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We systematically study the fundamental competition between quantum errorcorrection (QEC) and continuous symmetries, two key notions in quantuminformation and physics, in a quantitative manner. Three meaningful measures ofapproximate symmetries in quantum channels and in particular QEC codes,respectively based on the violation of covariance conditions over the entiresymmetry group or at a local point, and the violation of charge conservation,are introduced and studied. Each measure induces a correspondingcharacterization of approximately covariant codes. We explicate a host ofdifferent ideas and techniques that enable us to derive various forms oftrade-off relations between the QEC inaccuracy and all symmetry violationmeasures. More specifically, we introduce two frameworks for understanding andestablishing the trade-offs respectively based on the notions of chargefluctuation and gate implementation error, and employ methods including theKnill--Laflamme conditions as well as quantum metrology and quantum resourcetheory for the derivation. From the perspective of fault-tolerant quantumcomputing, our bounds on symmetry violation indicate limitations on theprecision or density of transversally implementable logical gates for generalQEC codes, refining the Eastin--Knill theorem. To exemplify nontrivialapproximately covariant codes and understand the achievability of the abovefundamental limits, we analyze the behaviors of two explicit types of codes: aparametrized extension of the thermodynamic code (which gives a construction ofa code family that continuously interpolates between exact QEC and exactsymmetry), and the quantum Reed--Muller codes. We show that both codes cansaturate the scaling of the bounds for group-global covariance and chargeconservation asymptotically, indicating the near-optimality of these bounds andcodes.

Quantum error correction meets continuous symmetries: fundamental trade-offs and case studies