A quantum algorithm to solve the parity problem is better than its most efficient classical counter- part with a separation that is polynomial in the number of queries. This was shown by E. Bernstein and U. Vazirani and was one of the earliest indications that the quantum information processing can outperform the classical one by a significant margin. The problem and its solution both is usually stated for a 2-level system since we generally work with bits/qubits. However, many works have been done generalizing known quantum computing techniques to higher level systems. Following this, we look at a generalization of the Bernstein-Vazirani algorithm implemented on a general qudit system.
A quantum algorithm to solve the parity problem is better than its most efficient classical counter- part with a separation that is polynomial in the number of queries. This was shown by E. Bernstein and U. Vazirani and was one of the earliest indications that the quantum information processing can outperform the classical one by a significant margin. The problem and its solution both is usually stated for a 2-level system since we generally work with bits/qubits. However, many works have been done generalizing known quantum computing techniques to higher level systems. Following this, we look at a generalization of the Bernstein-Vazirani algorithm implemented on a general qudit system.
First, we review the Bernstein-Vazirani algorithm for determining a bit string. Next, we discuss the generalized Bernstein-Vazirani algorithm for determining a natural number string. Finally, we discuss the generalized Bernstein-Vazirani algorithm for determining an integer string. All of the generalized algorithms presented here have the following structure. Given the set of real values {a1,a2,a3,…,aN} and a special function g, we determine N values of the function g(a1),g(a2),g(a3),…,g(aN) simultaneously. The speed of determining the strings is shown to outperform the best classical case by a factor of N in every case.
Quantum algorithms can outperform their classical counterparts in various tasks, the most prominent example being Shor's algorithm for efficient prime factorization on a quantum computer. It is clear that one of the reasons for the speedup is the superposition principle of quantum mechanics, which allows a quantum processor to be in a superposition of different states at the same time. While such a superposition can lead to entanglement across different qubits of the processors, there also exist quantum algorithms that outperform classical ones using superpositions of individual qubits without entangling them. As an example, the Bernstein-Vazirani algorithm allows one to determine a bit string encoded into an oracle. While the classical version of the algorithm requires multiple calls of the oracle to learn the bit string, a single query of the oracle is enough in the quantum case. In this article, we analyze in detail the quantum resources in the Bernstein-Vazirani algorithm. For this, we introduce and study its probabilistic version, where the goal is to guess the bit string after a single call of the oracle. We show that in the absence of entanglement, the performance of the algorithm is directly related to the amount of quantum coherence in the initial state. We further demonstrate that a large amount of entanglement in the initial state prevents the algorithm from achieving optimal performance. We also apply our methods to quantum computation with mixed states, proving that pseudopure states achieve optimal performance for a given purity in the Bernstein-Vazirani algorithm. We further investigate quantum resources in the one clean qubit model, showing that the model can exhibit speedup over any known classical algorithm even with an arbitrarily little amount of multipartite entanglement, general quantum correlations, and coherence.
This presentation gives a comprehensive overview of quantum query algorithms, beginning with an introduction to the query model of computation where input is accessed via an oracle function. It then details several foundational quantum algorithms that demonstrate a quantum advantage over classical approaches: Deutsch's problem is solved with a single query using phase kickback to determine if a function is constant or balanced; the Deutsch-Jozsa algorithm extends this to n-bit inputs, efficiently distinguishing constant from balanced functions with one query versus an exponential need classically; the Bernstein-Vazirani algorithm identifies a hidden string with one query, outperforming classical linear requirements; and Simon's algorithm, which finds a hidden period in a two-to-one function with a linear number of queries, showcases an exponential speedup over the best classical probabilistic algorithms.
Abstract This study presents practical experience implementing the Bernstein–Vazirani algorithm on the IBM Q quantum computing platform, focusing on energy-efficient and reversible computation guided by Landauer’s principle. The work highlights the role of fundamental quantum logic gates—including CNOT, NOT, CCNOT, OR, and XOR—in constructing scalable, reversible circuits. By applying these principles, the Bernstein–Vazirani algorithm demonstrates an efficient solution to the problem of determining a hidden binary vector with a minimal number of queries, showcasing the potential for quantum computational speedup and optimization of information processing. The research emphasizes the importance of reversible quantum operations in improving computational efficiency and expanding the capabilities of modern quantum systems.
Quantum entanglement and coherence are crucial resources in quantum information theory. In some scenarios, however, it is not necessary to directly estimate entanglement or coherence measures to quantify the capabilities of a state in quantum information processing. Instead, fully entangled fraction and coherence fraction are two alternatives for entanglement and coherence in specific quantum tasks. Here, we establish a link between the coherence fraction and the Bernstein-Vazirani algorithm, which has several potential applications including cryptography and database search. We show that the success probability of the generalized Bernstein-Vazirani algorithm depends only on the coherence fraction of the initial state rather than its entanglement or coherence. Moreover, we discuss the coherence fraction dynamics and establish a relation between the operator's coherence fraction and the algorithm's success probability. Our findings highlight how quantum coherence fraction influences the efficiency of quantum algorithms.
The complexity of certain classical computer algorithms can be reduced by a quantum algorithm completing the same task. We examine the implementation of the Bernstein–Vazirani algorithm. It finds the hidden n-bit string s from the function fs(x)=sx that takes an n-bit input string x and outputs one bit, the dot product. This algorithm reduces the complexity from n queries for the best classical algorithm to just one. Though the quantum algorithm theoretically always gives the correct string, in practice, factors such as decoherence of the physical qubits due to external noise reduce the algorithm’s accuracy. By implementing the algorithm on the IBM 5-qubit and 14-qubit quantum computers, we are able to analyze correlations between the accuracy of the results and the properties of the physical qubits used. Not all qubits are pairwise connected, nor do they have the same coherence times. Using Python to build the quantum circuits, we create several implementations of the circuit with different qubits. We establish that 1/T1, 1/T2, and gate errors are positively correlated with calculation errors. Consequently, we observe that the 14-qubit quantum computer, on average, is more prone to errors than the 5-qubit quantum computer.