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.
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.
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.
Quantum processors with sizes in the 10-100 qubit range are now increasingly common. However, with increased size comes increased complexity for benchmarking. The effectiveness of a given device may vary greatly between different tasks, and will not always be easy to predict from single and two qubit gate fidelities. For this reason, it is important to assess processor quality for a range of important tasks. In this work we propose and implement tests based on random quantum circuits. These are used to evaluate multiple different superconducting qubit devices, with sizes from 5 to 19 qubits, from two hardware manufacturers: IBM Research and Rigetti. The data is analyzed to give a quantitive description of how the devices perform. We also describe how it can be used for a qualititive description accessible to the layperson, by being played as a game.
Information scrambling, the process by which quantum information spreads and becomes effectively inaccessible, is central to modern quantum statistical physics and quantum chaos. These lecture notes provide an introduction to information scrambling from both static and dynamical perspectives. The spectral properties of reduced density matrices arising from Haar-random states are developed through the geometry of the unitary group and the universal results of random matrix theory. This geometric framework yields universal, model-independent predictions for entanglement and spectral statistics, capturing generic features of quantum chaos without reference to microscopic details. Dynamical diagnostics such as the spectral form factor and out-of-time-ordered correlators further reveal the onset of chaos in time-dependent evolution.
The notes are aimed at advanced undergraduate and graduate students in physics, mathematics, and computer science who are interested in the connections between quantum chaos, information dynamics, and quantum computing. Concepts such as entanglement growth, Haar randomness, random-matrix statistics, and unitary t-designs are introduced through their realization in random quantum circuits and circuit complexity. The same mathematical framework also underpins modern quantum-device benchmarking, where approximate unitary designs and random circuits translate geometric ideas into quantitative tools for assessing fidelity, noise, and scrambling efficiency in real quantum processors.
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.
We report the preparation and verification of a genuine 12-qubit entanglement in a superconducting processor. The processor that we designed and fabricated has qubits lying on a 1D chain with relaxation times ranging from 29.6 to 54.6 μs. The fidelity of the 12-qubit entanglement was measured to be above 0.5544±0.0025, exceeding the genuine multipartite entanglement threshold by 21 statistical standard deviations. After thermal cycling, the 12-qubit state fidelity was further improved to be above 0.707±0.008. Our entangling circuit to generate linear cluster states is depth invariant in the number of qubits and uses single- and double-qubit gates instead of collective interactions. Our results are a substantial step towards large-scale random circuit sampling and scalable measurement-based quantum computing.
Quantum computing holds the promise of a novel form of information processing in which data resides in quantum states. By exploiting superpositions and entanglement among these states, quantum logic operations are expected to far surpass conventional digital logic in certain classes of problems. To realize this prospect, we employ superconducting circuits at millikelvin temperatures. The nonlinearity of a Josephson tunnel junction enables a two-level quantum system, or qubit, with excitation energy in the microwave regime. Key recent advances in such devices are the demonstration of robust and simple multi-qubit gates, the integration of superconducting qubits into arrays of four or more, and gate fidelities reaching 99%. [1,2,3]