Overhead and noise threshold of fault-tolerant quantum error correction

Fault tolerant quantum error correction (QEC) networks are studied by a combination of numerical and approximate analytical treatments. The probability of failure of the recovery operation is calculated for a variety of CSS codes, including large block codes and concatenated codes. Recent insights into the syndrome extraction process, which render the whole process more efficient and more noise-tolerant, are incorporated. The average number of recoveries which can be completed without failure is thus estimated as a function of various parameters. The main parameters are the gate (gamma) and memory (epsilon) failure rates, the physical scale-up of the computer size, and the time t_m required for measurements and classical processing. The achievable computation size is given as a surface in parameter space. This indicates the noise threshold as well as other information. It is found that concatenated codes based on the [[23,1,7]] Golay code give higher thresholds than those based on the [[7,1,3]] Hamming code under most conditions. The threshold gate noise gamma_0 is a function of epsilon/gamma and t_m; example values are {epsilon/gamma, t_m, gamma_0} = {1, 1, 0.001}, {0.01, 1, 0.003}, {1, 100, 0.0001}, {0.01, 100, 0.002}, assuming zero cost for information transport. This represents an order of magnitude increase in tolerated memory noise, compared with previous calculations, which is made possible by recent insights into the fault-tolerant QEC process.Comment: 21 pages, 12 figures, minor mistakes corrected and layout improved, ref added; v4: clarification of assumption re logic gate

CamChoice: A Corpus of Multiple Choice Questions and Candidate Response Distributions

Multiple Choice examinations are a ubiquitous form of assessment that is used to measure the ability of candidates across various domains and tasks. Maintaining the quality of proposed questions is of great importance to test designers, and therefore newly proposed questions go through several pre-test evaluation stages before they can be deployed into real-world exams. This process is currently quite manual, which can lead to time lags in the question development cycle. Automating this process would lead to a large improvement in efficiency, however, current datasets do not contain sufficient pre-test analysis information. In this paper, we introduce CamChoice; a multiple-choice comprehension dataset with questions at different target levels, where questions have the true candidate selected options distributions. We introduce the task of candidate distribution matching, propose several evaluation metrics for the task, and demonstrate that automatic systems trained on RACE++ can be leveraged as baselines for our task. We further demonstrate that these automatic systems can be used for practical pre-test evaluation tasks such as detecting underperforming distractors, where our detection systems can automatically identify poor distractors that few candidates select. We release the data publicly for future research.Comment: 9 pages, 6 figures, 7 table

Exact Performance of Concatenated Quantum Codes

When a logical qubit is protected using a quantum error-correcting code, the net effect of coding, decoherence (a physical channel acting on qubits in the codeword) and recovery can be represented exactly by an effective channel acting directly on the logical qubit. In this paper we describe a procedure for deriving the map between physical and effective channels that results from a given coding and recovery procedure. We show that the map for a concatenation of codes is given by the composition of the maps for the constituent codes. This perspective leads to an efficient means for calculating the exact performance of quantum codes with arbitrary levels of concatenation. We present explicit results for single-bit Pauli channels. For certain codes under the symmetric depolarizing channel, we use the coding maps to compute exact threshold error probabilities for achievability of perfect fidelity in the infinite concatenation limit.Comment: An expanded presentation of the analytic methods and results from quant-ph/0111003; 13 pages, 6 figure

Topological quantum memory

We analyze surface codes, the topological quantum error-correcting codes introduced by Kitaev. In these codes, qubits are arranged in a two-dimensional array on a surface of nontrivial topology, and encoded quantum operations are associated with nontrivial homology cycles of the surface. We formulate protocols for error recovery, and study the efficacy of these protocols. An order-disorder phase transition occurs in this system at a nonzero critical value of the error rate; if the error rate is below the critical value (the accuracy threshold), encoded information can be protected arbitrarily well in the limit of a large code block. This phase transition can be accurately modeled by a three-dimensional Z_2 lattice gauge theory with quenched disorder. We estimate the accuracy threshold, assuming that all quantum gates are local, that qubits can be measured rapidly, and that polynomial-size classical computations can be executed instantaneously. We also devise a robust recovery procedure that does not require measurement or fast classical processing; however for this procedure the quantum gates are local only if the qubits are arranged in four or more spatial dimensions. We discuss procedures for encoding, measurement, and performing fault-tolerant universal quantum computation with surface codes, and argue that these codes provide a promising framework for quantum computing architectures.Comment: 39 pages, 21 figures, REVTe

Identifying an Experimental Two-State Hamiltonian to Arbitrary Accuracy

Precision control of a quantum system requires accurate determination of the effective system Hamiltonian. We develop a method for estimating the Hamiltonian parameters for some unknown two-state system and providing uncertainty bounds on these parameters. This method requires only one measurement basis and the ability to initialise the system in some arbitrary state which is not an eigenstate of the Hamiltonian in question. The scaling of the uncertainty is studied for large numbers of measurements and found to be proportional to one on the square-root of the number of measurements.Comment: Minor corrections, Accepted for publication in Physical Review

Biodegradation of the Alkaline Cellulose Degradation Products Generated during Radioactive Waste Disposal.

The anoxic, alkaline hydrolysis of cellulosic materials generates a range of cellulose degradation products (CDP) including α and β forms of isosaccharinic acid (ISA) and is expected to occur in radioactive waste disposal sites receiving intermediate level radioactive wastes. The generation of ISA's is of particular relevance to the disposal of these wastes since they are able to form complexes with radioelements such as Pu enhancing their migration. This study demonstrates that microbial communities present in near-surface anoxic sediments are able to degrade CDP including both forms of ISA via iron reduction, sulphate reduction and methanogenesis, without any prior exposure to these substrates. No significant difference (n = 6, p = 0.118) in α and β ISA degradation rates were seen under either iron reducing, sulphate reducing or methanogenic conditions, giving an overall mean degradation rate of 4.7×10−2 hr−1 (SE±2.9×10−3). These results suggest that a radioactive waste disposal site is likely to be colonised by organisms able to degrade CDP and associated ISA's during the construction and operational phase of the facility

Continuous quantum error correction via quantum feedback control

We describe a protocol for continuously protecting unknown quantum states from decoherence that incorporates design principles from both quantum error correction and quantum feedback control. Our protocol uses continuous measurements and Hamiltonian operations, which are weaker control tools than are typically assumed for quantum error correction. We develop a cost function appropriate for unknown quantum states and use it to optimize our state-estimate feedback. Using Monte Carlo simulations, we study our protocol for the three-qubit bit-flip code in detail and demonstrate that it can improve the fidelity of quantum states beyond what is achievable using quantum error correction when the time between quantum error correction cycles is limited.Comment: 12 pages, 6 figures, REVTeX; references fixe

High-fidelity indirect readout of trapped-ion hyperfine qubits

We propose and demonstrate a protocol for high-fidelity indirect readout of trapped ion hyperfine qubits, where the state of a $^9\text{Be}^+$ qubit ion is mapped to a $^{25}\text{Mg}^+$ readout ion using laser-driven Raman transitions. By partitioning the $^9\text{Be}^+$ ground state hyperfine manifold into two subspaces representing the two qubit states and choosing appropriate laser parameters, the protocol can be made robust to spontaneous photon scattering errors on the Raman transitions, enabling repetition for increased readout fidelity. We demonstrate combined readout and back-action errors for the two subspaces of $1.2^{+1.1}_{-0.6} \times 10^{-4}$ and $0^{+1.9}_{-0} \times 10^{-5}$ with 68% confidence while avoiding decoherence of spectator qubits due to stray resonant light that is inherent to direct fluorescence detection.Comment: 7 + 6 pages, 3 + 1 figure