Dynamic parameterized quantum circuits: expressive and barren-plateau free

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• ℲℲ𝕃ℶ⋗ℶ≈⅁≈ℶ⋊⋉∼ ⋊℧ ℕ⋊ℶ∼↷ ℝ⪸∖∼ℶ⋖ ℂ⋊⋗∣≊≈⅁≈ℶ⋊⋉⫌⫌∦ ⊮↛↛⋫

  𝔻∤ 𝔸≂⅁∖⋊⋉⋊⪸∦ 𝕄∤ 𝔹⋉∤𝕆∖∦ ℝ∤ 𝕀⋗∣⅁ð⋖ℶ⅁ϝϝ⋊ ⅁⋉ ℕ∤ ℕℶ∼⅁⋉

  Cited in the paper.

• ℲℲ𝔸 ℙ⋊⋖↷⋉⋊⋗ℶ⅁⋖∤𝕋ℶ⋗ ℂ⋖⅁∼∼ℶ⅁⋖ 𝔸⋖ð⋊∖ℶ≈≂⋗ ℧⋊∖ ℕ⋊ℶ∼↷ ℝ⅁⋉⋊⋗ ℂℶ∖≊ℶ≈ 𝕊⅁⋗∣⋖ℶ⋉ð⫌⫌

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  𝔻⋊∖ℶ≈ 𝔸≂⅁∖⋊⋉⋊⪸∦ 𝕏≊⋉ 𝔾⅁⋊∦ ℤ∣≂ 𝕃⅁⋉⅁≊∦ 𝕐≊⋉≂⅁⋊ 𝕃ℶ≊ ⅁⋉ 𝕌⋗∼≂ 𝕍⅁ϝℶ∖⅁⋉ℶ

  Cited in the paper.

• ℲℲ𝔼∥≊ℶ⪸⅁⋖⋉ ⋊℧ ℚ≊⅁⋉≈≊⋗ 𝔹⅁∖∖⋉ ℙ⋖⅁≈⅁≊∼ ≈⋊ ℂ⋊∼≈ ℂ⋊⋉⋉≈∖⅁≈ℶ⋊⋉ ⅁⋉ ℕ⅁∖∖⋊⪷ 𝔾⋊∖ð∼⫌⫌

  𝔸⋉∖⪷ 𝔸∖∖⅁∼⋗ℶ≈≂∦ ℤ⋊ë ℍ⋊⋖⋗∼∦ 𝕄 ℂ∖ϝ⋊ ⅁⋉ ℙ⅁≈∖ℶℸ ℂ⋊⋖∼

  Cited in the paper.

• ℲℲℂ⋖⅁∼∼ℶ⅁⋖⋖↷ 𝔼∼≈ℶ⋗⅁≈ℶ⋉ð 𝕆∼∖⪸⅁⋖∼ ⋊℧ ℕ⋊ℶ∼⋖∼∼ ℚ≊⅁⋉≈≊⋗ ℂℶ∖≊ℶ≈∼⫌⫌

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  𝔸∖⋗⅁⋉⋊ 𝔸⋉ð∖ℶ∼⅁⋉ℶ∦ 𝔸⋖↶⅁⋉∖ 𝕊≂⋗ℶ≂≊∖∦ 𝕄⅁⋉≊⋖∤ ℝ≊⋊⋖∣≂∦ 𝕄∤ ℂ∖ϝ⋊∦ ℤ⋊ë ℍ⋊⋖⋗∼ ⅁⋉ ℍ∼ℶ⋉∤𝕐≊⅁⋉ ℍ≊⅁⋉ð

  Cited in the paper.

• ℲℲℚ≊⅁⋉≈≊⋗ 𝔽⋊≊∖ℶ∖ 𝕋∖⅁⋉∼℧⋊∖⋗ 𝕌∼ℶ⋉ð 𝔻↷⋉⅁⋗ℶ ℂℶ∖≊ℶ≈∼⫌⫌

  Original

  𝔼⋖ℶ∼⅁ 𝔹ä≊⋗∖∦ 𝕍ℶ⋉⅁↷ 𝕋∖ℶ∣⅁≈≂ℶ∦ 𝔸⋖ℶ∖ϝ⅁ 𝕊ℶ℧∦ 𝔻⅁⋉ℶ⋖ 𝕃ℶ⅁∖ ⅁⋉ 𝔻∖ℸ∤ 𝕎⅁⋉ð

  Cited in the paper.

• ℲℲ𝔼℧℧ℶℶ⋉≈ 𝕃⋊⋉ð∤ℝ⅁⋉ð 𝔼⋉≈⅁⋉ð⋖⋗⋉≈ 𝕌∼ℶ⋉ð 𝔻↷⋉⅁⋗ℶ ℂℶ∖≊ℶ≈∼⫌⫌

  𝔼⋖ℶ∼⅁ 𝔹ä≊⋗∖∦ 𝕍ℶ⋉⅁↷ 𝕋∖ℶ∣⅁≈≂ℶ∦ 𝔻∖ℸ∤ 𝕎⅁⋉ð∦ ℙ⅁≈∖ℶℸ ℝ⅁⋖⋖∦ 𝔼⪷⅁∖∤ ℂ≂⋉∦ 𝕊⪷⅁∖⋉⅁∣ 𝕄⅁ℷ≊⋗∖∦ 𝔸⋖ℶ∖ϝ⅁ 𝕊ℶ℧ ⅁⋉ ℤ⋖⅁≈ℸ⋊∤ 𝕄ℶ⋉⪸

  Cited in the paper.

• ℲℲ𝕋∖⅁ℶ⋉ℶ⋉ð 𝔻∣ ℚ≊⅁⋉≈≊⋗ ℕ≊∖⅁⋖ ℕ≈⪷⋊∖ℸ∼⫌⫌

  𝕂∖∼≈ℶ⋉ 𝔹∖∦ 𝔻⋗↷≈∖⋊ 𝔹⋊⋉⅁∖⋉ℸ⋊∦ 𝕋∖∖↷ 𝔽⅁∖∖⋖⋖↷∦ 𝕋⋊ℶ⅁∼∤ 𝕆∼⋊∖⋉∦ ℝ⋊∖≈ 𝕊⅁⋖ϝ⋗⅁⋉⋉∦ 𝔻⅁⋉ℶ⋖ 𝕊≂ℶ∖⋗⅁⋉⋉ ⅁⋉ ℝ⅁⋗⋊⋉⅁ 𝕎⋊⋖℧

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• ℲℲℚ≊⅁⋉≈≊⋗ 𝔸≊≈⋊⋉⋊∖∼ ≈⋊ 𝔻⋉⋊ℶ∼ ℚ≊⅁⋉≈≊⋗ 𝔻⅁≈⅁⫌⫌

  𝔻⋗↷≈∖⋊ 𝔹⋊⋉⅁∖⋉ℸ⋊ ⅁⋉ ℙ⋊⋖ℶ⋉⅁ 𝔽⋖⋗⅁⋉⋉

  Cited in the paper.

• ℲℲ𝕊≈⅁≈ ℙ∖∣⅁∖⅁≈ℶ⋊⋉ ↷ 𝕊≂⅁⋖⋖⋊⪷ ℂℶ∖≊ℶ≈∼ 𝕌∼ℶ⋉ð 𝔽 𝔽⋊∖⪷⅁∖⫌⫌

  ℍ⅁∖∖↷ 𝔹≊≂∖⋗⅁⋉∦ 𝕄⅁∖≈⋉ 𝔽⋊⋖ℸ∖≈∼⋗⅁∦ 𝔹∖≊⋉⋊ 𝕃⋊℧℧ ⅁⋉ ℕℶ⋖∼∤∤ ℕ≊⋗⅁⋉⋉

  Cited in the paper.

• ℲℲℚ≊⅁⋉≈≊⋗ ℝ℧∖ℶð∖⅁≈⋊∖⫌⫌∦ ⊭⊬⊮⊯

  Original

  𝕄ℶ≂⅁⋖ 𝔹⋉∤𝕆∖∦ 𝔻⅁⋉ℶ⋖ 𝔾⋊≈≈∼⋗⅁⋉ ⅁⋉ 𝔸⪸ℶ⋉⅁≈⅁⋉ ℍ⅁∼∼ℶℶ⋗

  Cited in the paper.

• ℲℲ𝔸⅁∣≈ℶ⪸ ℂ⋊⋉∼≈⅁⋉≈∤𝔻∣≈≂ ℂℶ∖≊ℶ≈∼ ℧⋊∖ 𝕄⅁⋉ℶ∣≊⋖⅁≈ℶ⋉ð ℕ⋊⋉∤𝔸⋖ℶ⅁⋉ 𝔸⋉↷⋊⋉∼⫌⫌

  Original

  𝕊∖ð↷ 𝔹∖⅁⪸↷ℶ∦ 𝕀∼⅁⅁ 𝕂ℶ⋗∦ 𝔸⋖↶⅁⋉∖ 𝕂⋖ℶ∼≂ ⅁⋉ ℝ⋊∖≈ 𝕂⋊⋉ℶð

  Cited in the paper.

• ℲℲℙ⅁∖⅁⋗≈∖ℶϝ ∥≊⅁⋉≈≊⋗ ℶ∖≊ℶ≈∼ ⅁∼ ⋗⅁≂ℶ⋉ ⋖⅁∖⋉ℶ⋉ð ⋗⋊⋖∼⫌⫌

  𝕄⅁∖⋖⋖⋊ 𝔹⋉≈≈ℶ∦ 𝔼∖ℶℸ⅁ 𝕃⋖⋊↷∦ 𝕊≈℧⅁⋉ 𝕊⅁ℸ ⅁⋉ 𝕄⅁≈≈ℶ⅁ 𝔽ℶ⋊∖⋉≈ℶ⋉ℶ

  Cited in the paper.

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