A general error correction method is presented which is capable of correcting coherent errors originating from static residual inter-qubit couplings in a quantum computer. It is based on a randomization of static imperfections in a many-qubit system by the repeated application of Pauli operators which change the computational basis. This Pauli-Random-Error-Correction (PAREC)-method eliminates coherent errors produced by static imperfections and increases significantly the maximum time over which realistic quantum computations can be performed reliably. Furthermore, it does not require redundancy so that all physical qubits involved can be used for logical purposes.
A general error correction method is presented which is capable of correcting coherent errors originating from static residual inter-qubit couplings in a quantum computer. It is based on a randomization of static imperfections in a many-qubit system by the repeated application of Pauli operators which change the computational basis. This Pauli-Random-Error-Correction (PAREC)-method eliminates coherent errors produced by static imperfections and increases significantly the maximum time over which realistic quantum computations can be performed reliably. Furthermore, it does not require redundancy so that all physical qubits involved can be used for logical purposes.
The performance of a given quantum error correction (QEC) code depends upon the noise model that is assumed. Independent Pauli noise, applied after each quantum operation, is a simplistic noise model that is easy to simulate and understand in the context of stabilizer codes. Although such a noise model is artificial, it is equivalent to independent, random, unbiased qubit rotations. What about spatially or temporally correlated qubit rotations? Such a noise model is applicable to global operations (e.g., NMR or ESR), common control sources (e.g., lasers), or slow drift (e.g., charge or magnetic noise) in various qubit technologies. In the worst case, such errors can combine constructively and result in a post-correction failure rate that increases with the number of error correction cycles. However, we show that this worst case does not generally arise unless taking active corrective actions while performing QEC. That is, by employing virtual Pauli frame updates ("passive" error correction) rather than physical corrections ("active" error correction), coherent errors do not compound appreciably. Starting in a random Pauli frame is also advantageous. In fact, through perturbation theory arguments and supporting numerical simulations, we show that the logical qubit performance beyond distance 3 for correlated single-qubit Hamiltonian noise models (i.e., global errant qubit rotations), when employing these "lazy" strategies, essentially matches the performance of Pauli noise model with the same process fidelity (fidelity after one application). In a more general circuit model of noise, correlations may add constructively within syndrome extraction rounds but Pauli frame randomization from passive error correction mitigates this effect across multiple rounds.
An introduction to quantum error correction (QEC) is given, and some recent developments are described. QEC consists of two parts: the physics of error processes and their reversal, and the construction of quantum error–correcting codes. Errors are caused both by imperfect quantum operations, and by coupling between the quantum system and its environment. Any such process can be analysed into a sum of ‘error operators’, which are tensor products of Pauli spin operators. These are the analogues of classical error vectors. A quantum error correcting code is a set of orthogonal states, ‘quantum codewords’, which behave in a certain useful way under the action of the most likely error operators. A computer or channel which only uses such states can be corrected by measurements which determine the error while yielding no information about which codeword or superposition of codewords is involved. Powerful codes can be found using a construction based on classical error–correcting codes. An analysis which allows even the corrective operations themselves to be imperfect leads to powerful and counter–intuitive results: the quantum coherence of a long quantum computation can be preserved even though every qubit in the computer relaxes spontaneously many times before the computation is complete.
Studies of quantum error correction (QEC) typically focus on stochastic Pauli errors because the existence of a threshold error rate below which stochastic Pauli errors can be corrected implies that there exists a threshold below which generic errors can be corrected. However, rigorous estimates of the threshold for generic errors are typically orders of magnitude worse than the threshold for stochastic Pauli errors. Specifically, coherent errors have a particularly harmful impact on the encoded space because they can map encoded states to superpositions of logical and error states. Further, coherent errors can add up and interfere over multiple rounds of error correction or between syndrome measurements, which may result in significantly worse errors than expected under a stochastic Pauli error model. In this paper, we present an algorithm which decoheres noise at the logical level, projecting the state of the system onto a logical state with a well-defined error. The algorithm does not significantly increase the depth of the logical circuit (and usually does not lead to any increase in depth), and applies generally to most fault-tolerant gadgets and error correction steps.
We study the effects of static interqubit interactions on the accuracy of various quantum algorithms. Extensive numerical simulations show that their effect is stronger compared to external decoherence. Analytical approach based on Random Matrix Theory is developed. It gives universal law for fidelity decay induced by interqubit static interactions. This determines the time scale for reliable quantum computation in presence of realistic static imperfections and external decoherence. New polynomial algorithms are developed for simulation of complex dynamics in the regime of classical and quantum chaos, and Anderson metal- insulator transition. A generic quantum error correction method is developed; it allows to eliminate coherent effect of static errors. The theoretical results are confirmed by numerical computations with up to 28 qubits.
Errors in quantum computers are of two kinds: sudden perturbations to isolated qubits, and slow random drifts of all the qubits. The latter may be reduced, but not eliminated, by means of symmetrization, namely by using many replicas of the computer, and forcing their joint quantum state to be completely symmetric. On the other hand, isolated errors can be corrected by quantum codewords that represent a logical qubit in a redundant way, by several physical qubits. If one of the physical qubits is perturbed, for example if it gets entangled with an unknown environment, there still is enough information encoded in the other physical qubits to restore the logical qubit, and disentangle it from the environment. The recovery procedure may consist of unitary operations, without the need of actually identifying the error.
Quantum computing is entering a transitional regime between noisy intermediate-scale quantum (NISQ) processing and early fault-tolerant quantum computation (FTQC), in which increasingly capable hardware is beginning to support repeated syndrome measurements, partial error correction, and logical-qubit operations, while residual physical and logical errors remain non-negligible. In this regime, error suppression, error mitigation, and quantum error correction are increasingly better viewed as complementary layers of a unified error-reduction strategy rather than as separate approaches, with each acting at a different stage of the quantum computation to improve simulation reliability. Thus, in this review, we provide a practical and forward-looking overview of the principal hardware error sources and the corresponding error suppression and mitigation methods for reducing their impact across the current NISQ-FTQC transition. We discuss hardware-aware circuit design, coherent-error suppression, readout mitigation, noise extrapolation, classical inference, and software-supported workflows, with particular emphasis on their implementation on actual quantum processors. We further examine how error mitigation techniques can be adapted to encoded and logical-qubit settings so that they can operate alongside quantum error correction to suppress residual logical errors and improve the accuracy of computation in the early fault-tolerant regime.