![]() We find that while conventional on-site symmetries are not sufficient to allow for self-correction in commuting Hamiltonian models of this form, a generalized type of symmetry known as a 1-form symmetry is enough to guarantee self-correction. In particular, we investigate codes given by 2D symmetry-enriched topological (SET) phases that appear naturally on the boundary of 3D symmetry-protected topological (SPT) phases. We demonstrate that symmetry can lead to self-correction in 3D spin-lattice models. Finally, we also establish novel links between unitarity, complementary channels, and purity that are of independent interest.Ī self-correcting quantum memory can store and protect quantum information for a time that increases without bound with the system size and without the need for active error correction. In the latter case, we provide fundamental limits on how much a spin-jA system can be used to polarize a larger spin-jB system, and on how much one can invert spin polarization using a rotationally symmetric operation. In particular, we investigate in detail the U(1) and SU(2) symmetries related to energy and angular momentum conservation laws. It allows us to derive bounds on the deviation from conservation laws under any symmetric quantum channel in terms of the deviation from closed dynamics as measured by the unitarity of the channel E. In this analysis, the convex structure and extremal points of the set of quantum channels symmetric under the action of a Lie group G becomes essential. To what extent does Noether’s principle apply to quantum channels? Here, we quantify the degree to which imposing a symmetry constraint on quantum channels implies a conservation law and show that this relates to physically impossible transformations in quantum theory, such as time reversal and spin inversion. Finally, we discuss potential applications of our theory to several important topics in physics. For concrete examples, we showcase two explicit types of approximately covariant codes that nearly saturate certain bounds, respectively obtained from quantum Reed-Muller codes and thermodynamic codes. From a quantum computation perspective, our results indicate general limits on the precision and density of transversal logical gates. We first define meaningful measures of approximate symmetries based on the degree of covariance and charge conservation violation, which induce corresponding notions of approximately covariant codes, and then derive a series of trade-off bounds between these different approximate symmetry measures and QEC accuracy by leveraging insights and techniques from approximate QEC, quantum metrology, and resource theory. Here we systematically study the competition between continuous symmetries and QEC in a quantitative manner. Qubit codes to, for instance, oscillators and rotors.It is known that continuous symmetries induce fundamental restrictions on the accuracy of quantum error correction (QEC). We systematically construct codesĬovariant with respect to general groups, obtaining natural generalizations of ![]() Unitary group, achieving good accuracy for large $d$ (using random codes) or We construct codes covariant with respect to the full logical Physical qubits per subsystem that is inversely proportional to the error Theorem: If a code admits a universal set of transversal gates and correctsĮrasure with fixed accuracy, then, for each logical qubit, we need a number of Representation theory, we prove an approximate version of the Eastin-Knill Of infidelity with $n$ or $d$ as the lower bound. We exhibit codes achieving approximately the same scaling This boundĪpproaches zero when the number of subsystems $n$ or the dimension $d$ of each $G$-covariant code with $G$ a continuous group, we derive a lower bound on theĮrror correction infidelity following erasure of a subsystem. Realized by performing transformations on the individual subsystems. Symmetry group $G$ if a $G$ transformation on the logical system can be $n$ physical subsystems, we say that the code is covariant with respect to a If a logical quantum system is encoded into Including many-body systems, metrology in the presence of noise, fault-tolerantĬomputation, and holographic quantum gravity. Albert, Grant Salton, Fernando Pastawski, Patrick Hayden, John Preskill Download PDF Abstract: Quantum error correction and symmetry arise in many areas of physics, Authors: Philippe Faist, Sepehr Nezami, Victor V. ![]()
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