Accurate methods of assessing the performance of quantum gates are extremely important. Quantum process tomography and randomized benchmarking are the current favored methods. Quantum process tomography gives detailed information, but significant app
roximations must be made to reduce this information to a form quantum error correction simulations can use. Randomized benchmarking typically outputs just a single number, the fidelity, giving no information on the structure of errors during the gate. Neither method is optimized to assess gate performance within an error detection circuit, where gates will be actually used in a large-scale quantum computer. Specifically, the important issues of error composition and error propagation lie outside the scope of both methods. We present a fast, simple, and scalable method of obtaining exactly the information required to perform effective quantum error correction from the output of continuously running error detection circuits, enabling accurate prediction of large-scale behavior.
Topological quantum error correction codes are known to be able to tolerate arbitrary local errors given sufficient qubits. This includes correlated errors involving many local qubits. In this work, we quantify this level of tolerance, numerically st
udying the effects of many-qubit errors on the performance of the surface code. We find that if increasingly large area errors are at least moderately exponentially suppressed, arbitrarily reliable quantum computation can still be achieved with practical overhead. We furthermore quantify the effect of non-local two-qubit correlated errors, which would be expected in arrays of qubits coupled by a polynomially decaying interaction, and when using many-qubit coupling devices. We surprisingly find that the surface code is very robust to this class of errors, despite a provable lack of a threshold error rate when such errors are present.
The surface code is designed to suppress errors in quantum computing hardware and currently offers the most believable pathway to large-scale quantum computation. The surface code requires a 2-D array of nearest-neighbor coupled qubits that are capab
le of implementing a universal set of gates with error rates below approximately 1%, requirements compatible with experimental reality. Consequently, a number of authors are attempting to squeeze additional performance out of the surface code. We describe an optimal complexity error suppression algorithm, parallelizable to O(1) given constant computing resources per unit area, and provide evidence that this algorithm exploits correlations in the error models of each gate in an asymptotically optimal manner.
Many physical systems considered promising qubit candidates are not, in fact, two-level systems. Such systems can leak out of the preferred computational states, leading to errors on any qubits that interact with leaked qubits. Without specific metho
ds of dealing with leakage, long-lived leakage can lead to time-correlated errors. We study the impact of such time-correlated errors on topological quantum error correction codes, which are considered highly practical codes, using the repetition code as a representative case study. We show that, under physically reasonable assumptions, a threshold error rate still exists, however performance is significantly degraded. We then describe simple additional quantum circuitry that, when included in the error detection cycle, restores performance to acceptable levels.
Consider a 2-D square array of qubits of extent $Ltimes L$. We provide a proof that the minimum weight perfect matching problem associated with running a particular class of topological quantum error correction codes on this array can be exactly solv
ed with a 2-D square array of classical computing devices, each of which is nominally associated with a fixed number $N$ of qubits, in constant average time per round of error detection independent of $L$ provided physical error rates are below fixed nonzero values, and other physically reasonable assumptions. This proof is applicable to the fully fault-tolerant case only, not the case of perfect stabilizer measurements.
The surface code is highly practical, enabling arbitrarily reliable quantum computation given a 2-D nearest-neighbor coupled array of qubits with gate error rates below approximately 1%. We describe an open source library, Polyestimate, enabling a us
er with no knowledge of the surface code to specify realistic physical quantum gate error models and obtain logical error rate estimates. Functions allowing the user to specify simple depolarizing error rates for each gate have also been included. Every effort has been made to make this library user-friendly.
State distillation is the process of taking a number of imperfect copies of a particular quantum state and producing fewer better copies. Until recently, the lowest overhead method of distilling states |A>=(|0>+e^{ipi/4}|1>)/sqrt{2} produced a single
improved |A> state given 15 input copies. New block code state distillation methods can produce k improved |A> states given 3k+8 input copies, potentially significantly reducing the overhead associated with state distillation. We construct an explicit surface code implementation of block code state distillation and quantitatively compare the overhead of this approach to the old. We find that, using the best available techniques, for parameters of practical interest, block code state distillation does not always lead to lower overhead, and, when it does, the overhead reduction is typically less than a factor of three.
We present a comprehensive and self-contained simplified review of the quantum computing scheme of Phys. Rev. Lett. 98, 190504 (2007), which features a 2-D nearest neighbor coupled lattice of qubits, a threshold error rate approaching 1%, natural asy
mmetric and adjustable strength error correction and low overhead arbitrarily long-range logical gates. These features make it by far the best and most practical quantum computing scheme devised to date. We restrict the discussion to direct manipulation of the surface code using the stabilizer formalism, both of which we also briefly review, to make the scheme accessible to a broad audience.
Given any quantum error correcting code permitting universal fault-tolerant quantum computation and transversal measurement of logical X and Z, we describe how to perform time-optimal quantum computation, meaning the execution of an arbitrary Cliffor
d circuit followed by a layer of independent T gates and any necessary feedforward measurement determined corrective S gates all in the time of a single physical measurement. We assume fast classical processing and classical communication, and argue the reasonableness of this assumption. This enables fault-tolerant quantum computation to be performed orders of magnitude faster than previously thought possible, with the execution time independent of the error correction strength.
Topologically quantum error corrected logical gates are complex. Chains of errors can form in space and time and diagonally in spacetime. It is highly nontrivial to determine whether a given logical gate is free of low weight combinations of errors l
eading to failure. We report a new tool Nestcheck capable of analyzing an arbitrary topological computation and determining the minimum number of errors required to cause failure.
