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A quantum divide-and-conquer algorithm solves general TSP in O*(1.866^n) time, the first to beat the classical 2^n barrier.

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2026-07-12 14:52 UTC pith:2HPCT2T7

load-bearing objection Solid query-complexity improvement and a clean Ambainis correction; the O*(1.866^n) time claim is model-dependent on QRAM depth-only accounting. the 2 major comments →

arxiv 2606.07322 v2 pith:2HPCT2T7 submitted 2026-06-05 quant-ph cs.CC

Quantum Divide-and-Conquer for the Traveling Salesman Problem: Surpassing the 2^n Barrier

Xujun Bai , Yun Shang , Honghong Lin This is my paper
classification quant-ph cs.CC
keywords quantum divide-and-conquerset partitionquantum searchtraveling salesman problemDicke statequantum minimum findingHeld-Karp
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

For more than sixty years the exact classical algorithm for the traveling salesman problem has been stuck at O*(2^n). This paper claims that a hybrid quantum method finally breaks that barrier. The idea is to cut the n cities into a small number of subsets, pre-compute shortest paths inside each subset by ordinary dynamic programming, then use quantum minimum finding to search over all ways of stitching those paths into a tour. With four subsets of carefully chosen sizes the search costs O*(1.866^n) queries; the same bound holds for total gate complexity once a compact quantum state that encodes every labelled partition is prepared with only O(n^{2}) gates. The authors also show that an earlier claim of O*(1.727^n) rests on an under-count of the solution space and never actually falls below 2^n. Small Qiskit simulations recover the optimal tour with high probability, supporting the framework.

Core claim

The first quantum algorithm that solves general TSP with both query and time complexity O*(1.866^n), obtained by partitioning the vertex set into four subsets of optimal relative sizes, classically pre-computing all short paths inside subsets of that size, and searching the resulting labelled partitions by quantum minimum finding after preparing a structured set-partition superposition.

What carries the argument

The set-partition state: a uniform superposition over all ways to colour the n vertices with k labels and mark an origin and end inside each colour class, prepared by a sequence of Dicke-state circuits of O(n^{2}) gates and O(n) depth that lets the oracle evaluate tour lengths in quantum parallel without a classical bijection.

Load-bearing premise

The claim that a memory device needing exponentially many qubits can still be counted as having only polynomial cost because its circuit depth is linear in n.

What would settle it

Either a corrected counting argument that pushes the optimal four-subset exponent back to at least 1, or an explicit classical construction showing that the set-partition state preparation itself requires super-polynomial resources once realistic memory access is accounted for.

Watch this falsifier. Get emailed when new claim-graph text bears on it.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Referee Report

2 major / 4 minor

Summary. The manuscript corrects a structural counting error in the Ambainis et al. (SODA 2019) hybrid quantum algorithm for TSP, proving that the claimed O*(1.727^n) query complexity is in fact Omega*(2^n) once the oracle is required to concatenate complete solutions over the full recursive tree (Theorem 1). It then introduces a parameterized quantum divide-and-conquer framework: classically precompute shortest paths inside subsets of size at most alpha_1 n via Held-Karp DP, prepare a uniform superposition over all labeled k-partitions (the set-partition state), and run quantum minimum finding. Optimizing the classical-quantum trade-off yields O*(1.890^n) for k=3 and the global optimum O*(1.866^n) for k=4 with alpha_i approximately 0.3157 (Theorem 2). An explicit O(n^2)-gate, O(n)-depth circuit based on Dicke-state preparation constructs the set-partition state (Theorem 3); the oracle is realized via bucket-brigade QRAM, converting the query bound into a claimed total time complexity of O*(1.866^n). Qiskit simulations for n=6,7 (k=3) report high success probabilities.

Significance. A genuine asymptotic improvement over the classical Held-Karp O*(2^n) barrier for general TSP would be a landmark result in quantum algorithms for NP-hard problems. The correction of the SODA 2019 claim is carefully derived and valuable on its own. The optimization of the classical-quantum balance (balancing binary-entropy precomputation against the square-root of the multinomial partition count) is thorough and free of circularity, and the explicit Dicke-based circuit for the set-partition state is a concrete technical contribution that addresses a common oracle-construction bottleneck. The framework is cleanly parameterized and potentially portable to other partition-based combinatorial problems. These strengths stand even if the strongest time-complexity claim requires model clarification.

major comments (2)
  1. Abstract and Section III.C: the conversion of the O*(1.865666...^n) query bound into a total time complexity of O*(1.866^n) rests on counting only the O(n) circuit depth of each bucket-brigade QRAM query while explicitly acknowledging that the QRAM itself requires exponentially many qubits and gates. Under ordinary quantum circuit accounting (total gate count, or space-time product), that exponential resource is not absorbed into the O* notation; the claimed time advantage therefore holds only in a non-standard model that charges solely for depth. The manuscript should either adopt an explicit depth-only model with a clear statement of the exponential space cost, or weaken the headline claim to the (already solid) query complexity plus polynomial-depth oracles under exponential space.
  2. Section III.C (select(f_i) construction): the Boolean-function selectors that load the classical DP table are stated to use O*(2^{4n}) ancillary qubits. Even if each individual QRAM query has depth O(n), the preparation and uncomputation of these exponential ancillas must be accounted for in any total-resource claim; otherwise the same model inconsistency that affects the QRAM tree reappears. A precise resource table (qubits, gates, depth) for one full oracle call under the chosen model is needed.
minor comments (4)
  1. Table I and the abstract both list 1.865666...n / 1.866^n; a single consistent numerical presentation (or an explicit statement that the two differ only by rounding) would avoid reader confusion.
  2. Figure 1 caption and the surrounding text in Section II.A: the recursive tree is clear, but labeling the eight leaves that must be concatenated would make the counting argument of Theorem 1 immediately visual.
  3. Section IV: the simulations replace QRAM by a QFT shortcut and fix the starting vertex; a short remark that these experiments validate only the amplitude-amplification and state-preparation components, not the full QRAM oracle, would set expectations correctly.
  4. Equation (15) and the subsequent gradient calculation: the induction establishing g_k is clean, but a one-line reminder that the multinomial coefficient identity underlies the closed form would help readers who skip the induction.

Circularity Check

0 steps flagged

No circularity: the O*(1.866^n) bound is obtained by independently balancing classical binary-entropy precomputation against the square-root of a multinomial partition count and solving a low-dimensional numerical optimization.

full rationale

The paper's central claim (Theorem 2) equates the classical cost O*(2^{n H(\alpha1)}) with the quantum cost O*(sqrt(n! / \prod mi!)), rewrites the latter via the closed-form entropy identity proved by induction in Eq. (15), and numerically solves f(\alpha1) = g*_4(\alpha1) for the unique balancing point \alpha1 = \alpha2 = \alpha3 \approx 0.315742. Neither expression is defined in terms of the other; the numerical value is simply the intersection of two independently derived monotone functions. The correction of Ambainis et al. (Theorem 1) is a pure counting argument that does not feed into the new bound. Structured state preparation (Theorem 3) and the QRAM oracle are constructive circuit arguments that convert the query bound into a claimed time bound; they introduce no fitted parameters or self-referential definitions. Self-citations are limited to ordinary background (Dicke states, quantum minimum finding) and are not load-bearing for the complexity claim. The derivation is therefore self-contained against external benchmarks and exhibits no circular step of any of the seven enumerated kinds.

Axiom & Free-Parameter Ledger

1 free parameters · 5 axioms · 1 invented entities

The central claim rests on standard quantum-search primitives, classical Held-Karp DP, known Dicke-state preparation, and the modeling assumption that QRAM supplies polynomial-depth access to an exponential classical table. The only numerically fitted quantities are the optimal subset proportions that balance the two complexity expressions; they are not free parameters in the usual data-fitting sense. No new physical entities are postulated.

free parameters (1)
  • optimal subset proportions alpha_1=alpha_2=alpha_3 = 0.315742...
    Numerically solved balancing point of classical binary-entropy cost and quantum partition-search cost; value approximately 0.315742 that yields the exponent 0.899691. Not fitted to empirical data but chosen to minimize the max of two analytic functions.
axioms (5)
  • standard math Quantum minimum finding (Durr-Hoyer / Boyer et al.) finds the minimum among N candidates with O(sqrt(N)) queries to a coherent oracle.
    Invoked throughout Sections II and III as the black-box search primitive.
  • standard math Classical Held-Karp DP computes all shortest paths through subsets of size at most m in O*(2^{n H(m/n)}) time.
    Used for the classical precomputation step of Algorithm 3 (Eq. 11).
  • standard math Dicke states |D_n^k> can be prepared deterministically with O(kn) gates and O(n) depth without ancillas (Bartschi-Eidenbenz).
    Building block for the set-partition state circuit of Theorem 3.
  • domain assumption A bucket-brigade QRAM can load classical data into a quantum register with circuit depth linear in the address length, even though the number of qubits and gates is exponential.
    Explicitly used in Section III.C to convert the query bound into a claimed total time bound of O*(1.866^n).
  • domain assumption The optimal Hamiltonian cycle always arises as the concatenation of shortest paths on some labeled k-partition (Lemma 1).
    Correctness foundation of the whole divide-and-conquer search; proved by a standard exchange argument.
invented entities (1)
  • set partition state independent evidence
    purpose: Uniform superposition over all labeled k-partitions that encodes the search space of Algorithm 3 so that the oracle need not implement a bijection from sequential indices.
    Constructed explicitly from Dicke states and controlled-SWAPs (Algorithm 4 / Theorem 3); not a free-floating postulate but a concrete circuit object. Independent evidence is the small-n Qiskit simulations that prepare and search the state.

pith-pipeline@v1.1.0-grok45 · 29828 in / 3345 out tokens · 30894 ms · 2026-07-12T14:52:22.105021+00:00 · methodology

0 comments
read the original abstract

The traveling salesman problem (TSP) is a classic NP-hard problem. Held--Karp dynamic programming~\cite{held1962dynamic, bellman1962dynamic} solves it exactly in $O(n^2 2^n)$ time, a barrier that has stood for over six decades. Whether quantum computing can surpass $O^*(2^n)$ is a central open question. The authors of~\cite{ambainis2019quantum}\ (SODA~2019) claimed a query complexity of $O^*(1.727^n)$, but we identify a structural counting error: when corrected, their scheme requires $\Omega^*(2^n)$ queries and offers no advantage over classical Held--Karp. We design a quantum divide-and-conquer framework: partition $n$-vertex set into $k$ subsets, classically precompute shortest paths within each, then search over all $k$-partitions via quantum minimum finding. We prove $k=3$ achieves $O^*(1.890^n)$, and $k=4$ attains the global optimum $O^*(1.866^n)$, the first quantum algorithm to surpass $O^*(2^n)$ for general TSP. To convert the query bound into a time-complexity advantage, we overcome the oracle-construction bottleneck by preparing a set partition state, requiring $O(n^2)$ gates and $O(n)$ depth. Leveraging structured state preparation, we achieve a total time complexity of $O^*(1.866^n)$. Qiskit simulations on $n=6,7$ achieve $98.9\%$ and $100\%$ accuracy.

Figures

Figures reproduced from arXiv: 2606.07322 by Honghong Lin, Xujun Bai, Yun Shang.

Figure 1
Figure 1. Figure 1: FIG. 1: Address transmission for one vertex (four qubits) in our encoding method. For the branches [PITH_FULL_IMAGE:figures/full_fig_p009_1.png] view at source ↗
Figure 1
Figure 1. Figure 1: FIG. 1: Recursive tree of [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2: The framework of our TSP solver, where the [PITH_FULL_IMAGE:figures/full_fig_p010_2.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2: Example of 4-partition [PITH_FULL_IMAGE:figures/full_fig_p008_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3: Basic structures of the quantum circuits for preparing Dicke states, which are used to [PITH_FULL_IMAGE:figures/full_fig_p013_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4: Quantum circuit for preparing Dicke state of 4 qubits. [PITH_FULL_IMAGE:figures/full_fig_p013_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5: The binary tree produced in preparing [PITH_FULL_IMAGE:figures/full_fig_p014_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6: The binary tree produced in preparing a 4-partition state with size of 3 [PITH_FULL_IMAGE:figures/full_fig_p014_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: FIG. 7: Basic module to split the components in [PITH_FULL_IMAGE:figures/full_fig_p015_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: FIG. 8: Basic modified structures of the quantum circuits for preparing the set partition states, [PITH_FULL_IMAGE:figures/full_fig_p016_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: FIG. 9: Address transmission for one vertex (four qubits) in our encoding method. For the branches [PITH_FULL_IMAGE:figures/full_fig_p017_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: FIG. 10: The framework of our TSP solver, where the [PITH_FULL_IMAGE:figures/full_fig_p018_10.png] view at source ↗

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