Continuous-variable quantum state designs: theory and applications

Abstract

We generalize the notion of quantum state designs to infinite-dimensional spaces. We first prove that, under the definition of continuous-variable (CV) state $t$-designs from Comm. Math. Phys. 326, 755 (2014), no state designs exist for $t\ge 2$. Similarly, we prove that no CV unitary $t$-designs exist for $t\ge 2$. We propose an alternative definition for CV state designs, which we call rigged $t$-designs, and provide explicit constructions for $t=2$. As an application of rigged designs, we develop a design-based shadow-tomography protocol for CV states. Using energy-constrained versions of rigged designs, we define an average fidelity for CV quantum channels and relate this fidelity to the CV entanglement fidelity. As an additional result of independent interest, we establish a connection between torus $2$-designs and complete sets of mutually unbiased bases.

14+40 pages. V2 matches journal version. V3 minor typos fixed

References in corpus (59)

• Gaussian Quantum Information
• Encoding a qubit in an oscillator
• Black holes as mirrors: quantum information in random subsystems
• Error mitigation for short-depth quantum circuits
• Predicting Many Properties of a Quantum System from Very Few Measurements
• Continuous-variable optical quantum state tomography
• Fast Scramblers
• Symmetric Informationally Complete Quantum Measurements
• Randomized Benchmarking of Quantum Gates
• Resolving photon number states in a superconducting circuit
• Robust randomized benchmarking of quantum processes
• On mutually unbiased bases
• A simple formula for the average gate fidelity of a quantum dynamical operation
• Exact and Approximate Unitary 2-Designs: Constructions and Applications
• Scalable Noise Estimation with Random Unitary Operators
• Chaos and complexity by design
• Fault-Tolerant Measurement-Based Quantum Computing with Continuous-Variable Cluster States
• Randomizing quantum states: Constructions and applications
• Performance and structure of single-mode bosonic codes
• Information Scrambling in Computationally Complex Quantum Circuits
• Tight uniform continuity bounds for quantum entropies: conditional entropy, relative entropy distance and energy constraints
• Tight informationally complete quantum measurements
• Provably efficient machine learning for quantum many-body problems
• The role of the rigged Hilbert space in Quantum Mechanics
• Criteria for Continuous-Variable Quantum Teleportation
• Multiqubit Clifford groups are unitary 3-designs
• Robust encoding of a qubit in a molecule
• The power of random measurements: measuring Tr(ρ^n) on single copies of ρ
• Optimizing quantum process tomography with unitary 2-designs
• Operator-sum Representation for Bosonic Gaussian Channels
• Random bosonic states for robust quantum metrology
• Scalable randomized benchmarking of non-Clifford gates
• Quantum-limited loss sensing: Multiparameter estimation and Bures distance between loss channels
• Single-photon Resolved Cross-Kerr Interaction for Autonomous Stabilization of Photon-number States
• Weighted complex projective 2-designs from bases: optimal state determination by orthogonal measurements
• Scrambling and Complexity in Phase Space
• Experimental Estimation of Average Fidelity of a Clifford Gate on a 7-qubit Quantum Processor
• Mathematical surprises and Dirac's formalism in quantum mechanics
• Energy-constrained diamond norms and their use in quantum information theory
• Gate fidelity fluctuations and quantum process invariants
• Bounding the energy-constrained quantum and private capacities of phase-insensitive bosonic Gaussian channels
• Optimal design and quantum benchmarks for coherent state amplifiers
• Energy-constrained discrimination of unitaries, quantum speed limits and a Gaussian Solovay-Kitaev theorem
• On the complexity of quantum partition functions
• Constructing entanglement witnesses for infinite-dimensional systems
• Fidelity criterion for quantum-domain transmission and storage of coherent states beyond unit-gain constraint
• Implementation of a canonical phase measurement with quantum feedback
• Iso-entangled mutually unbiased bases, symmetric quantum measurements and mixed-state designs
• Convergence rates for quantum evolution & entropic continuity bounds in infinite dimensions
• Quantum data hiding with continuous variable systems
• Drive-induced nonlinearities of cavity modes coupled to a transmon ancilla
• The curious nonexistence of Gaussian 2-designs
• Designing Kerr interactions using multiple superconducting qubit types in a single circuit
• Optimal tests for continuous-variable quantum teleportation and photodetectors
• Coherent-state constellations and polar codes for thermal Gaussian channels
• Characterizing the performance of continuous-variable Gaussian quantum gates
• Test one to test many: a unified approach to quantum benchmarks
• Efficient verification of bosonic quantum channels via benchmarking
• Certification of continuous-variable gates using average channel-fidelity witnesses

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• Deep thermalization in Gaussian continuous-variable quantum systems
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• Projective toric designs, quantum state designs, and mutually unbiased bases
• Clifford operations and homological codes for rotors and oscillators
• Unification of Finite Symmetries in Simulation of Many-body Systems on Quantum Computers
• Bosonic randomized benchmarking with passive transformations
• Benchmarking the performance of a high-Q cavity qudit using random unitaries
• Continuous-variable designs and design-based shadow tomography from random lattices
• Chasing shadows with Gottesman-Kitaev-Preskill codes
• Single-Period Floquet Control of Bosonic Codes with Quantum Lattice Gates