Threshold theorem

In quantum computing, the threshold theorem (or quantum fault-tolerance theorem) states that a quantum computer with a physical error rate below a certain threshold can, through application of quantum error correction schemes, suppress the logical error rate to arbitrarily low levels. This shows that quantum computers can be made fault-tolerant, as an analogue to von Neumann's threshold theorem for classical computation. This result was proven (for various error models) by the groups of Dorit Aharanov and Michael Ben-Or; Emanuel Knill, Raymond Laflamme, and Wojciech Zurek; and Alexei Kitaev independently. These results built off a paper of Peter Shor, which proved a weaker version of the threshold theorem. The key question that the threshold theorem resolves is whether quantum computers in practice could perform long computations without succumbing to noise. Since a quantum computer will not be able to perform gate operations perfectly, some small constant error is inevitable; hypothetically, this could mean that quantum computers with imperfect gates can only apply a constant number of gates before the computation is destroyed by noise. Surprisingly, the quantum threshold theorem shows that if the error to perform each gate is a small enough constant, one can perform arbitrarily long quantum computations to arbitrarily good precision, with only some small added overhead in the number of gates. The formal statement of the threshold theorem depends on the types of error correction codes and error model being considered. Quantum Computation and Quantum Information, by Michael Nielsen and Isaac Chuang, gives the general framework for such a theorem: Threshold theorem for quantum computation: A quantum circuit on n qubits and containing p(n) gates may be simulated with probability of error at most ε using gates (for some constant c) on hardware whose components fail with probability at most p, provided p is below some constant threshold, , and given reasonable assumptions about the noise in the underlying hardware.