# Quantum lower bounds by quantum arguments

@article{Ambainis2000QuantumLB, title={Quantum lower bounds by quantum arguments}, author={Andris Ambainis}, journal={J. Comput. Syst. Sci.}, year={2000}, volume={64}, pages={750-767} }

We propose a new method for proving lower bounds on quantum query algorithms. Instead of a classical adversary that runs the algorithm with on input and then modifies the input, we use a quantum adversary that runs the algorithm with a superposition of inputs. If the algorithm works correctly, its state becomes entangled with the superposition over inputs. We bound the number of queries needed to achieve a sufficient entanglement and this implies a lower bound on the number of queries for the…

## 406 Citations

### Quantum Adversary (Upper) Bound

- Computer Science, MathematicsChic. J. Theor. Comput. Sci.
- 2013

A method for upper bounding the quantum query complexity of certain boolean formula evaluation problems, using fundamental theorems about the general adversary bound is described, which gives an upper bound on query complexity without producing an algorithm.

### An optimal adiabatic quantum query algorithm

- Computer Science, MathematicsArXiv
- 2014

This work revisits the quantum adversary bound result by providing a direct proof in the continuous-time model, and draws new connections between the adversary bound, a modern theoretical computer science technique, and early theorems of quantum mechanics.

### Explicit relation between all lower bound techniques for quantum query complexity

- Computer Science, MathematicsSTACS
- 2012

An explicit reduction from the polynomial method to the multiplicative adversary method is shown, which reveals a clear picture of the relation between the different lower bound techniques, as it implies that all known techniques reduce to the multiplier method.

### The Multiplicative Quantum Adversary

- Computer Science, Mathematics2008 23rd Annual IEEE Conference on Computational Complexity
- 2008

A new variant of the quantum adversary method, a method for proving lower bounds on the quantum query complexity of a function, is presented, rooted in the quantum lower-bound method by Ambainis, based on the analysis of eigenspaces of the density matrix.

### Applications of Adversary Method in Quantum Query Algorithms

- Computer Science, Mathematics
- 2014

A recently developed tight characterisation of quantum query complexity, the adversary bound, is used to develop new quantum algorithms and lower bounds, and a generalisation ofquantum walks that connects electrical properties of a graph and its quantum hitting time is developed.

### A Universal Adiabatic Quantum Query Algorithm

- Computer Science, PhysicsTQC
- 2015

This work revisits the result that quantum adversary bound holds for continuous-time quantum computation, and uses for the first time in the context of quantum computation a version of the adiabatic theorem that does not require a spectral gap.

### Negative weights make adversaries stronger

- Computer Science, MathematicsSTOC '07
- 2007

A stronger version of the adversary method which goes beyond this principle to make explicit use of the stronger condition that the algorithm actually computes the function, and which is a lower bound on bounded-error quantum query complexity.

### Efficient algorithms in quantum query complexity

- Computer Science, Mathematics
- 2014

These algorithms are a novel application of the quantum walk search framework and give improved upper bounds for several subgraph-finding problems and study the quantum query complexity of matrix multiplication and related problems over rings, semirings, and the Boolean semiring in particular.

### A quantum lower bound for distinguishing random functions from random permutations

- Computer Science, MathematicsQuantum Inf. Comput.
- 2014

The quantum query complexity of this problem is studied, and it is shown that any quantum algorithm that solves this problem with bounded error must make $\Omega(N^{1/5}/\log N)$ queries to the input function.

### A New Quantum Lower Bound Method, with an Application to a Strong Direct Product Theorem for Quantum Search

- Mathematics, Computer ScienceTheory Comput.
- 2010

A new method for proving lower bounds on quantum query al- gorithms is presented, an extension of the adversary method, by analyzing the eigenspace structure of the problem by proving a strong direct product theorem for quantum search.

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