Improved quantum circuits for elliptic curve discrete logarithms

We present improved quantum circuits for elliptic curve scalar multiplication, the most costly component in Shor's algorithm to compute discrete logarithms in elliptic curve groups. We optimize low-level components such as reversible integer and modular arithmetic through windowing techniques and more adaptive placement of uncomputing steps, and improve over previous quantum circuits for modular inversion by reformulating the binary Euclidean algorithm. Overall, we obtain an affine Weierstrass point addition circuit that has lower depth and uses fewer $T$ gates than previous circuits. While previous work mostly focuses on minimizing the total number of qubits, we present various trade-offs between different cost metrics including the number of qubits, circuit depth and $T$-gate count. Finally, we provide a full implementation of point addition in the Q# quantum programming language that allows unit tests and automatic quantum resource estimation for all components.Comment: 22 pages, to appear in: Int'l Conf. on Post-Quantum Cryptography (PQCrypto 2020

Improved techniques for preparing eigenstates of fermionic Hamiltonians

Modeling low energy eigenstates of fermionic systems can provide insight into chemical reactions and material properties and is one of the most anticipated applications of quantum computing. We present three techniques for reducing the cost of preparing fermionic Hamiltonian eigenstates using phase estimation. First, we report a polylogarithmic-depth quantum algorithm for antisymmetrizing the initial states required for simulation of fermions in first quantization. This is an exponential improvement over the previous state-of-the-art. Next, we show how to reduce the overhead due to repeated state preparation in phase estimation when the goal is to prepare the ground state to high precision and one has knowledge of an upper bound on the ground state energy that is less than the excited state energy (often the case in quantum chemistry). Finally, we explain how one can perform the time evolution necessary for the phase estimation based preparation of Hamiltonian eigenstates with exactly zero error by using the recently introduced qubitization procedure

OpenFermion: The Electronic Structure Package for Quantum Computers

Quantum simulation of chemistry and materials is predicted to be an important application for both near-term and fault-tolerant quantum devices. However, at present, developing and studying algorithms for these problems can be difficult due to the prohibitive amount of domain knowledge required in both the area of chemistry and quantum algorithms. To help bridge this gap and open the field to more researchers, we have developed the OpenFermion software package (www.openfermion.org). OpenFermion is an open-source software library written largely in Python under an Apache 2.0 license, aimed at enabling the simulation of fermionic models and quantum chemistry problems on quantum hardware. Beginning with an interface to common electronic structure packages, it simplifies the translation between a molecular specification and a quantum circuit for solving or studying the electronic structure problem on a quantum computer, minimizing the amount of domain expertise required to enter the field. The package is designed to be extensible and robust, maintaining high software standards in documentation and testing. This release paper outlines the key motivations behind design choices in OpenFermion and discusses some basic OpenFermion functionality which we believe will aid the community in the development of better quantum algorithms and tools for this exciting area of research.Comment: 22 page

Removing leakage-induced correlated errors in superconducting quantum error correction

Quantum computing can become scalable through error correction, but logical error rates only decrease with system size when physical errors are sufficiently uncorrelated. During computation, unused high energy levels of the qubits can become excited, creating leakage states that are long-lived and mobile. Particularly for superconducting transmon qubits, this leakage opens a path to errors that are correlated in space and time. Here, we report a reset protocol that returns a qubit to the ground state from all relevant higher level states. We test its performance with the bit-flip stabilizer code, a simplified version of the surface code for quantum error correction. We investigate the accumulation and dynamics of leakage during error correction. Using this protocol, we find lower rates of logical errors and an improved scaling and stability of error suppression with increasing qubit number. This demonstration provides a key step on the path towards scalable quantum computing

Palaeogenomic analysis of black rat (Rattus rattus) reveals multiple European introductions associated with human economic history

The distribution of the black rat (Rattus rattus) has been heavily influenced by its association with humans. The dispersal history of this non-native commensal rodent across Europe, however, remains poorly understood, and different introductions may have occurred during the Roman and medieval periods. Here, in order to reconstruct the population history of European black rats, we generated a de novo genome assembly of the black rat, 67 ancient black rat mitogenomes and 36 ancient nuclear genomes from sites spanning the 1st-17th centuries CE in Europe and North Africa. Analyses of mitochondrial DNA confirm that black rats were introduced into the Mediterranean and Europe from Southwest Asia. Genomic analyses of the ancient rats reveal a population turnover in temperate Europe between the 6th and 10th centuries CE, coincident with an archaeologically attested decline in the black rat population. The near disappearance and re-emergence of black rats in Europe may have been the result of the breakdown of the Roman Empire, the First Plague Pandemic, and/or post-Roman climatic cooling.Competing Interest StatementThe authors have declared no competing interest.- Results and Discussion -- The demographic history of Rattus rattus and its closely related species -- A global phylogeography of the black rat based on mitochondrial DNA -- Ancient genomes reveal the relationships of European black rats over space and time - Discussion - Method

Palaeogenomic analysis of black rat (Rattus rattus) reveals multiple European introductions associated with human economic history

The distribution of the black rat (Rattus rattus) has been heavily influenced by its association with humans. The dispersal history of this non-native commensal rodent across Europe, however, remains poorly understood, and different introductions may have occurred during the Roman and medieval periods. Here, in order to reconstruct the population history of European black rats, we first generate a de novo genome assembly of the black rat. We then sequence 67 ancient and three modern black rat mitogenomes, and 36 ancient and three modern nuclear genomes from archaeological sites spanning the 1st-17th centuries CE in Europe and North Africa. Analyses of our newly reported sequences, together with published mitochondrial DNA sequences, confirm that black rats were introduced into the Mediterranean and Europe from Southwest Asia. Genomic analyses of the ancient rats reveal a population turnover in temperate Europe between the 6th and 10th centuries CE, coincident with an archaeologically attested decline in the black rat population. The near disappearance and re-emergence of black rats in Europe may have been the result of the breakdown of the Roman Empire, the First Plague Pandemic, and/or post-Roman climatic cooling.Peer reviewe