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This multiplication can be implemented using the Sch¨ onhage-Strassen algorithm with circuit depth O(log n) and size O(n log n log logn)

Polynomial multiplication: Perform polynomial multiplication c′(x) · µ(x) to obtain the polynomial e(x) of degree at most (2n−2)+( n−2) = 3n−4

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Unitary designs in nearly optimal depth

quant-ph · 2025-07-08 · conditional · novelty 8.0

An ε-approximate unitary k-design on n qubits can be implemented in depth O(log k · log log(nk/ε)) using long-range two-qubit gates, with near-optimal ancilla and randomness costs.

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  • Unitary designs in nearly optimal depth quant-ph · 2025-07-08 · conditional · none · ref 2

    An ε-approximate unitary k-design on n qubits can be implemented in depth O(log k · log log(nk/ε)) using long-range two-qubit gates, with near-optimal ancilla and randomness costs.