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arXiv:0904.3660 · OASIcs : OpenAccess Series in Informatics

Exact Quantum Query Algorithm for Error Detection Code Verification

Alina Vasilieva

Published Jan 1, 2009• 5 citations
arXivHTMLPDF

Abstract

Quantum algorithms can be analyzed in a query model to compute Boolean functions. Function input is provided in a black box, and the aim is to compute the function value using as few queries to the black box as possible. A repetition code is an error detection scheme that repeats each bit of the original message r times. After a message with redundant bits is transmitted via a communication channel, it must be verified. If the received message consists of r-size blocks of equal bits, the conclusion is that there were no errors. The verification procedure can be interpreted as an application of a query algorithm, where input is a message to be checked. Classically, for N-bit message, values of all N variables must be queried. We demonstrate an exact quantum algorithm that uses only N/2 queries.

DOI
10.4230/drops.memics.2009.2343

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Quantum algorithms can be analyzed in a query model to compute Boolean functions. Function input is provided in a black box, and the aim is to compute the function value using as few queries to the black box as possible. A repetition code is an error detection scheme that repeats each bit of the original message r times. After a message with redundant bits is transmitted via a communication channel, it must be verified. If the received message consists of r-size blocks of equal bits, the conclusion is that there were no errors. The verification procedure can be interpreted as an application of a query algorithm, where input is a message to be checked. Classically, for N-bit message, values of all N variables must be queried. We demonstrate an exact quantum algorithm that uses only N/2 queries.
Quantum algorithms can be analyzed in a query model to compute Boolean functions where input is given in a black box and the aim is to compute function value for arbitrary input using as few queries as possible. We concentrate on quantum query algorithm designing tasks in this paper. The main aim of the research was to find new efficient algorithms and develop general algorithm designing techniques. First, we present several exact quantum query algorithms for certain problems that are better than classical counterparts. Next, we introduce algorithm transformation methods that allow significant enlarging of exactly computable functions sets. Finally, we propose quantum algorithm designing methods. Given algorithms for the set of sub-functions, our methods use them to design a more complex one, based on algorithms described before. Methods are applicable for input algorithms with specific properties and preserve acceptable error probability and number of queries. Methods offer constructions for computing AND, OR and MAJORITY kinds of Boolean functions.
Quantum algorithms can be analyzed in a query model to compute Boolean functions where input is given in a black box and the aim is to compute function value for arbitrary input using as few queries as possible. We concentrate on quantum query algorithm designing tasks in this paper. The main aim of the research was to find new efficient algorithms and develop general algorithm designing techniques. First, we present several exact quantum query algorithms for certain problems that are better than classical counterparts. Next, we introduce algorithm transformation methods that allow significant enlarging of exactly computable functions sets. Finally, we propose quantum algorithm designing methods. Given algorithms for the set of sub-functions, our methods use them to design a more complex one, based on algorithms described before. Methods are applicable for input algorithms with specific properties and preserve acceptable error probability and number of queries. Methods offer constructions for computing AND, OR and MAJORITY kinds of Boolean functions.
Quantum algorithms can be analyzed in a query model to compute Boolean functions. Function input is provided in a black box, and the aim is to compute the function value using as few queries to the black box as possible. The complexity of the algorithm is measured by the number of queries on the worst-case input. In this paper we consider computing AND Boolean function. First, we present a quantum algorithm for AND of two bits. Our algorithm uses one quantum query and correct result is obtained with a probability p=4/5, that improves previous results. The main result is generalization of our approach to design efficient quantum algorithms for computing composite function AND(f1,f2) where fi is a Boolean function. Finally, we demonstrate another kind of an algorithm for AND of two variables, that has a correct answer probability p=9/10, but cannot be extended to compute AND(f1,f2).
We consider the quantum query model for computing Boolean functions. The definition of the function is known, but a black box contains the input X = (x1, x2, ..., xn). Black box can be accessed by querying xivalues. The goal is to develop an algorithm, which would compute the function value for arbitrary input using as few queries to the black box as possible. We present two different quantum query algorithms for computing the basic Boolean function - logical AND of two bits. Both algorithms use only one query to determine the function value. Correct answer probability for the first algorithm is 80%, but for the second algorithm it is 90%. To compute this function with the same probability in the classical model, two queries are necessary. We analyze algorithms implementation differences and propose a way to extend the first algorithm to compute AND of two functions.
Quantum algorithms can be analyzed in a query model to compute Boolean functions where input is given in a black box, but the aim is to compute function value for arbitrary input using as few queries as possible. In this paper we concentrate on quantum query algorithm designing tasks. The main aim of research was to find new efficient algorithms and develop general algorithm designing techniques. We present several exact quantum query algorithms for certain problems that are better than classical counterparts. Next we introduce algorithm transformation methods that allow significant enlarging of sets of exactly computable functions. Finally, we propose quantum algorithm designing methods. Given algorithms for the set of sub-functions, our methods use them to design more complex one, based on algorithms described before. Methods are applicable for input algorithms with specific properties and preserve acceptable error probability and number of queries.
Let f=EXACT3,45f = \text{EXACT}^5_{3,4}f=EXACT3,45​ be the Boolean function that equals 1 exactly on inputs of Hamming weight 3 or 4. Earlier numerical studies reported success probabilities close to one using three queries, but the existence of an exact three-query algorithm had remained unresolved. This paper settles this problem by proving that the exact quantum query complexity of fff is 444. For the lower bound, we express every three-query algorithm through layer-by-layer Gram matrices of low-degree Fourier coefficients. We then construct a fixed rational dual certificate whose slack matrices are all positive definite. This certificate separates the necessary Gram-matrix region from the exact-output condition and therefore rules out every exact algorithm using at most three queries in the stated model. The positivity verification reduces to five small rational matrices and Sylvester's criterion. For the upper bound, we invoke the previously known four-query algorithm for the seven-bit function EXACT3,47\text{EXACT}^7_{3,4}EXACT3,47​ and apply the standard padding reduction. To quantify the near-exact behavior, we use the same certificate to derive a valid, though not claimed tight, lower bound 2560043779488192449>0\frac{25600}{43779488192449}>04377948819244925600​>0 on the worst-case error of every algorithm using at most three queries. This positive error gap is compatible with the near-unit success probabilities reported in earlier numerical studies, which do not establish exact computation.