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arXiv:1604.08688 · Physical Review A

Implementing the Deutsch-Jozsa algorithm with macroscopic ensembles

Henry Semenenko, Tim Byrnes

Published May 2, 2016• 8 citations

Abstract

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 quantum algorithm showing exponential speedup over classical algorithms is yet to be shown. Here, we show that the Deutsch-Jozsa algorithm can be implemented with macroscopic ensembles. The encoding that we use avoids the detrimental effects of decoherence that normally plagues macroscopic implementations. We discuss two mapping procedures which can be chosen depending upon the constraints of the oracle and the experiment. Both methods have an exponential speedup over the classical case, and only require control of the ensembles at the level of the total spin of the ensembles. It is shown that both approaches reproduce the qubit Deutsch-Jozsa algorithm, and are robust under decoherence.

DOI: 10.1103/physreva.93.052302

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The first optical proposal for the realization of the two-bit version of the Deutsch-Jozsa algorithm [D. Deutsch and R. Jozsa, Proc. R. Soc. London A {\bf 493}, 553 (1992)] is presented. The proposal uses Stark shifts in an ensemble of atoms and degenerate sources of photons. The photons interact dispersively with an atomic ensemble, leading to an effective Hamiltonian in atom-field basis, which is useful for performing the required two-qubit operations. Combining these with a set of one-qubit operations, the algorithm can be implemented. A discussion of the experimental feasibility of the proposal is given.

We present an optical scheme to implement the Deutsch-Jozsa algorithm using ac Stark shifts. The scheme uses an atomic ensemble consisting of four-level atoms interacting dispersively with a field. This leads to a Hamiltonian in the atom-field basis which is quite suitable for quantum computation. We show how one can implement the algorithm by performing proper one- and two-qubit operations. We emphasize that in our model the decoherence is expected to be minimal due to our usage of atomic ground states and freely propagating photon.

Quantum correlations have been pointed out as the most likely source of the speed-up in quantum computation. Here we study the presence of quantum correlations on the implementation of Deutsch- Jozsa algorithm running in the DQC1 model. We extend the analysis to a previous experimental realization of this computation presented in Phys. Rev. A 78, 022317 (2008). The Deutsch-Jozsa algorithm, an oracle based problem, can efficiently be solved in this model. Although there is no quantum correlations in the initial and final states of the computation, we find quantum correlations in intermediate steps of the algorithm. The presence of quantum correlations is not a sufficient property for computational gain in this case, since the performance of the classical probabilistic algorithm is better than the quantum one. The reason for this unexpected result relies on the fact that we have access to the outcomes of the control qubit only. A comparison between the efficiency of DQC1 and DQCp circuits to solve the Deustch-Jozsa problem is also presented.

A scheme to execute an \\eta-bit Deutsch-Jozsa (DJ) algorithm using \\eta qubits has been implemented for up to three qubits on an NMR quantum computer. For the one- and the two-bit Deutsch problem, the qubits do not get entangled, and the NMR implementation is achieved without using spin-spin interactions. It is for the three-bit case, that the manipulation of entangled states becomes essential. The interactions through scalar J-couplings in NMR spin systems have been exploited to implement entangling transformations required for the three bit DJ algorithm.

We discuss a new approach to simulate quantum algorithms using classical probabilistic bits and circuits. Each qubit (a two-level quantum system) is initially mapped to a vector in an eight dimensional probability space (equivalently, to a classical random variable with eight probabilistic outcomes). The key idea in this mapping is to store both the amplitude and phase information of the complex coefficients that describe the qubit state in the probabilities. Due to the identical tensor product structure of combining multiple quantum systems as well as multiple probability spaces,

n

qubits are then mapped to a tensor product of

n

8-dimensional probabilistic vectors (i.e., the Hilbert space of dimension

2^n

is mapped to a probability space of dimension

8^n

). After this initial mapping, we show how to implement the analogs of single-qubit and two-qubit gates in the probability space using correlation-inducing operations on these classical random variables. The key defining feature of both the mapping to the probability space and the transformations in this space (i.e., operations on the random variables) is that they are not linear, but instead affine. Using this architecture, the evolution of the

2^n

complex coefficients of the quantum system can be tracked in the joint fully-correlated probabilities of the polynomial number of random variables. We then give specific procedures for implementing (1) the Deutsch-Jozsa algorithm, and (2) the Quantum Fourier Transform in the probability space. Identical to the Quantum case, simulating the Quantum Fourier Transform in the probability space requires

O(n)

probabilistic bits and

O(n^2)

(i.e., quadratic in the number of quantum bits) operations.

Schemes of experimental realization of the main two-qubit processors for quantum computers and the Deutsch-Jozsa algorithm are derived in virtual spin representation. The results are applicable for every four quantum states allowing the required properties for quantum processor implementation if for qubit encoding, virtual spin representation is used. A four-dimensional Hilbert space of nuclear spin

3/2

is considered in detail for this aim.