We propose a novel dynamical method for beating decoherence and dissipation in open quantum systems. We demonstrate the possibility of filtering out the effects of unwanted (not necessarily known) system-environment interactions and show that the noise-suppression procedure can be combined with the capability of retaining control over the effective dynamical evolution of the open quantum system. Implications for quantum information processing are discussed.Comment: 4 pages, no figures; Plain ReVTeX. Final version to appear in Physical Review Letter

Quantum Error Correction will be necessary for preserving coherent states against noise and other unwanted interactions in quantum computation and communication. We develop a general theory of quantum error correction based on encoding states into larger Hilbert spaces subject to known interactions. We obtain necessary and sufficient conditions for the perfect recovery of an encoded state after its degradation by an interaction. The conditions depend only on the behavior of the logical states. We use them to give a recovery operator independent definition of error-correcting codes. We relate this definition to four others: The existence of a left inverse of the interaction, an explicit representation of the error syndrome using tensor products, perfect recovery of the completely entangled state, and an information theoretic identity. Two notions of fidelity and error for imperfect recovery are introduced, one for pure and the other for entangled states. The latter is more appropriate when using codes in a quantum memory or in applications of quantum teleportation to communication. We show that the error for entangled states is bounded linearly by the error for pure states. A formal definition of independent interactions for qubits is given. This leads to lower bounds on the number of qubits required to correct e errors and a formal proof that the classical bounds on the probability of error of e-error-correcting codes applies to e-errorcorrecting quantum codes, provided that the interaction is dominated by an identity component. *

We present a loophole-free violation of local realism using entangled photon pairs. We ensure that all relevant events in our Bell test are spacelike separated by placing the parties far enough apart and by using fast random number generators and high-speed polarization measurements. A high-quality polarization-entangled source of photons, combined with high-efficiency, low-noise, single-photon detectors, allows us to make measurements without requiring any fair-sampling assumptions. Using a hypothesis test, we compute p-values as small as 5.9 × 10−9 for our Bell violation while maintaining the spacelike separation of our events. We estimate the degree to which a local realistic system could predict our measurement choices. Accounting for this predictability, our smallest adjusted p-value is 2.3 × 10−7. We therefore reject the hypothesis that local realism governs our experiment.

In standard quantum computation, the initial state is pure and the answer is determined by making a measurement of some of the bits in the computational basis. What can be accomplished if the initial state is a highly mixed state and the answer is determined by measuring the expectation of σz on the first bit with bounded sensitivity? This is the situation in high temperature ensemble quantum computation. We show that in this model it is possible to perform interesting physics simulations which have no known efficient classical algorithms, even though the model is less powerful then standard quantum computing in the presence of oracles.Recent discoveries show that quantum computers can solve problems of practical interest much faster than known algorithms for classical computers [1,2]. This has lead to widespread recognition of the potential benefits of quantum computation. Where does the apparent power of quantum computers come from? This power is frequently attributed to "quantum parallelism" , interference phenomena derived from the superposition principle, and the ability to prepare and control pure states according to the Schrödinger equation. Real quantum computers are rarely in pure states and interact with their environments, which leads to non-unitary evolution. Furthermore, recent proposals and experiments using NMR at high temperature to study quantum computation involve manipulations of extremely mixed states. Recent research in error-correction and fault-tolerant computation has shown that non-unitary evolution due to weak interactions with the environment results in no loss of computational power, if sufficiently pure states can be prepared [3][4][5][6]. Here we consider the situation where there are no errors or interactions with the environment, but the initial state is highly mixed. We investigate the power of one bit of quantum information available for computing, by which we mean that the input state is equivalent to having one bit in a pure state and arbitrarily many additional bits in a completely random state. The model of computation which consists of a classical computer with access to a state of this form is called deterministic quantum computation with one quantum bit (DQC1). We demonstrate that in the presence of oracles, such a computer is less powerful than one with access to pure state bits. However, it can solve problems related to physics simulations for which no efficient classical algorithms are known. DQC1 is the first non-trivial entry in the class of models of computations which are between classical computation and standard quantum computation. Investigations of such models are expected to lead to a better understanding of the reasons for the power of quantum computation.There are many kinds of problems that one might like to solve using a computational device. The three main problems not involving communication are function evaluation, non-deterministic function evaluation and distribution sampling. Let S be the set of all binary strings and S n the set of binary strings of...

A key requirement for scalable quantum computing is that elementary quantum gates can be implemented with sufficiently low error. One method for determining the error behavior of a gate implementation is to perform process tomography. However, standard process tomography is limited by errors in state preparation, measurement and one-qubit gates. It suffers from inefficient scaling with number of qubits and does not detect adverse error-compounding when gates are composed in long sequences. An additional problem is due to the fact that desirable error probabilities for scalable quantum computing are of the order of 0.0001 or lower. Experimentally proving such low errors is challenging. We describe a randomized benchmarking method that yields estimates of the computationally relevant errors without relying on accurate state preparation and measurement. Since it involves long sequences of randomly chosen gates, it also verifies that error behavior is stable when used in long computations. We implemented randomized benchmarking on trapped atomic ion qubits, establishing a one-qubit error probability per randomized / 2 pulse of 0.00482͑17͒ in a particular experiment. We expect this error probability to be readily improved with straightforward technical modifications.

Quantum Error Correction will be necessary for preserving coherent states against noise and other unwanted interactions in quantum computation and communication. We develop a general theory of quantum error correction based on encoding states into larger Hilbert spaces subject to known interactions. We obtain necessary and sufficient conditions for the perfect recovery of an encoded state after its degradation by an interaction. The conditions depend only on the behavior of the logical states. We use them to give a recovery operator independent definition of error-correcting codes. We relate this definition to four others: The existence of a left inverse of the interaction, an explicit representation of the error syndrome using tensor products, perfect recovery of the completely entangled state, and an information theoretic identity. Two notions of fidelity and error for imperfect recovery are introduced, one for pure and the other for entangled states. The latter is more appropriate when using codes in a quantum memory or in applications of quantum teleportation to communication. We show that the error for entangled states is bounded linearly by the error for pure states. A formal definition of independent interactions for qubits is given. This leads to lower bounds on the number of qubits required to correct e errors and a formal proof that the classical bounds on the probability of error of e-error-correcting codes applies to e-errorcorrecting quantum codes, provided that the interaction is dominated by an identity component. *

Among the classes of highly entangled states of multiple quantum systems, the so-called 'Schrödinger cat' states are particularly useful. Cat states are equal superpositions of two maximally different quantum states. They are a fundamental resource in fault-tolerant quantum computing and quantum communication, where they can enable protocols such as open-destination teleportation and secret sharing. They play a role in fundamental tests of quantum mechanics and enable improved signal-to-noise ratios in interferometry. Cat states are very sensitive to decoherence, and as a result their preparation is challenging and can serve as a demonstration of good quantum control. Here we report the creation of cat states of up to six atomic qubits. Each qubit's state space is defined by two hyperfine ground states of a beryllium ion; the cat state corresponds to an entangled equal superposition of all the atoms in one hyperfine state and all atoms in the other hyperfine state. In our experiments, the cat states are prepared in a three-step process, irrespective of the number of entangled atoms. Together with entangled states of a different class created in Innsbruck, this work represents the current state-of-the-art for large entangled states in any qubit system.

In theory, quantum computers can efficiently simulate quantum physics, factor large numbers and estimate integrals, thus solving otherwise intractable computational problems. In practice, quantum computers must operate with noisy devices called "gates" that tend to destroy the fragile quantum states needed for computation. The goal of fault-tolerant quantum computing is to compute accurately even when gates have a high probability of error each time they are used. Here we give evidence that accurate quantum computing is possible with error probabilities above 3 % per gate, which is significantly higher than what was previously thought possible. However, the resources required for computing at such high error probabilities are excessive. Fortunately, they decrease rapidly with decreasing error probabilities. If we had quantum resources comparable to the considerable resources available in today's digital computers, we could implement non-trivial quantum computations at error probabilities as high as 1 % per gate.Research in quantum computing is motivated by the great increase in computational power offered by quantum computers. [1][2][3] There is a large and still growing number of experimental efforts whose ultimate goal is to demonstrate scalable quantum computing. Scalable quantum computing requires that arbitrarily large computations can be efficiently implemented with little error in the output. Criteria that need to be satisfied by devices used for scalable quantum computing have been specified by DiVincenzo. 4 One of the criteria is that the level of noise affecting the physical gates is sufficiently low. The type of noise affecting the gates in a given implementation is called the "error model". A scheme for scalable quantum computing in the presence of noise is called a "faulttolerant architecture". In view of the low-noise criterion, studies of scalable quantum computing involve constructing fault-tolerant architectures and providing answers to questions such as the following: Q1: Is scalable quantum computing possible for error model E? Q2: Can fault-tolerant architecture A be used for scalable quantum computing with error model E? Q3: What resources are required to implement quantum computation C using fault-tolerant architecture A with error model E?To obtain broadly applicable results, fault-tolerant architectures are constructed for generic error models. Here, the error model is parametrized by an error probability per gate (or simply error per gate, EPG), where the errors are unbiased and independent. The fundamental theorem of scalable quantum computing is the threshold theorem and answers question Q1 as follows: If 1