We study a comprehensive list of quantum codes as candidates of codes to be used at the bottom, physical, level in a fault-tolerant code architecture. Using the Aliferis-Gottesman-Preskill (AGP) ex-Rec method we calculate the pseudo-threshold for these codes against depolarizing noise at various levels of overhead. We estimate the logical noise rate as a function of overhead at a physical error rate of
p0=1×10−4. The Bacon-Shor codes and the Golay code are the best performers in our study.
Quantum computers have been shown to efficiently solve a class of problems for which no efficient solution is otherwise known. Physical systems can implement quantum computation, but devising realistic schemes is an extremely challenging problem largely due to the effect of noise. A quantum computer that is capable of correctly solving problems more rapidly than modern digital computers requires some use of so-called fault-tolerant components. Code-based fault-tolerance using quantum error-correcting codes is one of the most promising and versatile of the known routes for fault-tolerant quantum computation. This dissertation presents three main, new results about code-based fault-tolerant quantum computer architectures.
The first result is a large new family of quantum codes that go beyond stabilizer codes, the most well-studied family of quantum codes. Our new family of codeword stabilized codes contains all known codes with optimal parameters. Furthermore, we show how to systematically find, construct, and understand such codes as a pair of codes: an additive quantum code and a classical (nonlinear) code.
Second, we resolve an open question about universality of so-called transversal gates acting on stabilizer codes. Such gates are universal for classical fault-tolerant computation, but they were conjectured to be insufficient for universal fault-tolerant quantum computation. We show that transversal gates have a restricted form and prove that some important families of them cannot be quantum universal. This is strong evidence that so-called quantum software is necessary to achieve universality, and, therefore, fault-tolerant quantum computer architecture is fundamentally different from classical computer architecture.
Finally, we partition the fault-tolerant design problem into levels of a hierarchy of concatenated codes and present methods, compatible with rigorous threshold theorems, for numerically evaluating these codes. The methods are applied to measure inner error-correcting code performance, as a first step toward elucidation of an effective fault-tolerant quantum computer architecture that uses no more than a physical, inner, and outer level of coding. Of the inner codes, the Golay code gives the highest pseudothreshold of 2 × 10-3. A comparison of logical error rate and overhead shows that the Bacon-Shor codes are competitive with Knill's C 4/C6 scheme at a base error rate of 10 -4. (Copies available exclusively from MIT Libraries, Rm. 14-0551, Cambridge, MA 02139-4307. Ph. 617-253-5668; Fax 617-253-1690.)
We develop a scheme for fault-tolerant quantum computation based on asymmetric Bacon-Shor codes, which works effectively against highly biased noise dominated by dephasing. We find the optimal Bacon-Shor block size as a function of the noise strength and the noise bias, and estimate the logical error rate and overhead cost achieved by this optimal code. Our fault-tolerant gadgets, based on gate teleportation, are well suited for hardware platforms with geometrically local gates in two dimensions.
Quantum error correction is necessary to perform large-scale quantum computations in the presence of noise and decoherence. As a result, several aspects of quantum error correction have already been explored. These have been primarily studies of quantum memory[1, 2], an important first step towards quantum computation, where the objective is to increase the lifetime of the encoded quantum information. Additionally, several works have explored the implementation of logical gates[3-5]. In this work we study a next step - fault-tolerantly implementing quantum circuits. We choose the
[[4,1,2]] Bacon-Shor subsystem code, which has a particularly simple error-detection circuit. Through both numerics and site-counting arguments, we compute pseudo-thresholds for the Pauli error rate
p in a depolarizing noise model, below which the encoded circuits outperform the unencoded circuits. These pseudo-threshold values are shown to be as high as
p=3% for short circuits, and
p=0.6% for circuits of moderate depth. Additionally, we see that multiple rounds of stabilizer measurements give an improvement over performing a single round at the end. This provides a concrete suggestion for a small-scale fault-tolerant demonstration of a quantum algorithm that could be accessible with existing hardware.
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 Calderbank-Shor-Steane 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 that can be completed without failure is thus estimated as a function of various parameters. The main parameters are the gate $\ensuremath{\gamma}$ and memory $\ensuremath{\epsilon}$ failure rates, the physical scale-up of the computer size, and the time
tm 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 ${\ensuremath{\gamma}}_{0}$ is a function of $\ensuremath{\epsilon}/\ensuremath{\gamma}$ and
tm; example values are ${\ensuremath{\epsilon}/\ensuremath{\gamma}{,t}_{m},{\ensuremath{\gamma}}_{0}}={{1,1,10}^{\ensuremath{-}3}},$ ${0.01,1,3\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}3}},$ ${{1,100,10}^{\ensuremath{-}4}},$ ${0.01,100,2\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}3}},$ 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.