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Journal of Physics A Mathematical and Theoretical

The continuous-variable Deutsch–Jozsa algorithm using realistic quantum systems

Rob C. Wagner, Viv M. Kendon

Published May 30, 2012• 4 citations
Publisher

Abstract

This paper is a study of the continuous-variable Deutsch–Jozsa algorithm. First, we review an existing version of the algorithm for qunat states (Pati and Braunstein 2002 arXiv:0207108v1), and then, we present a realistic version of the Deutsch–Jozsa algorithm for continuous variables, which can be implemented in a physical quantum system given the appropriate oracle. Under these conditions, we have a probabilistic algorithm for deciding the function with a very high success rate with a single call to the oracle. Finally, we look at the effects of errors in both of these continuous-variable algorithms and how they affect the chances of success. We find that the algorithm is generally robust for errors in initialization and the oracle, but less so for errors in the measurement apparatus and the Fourier transform. This article is part of a special issue of Journal of Physics A: Mathematical and Theoretical devoted to 'Coherent states: mathematical and physical aspects'.

DOI
10.1088/1751-8113/45/24/244015

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This paper is a study of the continuous-variable Deutsch–Jozsa algorithm. First, we review an existing version of the algorithm for qunat states (Pati and Braunstein 2002 arXiv:0207108v1), and then, we present a realistic version of the Deutsch–Jozsa algorithm for continuous variables, which can be implemented in a physical quantum system given the appropriate oracle. Under these conditions, we have a probabilistic algorithm for deciding the function with a very high success rate with a single call to the oracle. Finally, we look at the effects of errors in both of these continuous-variable algorithms and how they affect the chances of success. We find that the algorithm is generally robust for errors in initialization and the oracle, but less so for errors in the measurement apparatus and the Fourier transform. This article is part of a special issue of Journal of Physics A: Mathematical and Theoretical devoted to 'Coherent states: mathematical and physical aspects'.
A hybrid model of the Deutsch-Jozsa algorithm is presented, inspired by the proposals of hybrid computation by S. Lloyd and P. van Loock et. al. The model is based on two observations made about both the discrete and continuous algorithms already available. First, the Fourier transform is a single-step operation in a continuous-variable (CV) setting. Additionally, any implementation of the oracle is nontrivial in both schemes. The steps of the computation are very similar to those in the CV algorithm, with the main difference being the way in which the qunats, or quantum units of analogic information, and the qubits interact in the oracle. Using both discrete and continuous states of light, linear devices, and photo-detection, an optical implementation of the oracle is proposed. For simplicity, infinitely squeezed states are used in the continuous register, whereas the optical qubit is encoded in the dual-rail logic of the KLM protocol. The initial assumption of ideal states as qunats will be dropped to study the effects of finite squeezing in the quality of the computation.
We reveal a close relationship between quantum metrology and the Deutsch-Jozsa algorithm on continuous-variable quantum systems. We develop a general procedure, characterized by two parameters, that unifies parameter estimation and the Deutsch-Jozsa algorithm. Depending on which parameter we keep constant, the procedure implements either the parameter-estimation protocol or the Deutsch-Jozsa algorithm. The parameter-estimation part of the procedure attains the Heisenberg limit and is therefore optimal. Due to the use of approximate normalizable continuous-variable eigenstates, the Deutsch-Jozsa algorithm is probabilistic. The procedure estimates a value of an unknown parameter and solves the Deutsch-Jozsa problem without the use of any entanglement.
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 a 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 the quantum procedure has one-sided error while the classical procedure has two-sided error. The role of coherence on the quantum success probabilities in both of these problems is examined.
A general framework for regarding oracle-assisted quantum algorithms as tools for discriminating among unitary transformations is described. This framework is applied to the Deutsch-Jozsa problem and all possible quantum algorithms which solve the problem with certainty using oracle unitaries in a particular form are derived. It is also used 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 classical algorithm when the problem size nnn exceeds  105.~{10}^{5}. 105.
Quantum information processing is appealing due its ability to solve certain problems quantitatively faster than classical information processing. Most quantum algorithms have been studied in discretely parameterized systems, but many quantum systems are continuously parameterized. The field of quantum optics in particular has sophisticated techniques for manipulating continuously parameterized quantum states of light, but the lack of a code-state formalism has hindered the study of quantum algorithms in these systems. To address this situation, a code-state formalism for the solution of oracle decision problems in continuously-parameterized quantum systems is developed. Quantum information processing is appealing due its ability to solve certain problems quantitatively faster than classical information processing. Most quantum algorithms have been studied in discretely parameterized systems, but many quantum systems are continuously parameterized. The field of quantum optics in particular has sophisticated techniques for manipulating continuously parameterized quantum states of light, but the lack of a code-state formalism has hindered the study of quantum algorithms in these systems. To address this situation, a code-state formalism for the solution of oracle decision problems in continuously-parameterized quantum systems is developed. In the infinite-dimensional case, we study continuous-variable quantum algorithms for the solution of the Deutsch–Jozsa oracle decision problem implemented within a single harmonic-oscillator. Orthogonal states are used as the computational bases, and we show that, contrary to a previous claim in the literature, this implementation of quantum information processing has limitations due to a position-momentum trade-off of the Fourier transform. We further demonstrate that orthogonal encoding bases are not unique, and using the coherent states of the harmonic oscillator as the computational bases, our formalism enables quantifying the relative performances of different choices of the encoding bases. We extend our formalism to include quantum algorithms in the continuously parameterized yet finite-dimensional Hilbert space of a coherent spin system. We show that the highest-squeezed spin state possible can be approximated by a superposition of two states thus transcending the usual model of using a single basis state as algorithm input. As a particular example, we show that the close Hadamard oracle-decision problem, which is related to the Hadamard codewords of digital communications theory, can be solved quantitatively more efficiently using this computational model than by any known classical algorithm.