The well-known Deutsch Algorithm (DA) and Deutsch-Jozsha Algorithm (DJA) both are used as an evidence to the power of quantum computers over classical computation mediums. In these theoretical experiments, it has been shown that a quantum computer can find the answer with certainty within a few steps although classical electronic systems must evaluate more iterations than quantum computer. In this paper, it is shown that a classical computation system formed by using ordinary electronic parts may perform the same task with equal performance than quantum computers. DA and DJA quantum circuits act like an analog computer, so it is unfair to compare the bit of classical digital computers with the qubit of quantum computers. An analog signal carrying wire will of course carry more information that a bit carrying wire without serial communication protocols.
We examine Ralph's equivalent circuit model of D-CTCs, which was proposed to derive Deutsch's maximum entropy rule of D-CTCs. By constructing counterexamples we show that the equivalent circuit model does not always reproduce the unique fixed state with the maximal von Neumann entropy. We speculate that the equivalent circuit model remains the correct description of D-CTCs and it can reproduce a revised maximum entropy rule of D-CTCs. We also suggest that the revised maximum entropy rule may eliminate the discontinuous quantum evolutions of D-CTCs.
We propose a protocol to construct the 35 $f$-controlled phase gates of a three-qubit refined Deutsch-Jozsa (DJ) algorithm, by using single-qubit $σ_z$ gates, two-qubit controlled phase gates, and two-target-qubit controlled phase gates. Using this protocol, we discuss how to implement the three-qubit refined DJ algorithm with superconducting transmon qutrits resonantly coupled to a single cavity. Our numerical calculation shows that implementation of this quantum algorithm is feasible within the present circuit QED technique. The experimental realization of this algorithm would be an importan…
We want in this article to show the usefulness of Quantum Turing Machine (QTM) in a high-level didactic context as well as in theoretical studies. We use QTM to show its equivalence with quantum circuit model for Deutsch and Deutsch-Jozsa algorithms. Further we introduce a strategy of translation from Quantum Circuit to Quantum Turing models by these examples. Moreover we illustrate some features of Quantum Computing such as superposition from a QTM point of view and starting with few simple examples very known in Quantum Circuit form.
It is known that a party with access to a Deutschian closed timelike curve (D-CTC) can perfectly distinguish multiple non-orthogonal quantum states. In this paper, we propose a practical method for discriminating multiple non-orthogonal states, by using a previously known quantum circuit designed to simulate D-CTCs. This method relies on multiple copies of an input state, multiple iterations of the circuit, and a fixed set of unitary operations. We first characterize the performance of this circuit and study its asymptotic behavior. We also show how it can be equivalently recast as a local, ad…
We analyse the design and optimisation of quantum logic circuits suitable for the experimental demonstration of a three-qubit quantum computation prototype based on optically-controlled, solid-state quantum logic gates. In these gates, the interaction between two qubits carried by the electron-spin of donors is mediated by the optical excitation of a control particle placed in their proximity. First, we use a geometrical approach for analysing the entangling characteristics of these quantum gates. Then, using a genetic programming algorithm, we develop circuits for the refined Deutsch-Jozsa al…
Design of a large-scale quantum computer has paramount importance for science and technologies. We investigate a scheme for realization of quantum algorithms using noncomposite quantum systems, i.e., systems without subsystems. In this framework, $n$ artificially allocated "subsystems" play a role of qubits in $n$-qubits quantum algorithms. With focus on two-qubit quantum algorithms, we demonstrate a realization of the universal set of gates using a $d=5$ single qudit state. Manipulation for an ancillary level in the systems allows effective implementation of operators from ${\rm U}(4)$ group …
In 1985, David Deutsch challenged the Church-Turing thesis by stating that his quantum model of computation "could, in principle, be built and would have many remarkable properties not reproducible by any Turing machine". While this is thought to be true in general, there is usually no way of knowing that the corresponding classical algorithms are the best possible solutions. Here we provide an efficient classical simulation of the Deutsch-Jozsa algorithm, which was one of the first examples of quantum computational speed-up. Our conclusion is that the Deutsch-Jozsa quantum algorithm owes its …
A long-standing aim of quantum information research is to understand what gives quantum computers their advantage. This requires separating problems that need genuinely quantum resources from those for which classical resources are enough. Two examples of quantum speed-up are the Deutsch-Jozsa and Simon's problem, both efficiently solvable on a quantum Turing machine, and both believed to lack efficient classical solutions. Here we present a framework that can simulate both quantum algorithms efficiently, solving the Deutsch-Jozsa problem with probability 1 using only one oracle query, and Sim…
We consider quantum circuits with time travel designed for distinguishing specific nonorthogonal quantum states in two most popular models: Deutschs and postselected. We modify them by a presence of weakly coupled thermal environment. Using the Davies approximation we study how the thermal noise affects an ability of the circuits to distinguish nonorthogonal quantum states. We show that for purely dephasing environment a paradoxial power of such circuits remains preserved. We also present a physics based argument for conditions of validity of the maximum entropy rule introduced by David Deutsc…
Manin, Feynman, and Deutsch have viewed quantum computing as a kind of universal physical simulation procedure. Much of the writing about quantum logic circuits and quantum Turing machines has shown how these machines can simulate an arbitrary unitary transformation on a finite number of qubits. The problem of universality has been addressed most famously in a paper by Deutsch, and later by Bernstein and Vazirani as well as Kitaev and Solovay. The quantum logic circuit model, developed by Feynman and Deutsch, has been more prominent in the research literature than Deutsch's quantum Turing mach…
Deutsch-Jozsa algorithm has been implemented via a quantum adiabatic evolution by S. Das et al. [Phys. Rev. A 65, 062310 (2002)]. This adiabatic algorithm gives rise to a quadratic speed up over classical algorithms. We show that a modified version of the adiabatic evolution in that paper can improve the performance to constant time.
That superpositions of states can be useful for performing tasks in quantum systems has been known since the early days of quantum information, but only recently has quantitative theory of quantum coherence been proposed. Here we apply that theory to an analysis of the Deutsch-Jozsa algorithm, which depends on quantum coherence for its operation. The Deutsch-Jozsa algorithm solves a decision problem, and we focus on a probabilistic version of that problem, comparing probability of being correct for both classical and quantum procedures. In addition, we study a related decision problem in which…
Probably the simplest and most frequently used way to illustrate the power of quantum computing is to solve the so-called {\it Deutsch's problem}. Consider a Boolean function $f: \{0,1\} \to \{0,1\}$ and suppose that we have a (classical) black box to compute it. The problem asks whether $f$ is constant (that is, $f(0) = f(1)$) or balanced ($f(0) \not= f(1)$). Classically, to solve the problem seems to require the computation of $f(0)$ and $ f(1)$, and then the comparison of results. Is it possible to solve the problem with {\em only one} query on $f$? In a famous paper published in 1985, Deut…
It is presently shown that the Deutsch-Jozsa algorithm is connected to the concept of bent function. Particularly, it is noticeable that the quantum circuit used to denote the well known quantum algorithm is by itself the quantum computer which performs the Walsh transform of a Boolean function. Consequently, the output from Deutsch-Jozsa algorithm, when the hidden function is bent, corresponds to a flat spectrum of quantum states.
We describe the experimental implementation of a recently proposed quantum algorithm involving quantum entanglement at the level of two qubits using NMR. The algorithm solves a generalisation of the Deutsch problem and distinguishes between even and odd functions using fewer function calls than is possible classically. The manipulation of entangled states of the two qubits is essential here, unlike the Deutsch-Jozsa algorithm and the Grover's search algorithm for two bits.
Quantum computing implementations under consideration today typically deal with systems with microscopic degrees of freedom such as photons, ions, cold atoms, and superconducting circuits. The quantum information is stored typically in low-dimensional Hilbert spaces such as qubits, as quantum effects are strongest in such systems. It has however been demonstrated that quantum effects can be observed in mesoscopic and macroscopic systems, such as nanomechanical systems and gas ensembles. While few-qubit quantum information demonstrations have been performed with such macroscopic systems, a quan…
We propose an optical implementation of the Deutsch-Jozsa Algorithm using classical light in a binary decision-tree scheme. Our approach uses a ring cavity and linear optical devices in order to efficiently quarry the oracle functional values. In addition, we take advantage of the intrinsic Fourier transforming properties of a lens to read out whether the function given by the oracle is balanced or constant.
Quantum parallelism is the main feature of quantum computation. In 1985 D. Deutsch showed that a single quantum computation may be sufficient to state whether a two-valued function of a two-valued variable is constant or not. Though the generalized problem with unconstrained domain and range size admits no deterministic quantum solution, a fully probabilistic quantum algorithm is presented in which quantum parallelism is harnessed to achieve a quicker exploration of the domain with respect to the classical ``sampling'' strategy.
Recently, [{arXiv:0810.3134}] is accepted and published. We show that the Bell inequalities lead to a new type of linear-optical Deutsch algorithms. We have considered a use of entangled photon pairs to determine simultaneously and probabilistically two unknown functions. The usual Deutsch algorithm determines one unknown function and exhibits a two to one speed up in a certain computation on a quantum computer rather than on a classical computer. We found that the violation of Bell locality in the Hilbert space formalism of quantum theory predicts that the proposed {\it probabilistic} Deutsch…
Deutsch's algorithm for two qubits (one control qubit plus one auxiliary qubit) is extended to two $d$-dimensional quantum systems or qudits for the case in which $d$ is equal to $2^n$, $n=1,2,...$ . This allows one to classify a certain oracle function by just one query, instead of the $2^{n-1}+1$ queries required by classical means. The given algorithm for two qudits also solves efficiently the Bernstein-Vazirani problem. Entanglement does not occur at any step of the computation.
We examine the "Guessing Secrets" problem arising in internet routing, in which the goal is to discover two or more objects from a known finite set. We propose a quantum algorithm using O(1) calls to an O(logN) oracle. This improves upon the best known classical result, which uses O(logN) questions and requires an additional O(logN^3) steps to produce the answer. In showing the possibilities of this algorithm, we extend the types of questions and function oracles that the Deutsch-Jozsa algorithm can be used to solve.
Deutsch-Jozsa (DJ) problem is one of the most important problems demonstrating the power of quantum algorithm. DJ problem can be described as a Boolean function $f$: $\{0,1\}^n\rightarrow \{0,1\}$ with promising it is either constant or balanced, and the purpose is to determine which type it is. DJ algorithm can solve it exactly with one query. In this paper, we first discover the inherent structure of DJ problem in distributed scenario by giving a number of equivalence characterizations between $f$ being constant (balanced) and some properties of $f$'s subfunctions, and then we propose three …
We describe a general framework for regarding oracle-assisted quantum algorithms as tools for discriminating between unitary transformations. We apply this to the Deutsch-Jozsa problem and derive all possible quantum algorithms which solve the problem with certainty using oracle unitaries in a particular form. We also use this to show that any quantum algorithm that solves the Deutsch-Jozsa problem starting with a quantum system in a particular class of initial, thermal equilibrium-based states of the type encountered in solution state NMR can only succeed with greater probability than a class…
We discuss the ensemble version of the Deutsch-Jozsa (DJ) algorithm which attempts to provide a "scalable" implementation on an expectation-value NMR quantum computer. We show that this ensemble implementation of the DJ algorithm is at best as efficient as the classical random algorithm. As soon as any attempt is made to classify all possible functions with certainty, the implementation requires an exponentially large number of molecules. The discrepancies arise out of the interpretation of mixed state density matrices.
Quantum computing takes fully advantage of the superposition principle to increase greatly (even exponentially) the speed of calculations, relative to the classical approach. The Deutsch-Jozsa algorithm is the simplest quantum algorithm illustrating this power. Unfortunately, the standard derivation involves several ingenious steps which usually leave students feeling that they could never have figured out the algorithm by themselves. We present here a different formulation of the problem which allows students to derive the algorithm using only basic knowledge of quantum mechanics and linear a…
We present a simple scheme to implement the Deutsch-Jozsa algorithm based on two-atom interaction in a thermal cavity. The photon-number-dependent parts in the evolution operator are canceled with the strong resonant classical field added. As a result, our scheme is immune to thermal field, and does not require the cavity to remain in the vacuum state throughout the procedure. Besides, large detuning between the atoms and the cavity is not necessary neither, leading to potential speed up of quantum operation. Finally, we show by numerical simulation that the proposed scheme is equal to demonst…
We show that it is possible to clone quantum states to arbitrary accuracy in the presence of a Deutschian closed timelike curve (D-CTC), with a fidelity converging to one in the limit as the dimension of the CTC system becomes large---thus resolving an open conjecture from [Brun et al., Physical Review Letters 102, 210402 (2009)]. This result follows from a D-CTC-assisted scheme for producing perfect clones of a quantum state prepared in a known eigenbasis, and the fact that one can reconstruct an approximation of a quantum state from empirical estimates of the probabilities of an informationa…
We investigate the creation of entanglement by the application of phases whose value depends on the state of a collection of qubits. First we give the necessary and sufficient conditions for a given set of phases to result in the creation of entanglement in a state comprising of an arbitrary number of qubits. Then we analyze the creation of entanglement between any two qubits in three qubit pure and mixed states. We use our result to prove that entanglement is necessary for Deutsch-Jozsa algorithm to have an exponential advantage over its classical counterpart.
Nuclear magnetic resonance (NMR) has been widely used as a demonstrative medium for showcasing the ability for quantum computations to outperform classical ones. A large number of such experiments performed have been implementations of the Deutsch-Jozsa algorithm. It is known, however, that in some cases the Deutsch-Jozsa problem can be solved classically using as many queries to the black-box as in the quantum solution. In this paper we describe experiments in which we take the contrasting approach of using NMR as a classical computing medium, treating the nuclear spin vectors classically and…
Quantum algorithms could efficiently solve certain classically intractable problems by exploiting quantum parallelism. To date, whether the quantum entanglement is useful or not for quantum computing is still a question of debate. Here, we present a new quantum algorithm to show that entanglement could help to gain advantage over classical algorithm and even the quantum algorithm without entanglement. Furthermore, we implement experiments to demonstrate our proposed algorithm using superconducting qubits. Our results show the viability of the algorithm and suggest that entanglement is essentia…
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 class…
Universal quantum gates and quantum error correction~(QEC) lie in the heart of quantum information science. Large-scale quantum computing depends on a universal set of quantum gates, in which some gates may be easily carried out, while others are hard with a certain physical system. There is a unique three-qubit quantum gate called the Deutsch gate~[$\mathbb{D}(θ)$], from which alone a circuit can be constructed so that any feasible quantum computing is attainable. As far as we know, however, $\mathbb{D}(θ)$ has not been demonstrated. Here we design an easily realizable $\mathbb{D}(θ)$ by usin…
We adapt the Deutsch-Josza algorithm to the context of formal language theory. Specifically, we use the algorithm to distinguish between trivial and nontrivial words in groups given by finite presentations, under the promise that a word is of a certain type. This is done by extending the original algorithm to functions of arbitrary length binary output, with the introduction of a more general concept of parity. We provide examples in which properties of the algorithm allow to reduce the number of oracle queries with respect to the deterministic classical case. This has some consequences for th…
In the rapidly growing area of quantum information, the Deutsch algorithm is ubiquitous and, in most cases, the first one to be introduced to any student of this relatively new field of research. The reason for this historical relevance stems from the fact that, although extremely simple, the algorithm conveys all the main features of more complex quantum computations. In spite of its simplicity, the uncountable experimental realizations of the algorithm in a broad variety of physical systems are in general quite involved. The aim of this work is two-fold: to introduce the basic concepts of qu…
Boolean functions are important building blocks in cryptography for their wide application in both stream and block cipher systems. For cryptanalysis of such systems one tries to find out linear functions that are correlated to the Boolean functions used in the crypto system. Let $f$ be an $n$-variable Boolean function and its Walsh spectra is denoted by $W_f(ω)$ at the point $ω\in \{0, 1\}^n$. The Boolean function is available in the form of an oracle. We like to find an $ω$ such that $W_f(ω) \neq 0$ as this will provide one of the linear functions which are correlated to $f$. We show that th…
This paper demonstrates the use of entanglement resources in quantum speedup by presenting an algorithm which is the generalization of an algorithm proposed by Goswami and Panigrahi [arXiv:1706.09489 (2017)]. We generalize the algorithm and show that it provides deterministic solutions having an advantage over classical algorithm. The algorithm answers the question of whether a given function is constant or balanced and whether two functions are equal or unequal. Finally, we experimentally verify the algorithm by using IBM's five-qubit quantum computer with a high fidelity.
Distributed quantum computing can give substantial noise reduction due to shallower circuits. An experiment illustrates the advantages in the case of Grover search. This motivates studying the quantum advantage of the distributed version of the Simon and Deutsch-Jozsa algorithm. We show that the distributed Simon algorithm retains the exponential advantage, but the complexity deteriorates from O(n) to O(n^2), where n = log2(N). The distributed Deutsch-Jozsa deteriorates to being probabilistic but retains a quantum advantage over classical random sampling.
We consider causality respecting (CR) quantum systems interacting with closed timelike curves (CTCs), within the Deutsch model. We introduce the concepts of popping up and elimination of quantum information and use them to show that no-cloning and no-deleting, which are true in CR quantum systems, are no more valid in the same that are interacting with CTCs. We also find limits on the possibility of creation of entanglement between a CR system and a CTC, and the same between two CR systems in the presence of a CTC. We prove that teleportation of quantum information, even in its approximate ver…
Quantum correlations have been pointed out as the most likely source of the speed-up in quantum computation. Here we analyzed the presence of quantum correlations in the implementation of Deutsch-Jozsa algorithm running in the DQC1 and DQCp models of quantum computing. For some balanced functions, the qubits in DQC1 model are quantum correlated just in the intermediate steps of the algorithm for a given decomposition into one and two qubits gates. In the DQCp model the final state is strongly quantum correlated for some balanced functions, so that the pairwise entanglement between blocks scale…
We propose a way to implement a three-qubit refined Deutsh-Jozsa (DJ) algorithm. The present proposal is based on the construction of the 35 $f$-controlled phase gates, which uses single-qubit $σ_z$ gates and two-qubit {\it standard} controlled-phase (CP) gates only. This proposal is implementable because a single-qubit $σ_z$ gate can be easily realized by applying a single classical pulse and a two-qubit CP gate has been experimentally demonstrated in various physical systems. Finally, it is noted that this proposal is quite general, and can be applied to implement a three-qubit refined DJ al…
Since Deutsch (1985), quantum computers have been modeled exclusively in the language of state vectors and the Schroedinger equation. We present a complementary view of quantum circuits inspired by the path integral formalism of quantum mechanics, and examine its application to some simple textbook problems.
The present work points out that the Deutsch-Jozsa algorithm was the first formal description of a quantum decider. In particular, it is studied here the class of languages whose indicator functions allow the Deutsch-Jozsa algorithm to output a monochromatic result, beyond the trivial case z = 0 for constant indicator functions. To illustrate examples of randomly balanced languages and some monochromatic cases, it was performed classical computational simulations of the Deutsch-Jozsa quantum algorithm for the specific cases of 4 and 6 qubits, respectively. The general case of the Deutsch-Jozsa…
We present an original model of paraconsistent Turing machines (PTMs), a generalization of the classical Turing machines model of computation using a paraconsistent logic. Next, we briefl y describe the standard models of quantum computation: quantum Turing machines and quantum circuits, and revise quantum algorithms to solve the so-called Deutsch's problem and Deutsch-Jozsa problem. Then, we show the potentialities of the PTMs model of computation simulating the presented quantum algorithms via paraconsistent algorithms. This way, we show that PTMs can resolve some problems in exponentially l…
We present a novel use case for quantum computation: the sensitivity analysis for a risk model used at Deutsche Börse Group. Such an analysis is computationally too expensive to perform on classical computers. We show in detail how the risk model and its analysis can be implemented as a quantum circuit. We test small scale versions of the model in simulation and find that the expected quadratic speedup compared to the classical implementation used at Deutsche Börse Group can be realized. Full scale production usage would be possible with less than 200 error corrected qubits. Our quantum algori…
It was shown by M.A. Nielsen and I.L. Chuang 1997, that it is impossible to build strictly universal programmable quantum gate array, that could perform any unitary operation precisely and it was suggested to use probabilistic gate arrays instead. In present work is shown, that if to use more physical and weak condition of universality (suggested already in earliest work by D.Deutsch 1985) and to talk about simulation with arbitrary, but finite precision, then it is possible to build universal programmable gate array. But now the same no-go theorem by Nielsen and Chuang will have new interesti…
We analyse an argument of Deutsch, which purports to show that the deterministic part of classical quantum theory together with deterministic axioms of classical decision theory, together imply that a rational decision maker behaves as if the probabilistic part of quantum theory (Born's law) is true. We uncover two missing assumptions in the argument, and show that the argument also works for an instrumentalist who is prepared to accept that the outcome of a quantum measurement is random in the frequentist sense: Born's law is a consequence of functional and unitary invariance principles belon…
In this article we present an ongoing effort to formalise quantum algorithms and results in quantum information theory using the proof assistant Isabelle/HOL. Formal methods being critical for the safety and security of algorithms and protocols, we foresee their widespread use for quantum computing in the future. We have developed a large library for quantum computing in Isabelle based on a matrix representation for quantum circuits, successfully formalising the no-cloning theorem, quantum teleportation, Deutsch's algorithm, the Deutsch-Jozsa algorithm and the quantum Prisoner's Dilemma. We di…
Deutsch has recently (in quant-ph/9906015) offered a justification, based only on the non-probabilistic axioms of quantum theory and of classical decision theory, for the use of the standard quantum probability rules. In this note, this justification is examined.
In a recent preprint by Deutsch et al. [1995] the authors suggest the possibility of polynomial approximability of arbitrary unitary operations on $n$ qubits by 2-qubit unitary operations. We address that comment by proving strong lower bounds on the approximation capabilities of g-qubit unitary operations for fixed g. We consider approximation of unitary operations on subspaces as well as approximation of states and of density matrices by quantum circuits in several natural metrics. The ability of quantum circuits to probabilistically solve decision problem and guess checkable functions is di…
We describe a method to axiomatize computations in deterministic Turing machines. When applied to computations in non-deterministic Turing machines, this method may produce contradictory (and therefore trivial) theories, considering classical logic as the underlying logic. By substituting in such theories the underlying logic by a paraconsistent logic we define a new computation model, the paraconsistent Turing machine. This model allows a partial simulation of superposed states of quantum computing. Such a feature allows the definition of paraconsistent algorithms which solve (with some restr…
There has been considerable recent interest in the consequences of closed timelike curves (CTCs) for the dynamics of quantum mechanical systems. A vast majority of research into this area makes use of the dynamical equations developed by Deutsch, which were developed from a consistency condition that assumes that mixed quantum states uniquely describe the physical state of a system. We criticise this choice of consistency condition from an epistemic perspective, i.e., a perspective in which the quantum state represents a state of knowledge about a system. We demonstrate that directly applying …
