We describe two implementations of the optimal error correction algorithm known as the maximum likelihood decoder (MLD) for the 2D surface code with a noiseless syndrome extraction. First, we show how to implement MLD exactly in time O(n 2 ), where n is the number of code qubits. Our implementation uses a reduction from MLD to simulation of matchgate quantum circuits. This reduction however requires a special noise model with independent bit-flip and phase-flip errors. Secondly, we show how to implement MLD approximately for more general noise models using matrix product states (MPS). Our implementation has running time O(nχ 3 ) where χ is a parameter that controls the approximation precision. The key step of our algorithm, borrowed from the DMRG method, is a subroutine for contracting a tensor network on the two-dimensional grid. The subroutine uses MPS with a bond dimension χ to approximate the sequence of tensors arising in the course of contraction. We benchmark the MPS-based decoder against the standard minimum weight matching decoder observing a significant reduction of the logical error probability for χ ≥ 4.

The Gottesman-Knill theorem allows for the efficient simulation of stabilizer-based quantum error-correction circuits. Errors in these circuits are commonly modeled as depolarizing channels by using Monte Carlo methods to insert Pauli gates randomly throughout the circuit. Although convenient, these channels are poor approximations of common, realistic channels like amplitude damping. Here we analyze a larger set of efficiently simulable error channels by allowing the random insertion of any one-qubit gate or measurement that can be efficiently simulated within the stabilizer formalism. Our new error channels are shown to be a viable method for accurately approximating real error channels.

Shor's quantum factoring algorithm exponentially outperforms known classical methods. Previous experimental implementations have used simplifications dependent on knowing the factors in advance. However, as we show here, all composite numbers admit simplification of the algorithm to a circuit equivalent to flipping coins. The difficulty of a particular experiment therefore depends on the level of simplification chosen, not the size of the number factored. Valid implementations should not make use of the answer sought.

We consider the problem of calculating the logical error probability for a stabilizer quantum code subject to random Pauli errors. To access the regime of large code distances where logical errors are extremely unlikely we adopt the splitting method widely used in Monte Carlo simulations of rare events and Bennett's acceptance ratio method for estimating the free energy difference between two canonical ensembles. To illustrate the power of these methods in the context of error correction, we calculate the logical error probability $P_L$ for the 2D surface code on a square lattice with a pair of holes for all code distances $d\le 20$ and all error rates $p$ below the fault-tolerance threshold. Our numerical results confirm the expected exponential decay $P_L\sim \exp{[-\alpha(p)d]}$ and provide a simple fitting formula for the decay rate $\alpha(p)$. Both noiseless and noisy syndrome readout circuits are considered.Comment: 16 pages, 11 figures. Version 3: added a new referenc