Quantum query complexity is known to be characterized by the so-called quantum adversary bound. While this result has been proved in the standard discrete-time model of quantum computation, it also holds for continuous-time (or Hamiltonian-based) quantum computation, due to a known equivalence between these two query complexity models. In this work, we revisit this result by providing a direct proof in the continuous-time model. One originality of our proof is that it draws new connections between the adversary bound, a modern theoretical computer science technique, and early theorems of quantum mechanics. Indeed, the proof of the lower bound is based on Ehrenfest's theorem, while the upper bound relies on the Adiabatic theorem, as we construct an optimal adiabatic quantum query algorithm.Quantum query complexity is known to be characterized by the so-called quantum adversary bound. While this result has been proved in the standard discrete-time model of quantum computation, it also holds for continuous-time (or Hamiltonian-based) quantum computation, due to a known equivalence between these two query complexity models. In this work, we revisit this result by providing a direct proof in the continuous-time model. One originality of our proof is that it draws new connections between the adversary bound, a modern theoretical computer science technique, and early theorems of quantum mechanics. Indeed, the proof of the lower bound is based on Ehrenfest's theorem, while the upper bound relies on the Adiabatic theorem, as we construct an optimal adiabatic quantum query algorithm.

Sep 11, 20140 citations

Efficient discrete-time simulations of continuous-time quantum query algorithms

Richard Cleve, Daniel Gottesman, Michele Mosca +2 more

The continuous-time query model is a variant of the discrete query model in which queries can be interleaved with known operations (called "driving operations") continuously in time. We show that any quantum algorithm in this model whose total query time is T can be simulated by a quantum algorithm in the discrete-time query model that makes O(T log T / loglog T) subset O~(T) queries. This is the first such upper bound that is independent of the driving operations (i.e., it holds even if the norm of the driving Hamiltonian is very large). A corollary is that any lower bound of T queries for a problem in the discrete-time query model immediately carries over to a lower bound of Omega(T loglog T / log T) subset Omega~(T) in the continuous-time query model.The continuous-time query model is a variant of the discrete query model in which queries can be interleaved with known operations (called "driving operations") continuously in time. We show that any quantum algorithm in this model whose total query time is T can be simulated by a quantum algorithm in the discrete-time query model that makes O(T log T / loglog T) subset O~(T) queries. This is the first such upper bound that is independent of the driving operations (i.e., it holds even if the norm of the driving Hamiltonian is very large). A corollary is that any lower bound of T queries for a problem in the discrete-time query model immediately carries over to a lower bound of Omega(T loglog T / log T) subset Omega~(T) in the continuous-time query model.

arXiv:0811.4428ACM Symposium on Theory of Computing (STOC)May 31, 200931 citations

The Multiplicative Quantum Adversary

Robert Špalek

We present a new variant of the quantum adversary method, a method for proving lower bounds on the quantum query complexity of a function. Adversary methods work as follows: one defines a progress function based on the state of the algorithm, and shows that for a successful algorithm there is a large gap between the initial and final value of the progress, and furthermore that the progress function cannot change by much with a single query. All known variants upper-bound the difference of the progress function, whereas our new variant upper-bounds the ratio and that is why we coin it the multiplicative adversary. Our new method is rooted in the quantum lower-bound method by Ambainis (2005, 2006), based on the analysis of eigenspaces of the density matrix. Ambainis's method is technically very complicated, it lacks intuition, and it only works for symmetric functions. Our method fits well into the adversary framework, has a simple formulation in terms of common block-diagonalization of two operators, and works for all functions. Furthermore, we prove an unconditional strong direct product theorem for the multiplicative quantum adversary bound.We present a new variant of the quantum adversary method, a method for proving lower bounds on the quantum query complexity of a function. Adversary methods work as follows: one defines a progress function based on the state of the algorithm, and shows that for a successful algorithm there is a large gap between the initial and final value of the progress, and furthermore that the progress function cannot change by much with a single query. All known variants upper-bound the difference of the progress function, whereas our new variant upper-bounds the ratio and that is why we coin it the multiplicative adversary. Our new method is rooted in the quantum lower-bound method by Ambainis (2005, 2006), based on the analysis of eigenspaces of the density matrix. Ambainis's method is technically very complicated, it lacks intuition, and it only works for symmetric functions. Our method fits well into the adversary framework, has a simple formulation in terms of common block-diagonalization of two operators, and works for all functions. Furthermore, we prove an unconditional strong direct product theorem for the multiplicative quantum adversary bound.

arXiv:quant-ph/0703237IEEE Conference on Computational Complexity (CCC)Jun 1, 200829 citations

A Brief Introduction to Quantum Query Complexity

Yassine Hamoudi

Quantum query complexity is a fundamental model for analyzing the computational power of quantum algorithms. It has played a key role in characterizing quantum speedups, from early breakthroughs such as Grover's and Simon's algorithms to more recent developments in quantum cryptography and complexity theory. This document provides a structured introduction to quantum query lower bounds, focusing on four major techniques: the hybrid method, the polynomial method, the recording method, and the adversary method. Each method is developed from first principles and illustrated through canonical problems. Additionally, the document discusses how the adversary method can be used to derive upper bounds, highlighting its dual role in quantum query complexity. The goal is to offer a self-contained exposition accessible to readers with a basic background in quantum computing, while also serving as an entry point for researchers interested in the study of quantum lower bounds.Quantum query complexity is a fundamental model for analyzing the computational power of quantum algorithms. It has played a key role in characterizing quantum speedups, from early breakthroughs such as Grover's and Simon's algorithms to more recent developments in quantum cryptography and complexity theory. This document provides a structured introduction to quantum query lower bounds, focusing on four major techniques: the hybrid method, the polynomial method, the recording method, and the adversary method. Each method is developed from first principles and illustrated through canonical problems. Additionally, the document discusses how the adversary method can be used to derive upper bounds, highlighting its dual role in quantum query complexity. The goal is to offer a self-contained exposition accessible to readers with a basic background in quantum computing, while also serving as an entry point for researchers interested in the study of quantum lower bounds.

arXiv:2508.08852Aug 12, 20250 citations

Lower bound on the time complexity of local adiabatic evolution

Zhenghao Chen, Pang Wei Koh, Yan Zhao

The adiabatic theorem of quantum physics has been, in recent times, utilized in the design of local search quantum algorithms, and has been proven to be equivalent to standard quantum computation, that is, the use of unitary operators [D. Aharonov et al. in Proceedings of the 45th Annual Symposium on the Foundations of Computer Science, 2004, Rome, Italy (IEEE Computer Society Press, New York, 2004), pp. 42--51]. Hence, the study of the time complexity of adiabatic evolution algorithms gives insight into the computational power of quantum algorithms. In this paper, we present two different approaches of evaluating the time complexity for local adiabatic evolution using time-independent parameters, thus providing effective tests (not requiring the evaluation of the entire time-dependent gap function) for the time complexity of newly developed algorithms. We further illustrate our tests by displaying results from the numerical simulation of some problems, viz. specially modified instances of the Hamming weight problem.The adiabatic theorem of quantum physics has been, in recent times, utilized in the design of local search quantum algorithms, and has been proven to be equivalent to standard quantum computation, that is, the use of unitary operators [D. Aharonov et al. in Proceedings of the 45th Annual Symposium on the Foundations of Computer Science, 2004, Rome, Italy (IEEE Computer Society Press, New York, 2004), pp. 42--51]. Hence, the study of the time complexity of adiabatic evolution algorithms gives insight into the computational power of quantum algorithms. In this paper, we present two different approaches of evaluating the time complexity for local adiabatic evolution using time-independent parameters, thus providing effective tests (not requiring the evaluation of the entire time-dependent gap function) for the time complexity of newly developed algorithms. We further illustrate our tests by displaying results from the numerical simulation of some problems, viz. specially modified instances of the Hamming weight problem.

Physical Review ANov 9, 20062 citations

Simulating quantum circuits by adiabatic computation: Improved spectral gap bounds

Shane Dooley, Graham Kells, Hosho Katsura +1 more

Adiabatic quantum computing is a framework for quantum computing that is superficially very different to the standard circuit model. However, it can be shown that the two models are computationally equivalent. The key to the proof is a mapping of a quantum circuit to an adiabatic evolution, and then showing that the minimum spectral gap of the adiabatic Hamiltonian is at least inverse polynomial in the number of computational steps $L$ . In this paper we provide two simplified proofs that the gap is inverse polynomial. Both proofs result in the same lower bound for the minimum gap, which for $L\ensuremath{\gg}1$ is ${min}_{s}{\mathrm{\ensuremath{\Delta}}}_{0}\ensuremath{\gtrsim}{\ensuremath{\pi}}^{2}/[8{(L+1)}^{2}]$, an improvement over previous bounds. Our first method is a direct approach based on an eigenstate ansatz, while the second uses Weyl's theorem to leverage known exact results into a bound for the gap. Our results suggest that it may be possible to use these methods to find bounds for spectral gaps of Hamiltonians in other scenarios.Adiabatic quantum computing is a framework for quantum computing that is superficially very different to the standard circuit model. However, it can be shown that the two models are computationally equivalent. The key to the proof is a mapping of a quantum circuit to an adiabatic evolution, and then showing that the minimum spectral gap of the adiabatic Hamiltonian is at least inverse polynomial in the number of computational steps $L$. In this paper we provide two simplified proofs that the gap is inverse polynomial. Both proofs result in the same lower bound for the minimum gap, which for $L\ensuremath{\gg}1$ is ${min}_{s}{\mathrm{\ensuremath{\Delta}}}_{0}\ensuremath{\gtrsim}{\ensuremath{\pi}}^{2}/[8{(L+1)}^{2}]$, an improvement over previous bounds. Our first method is a direct approach based on an eigenstate ansatz, while the second uses Weyl's theorem to leverage known exact results into a bound for the gap. Our results suggest that it may be possible to use these methods to find bounds for spectral gaps of Hamiltonians in other scenarios.

arXiv:1906.05233Physical Review AApr 1, 20205 citations

L

. In this paper we provide two simplified proofs that the gap is inverse polynomial. Both proofs result in the same lower bound for the minimum gap, which for $L\ensuremath{\gg}1$ is ${min}_{s}{\mathrm{\ensuremath{\Delta}}}_{0}\ensuremath{\gtrsim}{\ensuremath{\pi}}^{2}/[8{(L+1)}^{2}]$, an improvement over previous bounds. Our first method is a direct approach based on an eigenstate ansatz, while the second uses Weyl's theorem to leverage known exact results into a bound for the gap. Our results suggest that it may be possible to use these methods to find bounds for spectral gaps of Hamiltonians in other scenarios.