1.5-Query Setting in Quantum Complexity

• The 1.5-query setting is a quantum model allowing one full query plus a fractional query, enabling a nuanced approach to unitary synthesis.
• Methodologies like symmetrized linearization and chaining reduce deviation bounds from O(M log M/√K) to approximately O(1/√K) by leveraging structural properties.
• This framework enhances quantum cryptography, ensuring pseudorandom state ensembles are secure against adversaries with limited fractional-query access.

The “1.5-query setting” is an intermediate regime in quantum query complexity that specifically addresses resource requirements for the unitary synthesis problem when adversaries or algorithms are allowed fractional, rather than integral, numbers of queries. In this context, a 1.5-query algorithm is one that, operationally, may employ one full query alongside an additional fractional (amortized or distributed) query across computational branches—granting more power than the purely one-query model but falling short of unrestricted multi-query access. This scenario arises naturally in quantum information processing and reductions between oracle games and unitary implementation tasks, where fractional-query access provides a precise handle for delineating the advantages of limited adversaries in distinguishing, learning, or synthesizing quantum operations.

1.5-Query Regime: Definition and Motivation

The 1.5-query setting, as described in the paper, is not merely a theoretical convenience but is motivated by concrete operational scenarios. In particular, it characterizes a quantum adversary (or algorithm) that is allowed to make one standard oracle query plus a “half” (fractional, spread-out, or amortized) query per computational branch. This is not to be conflated with simply one or two queries; rather, it represents the case where the query access is slightly but not fully increased beyond the one-query regime, often as a byproduct of reduction analyses connecting oracle-state distinguishing to algorithmic unitary implementation.

This intermediary model enables fine-grained investigation into how a modest increase in query power affects the ability to distinguish, synthesize, or attack pseudorandom quantum operations. It is critical in theoretical quantum cryptography, where many lower bounds or security reductions must account for adversaries that are more powerful than “one shot” but still limited compared to fully adaptive multi-query algorithms.

Main Lower Bound Results

The central result of the cited work is a hierarchy of lower bounds for the unitary synthesis problem within the 1.5-query model. The deviation functional for all subsets, $\mathrm{Dev}_{1.5}$, is defined as:

$$\mathrm{Dev}_{1.5} := \sup_{S \subseteq [M]} \lVert \overline{Y}_S - \mu_S \rVert_\mathrm{op}$$

where $\overline{Y}_S$ represents the empirical outcome for subset $S$, and $\mu_S$ its mean. The main provable bounds (modulo polylogarithmic factors, denoted by $\tilde{\mathcal{O}}$) are as follows:

• Conservative (Assumption-Light):

$$\mathrm{Dev}_{1.5} = O\left(\frac{M \log M}{\sqrt{K}}\right)$$

• With block-orthogonality or “sum-to-max” structure:

$$\mathrm{Dev}_{1.5} = \tilde{\mathcal{O}}\left(\frac{(\log M)^{3/2}}{\sqrt{K}}\right)$$

• With chaining (sparsity $r = O(1)$ or effective dimension $d^* = O(\mathrm{polylog}(M))$):

$$\mathrm{Dev}_{1.5} = \tilde{\mathcal{O}}\left(\frac{1}{\sqrt{K}}\right)$$

Here, $M$ is the number of measurement outcome blocks, $K$ is the number of independent sample gadgets, and the bounds are stated up to polylogarithmic factors (i.e., $\tilde{\mathcal{O}}$ suppresses log terms).

These bounds are tight under mild per-sample operator norm constraints and demonstrate that the deviation in the 1.5-query regime interpolates between $O(M/\sqrt{K})$ (from naïve union bound arguments) and the optimal $O(1/\sqrt{K})$ when further structure is present.

Methodological Innovations

The proofs employ two principal methodologies:

• Conservative, Symmetrized Linearization: By symmetrizing the deviation functional with Rademacher variables and “linearizing” the supremum over subsets $S$, the deviation is expressed as a sum of independent block deviations. Application of a union bound and noncommutative Khintchine inequality yields the baseline $O(M \log M/\sqrt{K})$ bound.
• Chaining and Incremental Decomposition: For more structured settings (e.g., block-orthogonality or empirical sparsity), the supremum is analyzed via a chaining process, decomposing increments between index sets and defining suitable pseudometrics based on Hamming distance ($|S \Delta T|$) or square-root Hamming metrics. By Dudley’s entropy integral, and, in certain regimes, a “sum-to-max” trick, the linear dependence on $M$ is sharply reduced, enabling the achievement of $\tilde{\mathcal{O}}(1/\sqrt{K})$ deviation.

The use of chaining techniques—uncommon in the quantum query setting—provides a refined control over the power set of all outcome indices and hence the tightest known control of the fractional-query complexity regime.

Cryptographic Consequences

A key cryptographic application is the demonstration that pseudorandom quantum state ensembles and related primitives (such as quantum bit commitment schemes) attain security even against adversaries equipped with 1.5 queries. Specifically, previous one-query secure constructions [Lombardi, Ma, Wright 2023] are shown to retain their security when adversaries are allowed one full query plus a fractional (amortized) query across branches.

In parameter regimes relevant to cryptography (i.e., $K = \Theta(N)$ where $N$ is the security parameter), the distinguishing advantage is bounded by $\tilde{\mathcal{O}}(1/\sqrt{N})$ even when the adversary is upgraded to the 1.5-query setting—rendering the security advantage vanishingly small.

This clarifies that the transition from “one-shot” security to full multi-query adversary security is not abrupt; the 1.5-query model demonstrates strong resilience, provided the underlying structure (block-orthogonality/sparsity) is exploited.

Interpretive and Practical Significance

The work sheds light on the continuum between strict one-query and two-query regimes, underscoring that small increases in adversarial query power, modeled via fractional queries, yield intermediate but non-trivial lower bounds on query complexity. The 1.5-query framework is not an artificial interpolation but one that directly arises in quantum algorithm reductions and oracle-based constructions.

Practically, the results show that quantum protocol designers can obtain robust security guarantees against a spectrum of adversary models, provided the explicit dependence on $M$ (outcome block count) and $K$ (sample size) is controlled through block-structural or sparsity assumptions in the underlying primitive or protocol.

The chaining and entropy arguments developed in the paper add new tools for analyzing empirical process deviations in quantum information theory, with relevance for future protocols where adversaries have sub-integer query access.

Technical Table: Deviation Bounds by Structural Assumptions

Regime	Deviation Bound (up to logs)	Key Assumption
Conservative (no structure)	$O(\frac{M \log M}{\sqrt{K}})$	General, minimal extra assumptions
Block-orthogonality/sum-to-max	$$\tilde{\mathcal{O}}(\frac{(\log M)^{3/2}}{\sqrt{K}})$$	Block-orthogonal measurement or outcomes
Chaining with $r=O(1)$ or $d^*=O(\mathrm{polylog}(M))$	$\tilde{\mathcal{O}}(\frac{1}{\sqrt{K}})$	Empirical sparsity or low effective dimension

The table highlights the transition from linear dependence on $M$ in unstructured settings to poly-logarithmic or dimension-induced scaling under added structure.

Consequences and Future Directions

The clarification and tightening of lower bounds in the 1.5-query setting push the frontier of quantum query complexity. The direct implications for quantum cryptography—specifically, the security of pseudorandom quantum states against fractional-query adversaries—are immediate and strengthen previous claims. Future research directions include extending the chaining techniques to multi-level, arbitrary fractional-query settings, and exploiting the block-sparsity paradigm in the design of more efficient and robust quantum protocols. This work establishes the 1.5-query regime as a critical and natural point in quantum complexity theory and cryptography, both conceptually and technically.