Clifford Group and Unitary Designs under Symmetry

Abstract

We have generalized the well-known statement that the Clifford group is a unitary 3-design into symmetric cases by extending the notion of unitary design. Concretely, we have proven that a symmetric Clifford group is a symmetric unitary 3-design if and only if the symmetry constraint is described by some Pauli subgroup. We have also found a complete and unique construction method of symmetric Clifford groups with simple quantum gates for Pauli symmetries. For the overall understanding, we have also considered physically relevant U(1) and SU(2) symmetry constraints, which cannot be described by a Pauli subgroup, and have proven that the symmetric Clifford group is a symmetric unitary 1-design but not a 2-design under those symmetries. Our findings are numerically verified by computing the frame potentials, which measure the difference in randomness between the uniform ensemble on the symmetric group of interest and the symmetric unitary group. This work will open a new perspective into quantum information processing such as randomized benchmarking, and give a deep understanding to many-body systems such as monitored random circuits.

40 pages, 6 figures

References in corpus (36)

• Supplementary information for "Quantum supremacy using a programmable superconducting processor"
• Improved Simulation of Stabilizer Circuits
• Characterizing Quantum Supremacy in Near-Term Devices
• Predicting Many Properties of a Quantum System from Very Few Measurements
• The foundations of statistical mechanics from entanglement: Individual states vs. averages
• Randomized Benchmarking of Quantum Gates
• Reference frames, superselection rules, and quantum information
• Robust randomized benchmarking of quantum processes
• Measurement-driven entanglement transition in hybrid quantum circuits
• Quantum information can be negative
• Scalable Noise Estimation with Random Unitary Operators
• Chaos and complexity by design
• The thermodynamic meaning of negative entropy
• Evenly distributed unitaries: on the structure of unitary designs
• Verified Quantum Information Scrambling
• Quantum Data Hiding
• Qulacs: a fast and versatile quantum circuit simulator for research purpose
• Local random quantum circuits are approximate polynomial-designs
• Information Scrambling in Computationally Complex Quantum Circuits
• Multiqubit Clifford groups are unitary 3-designs
• Distinguishability of quantum states under restricted families of measurements with an application to quantum data hiding
• One-shot decoupling
• Reducing qubit requirements for quantum simulation using molecular point group symmetries
• Optimizing quantum process tomography with unitary 2-designs
• Relating quantum privacy and quantum coherence: an operational approach
• Hadamard-free circuits expose the structure of the Clifford group
• Standard forms of noisy quantum operations via depolarization
• Restrictions on realizable unitary operations imposed by symmetry and locality
• Unconditionally Secure Key Distribution In Higher Dimensions By Depolarization
• Generators and relations for n-qubit Clifford operators
• Quantum circuits for exact unitary $t$-designs and applications to higher-order randomized benchmarking
• Model-Independent Learning of Quantum Phases of Matter with Quantum Convolutional Neural Networks
• Non-malleable encryption of quantum information
• A generalization of Schur-Weyl duality with applications in quantum estimation
• Mixing and localisation in random time-periodic quantum circuits of Clifford unitaries
• Absence of localization in two-dimensional Clifford circuits

Cited by in corpus (12)

• Augmenting Density Matrix Renormalization Group with Clifford Circuits
• Nonstabilizerness in U(1) lattice gauge theory
• Hybrid Stabilizer Matrix Product Operator
• Efficient unitary designs and pseudorandom unitaries from permutations
• Efficient Unitary T-designs from Random Sums
• Designs from Local Random Quantum Circuits with SU(d) Symmetry
• Unitary Designs of Symmetric Local Random Circuits
• Symmetric Clifford twirling for cost-optimal quantum error mitigation in early FTQC regime
• Magic of discrete lattice gauge theories
• A quantum tug of war between randomness and symmetries on homogeneous spaces
• Error Crafting in Mixed Quantum Gate Synthesis
• Symmetric channel verification for purifying noisy quantum channels