 
Summary: FAULTTOLERANT QUANTUM COMPUTATION WITH CONSTANT ERROR RATE
DORIT AHARONOV y AND MICHAEL BENOR z
Key Words: Quantum computation, Noise and Decoherence, Density matrices, Concatenated
quantum error correcting codes, Polynomial codes, Universal quantum gates
Abstract. Shor has showed how to perform fault tolerant quantum computation when the probability for an error in a
qubit or a gate, , decays with the size of the computation polylogarithmically, an assumption which is physically unreasonable.
This paper improves this result and shows that quantum computation can be made robust against errors and inaccuracies,
when the error rate, , is smaller than a constant threshold, c . The cost is polylogarithmic in space and time. The result
holds for a very general noise model, which includes probabilistic errors, decoherence, amplitude damping, depolarization, and
systematic inaccuracies in the gates. Moreover, we allow exponentially decaying correlations between the errors both in space
and in time. Fault tolerant computation can be performed with any universal set of gates. The result also holds for quantum
particles with p > 2 states, namely qupits, and is also generalized to one dimensional quantum computers with only nearest
neighbor interactions. No measurements, or classical operations, are required during the quantum computation.
We use CalderbankShor Steane (CSS) quantum error correcting codes, generalized to qupits, and introduce a new class
of CSS codes over Fp , called polynomial codes. It is shown how to apply a universal set of gates fault tolerantly on states
encoded by general CSS codes, based on modications of Shor's procedures, and on states encoded by polynomial codes, where
the procedures for polynomial codes have a simple and systematic structure based on the algebraic properties of the code.
Geometrical and group theoretical arguments are used to prove the universality of the two sets of gates. Our key theorem
asserts that applying computation on encoded states recursively achieves fault tolerance against constant error rate. The
generalization to general noise models is done using the framework of quantum circuits with density matrices. Finally, we
