The one-clean-qubit model of quantum computation (DQC1) efficiently implements a computational task that is not known to have a classical alternative. During the computation, there is never more than a small but finite amount of entanglement present, and it is typically vanishingly small in the system size. In this paper, we demonstrate that there is nothing unexpected hidden within the DQC1 model---Grover's search, when acting on a mixed state, provably exhibits a speedup over classical, with guarantees as to the presence of only vanishingly small amounts of quantum correlations (entanglement and quantum discord)---while arguing that this is not an artifact of the oracle-based construction. We also present some important refinements in the evaluation of how much entanglement may be present in the DQC1 and how the typical entanglement of the system must be evaluated.
The one-clean-qubit model of quantum computation (DQC1) efficiently implements a computational task that is not known to have a classical alternative. During the computation, there is never more than a small but finite amount of entanglement present, and it is typically vanishingly small in the system size. In this paper, we demonstrate that there is nothing unexpected hidden within the DQC1 model---Grover's search, when acting on a mixed state, provably exhibits a speedup over classical, with guarantees as to the presence of only vanishingly small amounts of quantum correlations (entanglement and quantum discord)---while arguing that this is not an artifact of the oracle-based construction. We also present some important refinements in the evaluation of how much entanglement may be present in the DQC1 and how the typical entanglement of the system must be evaluated.
Deterministic quantum computation with one pure qubit (DQC1) is an efficient model of computation that uses highly mixed states. Unlike pure-state models, its power is not derived from the generation of a large amount of entanglement. Instead it has been proposed that other nonclassical correlations are responsible for the computational speedup, and that these can be captured by the quantum discord. In this Letter we implement DQC1 in an all-optical architecture, and experimentally observe the generated correlations. We find no entanglement, but large amounts of quantum discord-except in three cases where an efficient classical simulation is always possible. Our results show that even fully separable, highly mixed, states can contain intrinsically quantum mechanical correlations and that these could offer a valuable resource for quantum information technologies.
In this thesis, I look at the role of quantum entanglement in mixed-state quantum computation. The model we consider is the DQC1 or `power of one qubit' model. I show that there is minimal bipartite entanglement in a typical instance of the DQC1 circuit and even put an upper bound on the possible amount of entanglement. This limited amount of entanglement, however, does not imply that the system is classically simulatable. A matrix product state (MPS) algorithm for a typical instance of the DQC1 system requires exponential classical resources. This exposes a gap between the amount of entanglement and the amount of purely nonclassical correlations in a quantum system. This gap, I suggest, can be filled by quantum discord. I calculate it in a typical instance of the DQC1 circuit and find that the amount of discord is a constant fraction of the maximum possible discord for a system of that size. This allows an interpretation of quantum discord as the resource that drives mixed-state quantum computation. I also study quantum discord as a quantity of independent interest. Its role in the phenomenon of entanglement distribution is studied through an easily comprehensible example. This thesis also contains discussions on the relation between the complexity classes P, BQP and DQC1. Additional material is presented on the connections between the DQC1 model, Jones polynomials and statistical mechanics. The thesis concludes with a discussion of a few open problems related to the DQC1 model, the quantum discord and their scope in quantum information science.
In a quantum computation with pure states, the generation of large amounts of entanglement is known to be necessary for a speedup with respect to classical computations. However, examples of quantum computations with mixed states are known, such as the deterministic computation with one quantum qubit (DQC1) model [Knill and Laflamme, Phys. Rev. Lett. 81, 5672 (1998)], in which entanglement is at most marginally present, and yet a computational speedup is believed to occur. Correlations, and not entanglement, have been identified as a necessary ingredient for mixed-state quantum computation speedups. Here we show that correlations, as measured through the operator Schmidt rank, are indeed present in large amounts in the DQC1 circuit. This provides evidence for the preclusion of efficient classical simulation of DQC1 by means of a whole class of classical simulation algorithms, thereby reinforcing the conjecture that DQC1 leads to a genuine quantum computational speedup.
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.
We use quantum discord to characterize the correlations present in the model called deterministic quantum computation with one quantum bit (DQC1), introduced by Knill and Laflamme [Phys. Rev. Lett. 81, 5672 (1998)10.1103/PhysRevLett.81.5672]. The model involves a collection of qubits in the completely mixed state coupled to a single control qubit that has nonzero purity. The initial state, operations, and measurements in the model all point to a natural bipartite split between the control qubit and the mixed ones. Although there is no entanglement between these two parts, we show that the quantum discord across this split is nonzero for typical instances of the DQC1 ciruit. Nonzero values of discord indicate the presence of nonclassical correlations. We propose quantum discord as figure of merit for characterizing the resources present in this computational model.
Quantum entanglement is the most popular kind of quantum correlations. Its role in several tasks in quantum information theory like quantum cryptography, quantum dense coding and quantum teleportation is indisputable. The role of entanglement in quantum algorithms has also attracted enormous amount of attention in the literature. The research in this direction is motivated by the finding that a quantum computer can efficiently factorize arbitrary integer numbers. Since no classical algorithm is known today that can solve this problem efficiently, this example clearly demonstrates the superiority of a quantum computer compared to its classical counterpart. In this context, entanglement also plays a crucial role: if the quantum computer operates on a pure state without entanglement, then the computational process can be simulated efficiently on a classical computer. In this case entanglement is the key ingredient that makes the difference between a quantum and a classical computer.
However, the situation is more involved if the computational process is not perfect. In this case the quantum computer operates on a mixed state, and the role of entanglement is less obvious. A seminal result in this direction is a quantum algorithm known under the acronym DQC1. This algorithm operates on an almost maximally mixed state with vanishingly little entanglement. However, DQC1 can still perform tasks efficiently for which no efficient classical algorithm is known today. Thus, it is reasonable to assume that entanglement in general is not related to the efficiency of a quantum algorithm.
Triggered by this observation other kinds of quantum correlations have been studied. In this context a particular measure of quantum correlations known as quantum discord is referred to frequently in the literature. Quantum discord is a measure of quantum correlations beyond entanglement, i.e., a quantum state can have no entanglement, but nonvanishing quantum discord. Quantum discord is also regarded as the key resource in the DQC1 algorithm. Although this statement is still controversial, it was shown under very general assumptions that quantum discord is required if a quantum computer is to show an exponential speedup over any classical algorithm.
In this thesis we give a short introduction into the theory of entanglement and general quantum correlations, and further discuss our results. In particular, we show that two different kinds of entanglement measures can coincide. This result is used to build an algorithm for computing entanglement. We also consider the role of quantum correlations for the distribution of entanglement and for the quantum measurement process. We further discuss the behavior of quantum correlations under local noise, and their monogamy properties.