One of the most basic conjectures in quantum complexity theory states that every $t$-query quantum algorithm can be simulated on most inputs by a $\mathrm{poly}(t)$-query classical algorithm. If true, this would provide broad justification for the need for structure in quantum speedups.
We settle this conjecture for parallel quantum algorithms, showing that every $t$-query $d$-round quantum algorithm can be simulated on most inputs with $t^{O(d^2)}$ classical queries. This suggests that for unstructured problems, superpolynomial speedups would require quantum circuits of superconstant depth, and exponential speedups would further require polynomial depth. In contrast, most known speedups for structured problems are achieved by highly parallel, low-depth algorithms.
Our techniques also carry new implications for the status of $\mathsf{BPP}$ vs. $\mathsf{BQP}$ relative to a random oracle, a similarly longstanding problem.
One of the most basic conjectures in quantum complexity theory states that every t-query quantum algorithm can be simulated on most inputs by a poly(t)-query classical algorithm. If true, this would provide broad justification for the need for structure in quantum speedups.
We settle this conjecture for parallel quantum algorithms, showing that every t-query d-round quantum algorithm can be simulated on most inputs with tO(d2) classical queries. This suggests that for unstructured problems, superpolynomial speedups would require quantum circuits of superconstant depth, and exponential speedups would further require polynomial depth. In contrast, most known speedups for structured problems are achieved by highly parallel, low-depth algorithms.
Our techniques also carry new implications for the status of BPP vs. BQP relative to a random oracle, a similarly longstanding problem.
Aaronson and Ambainis (Theory of Computing, 2014) conjectured that quantum query algorithms admit efficient almost-everywhere classical simulation: for any T-query quantum algorithm, its acceptance probability can be approximated on a (1−δ) fraction of inputs, up to ε additive error, using poly(T,1/ε,1/δ) classical queries. At a high level, the conjecture suggests that exponential quantum speedups are possible only on sufficiently structured inputs.
In this work, we make progress on this conjecture by proving it for quantum algorithms that make massively parallel quantum queries. In contrast, Yamakawa and Zhandry (Journal of the ACM, 2024) showed that quantum algorithms restricted to parallel queries can still achieve exponential speedups over classical algorithms for sampling problems. We establish our simulation theorem by proving the stronger statement that parallel-query quantum algorithms cannot distinguish the uniform distribution over oracles from oracles drawn from so-called "dense distributions". Our main technical contribution is a coupling theorem that relates the uniform distribution over oracles to oracles drawn from dense distributions.
We further extend this approach beyond the purely parallel setting. We prove simulation theorems for quantum algorithms with constant rounds of adaptivity, and for quantum algorithms with a bounded quantum-query prefix followed by a massively parallel quantum-query stage.
Smoothed analysis is a central framework in classical algorithms for explaining the performance of algorithms beyond the worst case, often explaining why algorithms perform well in practice. We initiate a systematic study of its quantum counterpart and show the following results. (1) We show that there is a total function whose smoothed quantum query complexity is exponentially smaller than its classical query complexity. (2) We give near-tight characterizations of smoothed randomized and quantum query complexities for symmetric Boolean functions, unifying the worst-case complexity results of [Beals et al, FOCS'98] and average-case complexity results of [Ambainis and de Wolf, STACS'00]. (3) We study string problems such as pattern matching and edit distance and, in various regimes, give polynomial to superpolynomial quantum speedups. Our main technical ingredients include a near-tight quantum algorithm for ε-approximating the number of collisions between two non-repetitive strings, improving the result of Le Gall and Ng [QIC'22]. Together, our results show that smoothing can reveal larger quantum speedups than worst-case analysis suggests, opening a path towards quantum advantage on more realistic inputs.
Quantum query complexity underpins much of our understanding of quantum speedups, yet nearly all existing lower and upper bounds are derived for simple Boolean oracles. Many practical tasks-noisy database search, stochastic optimization, adaptive learning, and time-varying systems-are best modeled by richer oracle classes. This paper systematically extends quantum query complexity theory to four non-standard oracle models: (1) stochastic oracles that return random outputs drawn from a distribution; (2) multi-solution oracles encoding a solution set of size k; (3) time-dependent oracles whose internal state evolves between queries; and (4) relational oracles encoding structured constraints rather than simple membership. Using generalized adversary methods, polynomial method techniques, and hybrid arguments, we derive tight (up to polylog factors) lower bounds for each model and construct matching algorithms where possible. For instance, we show that searching for any one of k marked items in an unstructured database still admits a Grover-like quadratic speedup, but with optimal complexity Θ(N/k), and that stochastic noise in oracle outputs degrades the advantage gracefully as a function of the noise parameter. For certain time-dependent and relational models, we prove that no better-than-classical speedup is possible, clarifying the limits of quantum advantage in adaptive settings. These results provide a more realistic foundation for quantum algorithms targeting noisy search, stochastic optimization, and constraint satisfaction, and they highlight the precise structural features that enable or preclude query-based quantum speedups.
We study when partial Boolean functions can (and cannot) exhibit superpolynomial quantum query speedups, and develop a general framework for ruling out such speedups via two complementary lenses: promise-aware complexity measures and function completions. First, we introduce promise versions of standard combinatorial measures (including block sensitivity and related variants) and prove that if the relevant promise and completion measures ``collapse'', then deterministic and quantum query complexities are necessarily polynomially related, i.e. D(f)=poly(Q(f)). We then analyze structured families of promises, including symmetric partial functions and promises supported on Hamming slices, obtaining sharp (up to polynomial factors) characterizations in terms of a single gap parameter for the symmetric case and refined slice-dependent bounds for k-slice domains. Next, we formalize completion complexity as the minimum of a measure over total completions of a partial function, and show that completability of a measure captures the possibility of superpolynomial quantum speedups. Finally, we apply this viewpoint to derive broad non-speedup criteria for some classes of functions admitting well-behaved completions, such as functions defined on a single slice or a union of slices, functions with low maximum influence on both the standard and p-biased hypercubes and functions with efficiently identifiable domains, and then show some hardness results for general completion techniques.
Oracle-based quantum computation reframes efficiency from classical time/space costs to query complexity, i.e., the number of accesses to a black-box unitary oracle Uf needed to reveal a hidden property or target state. This study comparatively analyzes query complexity across four canonical oracle-driven algorithms—Deutsch’s, Bernstein–Vazirani, Grover’s search, and Amplitude Amplification—to clarify how superposition, entanglement, interference, and phase kickback translate into provable speedups independent of hardware details. The objective is to systematically connect oracle structure to achievable quantum advantage by contrasting exact, single-query interference algorithms with iterative, probabilistic rotation-based algorithms, and by benchmarking classical deterministic/probabilistic query requirements against quantum bounds. Methodologically, we (i) mathematically model Deutsch and Bernstein–Vazirani via stepwise state-vector evolution to expose phase kickback and phase-encoded answer extraction in one oracle call, (ii) geometrically track Grover and Amplitude Amplification as two-dimensional state-space rotations, quantifying how repeated oracle-driven reflections amplify success amplitude, and (iii) compute classical query baselines and compare them directly with quantum query scaling. Results show that Deutsch reduces classical 2 queries to exactly 1 quantum query for constant-versus-balanced testing, while Bernstein–Vazirani deterministically recovers an n-bit hidden string s with (O(1)) quantum queries versus (O(n)) classical queries. For unstructured search over (N=2n) items, Grover achieves a bounded-error quadratic improvement from (O(N)) to (O(√{N}) queries, and Amplitude Amplification generalizes this to boost success probability from (p) to near certainty in O (1/√p) oracle applications. The central implication is a structural divide: rigid algebraic structure permits maximal, single-query speedups, whereas unstructured “chaos” necessitates iterative geometric amplification, which is more broadly applicable but demands highly stable, error-corrected hardware; consequently, quantum systems are best viewed as specialized co-processors activated when oracle structure warrants a quantum shortcut.
Query complexity provides a basic framework to quantify the inherent computational cost of functions in the model where access to the input is limited to oracle queries. In this paper, we provide a detailed discussion of classical and quantum models of inquiry, emphasizing the structural differences and the common mathematical basis. We consider deterministic, randomized and quantum query complexities and highlight key techniques to prove lower bounds such as adversarial techniques, polynomial approximations and hybrid arguments . The main theme of this paper is the polynomial-spectral relation which underlies both the classical and the quantum models. In the classical setting, the decision tree complexity is closely related to the degree of real polynomials representing Boolean functions. In the quantum setting, the relation involves the spectral properties of the operators controlling quantum query algorithms, and gives deep insights into quantum accelerations. We study tight bounds for basic problems such as search and element distinctness, proving separations between classical and quantum query complexity. The present study uses methods from approximation theory, linear algebra and spectrum analysis to build a foundation for a complete understanding of the power and limitations of query-based computation.