There is, I think, something ethereal about i —the square root of minus one. I remember first hearing about it at school. It seemed an odd beast at that time—an intruder hovering on the edge of reality.

Usually familiarity dulls this sense of the bizarre, but in the case of i it was the reverse: over the years the sense of its surreal nature intensified. It seemed that it was impossible to write mathematics that described the real world in …

I and i

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.

/pdf/supplementary-information-for-quantum-supremacy-using-a-3tkhfafpzk.pdf

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

In this article, we present a small, complete, and efficient SAT-solver in the style of conflict-driven learning, as exemplified by Chaff. We aim to give sufficient details about implementation to enable the reader to construct his or her own solver in a very short time.This will allow users of SAT-solvers to make domain specific extensions or adaptions of current state-of-the-art SAT-techniques, to meet the needs of a particular application area. The presented solver is designed with this in mind, and includes among other things a mechanism for adding arbitrary boolean constraints. It also supports solving a series of related SAT-problems efficiently by an incremental SAT-interface.

/pdf/an-extensible-sat-solver-4r6o94uank.pdf

An Extensible SAT-solver

The promise of quantum computers is that certain computational tasks might be executed exponentially faster on a quantum processor than on a classical processor1. A fundamental challenge is to build a high-fidelity processor capable of running quantum algorithms in an exponentially large computational space. Here we report the use of a processor with programmable superconducting qubits2-7 to create quantum states on 53 qubits, corresponding to a computational state-space of dimension 253 (about 1016). Measurements from repeated experiments sample the resulting probability distribution, which we verify using classical simulations. Our Sycamore processor takes about 200 seconds to sample one instance of a quantum circuit a million times-our benchmarks currently indicate that the equivalent task for a state-of-the-art classical supercomputer would take approximately 10,000 years. This dramatic increase in speed compared to all known classical algorithms is an experimental realization of quantum supremacy8-14 for this specific computational task, heralding a much-anticipated computing paradigm.

/pdf/quantum-supremacy-using-a-programmable-superconducting-5182kq634u.pdf

Quantum supremacy using a programmable superconducting processor

The objective of this paper is to give a comprehensive introduction to applied cryptography with an engineer or computer scientist in mind. The emphasis is on the knowledge needed to create practical systems which supports integrity, confidentiality, or authenticity. Topics covered includes an introduction to the concepts in cryptography, attacks against cryptographic systems, key use and handling, random bit generation, encryption modes, and message authentication codes. Recommendations on algorithms and further reading is given in the end of the paper. This paper should make the reader able to build, understand and evaluate system descriptions and designs based on the cryptographic components described in the paper.

[서평]「Applied Cryptography」

https://web.eecs.umich.edu/~valeria/research/publications/Integration0708.pdf

SafeResynth: A new technique for physical synthesis

When minimizing a given objective function is challenging because of, for example, combinatorial complexity or points of nondifferentiability, one can apply more efficient and easier-to-implement algorithms to modified versions of the function. In the ideal case, one can employ known algorithms for the modified function that have a thorough theoretical and empirical record and for which public implementations are available. The main requirement here is that minimizers of the objective function do not change much through the modification, i.e., the modification must have a bounded effect on the quality of the solution. Review of classic and recent placement algorithms suggests a dichotomy between approaches that either: (a) heuristically minimize potentially irrelevant objective function (e.g., VLSI placement with quadratic wirelength) motivated by the simplicity and speed of a standard minimization algorithm; or (b) devise elaborate problem-specific minimization heuristics for more relevant objective functions (e.g., VLSI placement with linear wirelength). Smoothness and convexity of the objective functions typically enable efficient minimization. If either characteristic is not present in the objective function, one can modify and/or restrict the objective to special values of parameters to provide the missing properties. After the minimizers of the modified function are found, they can be further improved with respect to the original function by fast local search using only function evaluations. Thus, it is the modification step that deserves most attention. In this paper, we approximate convex nonsmooth continuous functions by convex differentiable functions which are parameterized by a scalar /spl beta/>0 and have convenient limit behavior as /spl beta//spl rarr/0. This allows the use of Newton-type algorithms for minimization and, for standard numerical methods, translates into a tradeoff between solution quality and speed. We prove that our methods apply to arbitrary multivariate convex piecewise-linear functions that are widely used in synthesis and analysis of electrical networks. The utility of our approximations is particularly demonstrated for wirelength and nonlinear delay estimations used by analytical placers for VLSI layout, where they lead to more "solvable" problems than those resulting from earlier comparable approaches [29]. For a particular delay estimate, we show that, while convexity is not straightforward to prove, it holds for a certain range of parameters, which, luckily, are representative of "real-world" technologies.

http://ieeexplore.ieee.org/iel5/81/20360/00940185.pdf

Efficient Optimization by Modifying the Objective Function: Applications to Timing-Driven

Quantum-mechanical phenomena are playing an increasing role in information processing, as transistor sizes approach the nanometer level, and quantum circuits and data encoding methods appear in the securest forms of communication. Simulating such phenomena efficiently is exceedingly difficult because of the vast size of the quantum state space involved. A major complication is caused by errors (noise) due to unwanted interactions between the quantum states and the environment. Consequently, simulating quantum circuits and their associated errors using the density matrix representation is potentially significant in many applications, but is well beyond the computational abilities of most classical simulation techniques in both time and memory resources. The size of a density matrix grows exponentially with the number of qubits simulated, rendering array-based simulation techniques that explicitly store the density matrix intractable. In this work, we propose a new technique aimed at efficiently simulating quantum circuits that are subject to errors. In particular, we describe new graph-based algorithms implemented in the simulator QuIDDPro/D. While previously reported graph-based simulators operate in terms of the state-vector representation, these new algorithms use the density matrix representation. To gauge the improvements offered by QuIDDPro/D, we compare its simulation performance with an optimized array-based simulator called QCSim. Empirical results, generated by both simulators on a set of quantum circuit benchmarks involving error correction, reversible logic, communication, and quantum search, show that the graph-based approach far outperforms the array-based approach.

Graph-based simulation of quantum computation in the density matrix representation

A central problem in built-in self test (BIST) is how to efficiently generate a small set of test vectors that detect all targeted faults. We propose a novel solution that uses linear algebraic concepts to partition the vector space of tests into subspaces (clusters). A subspace is defined by a compact set of basis vectors. We give an algorithm to compute sets of basis vectors defining the clusters. We also describe a low-cost logic circuit based on Gray codes that reproduces the subspaces from these basis vectors. Experimental results are presented which show that this approach reduces on-chip hardware overhead and test application time, while also guaranteeing full fault coverage.

/pdf/on-chip-test-generation-using-linear-subspaces-3bi95ohyuc.pdf

On-Chip Test Generation Using Linear Subspaces

Quantum algorithms and circuits can, in principle, outperform the best non-quantum (classical) techniques for some hard computational problems. However, this does not necessarily lead to useful applications. To gauge the practical significance of a quantum algorithm, one must weigh it against the best conventional techniques applied to useful instances of the same problem. Grover's quantum search algorithm is one of the most widely studied. 
We identify requirements for Grover's algorithm to be useful in practice: (1) a search application S where classical methods do not provide sufficient scalability; (2) an instantiation of Grover's algorithm Q(S) for S that has a smaller asymptotic worst-case runtime than any classical algorithm C(S) for S; (3) Q(S) with smaller actual runtime for practical instances of S than that of any C(S). 
We show that several commonly-suggested applications fail to satisfy these requirements, and outline directions for future work on quantum search.

Igor L. Markov

Papers

SafeResynth: A new technique for physical synthesis

Efficient Optimization by Modifying the Objective Function: Applications to Timing-Driven

Graph-based simulation of quantum computation in the density matrix representation

On-Chip Test Generation Using Linear Subspaces

Is Quantum Search Practical