Charge-conserving unitaries typically generate optimal covariant quantum error-correcting codes
arXiv:2102.11835
Abstract
Quantum error correction and symmetries play central roles in quantum information science and physics. It is known that quantum error-correcting codes covariant with respect to continuous symmetries cannot correct erasure errors perfectly (an important case being the Eastin-Knill theorem), in contrast to the case without symmetry constraints. Furthermore, there are fundamental limits on the accuracy of such covariant codes for approximate quantum error correction. Here, we consider the quantum error correction capability of random covariant codes. In particular, we show that -covariant codes generated by Haar random -symmetric unitaries, i.e. unitaries that commute with the charge operator (or conserve the charge), typically saturate the fundamental limits to leading order in terms of both the average- and worst-case purified distances against erasure noise. We note that the results hold for symmetric variants of unitary 2-designs, and comment on the convergence problem of charge-conserving random circuits. Our results not only indicate (potentially efficient) randomized constructions of optimal -covariant codes, but also reveal fundamental properties of random charge-conserving unitaries, which may underlie important models of complex quantum systems in wide-ranging physical scenarios where conservation laws are present, such as black holes and many-body spin systems.
References in corpus (33)
- Fault-tolerant quantum computation by anyons
- Black holes as mirrors: quantum information in random subsystems
- The String Landscape, Black Holes and Gravity as the Weakest Force
- Bulk Locality and Quantum Error Correction in AdS/CFT
- Operator Spreading in Random Unitary Circuits
- Chaos in quantum channels
- The operational meaning of min- and max-entropy
- Distance measures to compare real and ideal quantum processes
- Measurement-driven entanglement transition in hybrid quantum circuits
- Operator spreading and the emergence of dissipative hydrodynamics under unitary evolution with conservation laws
- Unitary-projective entanglement dynamics
- Chaos and complexity by design
- Restrictions on Transversal Encoded Quantum Gate Sets
- Quantum Error Correction in Scrambling Dynamics and Measurement-Induced Phase Transition
- Random Quantum Circuits are Approximate 2-designs
- Quantum Error Correction
- Duality Between Smooth Min- and Max-Entropies
- Constraints on symmetry from holography
- One-shot decoupling
- General conditions for approximate quantum error correction and near-optimal recovery channels
- Optimizing quantum process tomography with unitary 2-designs
- Decoupling with random quantum circuits
- Quantum Information Meets Quantum Matter -- From Quantum Entanglement to Topological Phase in Many-Body Systems
- Quantum Error Correcting Codes in Eigenstates of Translation-Invariant Spin Chains
- The Clifford group fails gracefully to be a unitary 4-design
- Entanglement, quantum randomness, and complexity beyond scrambling
- Using Quantum Metrological Bounds in Quantum Error Correction: A Simple Proof of the Approximate Eastin-Knill Theorem
- Continuous groups of transversal gates for quantum error correcting codes from finite clock reference frames
- No-go theorems for quantum resource purification II: new approach and channel theory
- Generalized Entanglement Entropies of Quantum Designs
- Soft mode and interior operator in Hayden-Preskill thought experiment
- Scrambling and decoding the charged quantum information
- Near-linear constructions of exact unitary 2-designs