Quantum error correction/detection (QEC/QED) and dynamical decoupling (DD) are tools for protecting quantum information. A natural goal is to combine them to outperform either approach alone. Such a benefit is not automatic: physical DD can conflict with encoded subspace, and QEC performance is governed by the errors that survive decoding, not just those DD suppresses. We analyze a hybrid memory cycle where DD is implemented logically (LDD) using normalizer elements of an
[[n,k,d]] stabilizer code, followed by a round of syndrome measurement and recovery (or, in the detection setting, postselection on a trivial syndrome). In an effective Pauli model with physical error probability
p, LDD suppression factor
pDD, and recovery imperfection rate
pQEC (or
pQED), we derive closed-form entanglement-fidelity expressions for QEC-only, QED-only, LDD-only, physical DD, and the hybrid LDD+QEC protocol. The formulas are expressed via a small set of code-dependent weight enumerator polynomials, making the role of the decoder and the LDD group explicit. For ideal recovery LDD+QEC outperforms QEC-only iff the conditional fraction of uncorrectable Pauli errors is larger in the LDD-suppressed sector than in the unsuppressed sector. In the low-noise regime, a sufficient design rule guaranteeing hybrid advantage is that LDD suppresses at least one minimum-weight uncorrectable Pauli error for the chosen recovery map. We show how stabilizer-equivalent choices of LDD generators can enforce this condition. We supplement our analysis with numerical results for the
[[7,1,3]] Steane code and a
[[13,1,3]] code, mapping regions of hybrid-protocol advantage in parameter space beyond the small-
p regime. Our work illustrates the need for co-design of the code, decoder, and logical decoupling group, and clarifies the conditions under which the hybrid LDD+QEC protocol is advantageous.
Proposals for quantum computing devices are many and varied. They each have unique noise processes that make none of them fully reliable at this time. There are several error correction/avoidance techniques which are valuable for reducing or eliminating errors, but not one, alone, will serve as a panacea. One must therefore take advantage of the strength of each of these techniques so that we may extend the coherence times of the quantum systems and create more reliable computing devices. To this end we give a general strategy for using dynamical decoupling operations on encoded subspaces. These encodings may be of any form; of particular importance are decoherence-free subspaces and quantum error correction codes. We then give means for empirically determining an appropriate set of dynamical decoupling operations for a given experiment. Using these techniques, we then propose a comprehensive encoding solution to many of the problems of quantum computing proposals which use exchange-type interactions. This uses a decoherence-free subspace and an efficient set of dynamical decoupling operations. It also addresses the problem of controllability in solid-state quantum dot devices.
Quantum systems can be used to measure various quantities in their environment with high precision. Often, however, their sensitivity is limited by the decohering effects of this same environment. Dynamical decoupling schemes are widely used to filter environmental noise from signals, but their performance is limited by the spectral properties of the signal and noise at hand. Quantum error correction schemes have therefore emerged as a complementary technique without the same limitations. To date, however, they have failed to correct the dominant noise type in many quantum sensors, which couples to each qubit in a sensor in the same way as the signal. Here we show how quantum error correction can correct for such noise, which dynamical decoupling can only partially address. Whereas dynamical decoupling exploits temporal noise correlations in signal and noise, our scheme exploits spatial correlations. We give explicit examples in small quantum devices and demonstrate a method by which error-correcting codes can be tailored to their noise.
Abstract Quantum systems can be used to measure various quantities in their environment with high precision. Often, however, their sensitivity is limited by the decohering effects of this same environment. Dynamical decoupling schemes are widely used to filter environmental noise from signals, but their performance is limited by the spectral properties of the signal and noise at hand. Quantum error correction schemes have therefore emerged as a complementary technique without the same limitations. To date, however, they have failed to correct the dominant noise type in many quantum sensors, which couples to each qubit in a sensor in the same way as the signal. Here we show how quantum error correction can correct for such noise, which dynamical decoupling can only partially address. Whereas dynamical decoupling exploits temporal noise correlations in signal and noise, our scheme exploits spatial correlations. We give explicit examples in small quantum devices and demonstrate a method by which error-correcting codes can be tailored to their noise.
Quantum control and fault-tolerant quantum computing (FTQC) are two of the cornerstones on which the hope of realizing a large-scale quantum computer is pinned, yet only preliminary steps have been taken towards formalizing the interplay between them. Here we explore this interplay using the powerful strategy of dynamical decoupling (DD), and show how it can be seamlessly and optimally integrated with FTQC. To this end we show how to find the optimal decoupling generator set (DGS) for various subspaces relevant to FTQC, and how to simultaneously decouple them. We focus on stabilizer codes, which represent the largest contribution to the size of the DGS, showing that the intuitive choice comprising the stabilizers and logical operators of the code is in fact optimal, i.e., minimizes a natural cost function associated with the length of DD sequences. Our work brings hybrid DD-FTQC schemes, and their potentially considerable advantages, closer to realization.