Scaling and renormalization in fault-tolerant quantum computers

This work is concerned with phrasing the concepts of fault-tolerant quantum computation within the framework of disordered systems, Bernoulli site percolation in particular. We show how the so-called "threshold theorems" on the possibility of fault-tolerant quantum computation with constant error rate can be cast as a renormalization (coarse-graining) of the site percolation process describing the occurrence of errors during computation. We also use percolation techniques to derive a trade-off between the complexity overhead of the fault-tolerant circuit and the threshold error rate.Comment: 4 pages, 2 eps figures; revtex4; based on talk given at the Simons Conference on Quantum and Reversible Computation, Stony Brook NY, May 28-31; minor typographical change

Quantum information and statistical mechanics: an introduction to frontier

This is a short review on an interdisciplinary field of quantum information science and statistical mechanics. We first give a pedagogical introduction to the stabilizer formalism, which is an efficient way to describe an important class of quantum states, the so-called stabilizer states, and quantum operations on them. Furthermore, graph states, which are a class of stabilizer states associated with graphs, and their applications for measurement-based quantum computation are also mentioned. Based on the stabilizer formalism, we review two interdisciplinary topics. One is the relation between quantum error correction codes and spin glass models, which allows us to analyze the performances of quantum error correction codes by using the knowledge about phases in statistical models. The other is the relation between the stabilizer formalism and partition functions of classical spin models, which provides new quantum and classical algorithms to evaluate partition functions of classical spin models.Comment: 15pages, 4 figures, to appear in Proceedings of 4th YSM-SPIP (Sendai, 14-16 December 2012

Fidelity of a t-error correcting quantum code with more than t errors

It is important to study the behavior of a t-error correcting quantum code when the number of errors is greater than t, because it is likely that there are also small errors besides t large correctable errors. We give a lower bound for the fidelity of a t-error correcting stabilizer code over a general memoryless channel, allowing more than t errors. We also show that the fidelity can be made arbitrary close to 1 by increasing the code length.Comment: 9 pages, ReVTeX4 beta 5. To be published in Phys. Rev. A. The lower bound is made tighter in version 4 and 5. All approximations in the first version are removed, and a lower bound for the average of the fidelity is given in the second version. A critical error is correcte

The Early Days of Quantum Computation

I recount some of my memories of the early development of quantum computation, including the discovery of the factoring algorithm, of error correcting codes, and of fault tolerance.Comment: 10 pages, Write-up of a talk given at QC40, the 40th anniversary of the 1981 Physics of Computation Conference at Endicott House, and at the 2022 Solvay Conference on Physic