of the art Õ(mr) time); the open question was whether quantum speedups are possible for hypergraph sparsification.
Approach
Compute hyperedge leverage score overestimates over sparse underlying graphs using quantum graph sparsification and quantum effective resistance, then sample hyperedges via multi-copy state preparation and reweight with quantum sum estimation, with a chaining argument for correctness.
Key results
- Quantum algorithm outputs an ε-spectral sparsifier with O(n log n log r/ε²) hyperedges in Õ(r√(mnr) + r√(mn)/ε) time.
- For constant rank r, the runtime matches the quantum lower bound Ω̃(√(mn)/ε) for graph-type sparsification, versus classical Ω(m) lower bound.
- For dense hypergraphs (m = Ω(n^r)), reduces time from Õ(rn^r) classically to Õ(rn^((r+1)/2)) quantumly.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Use quantum hypergraph sparsification instead of the state-of-the-art classical algorithm | Hypergraph sparsification · Hypergraph sparsification | time complexity: \widetilde O(mr) -> \widetilde O\parens{r\sqrt{mnr} +r\sqrt{mn}/\varepsilon} | Helped |
| Use quantum hypergraph sparsification | Hypergraph sparsification · Hypergraph sparsification | sparsifier size: O( n \log n \log r /\varepsilon^2) -> O\parens{n\log n \log r /\varepsilon^2} | No clear effect |
| Use quantum cut sparsification to approximate mincuts | Hypergraph mincut · Hypergraph mincut | time complexity | Helped |
| Use quantum cut sparsification to approximate s-t mincuts | Hypergraph s-t mincut · Hypergraph s-t mincut | time complexity | Helped |
Limitations
- Relies on QRAM, whose practical feasibility is speculative
- Conjectured tightness only; no hypergraph-specific quantum lower bound established
- Assumes epsilon >= sqrt(n/m) and m >= nr
Experimental setup
Metrics: time complexity sparsifier size
Baselines: classical Õ(mr) hypergraph sparsification algorithm
Builds on: quantum graph sparsification John ellipsoid approximation group leverage score overestimates quantum sum estimation preparing many copies of a quantum state
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