There are a lot of articles about quantum algorithms, which involve querying a black box function, often called an oracle.This paper proposes a way to combine two oracles and to determine how similar they are using the Deutsch-Jozsa algorithm.In this case being similar means, that the sets of marked states have just a few ones outside of their intersection.The new quantum algorithm is compared to its classical counterparts and a detailed analysis shows when it will outperform them.

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arXiv:quant-ph/0304017

Broken Promises and Quantum Algorithms

Adam Brazier, Martin Bodo Plenio

Published Apr 2, 2003• 0 citations

arXiv HTML PDF

Abstract

In the black-box model, problems constrained by a

promise' are the only ones that admit a quantum exponential speedup over the best classical algorithm in terms of query complexity. The most prominent example of this is the Deutsch-Jozsa algorithm. More recently, Wim van Dam put forward an algorithm for unstructured problems (i.e., those without a promise). We consider the Deutsch-Jozsa algorithm with a less restrictive (or

broken') promise and study the transition to an unstructured problem. We compare this to the success of van Dam's algorithm. These are both compared with a standard classical sampling algorithm. The Deutsch-Jozsa algorithm remains good as the problem initially becomes less structured, but the van Dam algorithm can be adapted so as to become superior to the Deutsch-Jozsa algorithm as the promise is weakened.

DOI: 10.48550/arxiv.quant-ph/0304017

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Broken promises and quantum algorithms

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In the black-box model, problems constrained by a "promise" are the only ones that admit a quantum exponential speedup over the best classical algorithm in terms of query complexity. The most prominent example of this is the Deutsch-Jozsa algorithm. More recently, Wim van Dam put forward an algorithm for unstrucred problems (i.e., those without a promise). We consider the Deutsch-Jozsa algorithm with a less restrictive (or "broken") promise and study the transition to an unstructured problem. We compare this to the success of van Dam's algorithm. These are both compared with a standard classical sampling algorithm. The Deutsch-Jozsa algorithm remains good as the problem initially becomes less structured, but the van Dam algorithm can be adapted so as to become superior to the Deutsch-Jozsa algorithm as the promise is weakened.In the black-box model, problems constrained by a "promise" are the only ones that admit a quantum exponential speedup over the best classical algorithm in terms of query complexity. The most prominent example of this is the Deutsch-Jozsa algorithm. More recently, Wim van Dam put forward an algorithm for unstrucred problems (i.e., those without a promise). We consider the Deutsch-Jozsa algorithm with a less restrictive (or "broken") promise and study the transition to an unstructured problem. We compare this to the success of van Dam's algorithm. These are both compared with a standard classical sampling algorithm. The Deutsch-Jozsa algorithm remains good as the problem initially becomes less structured, but the van Dam algorithm can be adapted so as to become superior to the Deutsch-Jozsa algorithm as the promise is weakened.

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In the black-box model, problems constrained by a ‘promise ’ are the only ones that admit a quantum exponential speedup over the best classical algorithm in terms of query complexity. The most prominent example of this is the Deutsch-Jozsa algorithm. More recently, Wim van Dam put forward an algorithm for unstructured problems (i.e., those without a promise). We consider the Deutsch-Jozsa algorithm with a less restrictive (or ‘broken’) promise and study the transition to an unstructured problem. We compare this to the success of van Dam’s algorithm. These are both compared with a standard classical sampling algorithm. The Deutsch-Jozsa algorithm remains good as the problem initially becomes less structured, but the van Dam algorithm can be adapted so as to become superior to the Deutsch-Jozsa algorithm as the promise is weakened. 1In the black-box model, problems constrained by a ‘promise ’ are the only ones that admit a quantum exponential speedup over the best classical algorithm in terms of query complexity. The most prominent example of this is the Deutsch-Jozsa algorithm. More recently, Wim van Dam put forward an algorithm for unstructured problems (i.e., those without a promise). We consider the Deutsch-Jozsa algorithm with a less restrictive (or ‘broken’) promise and study the transition to an unstructured problem. We compare this to the success of van Dam’s algorithm. These are both compared with a standard classical sampling algorithm. The Deutsch-Jozsa algorithm remains good as the problem initially becomes less structured, but the van Dam algorithm can be adapted so as to become superior to the Deutsch-Jozsa algorithm as the promise is weakened. 1

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{\text{DJ}}_{n}^{0}(x)=1

for

|x|=n/2, {\text{DJ}}_{n}^{0}(x)=0

for

|x|=0,n

, and it is undefined for the rest of the cases, where

n

is even, and

|x|

is the Hamming weight of

x

. The optimal quantum algorithm needs only one query to compute

{\text{DJ}}_{n}^{0}

but the classical deterministic algorithm requires ${2}^{n\ensuremath{-}1}+1$ queries to compute it in the worse case. In this article, we generalize the Deutsch-Jozsa problem as

{\text{DJ}}_{n}^{k}(x)=1

for

|x|=n/2, {\text{DJ}}_{n}^{k}(x)=0

for

|x|

in the set ${0,1,...,k,n\ensuremath{-}k,n\ensuremath{-}k+1,...,n}$, and it is undefined for the rest of the cases, where $0\ensuremath{\le}k<n/2$. In particular, we give and prove an optimal exact quantum query algorithm with complexity

k+1

for computing the generalized Deutsch-Jozsa problem

{\text{DJ}}_{n}^{k}

. It is clear that the case of

k=0

is in accordance with the Deutsch-Jozsa problem. Also, we give a method for finding the approximate and exact degrees of symmetric partial Boolean functions.

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.