Optimal Universal Quantum Error Correction via Bounded Reference Frames
arXiv:2007.09154 · doi:10.1103/PhysRevResearch.4.023107
Abstract
Error correcting codes with a universal set of transversal gates are a desideratum for quantum computing. Such codes, however, are ruled out by the Eastin-Knill theorem. Moreover, the theorem also rules out codes which are covariant with respect to the action of transversal unitary operations forming continuous symmetries. In this work, starting from an arbitrary code, we construct approximate codes which are covariant with respect to the entire group of local unitary gates in dimension , using quantum reference frames. We show that our codes are capable of efficiently correcting different types of erasure errors. When only a small fraction of the qudits upon which the code is built are erased, our covariant code has an error that scales as , which is reminiscent of the Heisenberg limit of quantum metrology. When every qudit has a chance of being erased, our covariant code has an error that scales as . We show that the error scaling is optimal in both cases. Our approach has implications for fault-tolerant quantum computing, reference frame error correction, and the AdS-CFT duality.
10 + 23. New results added including numerical analysis of performance for various different error models (including non-erasure errors such as dephasing)
References in corpus (73)
- Gaussian Quantum Information
- Universal Quantum Computation with ideal Clifford gates and noisy ancillas
- Error mitigation for short-depth quantum circuits
- Reference frames, superselection rules, and quantum information
- Bulk Locality and Quantum Error Correction in AdS/CFT
- Stabilizer Codes and Quantum Error Correction
- Exact and Approximate Unitary 2-Designs: Constructions and Applications
- Distance measures to compare real and ideal quantum processes
- Roads towards fault-tolerant universal quantum computation
- Practical Quantum Error Mitigation for Near-Future Applications
- Restrictions on Transversal Encoded Quantum Gate Sets
- Capacities of Quantum Erasure Channels
- Random Quantum Circuits are Approximate 2-designs
- Local random quantum circuits are approximate polynomial-designs
- Quantum mechanics and the covariance of physical laws in quantum reference frames
- Universal fault-tolerant quantum computation with only transversal gates and error correction
- A perspective on multiparameter quantum metrology: from theoretical tools to applications in quantum imaging
- Constraints on symmetry from holography
- Optimal Quantum Clocks
- The theory of manipulations of pure state asymmetry: basic tools and equivalence classes of states under symmetric operations
- Topological Computation without Braiding
- The Trinity of Relational Quantum Dynamics
- Efficient Quantum Circuits for Schur and Clebsch-Gordan Transforms
- Superselection rules and quantum protocols
- Approximation by Quantum Circuits
- A scheme for direct detection of qubit-environment entanglement generated during qubit pure dephasing
- A change of perspective: switching quantum reference frames via a perspective-neutral framework
- Efficient use of quantum resources for the transmission of a reference frame
- The Spectra of Density Operators and the Kronecker Coefficients of the Symmetric Group
- Efficient Generation of Generic Entanglement
- Fault-tolerant quantum computation
- A variational toolbox for quantum multi-parameter estimation
- Optimal estimation of group transformations using entanglement
- Threshold Accuracy for Quantum Computation
- Using concatenated quantum codes for universal fault-tolerant quantum gates
- Optimal estimation of quantum dynamics
- Continuous symmetries and approximate quantum error correction
- Physics within a quantum reference frame
- Efficient unitary designs with nearly time-independent Hamiltonian dynamics
- A fidelity measure for quantum channels
- Quantum reference frames for general symmetry groups
- Relativistic quantum reference frames: the operational meaning of spin
- Transmission of a Cartesian Frame by a Quantum System
- A fault-tolerant non-Clifford gate for the surface code in two dimensions
- Universal fault-tolerant gates on concatenated stabilizer codes
- Emergence of typical entanglement in two-party random processes
- Ultimate precision of multi-parameter quantum magnetometry under the parallel scheme
- Aligning Reference Frames Using Quantum States
- Efficient Quantum Polar Coding
- Dialogue Concerning Two Views on Quantum Coherence: Factist and Fictionist
- Quantum Relativity of Subsystems
- Switching quantum reference frames in the N-body problem and the absence of global relational perspectives
- Polar codes for private and quantum communication over arbitrary channels
- Using Quantum Metrological Bounds in Quantum Error Correction: A Simple Proof of the Approximate Eastin-Knill Theorem
- Optimal probes and error-correction schemes in multi-parameter quantum metrology
- Quantum reference frame transformations as symmetries and the paradox of the third particle
- Continuous groups of transversal gates for quantum error correcting codes from finite clock reference frames
- Intrinsic Sensitivity Limits for Multiparameter Quantum Metrology
- An efficient high dimensional quantum Schur transform
- Error Correction of Quantum Reference Frame Information
- Comment on the paper "Random Quantum Circuits are Approximate 2-designs"
- New perspectives on covariant quantum error correction
- Optimal universal programming of unitary gates
- Efficiency of Producing Random Unitary Matrices with Quantum Circuits
- Toy Models of Holographic Duality between local Hamiltonians
- Kinematics and dynamics in noninertial quantum frames of reference
- Optimal compression for identically prepared qubit states
- Efficient Quantum Compression for Ensembles of Identically Prepared Mixed States
- Building all Time Evolutions with Rotationally Invariant Hamiltonians
- Quasi-exact quantum computation
- Practical and reliable error bars for quantum process tomography
- Compression for quantum population coding
- Quantum error correction and large
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- Wavefunction branching: when you can't tell pure states from mixed states
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- SU(d)-Symmetric Random Unitaries: Quantum Scrambling, Error Correction, and Machine Learning
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- Quantum geometric tensor determines the pure-state i.i.d. conversion rate in the resource theory of asymmetry for any compact Lie group