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arxiv: 2604.10881 · v1 · submitted 2026-04-13 · 🪐 quant-ph · cs.CR

Answering Counting Queries with Differential Privacy on a Quantum Computer

Arghya Mukherjee , Hassan Jameel Asghar , Gavin K. Brennen This is my paper

Pith reviewed 2026-05-10 16:20 UTC · model grok-4.3

classification 🪐 quant-ph cs.CR
keywords differential privacyquantum computingcounting queriesamplitude estimationprivacy amplificationquantum data encoding
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The pith

Counting queries on a quantum-encoded dataset reduce to measuring the amplitude of one of two orthogonal states while preserving differential privacy.

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

The paper shows that basic counting queries on private data can be answered on a quantum computer while satisfying differential privacy. By encoding the dataset in a quantum state, the query result becomes the amplitude of a particular basis state. Two measurement approaches are analyzed in detail: repeated measurements in the computational basis, which strengthen privacy through their built-in randomness, and amplitude estimation, for which a precise bound on amplitude sensitivity to single-record changes is derived. These techniques also support outsourcing the computation to a quantum server that returns private answers without seeing the raw data.

Core claim

We show that answering these queries on a quantum encoded dataset reduces to measuring the amplitude of one of two orthogonal states. We then analyze the differential privacy properties of two algorithms from literature to measure amplitude: one which performs repeated measurements in the computational basis, and the other which utilizes the classic amplitude estimation algorithm. For the first technique, we prove privacy results for the case of counting queries that improve on previously known results on general queries, and show that the mechanism in fact amplifies privacy due to inherent randomness. For the second method, we derive a tight bound on maximum possible change in the amplitude

What carries the argument

Reduction of a counting query to the amplitude of one of two orthogonal states in a quantum superposition that encodes the dataset, so that measurement statistics directly yield the count while allowing sensitivity analysis for privacy.

If this is right

  • Repeated measurements on counting queries amplify privacy beyond the bounds known for general queries.
  • The tight sensitivity bound on amplitude allows construction of a differentially private amplitude estimation algorithm.
  • Both methods support outsourcing counting queries to a quantum server while maintaining differential privacy.
  • The privacy analysis is specific to counting queries and yields stronger guarantees than those available for arbitrary queries.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The amplitude-reduction approach might extend to other linear statistics if analogous quantum encodings can be found.
  • Practical deployment would require quantum hardware that maintains coherence for datasets large enough to be useful.
  • The observed privacy amplification could motivate new classical sampling mechanisms that deliberately inject similar randomness.

Load-bearing premise

The dataset can be faithfully encoded into a quantum state and the quantum computer can perform the required measurements without significant noise or decoherence.

What would settle it

Measure the actual change in amplitude after adding or removing one record from the encoded dataset and check whether it exceeds the derived tight global sensitivity bound; a larger change would invalidate the privacy guarantee for amplitude estimation.

Figures

Figures reproduced from arXiv: 2604.10881 by Arghya Mukherjee, Gavin K. Brennen, Hassan Jameel Asghar.

Figure 1
Figure 1. Figure 1: Circuit to implement canonical quantum amplitude estimation [PITH_FULL_IMAGE:figures/full_fig_p013_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: An illustration of the fact that if |sin2 θ1 − sin2 θ2| = 1/n, for any two angles θ1, θ2 ∈ [0, π/2], then the maximum possible absolute distance between the angles is achieved in the period where the slope of sin2 θ is the flattest. This corresponds with one of the end-points of the period being either π/2 or 0 due to the symmetry of the function around π/4. Let θα = π/2 and θα′ = π/2 − β, where 0 < β ≤ π/… view at source ↗
Figure 3
Figure 3. Figure 3: The function f(x) = sin2 x over the interval [0, π/2]. The function is concave on the interval [π/4, π/2]. Further note that sin2 (π/2 − x ′ ) − sin2 (π/4) = sin2 π/4 − sin2 x ′ . Proof. Let f(x) = sin2 x. We first assume that θα, θα′ ∈ [π/4, π/2]. Without loss of generality, assume that θα > θα′ . Set θα = π 2 and θα′ = π 2 − β. Then |θα − θα′ | = β, and |sin2 θα − sin2 θα′ | = [PITH_FULL_IMAGE:figures/f… view at source ↗
read the original abstract

Differential privacy is a mathematical notion of data privacy that has fast become the de facto standard in privacy-preserving data analysis. Recently a lot of work has focused on differential privacy in the quantum setting. Continuing on this line of study, we investigate how to answer counting queries on a quantum encoded dataset with differential privacy. An example of a counting query is ``How many people in the dataset are over the age of 25 and with a university education?'' Counting queries form the most basic but nonetheless rich set of statistics extractable from a dataset. We show that answering these queries on a quantum encoded dataset reduces to measuring the amplitude of one of two orthogonal states. We then analyze the differential privacy properties of two algorithms from literature to measure amplitude: one which performs repeated measurements in the computational basis, and the other which utilizes the classic amplitude estimation algorithm. For the first technique, we prove privacy results for the case of counting queries that improve on previously known results on general queries, and show that the mechanism in fact \emph{amplifies} privacy due to inherent randomness. For the second method, we derive a tight bound on maximum possible change in the amplitude if we add or remove a single item in the dataset, a quantity called global sensitivity which is central in making an algorithm differentially private. We then show a differentially private version of the amplitude estimation algorithm for counting queries. We also discuss how these methods can be outsourced to a quantum server to blindly compute counting queries with differential privacy.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 4 minor

Summary. The paper investigates answering counting queries with differential privacy on a quantum computer. It claims that such queries on a quantum-encoded dataset reduce to measuring the amplitude of one of two orthogonal states. The authors analyze the differential privacy properties of repeated computational-basis measurements, showing improved privacy results and privacy amplification for counting queries, and of amplitude estimation, for which they derive a tight global sensitivity bound on amplitude change due to single item addition/removal, enabling a differentially private version. They also discuss outsourcing these computations to a quantum server.

Significance. If the results hold, this work establishes a direct connection between quantum amplitude techniques and differential privacy for basic counting queries, which are fundamental in data analysis. Notable strengths include the self-contained privacy analyses, the explicit tight global sensitivity bound for the amplitude, and the demonstration of privacy amplification due to quantum randomness. This could advance the field of quantum differential privacy by providing mechanisms that leverage quantum properties for better privacy-utility tradeoffs.

minor comments (4)
  1. The claim that the mechanism 'amplifies privacy due to inherent randomness' (abstract) would benefit from a brief explanation or reference to how this amplification is quantified compared to classical mechanisms for general queries.
  2. The encoding of the dataset into a quantum state (central to the reduction to amplitude measurement) relies on preparing a uniform superposition; the assumptions on this encoding circuit and its complexity should be stated more explicitly in the preliminaries to clarify applicability.
  3. The discussion on outsourcing to a quantum server would be strengthened by citing related work on blind quantum computation.
  4. Notation for the two orthogonal states and the predicate circuit could be introduced with explicit definitions earlier to improve readability for readers unfamiliar with the quantum encoding.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive summary and recommendation of minor revision. The review correctly captures the core contributions of our work on reducing counting queries to amplitude measurements and deriving privacy guarantees for both repeated measurements and amplitude estimation. Since no specific major comments were raised, we provide a brief overall response below and will incorporate any minor editorial suggestions in the revised version.

Circularity Check

0 steps flagged

No significant circularity; derivation is self-contained

full rationale

The central reduction of counting queries to amplitude measurement on orthogonal states follows directly from applying a predicate circuit to the uniform superposition over the quantum-encoded dataset. The global sensitivity bound is obtained by explicit calculation of the amplitude change under addition or removal of a single record, without any fitting, renaming, or self-referential definitions. Subsequent privacy analyses for repeated measurements and amplitude estimation apply standard differential-privacy calibration to this independently derived sensitivity; no load-bearing step reduces to a fitted input, self-citation chain, or ansatz smuggled via prior work. The paper is self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The work rests on standard quantum mechanics and differential privacy definitions with no free parameters, new entities, or ad-hoc axioms introduced.

axioms (2)
  • domain assumption A classical dataset can be encoded into a quantum state such that counting queries correspond to amplitude measurements
    Stated directly in the abstract as the starting point for both algorithms
  • domain assumption Standard differential privacy definitions and sensitivity analysis extend to quantum amplitude measurements
    Used to prove privacy for repeated measurements and to bound amplitude change for the estimation algorithm

pith-pipeline@v0.9.0 · 5570 in / 1203 out tokens · 45314 ms · 2026-05-10T16:20:33.662143+00:00 · methodology

discussion (0)

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Reference graph

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