[Paper Review] A sharp phase transition in linear cross-entropy benchmarking

Brayden Ware, Abhinav Deshpande+5 more|arXiv (Cornell University)|May 8, 2023

Neural Networks and Reservoir Computing9 citations

TL;DR

The paper demonstrates a sharp phase transition in XEB as a fidelity proxy, occurring at a critical εN that depends on circuit architecture and gate entangling power, analyzed via a two-copy statistical mechanics mapping for all-to-all and 1D circuit geometries.

ABSTRACT

Demonstrations of quantum computational advantage and benchmarks of quantum processors via quantum random circuit sampling are based on evaluating the linear cross-entropy benchmark (XEB). A key question in the theory of XEB is whether it approximates the fidelity of the quantum state preparation. Previous works have shown that the XEB generically approximates the fidelity in a regime where the noise rate per qudit $\varepsilon$ satisfies $\varepsilon N \ll 1$ for a system of $N$ qudits and that this approximation breaks down at large noise rates. Here, we show that the breakdown of XEB as a fidelity proxy occurs as a sharp phase transition at a critical value of $\varepsilon N$ that depends on the circuit architecture and properties of the two-qubit gates, including in particular their entangling power. We study the phase transition using a mapping of average two-copy quantities to statistical mechanics models in random quantum circuit architectures with full or one-dimensional connectivity. We explain the phase transition behavior in terms of spectral properties of the transfer matrix of the statistical mechanics model and identify two-qubit gate sets that exhibit the largest noise robustness.

Motivation & Objective

• Clarify under what noise conditions XEB approximates state fidelity in noisy quantum circuits.
• Characterize how circuit geometry and gate entangling power affect the XEB fidelity proxy.
• Develop a transfer-matrix based statistical model to describe two-copy observables like fidelity and XEB.
• Identify gate sets and architectures that maximize noise robustness of the XEB proxy.

Proposed method

• Model two-copy density matrices averaged over random gates to derive a transfer matrix T(α, β, γ).
• Represent the two-copy state in a basis {I/q^2, S/q} and study the evolution via a (N+1)x(N+1) transfer matrix in the all-to-all geometry.
• Use Haar-random two-qudit gates to set α=1, β=0 and derive parameter mappings for other gate sets from Appendix A.
• Analyze spectral properties of T, including leading and subleading eigenvalues Λa, to explain the XEB phase transition.
• Employ matrix-product state techniques (TEBD) and MPS-Krylov methods to study 1D circuit dynamics and spectra.
• Reduce the problem to a permutation-symmetric transfer matrix in the all-to-all geometry to obtain analytical insights (Eq. 33).

Experimental results

Research questions

• RQ1 What is the critical value of εN where XEB stops tracking fidelity, and how does it depend on gate set and circuit geometry?
• RQ2 How does the spectral structure of the transfer matrix govern the observed phase transition in XEB versus fidelity?
• RQ3 To what extent is the phase transition universal across circuit architectures (all-to-all and 1D)?
• RQ4 Which gate sets maximize noise robustness of XEB as a fidelity proxy?
• RQ5 How do Haar-random gates compare to fixed two-qubit gates dressed by Haar-random single-qubit gates in determining the transition point?

Key findings

• XEB decays with a rate matching global white-noise predictions at low εN but saturates to a constant at higher εN, signaling a phase transition where XEB ceases to proxy fidelity.
• The transition occurs at a critical εN determined by an eigenvalue crossing between Λv and Λg of the transfer matrix, i.e., when (1−ε)N equals the gap Λg.
• In all-to-all and 1D geometries, the leading behavior is captured by a transfer-matrix spectral analysis, indicating a universal transition feature of random quantum circuits.
• For Haar-random two-qudit gates, the critical value follows Λg ≈ 1 − 3α/5 with α=1, and more generally the transition location can be tuned by gate entangling power (α) and swapping power (β).
• Gate sets with higher entangling power can increase the noise robustness of the XEB proxy, with the all-to-all case allowing εN up to about ln 3 under certain parameter choices.
• The phase transition manifests as a kink in the asymptotic decay rate of XEB and a crossing of the leading nontrivial transfer-matrix eigenvalues (Λv, Λg).