The task of sampling the output distribution of random quantum circuits will drive a multiqubit system into chaos, reaching the Porter-Thomas distribution where small multiqubit probabilities appear most often. Such a task is hard to be computed classically and has been proposed as an important measure for quantum supremacy. An equivalent route to reach chaos is the controlled dynamics uniformly exploring the large Hilbert space of coupled qubits, and a proof-of-concept experiment was recently demonstrated by Neill et al. [Science 360, 195 (2018)] in a chain of nine qubits featuring only neighboring couplings. Here, we show numerically that a faster convergence to chaos is possible when full connectivity, instead of partial connectivity, is present among the qubits, i.e., cycles or time required to reach chaotic regime increase only slightly with the number of qubits. We validate the numerical result by carrying out an experiment using up to 12 qubits in a fully connected superconducting quantum processor. Based on a machine learning model, we quantify the evolution error rate to be less than (0.5%/cycle)/qubit, which is mostly determined by qubit dephasing.
The task of sampling the output distribution of random quantum circuits will drive a multiqubit system into chaos, reaching the Porter-Thomas distribution where small multiqubit probabilities appear most often. Such a task is hard to be computed classically and has been proposed as an important measure for quantum supremacy. An equivalent route to reach chaos is the controlled dynamics uniformly exploring the large Hilbert space of coupled qubits, and a proof-of-concept experiment was recently demonstrated by Neill et al. [Science 360, 195 (2018)] in a chain of nine qubits featuring only neighboring couplings. Here, we show numerically that a faster convergence to chaos is possible when full connectivity, instead of partial connectivity, is present among the qubits, i.e., cycles or time required to reach chaotic regime increase only slightly with the number of qubits. We validate the numerical result by carrying out an experiment using up to 12 qubits in a fully connected superconducting quantum processor. Based on a machine learning model, we quantify the evolution error rate to be less than (0.5%/cycle)/qubit, which is mostly determined by qubit dephasing.
Sampling from the output distribution of chaotic quantum evolutions, and of pseudo-random universal quantum circuits in particular, has been proposed as a prominent milestone for near-term quantum supremacy. The same paper notes that chaotic distributions are very sensitive to noise, and under quite general noise models converge to the uniform distribution over bit-strings exponentially in the number of gates. On the one hand, for increasing number of gates, it suffices to choose bit-strings at random to approximate the noisy distribution with fixed statistical distance. On the other hand, cross-entropy benchmarking can be used to gauge the fidelity of an experiment, and the distance to the uniform distribution. We estimate that state-of-the-art classical supercomputers would fail to simulate high-fidelity chaotic quantum circuits with approximately fifty qubits and depth forty. A recent interesting paper proposed a different approximation algorithm to a noisy distribution, extending previous results on the Fourier analysis of commuting quantum circuits. Using the statistical properties of the Porter-Thomas distribution, we show that this new approximation algorithm does not improve random guessing, in polynomial time. Therefore, it confirms previous results and does not represent an additional challenge to the suggested failure stated above.
Universal statistical laws are a hallmark of quantum chaos. A natural route to probing them in quantum circuits is offered by the statistics of output probabilities, which for sufficiently deep chaotic circuits approach the Porter--Thomas distribution. Here we show, theoretically and experimentally, that the approach to this limit is itself universal, and is organized into geometry-dependent universality classes. At shallow depths, growing only logarithmically with system size, we derive universal output-probability distributions that differ markedly from Porter--Thomas statistics, and we test them on a programmable superconducting processor. Two-dimensional lattices, open one-dimensional chains, and periodic rings exhibit distinct universal forms, retaining signatures of geometry and boundary topology within the same logarithmic-depth regime. Within each class, the distributions depend on only two parameters: an effective many-body Thouless scale and the accumulated number of incoherent errors. Extracting these directly from measured bitstring frequencies provides access to scrambling, entanglement growth of the target state, and global fidelity, without reconstructing or classically simulating the implemented circuit. We also identify a noise-driven transition between output statistics retaining system-wide quantum correlations and those reproducible by independent small patches. Our results establish universality beyond Porter--Thomas statistics as a quantitative tool for characterizing noisy quantum processors.
A key step toward demonstrating a quantum system that can address difficult problems in physics and chemistry will be performing a computation beyond the capabilities of any classical computer, thus achieving so-called quantum supremacy. In this study, we used nine superconducting qubits to demonstrate a promising path toward quantum supremacy. By individually tuning the qubit parameters, we were able to generate thousands of distinct Hamiltonian evolutions and probe the output probabilities. The measured probabilities obey a universal distribution, consistent with uniformly sampling the full Hilbert space. As the number of qubits increases, the system continues to explore the exponentially growing number of states. Extending these results to a system of 50 qubits has the potential to address scientific questions that are beyond the capabilities of any classical computer.
Random quantum states serve as a powerful tool in various scientific fields, including quantum supremacy and black hole physics. It has been theoretically predicted that entanglement transitions may happen for different partitions of multipartite random quantum states; however, the experimental observation of these transitions is still absent. Here, we experimentally demonstrate the entanglement transitions witnessed by negativity on a fully connected superconducting processor. We apply parallel entangling operations, that significantly decrease the depth of the pseudo-random circuits, to generate pseudo-random pure states of up to 15 qubits. By quantum state tomography of the reduced density matrix of six qubits, we measure the negativity spectra. Then, by changing the sizes of the environment and subsystems, we observe the entanglement transitions that are directly identified by logarithmic entanglement negativities based on the negativity spectra. In addition, we characterize the randomness of our circuits by measuring the distance between the distribution of output bit-string probabilities and the Porter-Thomas distribution. Our results show that superconducting processors with all-to-all connectivity constitute a promising platform for generating random states and understanding the entanglement structure of multipartite quantum systems.
A common task in quantum information science is to embed classical data into entangled multiqubit states. The author offers a practical embedding circuit for a chain geometry that is especially well suited for first-generation superconducting architectures. Measuring a variety of fidelity and information-theoretic quantities provides a detailed, quantitative assessment of actual performance using the IBM Quantum Experience ibmqx5 device. To check the expressiveness of this general-purpose circuit, the author also measures its ability to generate Haar random unitaries and quantum chaos, as reckoned by Porter-Thomas statistics and out-of-time-order correlation functions.
Quantum supremacy is the ability of quantum processors to outperform classical computers at certain tasks. In digital random quantum circuit approaches for supremacy, the output distribution produced is described by the Porter-Thomas (PT) distribution. In this regime, the system uniformly explores its entire Hilbert space, which makes simulating such quantum dynamics with classical computational resources impossible for large systems. However, the latter has no direct application so far in solving a specific problem. In this work, we show that the same sampling complexity can be achieved from driven analog quantum processors, with less stringent requirements for coherence and control. More importantly, we discuss how to apply this approach to solve problems in quantum simulations of phases of matter and machine learning. Specifically, we consider a simple quantum spin chain with nearest-neighbor interactions driven by a global magnetic field. We show how quantum supremacy is achieved as a consequence of the thermalization due to the interplay between the disorder and the driven many-body dynamics. We analyze how the achieved PT distribution can be used as an accessible reference distribution to probe the many-body localization (MBL) phase transition. In the second part of our work, we show how our setup can be used for generative modeling machine learning tasks. We propose a novel variational hybrid quantum-classical approach, exploiting the system's inherent tunable MBL dynamics, to train the device to learn distributions of complex classical data. The performance of our training protocol depends solely on the phase that the quantum system is in, which makes fine-tuning of local parameters not necessary. The protocol is implementable in a range of driven quantum many-body systems, compatible with noisy intermediate-scale quantum devices.