Phase transition in Random Circuit Sampling (2304.11119v2)

Abstract: Undesired coupling to the surrounding environment destroys long-range correlations on quantum processors and hinders the coherent evolution in the nominally available computational space. This incoherent noise is an outstanding challenge to fully leverage the computation power of near-term quantum processors. It has been shown that benchmarking Random Circuit Sampling (RCS) with Cross-Entropy Benchmarking (XEB) can provide a reliable estimate of the effective size of the Hilbert space coherently available. The extent to which the presence of noise can trivialize the outputs of a given quantum algorithm, i.e. making it spoofable by a classical computation, is an unanswered question. Here, by implementing an RCS algorithm we demonstrate experimentally that there are two phase transitions observable with XEB, which we explain theoretically with a statistical model. The first is a dynamical transition as a function of the number of cycles and is the continuation of the anti-concentration point in the noiseless case. The second is a quantum phase transition controlled by the error per cycle; to identify it analytically and experimentally, we create a weak link model which allows varying the strength of noise versus coherent evolution. Furthermore, by presenting an RCS experiment with 67 qubits at 32 cycles, we demonstrate that the computational cost of our experiment is beyond the capabilities of existing classical supercomputers, even when accounting for the inevitable presence of noise. Our experimental and theoretical work establishes the existence of transitions to a stable computationally complex phase that is reachable with current quantum processors.

• The paper demonstrates a dynamical phase transition where increased circuit cycles shift the output from concentrated to anti-concentrated states.
• The paper identifies a noise-induced phase transition driven by error per cycle that distinguishes between globally entangled quantum states and classically spoofer regimes.
• The study’s 67-qubit, 32-cycle experiment underscores quantum supremacy and provides benchmarks for enhancing noise mitigation and processor calibration.

Phase Transitions in Random Circuit Sampling

In the pursuit of fully realizing the computational capabilities of near-term quantum processors, the intersection of quantum dynamics and noise forms a critical research frontier. This paper presents a thorough investigation into the phase transitions within Random Circuit Sampling (RCS) as a method to benchmark the coherent computational space available on quantum processors. The discussion emphasizes two key phase transitions observable via Cross-Entropy Benchmarking (XEB): a dynamical transition dependent on circuit cycles and a quantum phase transition influenced by noise or error per cycle.

Summary of Findings

1. Dynamical Transition: This phase transition emerges as a function of the number of circuit cycles. Initially, as cycles increase, there exists a crossing point where the system transitions from a concentrated output distribution to an anti-concentrated one. This transition is essential for ensuring that the output distribution has adequately delocalized from the computational basis, which is pivotal for XEB to accurately estimate quantum state fidelity. The experimental data support this transition, showing that beyond a certain number of cycles, XEB measurements stabilize, aligning with theoretical fidelity predictions.
2. Noise-Induced Phase Transition: Noise represents a formidable challenge in exploiting the computational power of quantum devices. The paper identifies a second phase transition driven by noise, specifically by the error rate per cycle. A critical finding is the delineation of two regimes: one in which the system's quantum correlations are globally entangled (and computationally intricate), and another where correlations remain localized (hence might be spoofed by classical algorithms). The threshold of error per cycle marking this transition dictates whether a quantum processor can harness the full expanse of its Hilbert space. Experiments and constructs like the weak-link model elucidate the physics underlying this noise-induced phase transition.

Furthermore, the researchers conducted a 67-qubit RCS experiment at 32 cycles, dramatically surpassing the limits faced by existing classical supercomputing capabilities. This not only strengthens the argument for quantum supremacy but also highlights the computational complexity intrinsic to high-dimensional quantum states.

Implications and Future Directions

The results pave the way for more nuanced understanding and improved calibration of quantum processors. They highlight the delicate balance between circuit depth and noise that must be managed to leverage quantum supremacy effectively. These phase transitions serve as a benchmark for setting experimental protocols ensuring the robustness of quantum computations, even amidst unavoidable noise. In practice, these insights can be crucial for enhancing noise mitigation strategies and refining quantum algorithms.

While Random Circuit Sampling currently serves as a primary exhibiting quantum supremacy, the implications of these findings stretch beyond theoretical benchmarking. They encourage exploration into practical applications such as certified randomness generation and further motivations for investigating quantum error correction mechanisms. Additionally, having established the critical noise levels permissible for leveraging computational quantum phases, the paper posits a hopeful landscape for future developments in quantum computing architectures.

The theoretical and experimental advances detailed in this paper contribute significantly to quantum information science, providing a more structured framework for understanding how quantum coherence and noise dynamically interact in complex quantum systems.