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The successful operation of quantum computers relies on protecting qubits from decoherence and noise, which--if uncorrected--will lead to erroneous results. Because these errors accumulate during an algorithm, correcting them is a key requirement for large-scale and fault-tolerant quantum information processors. Besides computational errors, which can be addressed by quantum error correction.sup.1-9, the carrier of the information can also be completely lost or the information can leak out of the computational space.sup.10-14. It is expected that such loss errors will occur at rates that are comparable to those of computational errors. Here we experimentally implement a full cycle of qubit loss detection and correction on a minimal instance of a topological surface code.sup.15,16 in a trapped-ion quantum processor. The key technique used for this correction is a quantum non-demolition measurement performed via an ancillary qubit, which acts as a minimally invasive probe that detects absent qubits while imparting the smallest quantum mechanically possible disturbance to the remaining qubits. Upon detecting qubit loss, a recovery procedure is triggered in real time that maps the logical information onto a new encoding on the remaining qubits. Although the current demonstration is performed in a trapped-ion quantum processor.sup.17, the protocol is applicable to other quantum computing architectures and error correcting codes, including leading two- and three-dimensional topological codes. These deterministic methods provide a complete toolbox for the correction of qubit loss that, together with techniques that mitigate computational errors, constitute the building blocks of complete and scalable quantum error correction. A deterministic correction of errors caused by qubit loss or leakage outside the computational space is demonstrated in a trapped-ion experiment by using a minimal instance of the topological surface code., Author(s): Roman Stricker [sup.1] , Davide Vodola [sup.2] [sup.3] [sup.4] , Alexander Erhard [sup.1] , Lukas Postler [sup.1] , Michael Meth [sup.1] , Martin Ringbauer [sup.1] , Philipp Schindler [sup.1] [...]

The accuracy of logical operations on quantum bits (qubits) must be improved for quantum computers to outperform classical ones in useful tasks. One method to achieve this is quantum error correction (QEC), which prevents noise in the underlying system from causing logical errors. This approach derives from the reasonable assumption that noise is local, that is, it does not act in a coordinated way on different parts of the physical system. Therefore, if a logical qubit is encoded non-locally, we can--for a limited time--detect and correct noise-induced evolution before it corrupts the encoded information.sup.1. In 2001, Gottesman, Kitaev and Preskill (GKP) proposed a hardware-efficient instance of such a non-local qubit: a superposition of position eigenstates that forms grid states of a single oscillator.sup.2. However, the implementation of measurements that reveal this noise-induced evolution of the oscillator while preserving the encoded information.sup.3-7 has proved to be experimentally challenging, and the only realization reported so far relied on post-selection.sup.8,9, which is incompatible with QEC. Here we experimentally prepare square and hexagonal GKP code states through a feedback protocol that incorporates non-destructive measurements that are implemented with a superconducting microwave cavity having the role of the oscillator. We demonstrate QEC of an encoded qubit with suppression of all logical errors, in quantitative agreement with a theoretical estimate based on the measured imperfections of the experiment. Our protocol is applicable to other continuous-variable systems and, in contrast to previous implementations of QEC.sup.10-14, can mitigate all logical errors generated by a wide variety of noise processes and facilitate fault-tolerant quantum computation. Quantum error correction of Gottesman-Kitaev-Preskill code states is realized experimentally in a superconducting quantum device., Author(s): P. Campagne-Ibarcq [sup.1] , A. Eickbusch [sup.1] , S. Touzard [sup.1] , E. Zalys-Geller [sup.1] , N. E. Frattini [sup.1] , V. V. Sivak [sup.1] , P. Reinhold [sup.1] [...]

Quantum computing becomes viable when a quantum state can be protected from environment-induced error. If quantum bits (qubits) are sufficiently reliable, errors are sparse and quantum error correction (QEC) (1-6) [...]

Error correction is central to fault-tolerant quantum computation, but although various schemes have been developed in theory, there are few experimental realizations; a quantum error correction process is now reported for a single system of electron and nuclear spins residing in a diamond crystal. Practical quantum error correction Quantum information processing has the potential to be very powerful, solving problems that classical computing cannot even address. One drawback is its reliance on fragile resources -- quantum superposition and entanglement -- that are easily perturbed. Error correction is therefore central to fault-tolerant quantum computation and although various schemes have been proposed, there are few experimental realizations. Gerald Waldherr et al. successfully demonstrate a quantum error correction process on a system of electron and nuclear spins residing in a diamond crystal. Three nearby nuclear spins form the three entangled quantum bits (qubits) that are necessary for a quantum error correction protocol and the interaction with an electron spin enables readout. This new approach is applicable to other solid-state hybrid quantum spin systems such as those based on dopants in silicon. Error correction is important in classical and quantum computation. Decoherence caused by the inevitable interaction of quantum bits with their environment leads to dephasing or even relaxation. Correction of the concomitant errors is therefore a fundamental requirement for scalable quantum computation.sup.1,2,3,4,5,6,7. Although algorithms for error correction have been known for some time, experimental realizations are scarce.sup.2,3,4,5,6,7. Here we show quantum error correction in a heterogeneous, solid-state spin system.sup.8,9,10,11,12,13,14,15,16,17,18,19,20,21. We demonstrate that joint initialization, projective readout and fast local and non-local gate operations can all be achieved in diamond spin systems, even under ambient conditions. High-fidelity initialization of a whole spin register (99 per cent) and single-shot readout of multiple individual nuclear spins are achieved by using the ancillary electron spin of a nitrogen-vacancy defect. Implementation of a novel non-local gate generic to our electron-nuclear quantum register allows the preparation of entangled states of three nuclear spins, with fidelities exceeding 85 per cent. With these techniques, we demonstrate three-qubit phase-flip error correction. Using optimal control, all of the above operations achieve fidelities approaching those needed for fault-tolerant quantum operation, thus paving the way to large-scale quantum computation. Besides their use with diamond spin systems, our techniques can be used to improve scaling of quantum networks relying on phosphorus in silicon.sup.19, quantum dots.sup.22, silicon carbide.sup.11 or rare-earth ions in solids.sup.20,21., Author(s): G. Waldherr [sup.1] , Y. Wang [sup.1] , S. Zaiser [sup.1] , M. Jamali [sup.1] , T. Schulte-Herbrüggen [sup.2] , H. Abe [sup.3] , T. Ohshima [sup.3] , J. [...]

This paper considers constrained [l.sub.1] minimization methods in a unified framework for the recovery of high-dimensional sparse signals in three settings: noiseless, bounded error, and Gaussian noise. Both [l.sub.1] minimization with an [l.sub.[infinity]] constraint (Dantzig selector) and [l.sub.1] minimization under an [l.sub.2] constraint are considered. The results of this paper improve the existing results in the literature by weakening the conditions and tightening the error bounds. The improvement on the conditions shows that signals with larger support can be recovered accurately. In particular, our results illustrate the relationship between [l.sub.1] minimization with an [l.sub.2] constraint and [l.sub.1] minimization with an [l.sub.[infinity]] constraint. This paper also establishes connections between restricted isometry property and the mutual incoherence property. Some results of Candes, Romberg, and Tao (2006), Candes and Tao (2007), and Donoho, Elad, and Temlyakov (2006) are extended. Index Terms--Dantzig selector [l.sub.1] minimization, restricted isometry property, sparse recovery, sparsity.

We establish random-coding lower bounds to the error exponent of discrete and Gaussian joint quantization and private watermarking systems. In the discrete system, both the covertext and the attack channel are memoryless and have finite alphabets. In the Gaussian system, the covertext is memoryless Gaussian and the attack channel has additive memoryless Gaussian noise. In both cases, our bounds on the error exponent are positive in the interior of the achievable quantization and watermarking rate region. Index Terms--Capacity region, error exponent, Gaussian-type class, information hiding, joint quantization and watermarking, private watermarking, random-coding lower bound.

Randomized linear network code for single source multicast was introduced and analyzed in Ho et al. (IEEE Transactions on Information Theory, October 2006) where the main results are upper bounds for the failure probability of the code. In this paper, these bounds are improved and tightness of the new bounds is studied by analyzing the limiting behavior of the failure probability as the field size goes to infinity. In the linear random coding setting for single source multicast, the minimum distance of the code defined in Zhang, (IEEE Transactions on lnformation Theory, January 2008) is a random variable taking nonnegative integer values that satisfy the inequality in the Singleton bound recently established in Yeung and Cai (Communications in Information and Systems, 2006) for network error correction codes. We derive a bound on the probability mass function of the minimum distance of the random linear network code based on our improved upper bounds for the failure probability. Codes having the highest possible minimum distance in the Singleton bound are called maximum distance separable (MDS). The bound on the field size required for the existence of MDS codes reported in Zhang, (IEEE Transactions on Information Theory, January 2008) and Matsumoto (arXiv:cs.lT/06lO121, Oct. 2006) suggests that such codes exist only when field size is large. Define the degradation of a code as the difference between the highest possible minimum distance in the Singleton bound and the actual minimum distance of the code. The bound for the probability mass function of the minimum distance leads to a bound on the field size required for the existence of network error correction codes with a given maximum degradation. The result shows that allowing minor degradation reduces the field size required dramatically. Index Terms--Random network code, network error correction code, maximum distance separable (MDS) code, the Singleton bound, random network error correction code, failure probability of random network code.