A leading proposal for verifying near-term quantum supremacy experiments on noisy random quantum circuits is linear cross-entropy benchmarking. For a quantum circuit C on n qubits and a sample z ∈ {0,1}ⁿ, the benchmark involves computing |⟨z|C|0ⁿ⟩|², i.e. the probability of measuring z from the output distribution of C on the all zeros input. Under a strong conjecture about the classical hardness of estimating output probabilities of quantum circuits, no polynomial-time classical algorithm given C can output a string z such that |⟨z|C|0ⁿ⟩|² is substantially larger than 1/(2ⁿ) (Aaronson and Gunn, 2019). On the other hand, for a random quantum circuit C, sampling z from the output distribution of C achieves |⟨z|C|0ⁿ⟩|² ≈ 2/(2ⁿ) on average (Arute et al., 2019). In analogy with the Tsirelson inequality from quantum nonlocal correlations, we ask: can a polynomial-time quantum algorithm do substantially better than 2/(2ⁿ)? We study this question in the query (or black box) model, where the quantum algorithm is given oracle access to C. We show that, for any ε ≥ 1/poly(n), outputting a sample z such that |⟨z|C|0ⁿ⟩|² ≥ (2 + ε)/2ⁿ on average requires at least Ω((2^{n/4})/poly(n)) queries to C, but not more than O (2^{n/3}) queries to C, if C is either a Haar-random n-qubit unitary, or a canonical state preparation oracle for a Haar-random n-qubit state. We also show that when C samples from the Fourier distribution of a random Boolean function, the naive algorithm that samples from C is the optimal 1-query algorithm for maximizing |⟨z|C|0ⁿ⟩|² on average.
@InProceedings{kretschmer:LIPIcs.ITCS.2021.13, author = {Kretschmer, William}, title = {{The Quantum Supremacy Tsirelson Inequality}}, booktitle = {12th Innovations in Theoretical Computer Science Conference (ITCS 2021)}, pages = {13:1--13:13}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-177-1}, ISSN = {1868-8969}, year = {2021}, volume = {185}, editor = {Lee, James R.}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2021.13}, URN = {urn:nbn:de:0030-drops-135524}, doi = {10.4230/LIPIcs.ITCS.2021.13}, annote = {Keywords: quantum supremacy, quantum query complexity, random circuit sampling} }
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