Multiqubit Clifford groups are unitary 3-designs

Abstract

Unitary $t$-designs are a ubiquitous tool in many research areas, including randomized benchmarking, quantum process tomography, and scrambling. Despite the intensive efforts of many researchers, little is known about unitary $t$-designs with $t\ge 3$ in the literature. We show that the multiqubit Clifford group in any even prime-power dimension is not only a unitary 2-design, but also a 3-design. Moreover, it is a minimal 3-design except for dimension~4. As an immediate consequence, any orbit of pure states of the multiqubit Clifford group forms a complex projective 3-design; in particular, the set of stabilizer states forms a 3-design. In addition, our study is helpful to studying higher moments of the Clifford group, which are useful in many research areas ranging from quantum information science to signal processing. Furthermore, we reveal a surprising connection between unitary 3-designs and the physics of discrete phase spaces and thereby offer a simple explanation of why no discrete Wigner function is covariant with respect to the multiqubit Clifford group, which is of intrinsic interest to studying quantum computation.

arXiv comment7 pages, published in Phys. Rev. A

References in corpus (41)

• Black holes as mirrors: quantum information in random subsystems
• Quantum state tomography via compressed sensing
• Fast Scramblers
• Symmetric Informationally Complete Quantum Measurements
• Randomized Benchmarking of Quantum Gates
• Robust randomized benchmarking of quantum processes
• Stabilizer Codes and Quantum Error Correction
• Chaos in quantum channels
• Exact and Approximate Unitary 2-Designs: Constructions and Applications
• Contextuality supplies the magic for quantum computation
• Chaos and complexity by design
• Negative Quasi-Probability as a Resource for Quantum Computation
• Evenly distributed unitaries: on the structure of unitary designs
• Quantum Data Hiding
• Hudson's Theorem for finite-dimensional quantum systems
• Random Quantum Circuits are Approximate 2-designs
• Tight informationally complete quantum measurements
• Randomized Benchmarking with Confidence
• Distinguishability of quantum states under restricted families of measurements with an application to quantum data hiding
• Wigner function negativity and contextuality in quantum computation on rebits
• Optimizing quantum process tomography with unitary 2-designs
• Unitary designs and codes
• A Partial Derandomization of PhaseLift using Spherical Designs
• SIC~POVMs and Clifford groups in prime dimensions
• Efficient unitary designs with nearly time-independent Hamiltonian dynamics
• Reexamination of optimal quantum state estimation of pure states
• Multipartite Entanglement in Stabilizer Tensor Networks
• Unconditionally Secure Key Distribution In Higher Dimensions By Depolarization
• The Clifford group fails gracefully to be a unitary 4-design
• Efficient Quantum Pseudorandomness
• Permutation Symmetry Determines the Discrete Wigner Function
• Super-symmetric informationally complete measurements
• Properties of the extended Clifford group with applications to SIC-POVMs and MUBs
• Efficient Simulation of Random Quantum States and Operators
• Non-malleable encryption of quantum information
• Spectra of phase point operators in odd prime dimensions and the extended Clifford group
• Mutually unbiased bases as minimal Clifford covariant 2-designs
• Local random quantum circuits are approximate polynomial-designs - numerical results
• Low rank matrix recovery from Clifford orbits
• Distinguishing quantum states using Clifford orbits
• Sharply covariant mutually unbiased bases

Cited by in corpus (134)

• Random Quantum Circuits
• Chaos and complexity by design
• Security in Quantum Cryptography
• Linear growth of quantum circuit complexity
• Robust shadow estimation
• Introduction to Haar Measure Tools in Quantum Information: A Beginner's Tutorial
• Theory of quantum system certification: a tutorial
• A general framework for randomized benchmarking
• Avoiding barren plateaus using classical shadows
• Robust Dynamic Hamiltonian Engineering of Many-Body Spin Systems
• Pseudorandom States, Non-Cloning Theorems and Quantum Money
• Models of quantum complexity growth
• Contextuality and Wigner function negativity in qubit quantum computation
• Robustness of Magic and Symmetries of the Stabiliser Polytope
• Schur-Weyl Duality for the Clifford Group with Applications: Property Testing, a Robust Hudson Theorem, and de Finetti Representations
• Classical Shadows With Noise
• Hadamard-free circuits expose the structure of the Clifford group
• Single-copies estimation of entanglement negativity
• Measurement-induced phase transitions in $(d+1)$-dimensional stabilizer circuits
• Quantum Chaos is Quantum
• Random quantum circuits are approximate unitary $t$-designs in depth $O\left(n{t}^{5+o(1)}\right)$
• Matchgate Shadows for Fermionic Quantum Simulation
• Experimental single-setting quantum state tomography
• Fast quantum circuit cutting with randomized measurements
• Multipartite Entanglement in Stabilizer Tensor Networks
• Scrambling and Complexity in Phase Space
• Correlation measures and the entanglement wedge cross-section after quantum quenches in two-dimensional conformal field theories
• Measuring nonstabilizerness via multifractal flatness
• Many-body quantum teleportation via operator spreading in the traversable wormhole protocol
• Unitary designs from statistical mechanics in random quantum circuits
• Recovering quantum gates from few average gate fidelities
• Shallow shadows: Expectation estimation using low-depth random Clifford circuits
• Scrambling Transition in a Radiative Random Unitary Circuit
• Universal cost bound of quantum error mitigation based on quantum estimation theory
• Entanglement, quantum randomness, and complexity beyond scrambling
• Hamiltonian simulation with random inputs
• Real Randomized Benchmarking
• Efficient unitary designs with a system-size independent number of non-Clifford gates
• Probing post-measurement entanglement without post-selection
• Permutation Symmetry Determines the Discrete Wigner Function
• Multi-qubit Randomized Benchmarking Using Few Samples
• Transitions in entanglement complexity in random quantum circuits by measurements
• Detecting entanglement in quantum many-body systems via permutation moments
• Mitigating Barren Plateaus with Transfer-learning-inspired Parameter Initializations
• A Bayesian analysis of classical shadows
• Quantum authentication with key recycling
• Random Matrix Theory of the Isospectral twirling
• Spectral decoupling in many-body quantum chaos
• Character randomized benchmarking for non-multiplicity-free groups with applications to subspace, leakage, and matchgate randomized benchmarking
• Informational power of the Hoggar SIC-POVM
• Crystalline Quantum Circuits
• Estimating gate-set properties from random sequences
• Retrieving information from a black hole using quantum machine learning
• Phase transition in Stabilizer Entropy and efficient purity estimation
• Closed-form analytic expressions for shadow estimation with brickwork circuits
• Efficient quantum pseudorandomness with simple graph states
• Markovianization with approximate unitary designs
• Efficient Unitarity Randomized Benchmarking of Few-qubit Clifford Gates
• Schrödinger-Heisenberg Variational Quantum Algorithms
• Realizing topologically ordered states on a quantum processor
• A Practical Introduction to Benchmarking and Characterization of Quantum Computers
• Improved Stabilizer Estimation via Bell Difference Sampling
• Evaluating a quantum-classical quantum Monte Carlo algorithm with Matchgate shadows
• The role of cohomology in quantum computation with magic states
• Randomized benchmarking in measurement-based quantum computing
• Assessing and Advancing the Potential of Quantum Computing: A NASA Case Study
• Representations of the multi-qubit Clifford group
• Improving compressed sensing with the diamond norm
• Quantifying scrambling in quantum neural networks
• Sample-optimal classical shadows for pure states
• Spectral Properties Versus Magic Generation in $T$-doped Random Clifford Circuits
• Magic of Random Matrix Product States
• Efficient Unitary T-designs from Random Sums
• Efficient unitary designs and pseudorandom unitaries from permutations
• Stabilizer Testing and Magic Entropy via Quantum Fourier Analysis
• Continuous-variable quantum state designs: theory and applications
• Quantum Measurements in the Light of Quantum State Estimation
• Optimising quantum tomography via shadow inversion
• Mixing and localisation in random time-periodic quantum circuits of Clifford unitaries
• On the explicit constructions of certain unitary $t$-designs
• Clifford Group and Unitary Designs under Symmetry
• Weak approximate unitary designs and applications to quantum encryption
• Designs from Local Random Quantum Circuits with SU(d) Symmetry
• Absence of localization in two-dimensional Clifford circuits
• Extracting randomness from magic quantum states
• Certifying Quantum Temporal Correlation via Randomized Measurements: Theory and Experiment
• NoRA: A Tensor Network Ansatz for Volume-Law Entangled Equilibrium States of Highly Connected Hamiltonians
• Complexity Classification of Conjugated Clifford Circuits
• Rank-deficient representations in the Theta correspondence over finite fields arise from quantum codes
• Efficient Classical Shadow Tomography through Many-body Localization Dynamics
• Scalable evaluation of quantum-circuit error loss using Clifford sampling
• Predicting Features of Quantum Systems from Very Few Measurements
• Fast and robust quantum state tomography from few basis measurements
• Qudit Shadow Estimation Based on the Clifford Group and the Power of a Single Magic Gate
• Universal Spreading of Conditional Mutual Information in Noisy Random Circuits
• Measurement Contextuality and Planck's Constant
• Estimation of correlations and non-separability in quantum channels via unitarity benchmarking
• Learning Properties of Quantum States Without the I.I.D. Assumption
• Dynamical phase transitions of information flow in random quantum circuits
• New security notions and feasibility results for authentication of quantum data
• Fundamental limits on quantum cloning from the no-signalling principle
• Non-Clifford Cost of Random Unitaries
• Non-Haar random circuits form unitary designs as fast as Haar random circuits
• Lower Bounds for Learning Quantum States with Single-Copy Measurements
• Morphophoric POVMs, generalised qplexes, and 2-designs
• Randomized measurements for multi-parameter quantum metrology
• Learning Quantum Processes with Quantum Statistical Queries
• Estimating many properties of a quantum state via quantum reservoir processing
• Robust Estimation of Nonlinear Properties of Quantum Processes
• Wigner's Theorem for stabilizer states and quantum designs
• Impact of dynamics, entanglement, and Markovian noise on the fidelity of few-qubit digital quantum simulation
• Nearly query-optimal classical shadow estimation of unitary channels
• Advancing Quantum Networking: Some Tools and Protocols for Ideal and Noisy Photonic Systems
• Multipartite Greenberger-Horne-Zeilinger Entanglement in Monitored Random Clifford Circuits
• Agnostic Process Tomography
• Clifford Circuits can be Properly PAC Learned if and only if $\text{RP}=\text{NP}$
• Invested and Potential Magic Resources in Measurement-Based Quantum Computation
• Equilibration and Typicality in Quantum Processes
• Generalized group designs: constructing novel unitary 2-, 3- and 4-designs
• Resource-efficient shadow tomography using equatorial stabilizer measurements
• Theory of mirror benchmarking and demonstration on a quantum computer
• On character table of Clifford groups
• Approximate 3-designs and partial decomposition of the Clifford group representation using transvections
• Partial randomized benchmarking
• Generalized Entanglement of Purification Criteria for 2-Producible States in Multipartite Systems
• Adaptively secure unitary designs with constant non-Clifford cost
• Measurement-induced entanglement in noisy 2D random circuits
• High-expressibility Quantum Neural Networks using only classical resources
• Optimal randomized measurements for a family of non-linear quantum properties
• Approximate Quantum Error Correction with 1D Log-Depth Circuits
• Certifying entanglement dimensionality by $k$-reduction moments
• Optimal estimation of three parallel spins with genuine and restricted collective measurements
• Duality theory for Clifford tensor powers
• Quantum Advantage via Efficient Post-processing on Qudit Classical Shadow tomography