We present a novel approach to quantum algorithms, by taking advantage of modular values, i.e., complex and unbounded quantities resulting from specific post-selected measurement scenarios. Our focus is on the problem of ascertaining whether a given function acting on a set of binary values is constant (uniformly yielding outputs of either all 0 or all 1), or balanced (a situation wherein half of the outputs are 0 and the other half are 1). Such problem can be solved by relying on the Deutsch-Jozsa algorithm. The proposed method, relying on the use of modular values, provides a high number of degrees of freedom for optimizing the new algorithm inspired from the Deutsch-Jozsa one. In particular, we explore meticulously the choices of the pre- and post-selected states. We eventually test the novel theoretical algorithm on a quantum computing platform. While the outcomes are currently not on par with the conventional approach, they nevertheless shed light on potential for future improvements, especially with less-optimized algorithms. We are thus confidend that the proposed proof of concept could prove its validity in bridging quantum algorithms and modular values research fields.
Recent advances in quantum computational science promise substantial improvements in the speed with which certain classes of problems can be computed. Various algorithms that utilize the distinctively non-classical characteristics of quantum mechanics have been formulated to take advantage of this promising new approach to computation. One such algorithm was formulated by David Deutsch and Richard Jozsa. By measuring the output of a quantum network that implements this algorithm, it is possible to determine with N – 1 measurements certain global properties of a function f(x), where N is the number of network inputs. Classically, it may not be possible to determine these same properties without evaluating f(x) a number of times that rises exponentially as N increases. Hitherto, the potential power of this algorithm has been explored in the context of qubits, the quantum computational analogue of classical bits. However, just as one can conceive of classical computation in the context of non-binary logic, such as ternary or quaternary logic, so also can one conceive of corresponding higher-order quantum computational equivalents.This thesis investigates the behaviour of the Deutsch-Jozsa algorithm in the context of these higher-order quantum computational forms of logic and explores potential applications for this algorithm. An important conclusion reached is that, not only can the Deutsch-Jozsa algorithm’s known computational advantages be formulated in more general terms, but also a new algorithmic property is revealed with potential practical applications.
The Deutsch-Jozsa algorithm is one of the first examples of a quantum algorithm that is exponentially faster than any possible deterministic classical algorithm. It was proposed by Deutsch and Jozsa in 1992 with improvements by Cleve, Ekert, Macchiavello, and Mosca in 1998. The Deutsch-Jozsa problem is a promise problem and we can equivalently describe it as a partial function ${\text{DJ}}_{n}^{0}:{{0,1}}^{n}\ensuremath{\rightarrow}{0,1}$ defined as
DJn0(x)=1 for
∣x∣=n/2,DJn0(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
DJn0 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
DJnk(x)=1 for
∣x∣=n/2,DJnk(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
DJnk. 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.
Deutsch algorithm or Deutsch–Jozsa algorithm only needs one query to determine whether the function f(x) is constant or balanced. That is, the global property of the function f(x) is rapidly known. But these algorithms do not give the concrete value of function f(x). For the f(x), its value is also of interest. Here we propose a scheme in which the value of function f(x) can be determined. We use an entangled state as the input state and send it into quantum black box. Through performing measurement on the output state, we can exactly determine the value of f(x). Its value plus its global property enables us to thoroughly understand elementary information of f(x). Based on these informations, we can determine unitary transformations on the input state walking through quantum black box. The quantum black box becomes quantum white box to a certain extent.
Deutsch's algorithm and Grover's quantum search are theoretically interesting, but it is not obvious that these two algorithms are actually useful for anything important. We next consider a selection of more advanced algorithms, several of which may have real-life applications. Some of these will be too complicated to explain fully, and their properties will only be sketched briefly. The Deutsch–Jozsa algorithm Deutsch's algorithm is simple, but important, as it shows that a quantum algorithm can find a property of an unknown function (its parity) with a smaller number of queries than any possible classical algorithm (one rather than two). For this reason we can say that quantum computing is more efficient than classical computing within the oracle model of function evaluation. (It is widely believed that quantum computing is more efficient than classical computing in general, but this is a surprisingly hard thing to prove.) The simplicity of the algorithm is also an advantage, as it can be implemented on very primitive quantum computers. Beyond this, however, Deutsch's algorithm is also important as the simplest member of a large family of quantum algorithms, including most notably Shor's quantum factoring algorithm. The second simplest algorithm in the family is the Deutsch–Jozsa algorithm, which solves a very closely related problem. Consider an unknown binary function with n input bits, giving N = 2 n possible inputs, and a single output bit.
We present a generalized Deutsch-Jozsa (DJ) quantum algorithm that not only determines both the global type of an unknown Boolean function (constant or balanced) but also determines explicit output values of the function in a single oracle query. Unlike the original DJ algorithm, which identifies only whether a function is constant or balanced, our generalization retrieves actual function output values at the same time with using a Bell state as ancilla. This makes a richer function characterization with minimal queries to have practical quantum advantages, e.g. data classification, logistic regression, and quantum cryptography.