Stabilizer testing requires Θ(n-k) copies and non-adaptive learning Θ(n²/k) copies with k-qubit memory, removing the testing-learning separation.
Qubit stabilizer states are complex projective 3-designs
9 Pith papers cite this work. Polarity classification is still indexing.
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abstract
A complex projective $t$-design is a configuration of vectors which is ``evenly distributed'' on a sphere in the sense that sampling uniformly from it reproduces the moments of Haar measure up to order $2t$. We show that the set of all $n$-qubit stabilizer states forms a complex projective $3$-design in dimension $2^n$. Stabilizer states had previously only been known to constitute $2$-designs. The main technical ingredient is a general recursion formula for the so-called frame potential of stabilizer states. To establish it, we need to compute the number of stabilizer states with pre-described inner product with respect to a reference state. This, in turn, reduces to a counting problem in discrete symplectic vector spaces for which we find a simple formula. We sketch applications in quantum information and signal analysis.
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Proves Θ(d²) bases are necessary and sufficient for worst-case optimal shadow estimation while 2-designs achieve average-case optimality with universal constant bounds.
Uniform superpositions with random binary phases are exponentially close in trace distance to Haar-random states for polynomially many copies, enabling pseudorandom quantum state constructions from post-quantum PRFs and low-depth t-designs.
Log-depth circuits suffice for average-case single-copy stabilizer learning with t=O(log n), but worst-case adaptive single-copy learning requires exp(t) samples.
A unified structured factorization framework for quantum state tomography that parametrizes the density matrix as FF^dagger, supports multiple priors, provides sample complexity bounds, and introduces projected gradient descent and power-method algorithms.
Noisy 2-design ensembles show a conditional-entropy-governed threshold for distinguishability preservation while post-measured versions collapse exponentially with no protected regime.
A survey of structured quantum state tomography covering compact representations, measurement design, and optimization algorithms, connected to compressive sensing for sample efficiency.
This review compiles fourteen equivalent formulations of the open existence problem for maximal mutually unbiased bases in composite dimensions and summarizes known analytic, computer-aided and numerical results along with potential solution strategies.
A review of how quantum information science is expected to provide new tools and insights for nuclear and high-energy physics phenomenology and quantum simulations.
citing papers explorer
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Optimal Stabilizer Testing and Learning with Limited Quantum Memory
Stabilizer testing requires Θ(n-k) copies and non-adaptive learning Θ(n²/k) copies with k-qubit memory, removing the testing-learning separation.
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Optimal Shadow Estimation with Minimal Measurement Settings
Proves Θ(d²) bases are necessary and sufficient for worst-case optimal shadow estimation while 2-designs achieve average-case optimality with universal constant bounds.
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(Pseudo) Random Quantum States with Binary Phase
Uniform superpositions with random binary phases are exponentially close in trace distance to Haar-random states for polynomially many copies, enabling pseudorandom quantum state constructions from post-quantum PRFs and low-depth t-designs.
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Single-copy stabilizer learning: average case and worst case
Log-depth circuits suffice for average-case single-copy stabilizer learning with t=O(log n), but worst-case adaptive single-copy learning requires exp(t) samples.
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Structured Factorization Approaches for Quantum State Tomography
A unified structured factorization framework for quantum state tomography that parametrizes the density matrix as FF^dagger, supports multiple priors, provides sample complexity bounds, and introduces projected gradient descent and power-method algorithms.
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Can scrambling protect quantum state distinguishability under noise?
Noisy 2-design ensembles show a conditional-entropy-governed threshold for distinguishability preservation while post-measured versions collapse exponentially with no protected regime.
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Statistical and Algorithmic Foundations of Probing Quantum Systems with Compressive Measurements: A Review
A survey of structured quantum state tomography covering compact representations, measurement design, and optimization algorithms, connected to compressive sensing for sample efficiency.
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Mutually Unbiased Bases in Composite Dimensions -- A Review
This review compiles fourteen equivalent formulations of the open existence problem for maximal mutually unbiased bases in composite dimensions and summarizes known analytic, computer-aided and numerical results along with potential solution strategies.
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Quantum Complexity and New Directions in Nuclear Physics and High-Energy Physics Phenomenology
A review of how quantum information science is expected to provide new tools and insights for nuclear and high-energy physics phenomenology and quantum simulations.