Queasy Instances: Quantum and Classical Complexity

• Queasy instances are individual inputs where quantum instance complexity is significantly lower than classical, highlighting clear quantum advantage.
• The framework integrates classical and quantum Kolmogorov complexities with specific time bounds to quantitatively assess instance difficulty.
• Reductions from problems like FACTORING and SAT provide canonical examples, guiding practical search for quantum advantage in structured instances.

Searching arXiv for recent and related papers on "queasy instances" and instance-level notions of computational advantage. Queasy instances are individual problem instances for which the quantum instance complexity is significantly smaller than the classical instance complexity. In the formal framework for quantum advantage, they are meant to isolate inputs that are quantum-easy but classically hard, thereby shifting attention from whole languages and worst-case hardness to single instances and their algorithmic descriptions. This instance-level viewpoint is motivated by quantum heuristics and by average-case rather than worst-case algorithmic analysis, and it introduces quantitative notions of queasiness that can be used to classify, compare, and search for quantum advantage on concrete inputs (Buhrman et al., 2 Oct 2025).

1. Conceptual setting

The formal notion of a queasy instance arises from combining classical instance complexity with quantum analogues of Kolmogorov complexity and distinguishing complexity. The central idea is that a computational advantage may be visible most sharply not at the level of an entire problem class, but at the level of a particular input $x$, together with the size of the shortest classical or quantum procedure that can correctly decide $x$ while remaining consistent with the language $L$ (Buhrman et al., 2 Oct 2025).

This perspective differs from the traditional worst-case analysis of computational complexity. A related shift appears in work on practically interesting MAXCUT instances, which argues that worst-case hardness can overstate practical difficulty and that structurally stable instances may be efficiently solvable even though MAXCUT remains NP-hard in general. In particular, the paper shows polynomial-time solvability for dense, metric, distinguished, expanding, and $2^{\sqrt n}$-stable instances under stated stability conditions, thereby formalizing a gap between worst-case complexity and practically relevant instance families (Bilu et al., 2012). This suggests a broader methodological alignment: queasy instances belong to a larger research program in which structure at the level of individual instances, rather than asymptotic hardness of the ambient problem, is the primary object of study.

A common misconception is to treat queasiness as a property of a language such as SAT or FACTORING as a whole. In the formal framework, queasiness is instead attached to a specific instance $x$ relative to a language $L$, under explicit time bounds for the classical and quantum descriptions. The same problem can therefore contain easy instances, hard instances, and queasy instances.

2. Complexity-theoretic definitions

The framework begins with classical Kolmogorov complexity $C(x)$, the size of the shortest program that outputs a string $x$, and its time-bounded version

$$C^t(x) = \min \{ |P| \mid P \text{ outputs } x \text{ in time } t(\lvert x \rvert) \}.$$

Its quantum analogue is the quantum time-bounded Kolmogorov complexity $QC^{t,\varepsilon}(x)$, defined as the size of the shortest classical program that outputs a quantum circuit $x$0 such that $x$1 outputs $x$2 with probability $x$3 in time $x$4. The dependence on $x$5 is stated to be non-essential by amplification (Buhrman et al., 2 Oct 2025).

The framework also uses distinguishing complexity. Classical distinguishing complexity $x$6 is the shortest program that accepts $x$7 and rejects all other inputs within time bound $x$8. Quantum distinguishing complexity $x$9 is the shortest program outputting a quantum circuit that accepts $L$0 with high confidence and rejects $L$1 (Buhrman et al., 2 Oct 2025).

For a language $L$2, the classical instance complexity of $L$3 is

$L$4

Here $L$5 is the characteristic function of $L$6, and $L$7 denotes “I don’t know.” The quantum version $L$8 is defined similarly, but with a quantum circuit $L$9 described by a classical program $2^{\sqrt n}$0, a flag qubit that signals “I know” or “I don’t know,” and correctness with high confidence whenever the circuit signals that it knows. The framework states that for any $2^{\sqrt n}$1, $2^{\sqrt n}$2 for suitable, slightly larger $2^{\sqrt n}$3 (Buhrman et al., 2 Oct 2025).

These definitions make queasiness a statement about succinctness of decision procedures for a single instance. The comparison is not between two runtimes alone, but between the sizes of the shortest classical and quantum descriptions that decide the instance under the relevant consistency conditions.

3. Measures of queasiness

For a problem $2^{\sqrt n}$4 and instance $2^{\sqrt n}$5, queasiness is defined by comparing quantum polynomial-time instance complexity with classical subexponential-time instance complexity. Formally, with $2^{\sqrt n}$6, $2^{\sqrt n}$7, and $2^{\sqrt n}$8,

$2^{\sqrt n}$9

The framework introduces two quantitative measures. The first is the queasiness value,

$x$0

and the second is the queasiness factor,

$x$1

The factor $x$2 lies in $x$3. Instances with $x$4 are described as maximally queasy; instances with $x$5 are not queasy (Buhrman et al., 2 Oct 2025).

Within this classification, the framework distinguishes three broad types of inputs. Easy instances have small quantum and small classical instance complexity. Hard instances have large quantum and large classical instance complexity. Queasy instances have small quantum and large classical instance complexity. The usefulness of the definition is that it quantitatively separates “quantum-easy, classically-hard” instances from both jointly easy and jointly hard instances, rather than folding them all into a single asymptotic complexity class.

This measure-based formulation also makes clear that queasiness is relative to time bounds. The same string may fail to be queasy under one pair of bounds and become queasy under another. The notion therefore belongs to time-bounded instance complexity rather than to absolute decidability.

4. Existence results and canonical examples

FACTORING provides the canonical example. On the quantum side, Shor’s algorithm implies

$x$6

because a universal quantum circuit can solve factoring with a description size that is constant. On the classical side, the framework states that all known algorithms have subexp exponential time, and that $x$7 is large for hard instances (Buhrman et al., 2 Oct 2025).

The main transfer result is a reduction to SAT. The framework states that if FACTORING requires exponential time classically, for example $x$8, then SAT contains infinitely many queasy instances. More specifically,

$x$9

for suitable $L$0, with $L$1. These SAT instances are described as maximally queasy under reasonable complexity-theoretic assumptions. The underlying intuition is that a poly-time invertible, one-to-one reduction from a variant of FACTORING transfers hard-but-quantum-easy instances into SAT (Buhrman et al., 2 Oct 2025).

The framework also gives density-style statements under additional assumptions. Under assumptions such as $L$2 or Quantum SETH, it proves strong lower bounds for the quantum instance complexity of hard SAT instances. Under reasonable assumptions on advice and verification complexity classes, such as $L$3, it further states that maximally queasy instances are exponentially dense (Buhrman et al., 2 Oct 2025).

These results matter because they place queasy instances beyond anecdotal examples. FACTORING supplies the intuitive prototype, but the reduction to SAT shows that the phenomenon is not confined to algebraic number-theoretic problems; it can be inherited by canonical NP problems at the instance level.

5. Algorithmic utility and the search for quantum advantage

The framework does not treat queasiness only as a classificatory notion. It also states an algorithmic utility theorem: if an instance $L$4 is maximally queasy, and the difference

$L$5

is large, then the associated quantum program $L$6 can decide an exponentially large set $L$7 of other instances, with

$L$8

The interpretation given is that a very small quantum instance complexity, combined with high distinguishing complexity, implies that the corresponding quantum program has substantial algorithmic reuse beyond the original instance (Buhrman et al., 2 Oct 2025).

This supports an instance-by-instance approach to quantum algorithm design. The framework explicitly suggests that one should search for and characterize queasy instances in relevant domains, because quantum heuristics with maximal queasiness are expected to solve broad families of classically hard instances. The queasiness factor $L$9 is proposed as a quantitative guide for selecting and classifying benchmark instances, and for the development of quantum heuristics, or “quristics” (Buhrman et al., 2 Oct 2025).

The same instance-centric logic appears, from the opposite direction, in benchmark engineering for optimization. “Practical engineering of hard spin-glass instances” treats the generation of hard Ising instances as an optimization problem, introducing Random Adaptive Optimization (RAO) and Loop Adaptive Optimization (LAO) to produce instances with large Parallel Tempering mixing times $C(x)$0, large Hamze-de Freitas-Selby runtimes $C(x)$1, temperature chaos, and large Hamming distance between the ground state and low-lying excited states. The purpose is to construct discriminative benchmarks without cherry picking from large random ensembles (Marshall et al., 2016). Queasy instances and RAO-generated hard instances occupy different regions of the algorithmic landscape, but both rely on the premise that carefully characterized single instances can reveal phenomena that average random benchmarks conceal.

6. Related uses of instance-centered analysis

The formal notion of queasy instances is specific to quantum advantage, but the broader practice of isolating structurally informative instances appears in several adjacent areas. In MAXCUT, the language of practically interesting instances is organized around stability and distinction. A $C(x)$2-stable instance preserves its optimal cut under multiplicative perturbations of the edge weights, and the paper gives polynomial-time algorithms for several structured families, including metric and dense instances under $C(x)$3-local stability, distinguished and expanding instances under appropriate conditions, and $C(x)$4-stable instances via repeated merging of vertices (Bilu et al., 2012). This is not the same definition as quantum queasiness, but it similarly replaces undifferentiated worst-case analysis with a taxonomy of instance classes.

In databases, “Understanding Queries by Conditional Instances” introduces c-instances as abstract representative instances with labeled nulls and global conditions. A c-instance denotes all ground instances obtained by substituting the labeled nulls with values satisfying the condition, and a universal solution is a set of minimal c-instances that collectively cover all possible coverages by which a query can be satisfied. For general domain relational calculus, the existence problem is undecidable; for restricted fragments such as conjunctive queries with safe negation, universal solutions are computable in polynomial time (Gilad et al., 2022). This is again a different formal program, but it shows how single instances can function as canonical representatives of behavior rather than merely as samples.

Other literatures use “instance” in operational senses that should not be conflated with queasy instances in the complexity-theoretic sense. In machine learning, Reduced Detrimental Instance Learning (RDIL) estimates how detrimental a data instance may be by approximating $C(x)$5 from an ensemble of classifiers and then weighting each instance continuously during training; the strongest empirical benefit is reported for multilayer perceptrons (Smith et al., 2014). In distributed SGD, volatile cloud instances are cheaper but preemptible worker nodes, and the main convergence bound depends on $C(x)$6, the expected reciprocal of the number of active workers (Zhang et al., 2020). In video see-through systems, geometry aware passthrough is reported to reduce nausea, disorientation, and total cybersickness relative to direct passthrough, even though warping artifacts increase (Chemaly et al., 17 Feb 2025). These works are instance-centered, but the “instance” being analyzed is respectively a training example, a compute node, or a rendering condition, not an input whose quantum and classical instance complexities are compared.

The technical meaning of queasy instances is therefore narrow and formal: it belongs to time-bounded quantum and classical instance complexity. Its broader significance is methodological. Across optimization, learning, databases, and quantum computing, researchers repeatedly turn to specially structured instances when worst-case statements fail to explain practical behavior. In the quantum-advantage setting, queasy instances are the precise expression of that turn: they identify where the shortest quantum description of a correct decision is much smaller than the shortest classical one, and thus where single-instance quantum advantage is most sharply exposed (Buhrman et al., 2 Oct 2025).