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.
Qubit stabilizer states are complex projective 3-designs
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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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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.
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.
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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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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
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