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existence of a threshold for quantum error correction was shown in Aharonov and. The threshold theorem states that it is possible to create a quantum computer to perform an arbitrary quantum computation provided the error rate per physical gate or time step is below some constant threshold value. Fault-tolerance thresholds for the surface code with fabrication errors. Fundamental thresholds of realistic quantum error correction circuits from classical spin models DavideVodola ,1,2,ManuelRispler 3,SeyongKim4,andMarkusMüller5,6. If you have the right tools available, the simplest way to estimate the threshold is to just simulate running the circuit under various levels of noise and see how often decoding. In general you can't compute it analytically, you can only estimate it numerically. To build a quantum computer which behaves correctly in the presence of errors, we also need a theory of fault-tolerant quantum computation, instructing us how to perform quantum gates on qubits which are encoded in a quantum error-correcting code. The threshold is relative to the noise model and the decoder that you are using. The stabilizer formalism for quantum codes also illustrates the relationships to classical coding theory, particularly classical codes over GF(4), the finite field with four elements. 1) Consistency checks Doesn’t get you very far (quantum or classical) 2) Checkpointing Doesn’t work in quantum case 3) Error-correcting codes These work 4) Massive redundancy Doesn’t seem to get you very far in quantum case. The stabilizer is a finite Abelian group, and allows a straightforward characterization of the error-correcting properties of the code. Many quantum codes can be described in terms of the stabilizer of the codewords. We demonstrate ultrahigh thresholds for the tailored surface code: 39 with a realistic bias of 100, and 50 with pure Z noise, far exceeding known thresholds for the standard surface code: 11 with. The theory of quantum error-correcting codes has some close ties to and some striking differences from the theory of classical error-correcting codes. We find that the surface code is highly resilient to Y-biased noise, and tailor it to Z-biased noise, whilst retaining its practical features. Quantum states are very delicate, so it is likely some sort of quantum error correction will be necessary to build reliable quantum computers.