We study the complexity of quantum query algorithms that make p queries in parallel in each timestep. This model is in part motivated by the fact that decoherence times of qubits are typically small, so it makes sense to parallelize quantum algorithms as much as possible. We show tight bounds for a number of problems, specifically Theta((n/p)^{2/3}) p-parallel queries for element distinctness and Theta((n/p)^{k/(k+1)} for k-sum. Our upper bounds are obtained by parallelized quantum walk algorithms, and our lower bounds are based on a relatively small modification of the adversary lower bound method, combined with recent results of Belovs et al. on learning graphs. We also prove some general bounds, in particular that quantum and classical p-parallel complexity are polynomially related for all total functions f when p is small compared to f's block sensitivity.We study the complexity of quantum query algorithms that make p queries in parallel in each timestep. This model is in part motivated by the fact that decoherence times of qubits are typically small, so it makes sense to parallelize quantum algorithms as much as possible. We show tight bounds for a number of problems, specifically Theta((n/p)^{2/3}) p-parallel queries for element distinctness and Theta((n/p)^{k/(k+1)} for k-sum. Our upper bounds are obtained by parallelized quantum walk algorithms, and our lower bounds are based on a relatively small modification of the adversary lower bound method, combined with recent results of Belovs et al. on learning graphs. We also prove some general bounds, in particular that quantum and classical p-parallel complexity are polynomially related for all total functions f when p is small compared to f's block sensitivity.

arXiv:1309.6116Sep 24, 20137 citations

Quantum State Superposition-Based Database Query Optimization

Jincheng Zhang

This paper investigates the potential of leveraging quantum state superposition to optimize database queries. Traditional database queries often face bottlenecks due to sequential processing and limitations in parallelization. We propose a novel approach that translates database queries into quantum state superposition problems, enabling parallel exploration of multiple query branches simultaneously. The core mechanism involves constructing a quantum circuit designed to exploit superposition and implementing algorithms for efficient result merging. We present a theoretical framework and a conceptual algorithm demonstrating the feasibility of this approach, highlighting the potential for significant speedups in query execution. The focus is on the conceptual design and theoretical considerations, without requiring experimental validation. The key innovation lies in directly mapping database query operations onto quantum superposition and leveraging quantum parallelism. This work represents a foundational step toward exploring quantum computing's role in database optimization.This paper investigates the potential of leveraging quantum state superposition to optimize database queries. Traditional database queries often face bottlenecks due to sequential processing and limitations in parallelization. We propose a novel approach that translates database queries into quantum state superposition problems, enabling parallel exploration of multiple query branches simultaneously. The core mechanism involves constructing a quantum circuit designed to exploit superposition and implementing algorithms for efficient result merging. We present a theoretical framework and a conceptual algorithm demonstrating the feasibility of this approach, highlighting the potential for significant speedups in query execution. The focus is on the conceptual design and theoretical considerations, without requiring experimental validation. The key innovation lies in directly mapping database query operations onto quantum superposition and leveraging quantum parallelism. This work represents a foundational step toward exploring quantum computing's role in database optimization.

Sep 5, 20260 citations

Parallel Quantum Advantage with Limited Adaptivity Requires Structure

Qipeng Liu, Saachi Mutreja

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 $\mathrm{poly}(T, 1/ε, 1/δ)$ 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.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 $\mathrm{poly}(T, 1/ε, 1/δ)$ 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.

quant-pharXiv:2608.20297Aug 21, 20260 citations

Quantum query algorithms and lower bounds

Andris Ambainis

Springer eBooksNov 2, 200726 citations

Span programs and optimal quantum query algorithms

Ben W. Reichardt

Bulletin of the American Physical SocietyMar 15, 20100 citations

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

\mathrm{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.