Quantum verification gives exponential edge for TreeEQμ graph property
arXiv: 2611.08076v1
Recent work shows that a one‑round distributed quantum Merlin–Arthur protocol can certify the TreeEQμ property with only O( μ log n ) qubits per node, while any classical μ‑round distributed MA protocol needs at least Ω( n /( μ log² n ) ) bits per node. This is an exponential quantum advantage that applies to all network sizes. The same study presents a one‑round quantum protocol for the graph symmetry problem (SYM). It certifies the existence of a nontrivial automorphism using about O~( n5⁄3 ) qubits per node, compared to the classical lower bound of Ω( n² / μ ) bits per node for μ rounds. These results come from a detailed comparison shown in a table that lists the qubit and bit bounds for each protocol. The authors also note lower limits for related problems: a one‑round quantum protocol for equality on a path still requires Ω( log N ) qubits at some node, and any classical μ‑round protocol for SYM requires at least Ω( n² / μ ) bits per node. Together, these findings highlight a clear separation between quantum and classical distributed verification for network‑topology–dependent graph properties.
Table showing, for TreeEQμ, a quantum one‑round qubit bound of O(μ log n) versus a classical μ‑round bit bound of Ω(n/(μ log²n)); for SYM, a quantum bound of O~(n^{5/3}) versus a classical bound of Ω(n²/μ).
Property
One-round 𝖽𝖰𝖬𝖠: qubits per node and message
μ-round 𝖽𝖬𝖠: certificate bits per node
TreeEQμ
O(μlogn)
Ω(n/(μlog2n)); fixed μ≥1
SYM
O~(n5/3)
Ω(n2/μ); μ=o(n)
SYMD
O(nDlogn)
Ω(n2/μ); D≥5, μ≤D−4, μ=o(n)
EQN2
O(min{rlogN,N2/3(logN)5/3})
Ω(N/μ) if r≥2μ+1 [25, Theorem 9]
Table 1. Table 1 compares the one‑round quantum certificate sizes to the classical μ‑round lower bounds for TreeEQμ and SYM, showing exponential quantum advantage.Jáuregui, Benjamín, Miyamoto, Masayuki · arXiv:2611.08076v1 · Table 1: · CC BY 4.0