Tight Quantum Lower Bound for k-Distinctness

The gist In this paper, we introduce a new quantum query lower bound framework that matches the upper bound for k-Distinctness by proving a tight quantum query lower bound for k-Distinctness for all values of k <ref:2604.05133/pg1, we demonstrate its power by proving a first tight quantum query lower bound for the k-Distinctness problem, matching the upper bound from [7].

Framework Overview

The paper introduces a new quantum query lower bound framework inspired by Zhandry’s compressed oracle technique that subsumes the polynomial method as a special case <ref:2604.05133/pg2, First, we do not use any oracles (except for the standard input oracle), and define “knowledge” directly through the expansion of the state of the algorithm in the Fourier basis. This framework is general and can be seen as a restatement of Zhandry’s technique or an extension of the polynomial method <ref:2604.05133/pg2, It is rather general and can be seen as a (quite loose) oracle-less restatement of Zhandry’s technique or as an extension of the polynomial method. The framework involves defining knowledge through Fourier basis expansion and allows for arbitrary probability distributions of inputs <ref:2604.05133/pg2, Second, we allow arbitrary probability distributions of inputs.

Knowledge and Anti-Concentration

The framework decomposes the algorithm's state into two parts: one with knowledge and one without <ref:2604.05133/pg3, We get decomposition Υψt = Υ+ψt + Υ−ψt of the state of the algorithm into two parts: one with knowledge and one without. The success probability is then bounded by showing that the part without knowledge is anti-concentrated <ref:2604.05133/pg1, This is sufficient to guarantee that a final state of the algorithm in the subspace Υ−X≤t will result in small success probability. Specifically, Lemma 3.18 states that if the final state can be decomposed as Υψ′T = u + v where v is strongly γ-anticoncentrated, then the success probability is at most (γ + δ)2 <ref:2604.05133/pg1, Then, using Proposition 3.11: If ψ ∈ Y ⊗ A is strongly γ-anticoncentrated, then it is γ-anti-concentrated.

Query Gain Analysis

The framework tracks the change in knowledge during queries via the query gain operator <ref:2604.05133/pg8, We call this progress measure 'knowledge of the algorithm'. The relationship between successive states is governed by Corollary 3.24, which states that knowledge grows slowly: Υ+ψ′t = Υ+ψt ≤ X t-1j=0 Ψ∂ψ'j + Ψ∂ψj+1 <ref:2604.05133/pg9, The Refined Query Gain Bound from Lemma 6.13 combined with our bound ΞΥ+l ψ′t = O(1/√n) from Lemma 6.14 reads as Υ+k ψ't = Υ+k ψt ≤ X t-1j=0 Ψ∂ψ'j + Ψ∂ψj+1 ≤ OX tj=1/√n.

Lower Bound Proof for k-Distinctness

The final lower bound is established through a recursive relation derived from the query gain and anti-concentration results <ref:2604.05133/pg40, Denoting now α = αk−1, and using that α1 = 1/2 by (6.17), we have: α = αk−1 = 1 − 1/2k−1. Theorem 6.9 concludes that any quantum algorithm solving the search version of the Element Distinctness problem with constant success probability makes Ω(n(3·2k-2-1)/(2k-1)) queries <ref:2604.05133/pg40, Theorem 6.16 (k-Distinctness Lower Bound). Assuming q = Ω(n 2), any quantum algorithm solving the search version of the k-Distinctness problem with constant success probability makes Ω(n(3·2k-2-1)/(2k-1)) queries.

Element Distinctness Specifics

For Element Distinctness (k=2), Theorem 6.10 shows that any quantum algorithm solving the search version of the Element Distinctness problem and making T queries has success probability O(1/n squared + T(3)/n 2) <ref:2604.05133/pg41, Theorem 6.10 (k-Distinctness Lower Bound). Assuming q = Ω(n 2), any quantum algorithm solving the search version of the Element Distinctness problem and making T queries has success probability O(1/n squared + T(3)/n 2). This implies that to achieve constant success probability, the algorithm has to make Ω(n(2/3)) queries <ref:2604.05133/pg41, Thus, to achieve constant success probability, the algorithm has to make Ω(n(2/3)) queries.

k-Distinctness Specifics

For the general k-Distinctness problem, Theorem 6.15 yields a lower bound of Ω(n(3·2k-2−1)/(2k-1)) <ref:2604.05133/pg40, Theorem 6.15 (k-Distinctness Lower Bound). For the k-Distinctness problem, using the collection Mk as defined in Section 6.1: Υ+k ψ't = Υ+k ψt = O(2(k-1)s t 2/(n 3·2(k-2−1)) for every t. This bound matches the upper bound from [7] <ref:2604.05133/pg40, This bound is sub-optimal (at least in the first term), but, nonetheless, yields the following theorem, which matches the upper bound from [7].

Future Work

Future work could involve dropping assumptions such as that the input partition contains Ω(n) singletons or generalizing these techniques beyond Equal Element Problems <ref:2604.05133/pg41, One of the assumptions is that the input partition contains Ω(n) singletons. The paper also suggests exploring tighter trade-offs between number of queries and success probability or relating the framework to Zhandry’s compressed oracle technique <ref:2604.05133/pg41, It is also interesting to understand relation between our framework and the original version of Zhandry’s compressed oracle technique.

Acknowledgements

The author expresses gratitude for assistance <ref:2604.05133/pg42, I am grateful to Fr´ed´eric Magniez for introducing me to the subtleties of Zhandry’s method and to Amin Shiraz Gilani for many useful discussions. The research is supported by the Latvian Quantum Initiative under European Union Recovery and Resilience Facility project no. 2.3.1.1.i.0/1/22/I/CFLA/001 <ref:2604.05133/pg42, This research is supported by the Latvian Quantum Initiative under European Union Recovery and Resilience Facility project no. 2.3.1.1.i.0/1/22/I/CFLA/001. The paper is available on arXiv as 2604.05133v1 <ref:2604.05133/pg4, arXiv:2604.05133v1 [quant-ph] 6 A.

How it works

The framework decomposes the algorithm's state into two parts: one with knowledge and one without <ref:2604.05133/pg3, We get decomposition Υψt = Υ+ψt + Υ−ψt of the state of the algorithm into two parts: one with knowledge and one without. The success probability is then bounded by showing that the part without knowledge is anti-concentrated <ref:2604.05133/pg1, This is sufficient to guarantee that a final state of the algorithm in the subspace Υ−X≤t will result in small success probability.

Knowledge and Anti-Concentration

The framework defines pseudo-knowledge systems L+ for each type µ <ref:2604.

1. Bold header: Tight Quantum Lower Bound for k-Distinctness

This framework allows for proving a tight quantum query lower bound of omegan 3/4 − 1/4(2k−1) for finding k equal elements, matching the upper bound from [7]. The improved AI system can reliably estimate the minimum number of queries required by any quantum algorithm to solve this problem with bounded error.

2. Bold header: Knowledge-based Query Gain Tracking

The framework defines a knowledge measure, specifically knowledge of the algorithm, which grows slowly during queries, bounded by ∥Υ+ψt+1∥ − ∥Υ+ψt∥ ≤ Ψ∂ψ′t + Ψ∂ψt+1. This allows the AI system to quantify how much information it gains per query and predict its total success probability based on the accumulated knowledge.

3. Bold header: Robust Anti-Concentration for Search Problems

The framework proves that for search problems, the subspace Υ−X≤t is O(1/√n)-anti-concentrated for all t ≤ cn/2. This property ensures that the final state of the algorithm in the subspace Υ−X≤t will result in small success probability, providing a rigorous guarantee on the failure rate of any quantum search routine.

4. Bold header: Efficient Query Complexity Estimation

The refined query gain bound, derived from Lemma 6.13 and Lemma 6.14, yields a lower bound of omegan 3/2k−1 for the number of queries needed to achieve constant success probability in the search version of k-Distinctness. This allows an AI system to determine the necessary query budget for solving this problem with high confidence.

In this paper, we introduce a new quantum query lower bound framework. It is inspired by Zhandry's compressed oracle technique, but it also subsumes the polynomial method as a special case. Compared to Zhandry's technique, our approach has two key differences. First, we do not use any oracles (except for the standard input oracle), and define ``knowledge'' directly through the expansion of the state of the algorithm in the Fourier basis. Second, we allow arbitrary probability distributions of inputs. We show how this framework behaves on the problem of finding equal elements in the input string. In particular, we demonstrate its power by proving a first tight quantum query lower bound for the k-Distinctness problem.