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Scrambling conjecture for reducing adaptive queries to distinct inputs

Prove the Scrambling Conjecture: construct an ensemble of efficiently implementable n-qubit unitaries W such that for all polynomial t and all initial states |B_0⟩ and interleaved unitaries B_1, …, B_t on n + poly(n) qubits, the trace-norm distance between E_{W} |B(W)⟩⟨B(W)| and E_{W} |B(W)_Λ⟩⟨B(W)_Λ| is negligible in n, where |B(W)⟩ is the adaptive output state with W inserted between the B_i operations and |B(W)_Λ⟩ is the corresponding state restricted to distinct-string inputs.

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Background

To extend non-adaptive PRU security to adaptive security, the authors seek a reduction that effectively limits adaptive queries to distinct computational basis inputs, enabling their relative-error analysis on the distinct subspace to apply.

They formalize this requirement as a conjecture asserting the existence of an efficient scrambling ensemble W that makes the expected output of any adaptive algorithm indistinguishable (up to negligible error) from its restriction to distinct inputs. Proving this conjecture would likely yield adaptively secure PRUs via their PF-with-Clifford framework.

References

We make the following conjecture. There exists an ensemble of efficient n-qubits unitaries W \sim such that for all t = poly(n) and all initial states |B_0⟩ and sequences of unitaries B_1, \dots, B_t on n + poly(n) qubits, \begin{align*} \norm{{W \in } |B(W)⟩⟨B(W)| - _{W \in } |B(W)\Lambda⟩⟨B(W)_\Lambda|}_1 = negl(n) \,. \end{align*}

— Simple constructions of linear-depth t-designs and pseudorandom unitaries (2404.12647 - Metger et al., 19 Apr 2024) in Section 6.2 (Towards PRUs with adaptive security)