This is an updated version of supplementary information to accompany "Quantum supremacy using a programmable superconducting processor", an article published in the October 24, 2019 issue of Nature. The main article is freely available at this https URL. Summary of changes since arXiv:1910.11333v1 (submitted 23 Oct 2019): added URL for qFlex source code; added Erratum section; added Figure S41 comparing statistical and total uncertainty for log and linear XEB; new References [1,65]; miscellaneous updates for clarity and style consistency; miscellaneous typographical and formatting corrections.

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Supplementary information for "Quantum supremacy using a programmable superconducting processor"

Noisy Intermediate-Scale Quantum (NISQ) technology will be available in the near future. Quantum computers with 50-100 qubits may be able to perform tasks which surpass the capabilities of today's classical digital computers, but noise in quantum gates will limit the size of quantum circuits that can be executed reliably. NISQ devices will be useful tools for exploring many-body quantum physics, and may have other useful applications, but the 100-qubit quantum computer will not change the world right away --- we should regard it as a significant step toward the more powerful quantum technologies of the future. Quantum technologists should continue to strive for more accurate quantum gates and, eventually, fully fault-tolerant quantum computing.

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Quantum Computing in the NISQ era and beyond

The purpose of this paper is to provide a comprehensive presentation and interpretation of the Ensemble Kalman Filter (EnKF) and its numerical implementation. The EnKF has a large user group, and numerous publications have discussed applications and theoretical aspects of it. This paper reviews the important results from these studies and also presents new ideas and alternative interpretations which further explain the success of the EnKF. In addition to providing the theoretical framework needed for using the EnKF, there is also a focus on the algorithmic formulation and optimal numerical implementation. A program listing is given for some of the key subroutines. The paper also touches upon specific issues such as the use of nonlinear measurements, in situ profiles of temperature and salinity, and data which are available with high frequency in time. An ensemble based optimal interpolation (EnOI) scheme is presented as a cost-effective approach which may serve as an alternative to the EnKF in some applications. A fairly extensive discussion is devoted to the use of time correlated model errors and the estimation of model bias.

The Ensemble Kalman Filter: Theoretical formulation and practical implementation

This monograph provides an overview of general deep learning methodology and its applications to a variety of signal and information processing tasks. The application areas are chosen with the following three criteria in mind: (1) expertise or knowledge of the authors; (2) the application areas that have already been transformed by the successful use of deep learning technology, such as speech recognition and computer vision; and (3) the application areas that have the potential to be impacted significantly by deep learning and that have been experiencing research growth, including natural language and text processing, information retrieval, and multimodal information processing empowered by multi-task deep learning.

/pdf/deep-learning-methods-and-applications-5cyi610sjz.pdf

Deep Learning: Methods and Applications

Noisy Intermediate-Scale Quantum (NISQ) technology will be available in the
near future. Quantum computers with 50-100 qubits may be able to perform tasks
which surpass the capabilities of today's classical digital computers, but
noise in quantum gates will limit the size of quantum circuits that can be
executed reliably. NISQ devices will be useful tools for exploring many-body
quantum physics, and may have other useful applications, but the 100-qubit
quantum computer will not change the world right away --- we should regard it
as a significant step toward the more powerful quantum technologies of the
future. Quantum technologists should continue to strive for more accurate
quantum gates and, eventually, fully fault-tolerant quantum computing.

/pdf/quantum-computing-in-the-nisq-era-and-beyond-3utyn832f7.pdf

While an experimental design for well-posed inverse linear problems has been well studied, covering a vast range of well-established design criteria and optimization algorithms, its ill-posed counterpart is a rather new topic The ill-posed nature of the problem entails the incorporation of regularization techniques The consequent non-stochastic error introduced by regularization needs to be taken into account when choosing an experimental design criterion We discuss different ways to define an optimal design that controls both an average total error of regularized estimates and a measure of the total cost of the design We also introduce a numerical framework that efficiently implements such designs and natively allows for the solution of large-scale problems To illustrate the possible applications of the methodology, we consider a borehole tomography example and a two-dimensional function recovery problem

http://www.mathcs.emory.edu/~horesh/publications/journal/Haber%202008%20-%20Numerical%20methods%20for%20experimental%20design%20of%20large-scale%20linear%20ill-posed%20inverse%20problems%20preprint.pdf

Numerical methods for experimental design of large-scale linear ill-posed inverse problems

With the current rate of progress in quantum computing technologies, systems with more than 50 qubits will soon become reality. Computing ideal quantum state amplitudes for devices of such and larger sizes is a fundamental step to assess their fidelity, but memory requirements for such calculations on classical computers grow exponentially. In this study, we present a new approach for this task that extends the boundaries of what can be computed on a classical system. We present results obtained from a calculation of the complete set of output amplitudes of a universal random circuit with depth 27 in a 2D lattice of $7 \times 7$ qubits, and an arbitrarily selected slice of $2^{37}$ amplitudes of a universal random circuit with depth 23 in a 2D lattice of $8 \times 7$ qubits. Combining our methodology with other decomposition techniques found in the literature, we show that we can simulate $7 \times 7$-qubit random circuits to arbitrary depth by leveraging secondary storage. These calculations were thought to be impossible due to memory requirements; our methodology requires memory within the limits of existing classical computers.

Breaking the 49-Qubit Barrier in the Simulation of Quantum Circuits

In a recent paper, we showed that secondary storage can extend the range of quantum circuits that can be practically simulated with classical algorithms. Here we refine those techniques and apply them to the simulation of Sycamore circuits with 53 and 54 qubits, with the entanglement pattern ABCDCDAB that has proven difficult to classically simulate with other approaches. Our analysis shows that on the Summit supercomputer at Oak Ridge National Laboratories, such circuits can be simulated with high fidelity to arbitrary depth in a matter of days, outputting all the amplitudes.

Leveraging Secondary Storage to Simulate Deep 54-qubit Sycamore Circuits

MFEIT (multi-frequency electrical impedance tomography) could distinguish between ischaemic and haemorrhagic stroke and permit the urgent use of thrombolytic drugs in patients with ischaemic stroke. The purpose of this study was to characterize the UCLH Mk 2 MFEIT system, designed for this purpose, with 32 electrodes and a multiplexed 2 kHz to 1.6 MHz single impedance measuring circuit. Data were collected in seven subjects with brain tumours, arteriovenous malformations or chronic stroke, as these resembled the changes in haemorrhagic or ischaemic stroke. Calibration studies indicated that the reliable bandwidth was only 16-64 kHz because of front-end components placed to permit simultaneous EEG recording. In raw in-phase component data, the SD of 16-64 kHz data for one electrode combination across subjects was 2.45 +/- 0.9%, compared to a largest predicted change of 0.35% estimated using the FEM of the head. Using newly developed methods of examining the most sensitive channels from the FEM, and nonlinear imaging constrained to the known site of the lesion, no reproducible changes between pathologies were observed. This study has identified a specification for accuracy in EITS in acute stroke, identified the size of variability in relation to this in human recordings, and presents new methods for analysis of data. Although no reproducible changes were identified, we hope this will provide a foundation for future studies in this demanding but potentially powerful novel application.

https://iopscience.iop.org/article/10.1088/0967-3334/27/5/S13/pdf

Multi-frequency electrical impedance tomography (EIT) of the adult human head: initial findings in brain tumours, arteriovenous malformations and chronic stroke, development of an analysis method and calibration

A new, compact UCLH Mk 2.5 EIT system has been developed and calibrated for EIT imaging of the head. Improvements include increased input and output impedances, increased bandwidth and improved CMRR (80 dB) and linearity over frequencies and load (0.2% on a single channel, +/-0.7% on a saline tank over 20 Hz-256 kHz and 10-65 Omega). The accuracy of the system is sufficient to image severe acute stroke according to the specification from recent detailed anatomical modelling (Horesh et al 2005 3rd European Medical and Biological Engineering Conference EMBEC'05). A preliminary human study has validated the main specifications of the modelling, the range of trans-impedance from the head (8-70 Omega) using a 32 electrode, 258 combination protocol and contact impedances of 300 Omega to 2.7 kOmega over 20 Hz to 256 kHz.

http://www.mathcs.emory.edu/~horesh/publications/journal/McEwan%20-%20Design%20and%20calibration%20of%20a%20compact%20multi-frequency%20EIT%20system%20for%20acute%20stroke%20imaging%20-%20pm6_5_s17.pdf

Lior Horesh

Papers

Numerical methods for experimental design of large-scale linear ill-posed inverse problems

Breaking the 49-Qubit Barrier in the Simulation of Quantum Circuits

Leveraging Secondary Storage to Simulate Deep 54-qubit Sycamore Circuits

Multi-frequency electrical impedance tomography (EIT) of the adult human head: initial findings in brain tumours, arteriovenous malformations and chronic stroke, development of an analysis method and calibration

Design and calibration of a compact multi-frequency EIT system for acute stroke imaging