Quantum computers are expected to break today’s public key cryptography within a few decades. New cryptosystems are being designed and standardized for the post-quantum era, and a significant proportion of these rely on the hardness of problems like the Shortest Vector Problem to a quantum adversary. In this paper, we describe two variants of a quantum Ising algorithm to solve this problem. One variant is spatially efficient, requiring only O(N log N) qubits where N is the lattice dimension, while the other variant is more robust to noise. Analysis of the algorithms’ performance on a quantum annealer and in numerical simulations show that the more qubit-efficient variant will outperform in the long run, while the other variant is more suitable for near-term implementation.
Scientific Publications
D-Wave is the world’s first commercial supplier of quantum computers. Learn more about the science behind our cutting-edge quantum technology and the use cases it can be applied to. Visit our resource library to read our technical white papers.
2024
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Computational supremacy in quantum simulation
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Quantum fluctuations drive nonmonotonic correlations in a qubit lattice
2023
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Kagome qubit ice
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Quantum error mitigation in quantum annealing
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Tutorial: calibration refinement in quantum annealing
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Milestones on the Quantum Utility Highway
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Quantum Critical Dynamics in a 5,000-Qubit Programmable Spin Glass
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Probing Flux and Charge Noise with Macroscopic Resonant Tunneling
2022
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Multilevel Resonant Tunneling in the Presence of Flux and Charge Noise
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Coherent Quantum Annealing in a Programmable 2,000 Qubit Ising Chain
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Annealing-based Quantum Computing for Combinatorial Optimal Power Flow
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Hybrid Quantum Annealing for Larger-than-QPU Lattice-Structured Problems
2021
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Community Detection in Electrical Grids Using Quantum Annealing
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Quantum Permutation Synchronization
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Multi-qubit Correction for Quantum Annealers
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Essentially Exact Numerical Modelling of Flux Qubit Chains Subject to Charge and Flux Noise
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Architectural Considerations in the Design of a Third-Generation Superconducting Quantum Annealing Processor
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Qubit Spin Ice
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Improving Nonstoquastic Quantum Annealing with Spin-Reversal Transformations
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Computing Molecular Excited States on a D-Wave Quantum Annealer
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Quantum Computing in Green Energy Production
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Scaling Advantage Over Path-Integral Monte Carlo in Quantum Simulation of Geometrically Frustrated Magnets
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Two Quantum Ising Algorithms for the Shortest Vector Problem: One for Now and One for Later
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Quantum Annealing Simulation of Out-of-Equilibrium Magnetization in a Spin-Chain Compound
2020
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Simulating the Shastry-Sutherland Ising Model Using Quantum Annealing
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Improving Performance of Logical Qubits by Parameter Tuning and Topology Compensation
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Performance Benefits of Increased Qubit Connectivity in Quantum Annealing 3-Dimensional Spin Glasses
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De-Signing Hamiltonians for Quantum Adiabatic Optimization
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Coupler-Induced Errors for Flux-qubit Quantum Annealer: Measurements for One-Dimensional Ising Model with Twisted Boundary Fields
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Theory Versus Practice in Annealing-Based Quantum Computing
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Hybrid Quantum Annealing via Molecular Dynamics
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Probing Environmental Spin Polarization with Superconducting Flux Qubits
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Accuracy and Minor Embedding in Subqubo Decomposition with Fully Connected Large Problems
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Demonstration of a Nonstoquastic Hamiltonian in Coupled Superconducting Flux Qubits
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Traffic Signal Optimization on a Square Lattice using the D-Wave Quantum Annealer
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Designing Metamaterials with Quantum Annealing and Factorization Machines
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Reinforcement Quantum Annealing: A Quantum-Assisted Learning Automata Approach
2019
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A Path Towards Quantum Advantage in Training Deep Generative Models with Quantum Annealers
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Unconventional Machine Learning of Genome-Wide Human Cancer Data
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PixelVAE++: Improved PixelVAE with Discrete Prior
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Quantum-Assisted Genetic Algorithm
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The Mathematics of Quantum-Enabled Applications on the D-Wave Quantum Computer
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Demonstration of Nonstoquastic Hamiltonian in Coupled Superconducting Flux Qubits
2018
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Flight Gate Assignment with a Quantum Annealer
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Solving SAT and MaxSAT with a Quantum Annealer: Foundations, Encodings, and Preliminary Results
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Computing Wasserstein Distance for Persistence Diagrams on a Quantum Computer
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Theory of Open Quantum Dynamics with Hybrid Noise
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Observation of Topological Phenomena in a Programmable Lattice of 1,800 Qubits
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Phase Transitions in a Programmable Quantum Spin Glass Simulator
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Mathematical Methods for a Quantum Annealing Computer
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Quantum-Assisted Cluster Analysis on a Quantum Annealing Device
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GumBolt: Extending Gumbel Trick to Boltzmann Priors
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DVAE#: Discrete Variational Autoencoders with Relaxed Boltzmann Priors
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Toward Robustness Against Label Noise in Training Deep Discriminative Neural Networks
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Quantum Variational Autoencoder
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Quantum Annealing Versus Classical Machine Learning Applied to a Simplified Computational Biology Problem
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DVAE++: Discrete Variational Autoencoders with Overlapping Transformations
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Leveraging Adiabatic Quantum Computation for Election Forecasting
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Efficient Combinatorial Optimization Using Quantum Annealing
2017
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Traffic Flow Optimization using a Quantum Annealer
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From Near to Eternity: Spin-Glass Planting, Tiling Puzzles, and Constraint Satisfaction Problems
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A Deceptive Step Towards Quantum Speedup Detection
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Solving a Higgs Optimization Problem with Quantum Annealing for Machine Learning
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Experimental Demonstration of Perturbative Anticrossing Mitigation using Non-Uniform Driver Hamiltonians
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Graph Partitioning using Quantum Annealing on the D-Wave System
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Nonnegative/Binary Matrix Factorization with a D-Wave Quantum Annealer
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Can Quantum Monte Carlo Simulate Quantum Annealing?
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A Study of Complex Deep Learning Networks on High Performance, Neuromorphic, and Quantum Computers
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Quantum Eigenstate Tomography with Qubit Tunneling Spectroscopy
2016
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Benchmarking Quantum Hardware for Training of Fully Visible Boltzmann Machines
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Discrete Variational Autoencoders
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Not Magic…Quantum
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Global Warming: Temperature Estimation in Annealers
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Mapping Constrained Optimization Problems to Quantum Annealing with Application to Fault Diagnosis
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Spanning Tree Calculations on D-Wave 2 Machines
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Quantum Boltzmann Machine
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A Frequency and Sensitivity Tunable Microresonator Array for High-Speed Quantum Processor Readout
2015
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Degeneracy, Degree, and Heavy Tails in Quantum Annealing
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Fast Clique Minor Generation in Chimera Qubit Connectivity Graphs
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Constructing SAT Filters with a Quantum Annealer
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Performance of a Quantum Annealer on Range-Limited Constraint Satisfaction Problems
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Benchmarking a Quantum Annealing Processor with the Time-to-Target Metric
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Searching for Quantum Speedup in Quasistatic Quantum Annealers
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Adiabatic Quantum Computing Challenges
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Computational Role of Collective Tunneling in a Quantum Annealer
2014
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Discrete Optimization Using Quantum Annealing on Sparse Ising Models
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Reexamining Classical and Quantum Models for the D-Wave One Processor
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Quantum Annealing Correction for Random Ising Problems
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A Quantum Annealing Approach for Fault Detection and Diagnosis of Graph-Based Systems
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Quantum Optimization of Fully-Connected Spin Glasses
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A Practical Heuristic for Finding Graph Minors
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Entanglement in a Quantum Annealing Processor
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Consistency Tests of Classical and Quantum Models for a Quantum Annealer
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Architectural Considerations in the Design of a Superconducting Quantum Annealing Processor
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Glassy Chimeras Could be Blind to Quantum Speedup: Designing Better Benchmarks for Quantum Annealing Machines
2013
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Evidence for Temperature Dependent Spin-Diffusion as a Mechanism of Intrinsic Flux Noise in SQUIDs
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Experimental Determination of Ramsey Numbers
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Error Corrected Quantum Annealing with Hundreds of Qubits
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MAX 2-SAT with up to 108 Qubits
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Hearing the Shape of Ising Models: On the Distinguishability Power of Physics
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Experimental Signature of Programmable Quantum Annealing
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Tunneling Spectroscopy Using a Probe Qubit
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Thermally Assisted Quantum Annealing of a 16-Qubit Problem
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Quantum Annealing with More Than One Hundred Qubits
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Adiabatic Quantum Optimization with Qudits
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How Fast Can Quantum Annealers Count?
2012
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Construction of Energy Functions for Lattice Heteropolymer Models: A Case Study in Constraint Satisfaction Programming and Adiabatic Quantum Optimization
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Solving the Graph Isomorphism Problem with a Quantum Annealer
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Algorithmic Approach to Adiabatic Quantum Optimization
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Approximate Diagonalization Method for Large-Scale Hamiltonian
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Robust Classification with Adiabatic Quantum Optimization
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A Near-Term Quantum Computing Approach for Hard Computational Problems in Space Exploration
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Quantum Speedup by Quantum Annealing
2011
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Probing High-Frequency Noise with Macroscopic Resonant Tunneling
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Does Adiabatic Quantum Optimization Fail for NP-Complete Problems?
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Importance of Explicit Vectorization for CPU and GPU Software Performance
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The Ising Model: Teaching an Old Problem New Tricks
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Investigating the Performance of an Adiabatic Quantum Optimization Processor
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Quantum Annealing with Manufactured Spins
2010
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A Scalable Control System for a Superconducting Adiabatic Quantum Optimization Processor
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Experimental Demonstration of a Robust and Scalable Flux Qubit
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High-Performance Physics Simulations Using Multi-Core CPUs and GPGPUs in a Volunteer Computing Context
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Robust Parameter Selection for Parallel Tempering
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Experimental Investigation of an Eight Qubit Unit Cell in a Superconducting Optimization Processor
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Cotunneling in Pairs of Coupled Flux Qubits
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A Scalable Readout System for a Superconducting Adiabatic Quantum Optimization System
2009
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Training a Large Scale Classifier with the Quantum Adiabatic Algorithm
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Geometrical Dependence of the Low-Frequency Noise in Superconducting Flux Qubits
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Non-Markovian Incoherent Quantum Dynamics of a Two-State System
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Decoherence in Adiabatic Quantum Computation
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A Compound Josephson Junction Coupler for Flux Qubits With Minimal Crosstalk
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Landau-Zener Transitions in an Adiabatic Quantum Computer
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Consistency of the Adiabatic Theorem
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Landau-Zener Transitions in the Presence of Spin Environment
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First Order Quantum Phase Transition in Adiabatic Quantum Computation
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The Role of Single Qubit Decoherence Time in Adiabatic Quantum Computation
2008
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Training a Binary Classifier with the Quantum Adiabatic Algorithm
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Minor-Embedding in Adiabatic Quantum Computation: I. The Parameter Setting Problem
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Macroscopic Resonant Tunneling in the Presence of Low Frequency Noise
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On the Construction of Model Hamiltonians for Adiabatic Quantum Computing and its Application to Finding Low Energy Conformations of Lattice Protein Models
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Effect of Local Minima on Adiabatic Quantum Optimization
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Thermally Assisted Adiabatic Quantum Computation
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Probing Noise in Flux Qubits via Macroscopic Resonant Tunneling
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Realizable Hamiltonians for Universal Adiabatic Quantum Computers
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Image Recognition with an Adiabatic Quantum Computer I. Mapping to Quadratic Unconstrained Binary Optimization
2007
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Sign- and Magnitude-Tunable Coupler for Superconducting Flux Qubits
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A Characterization of Global Entanglement
2006
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Rabi Oscillations in Systems with Small Anharmonicity
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Four-Qubit Device with Mixed Couplings
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Adiabatic Quantum Computation with Flux Qubits, First Experimental Results
2005
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Silent Phase Qubit Based on D-Wave Josephson Junctions
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Flux Qubit in Charge-Phase regime
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Mediated Tunable Coupling of Flux Qubits
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Direct Josephson Coupling Between Superconducting Flux Qubits
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Simulated Quantum Computation of Molecular Energies
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Hamiltonian for Coupled Flux Qubits
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Quantum Nondemolition Charge Measurement of a Josephson Qubit
2004
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Evidence for Entangled States of Two Coupled Flux Qubits
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Low-Frequency Measurement of the Tunneling Amplitude in a Flux Qubit
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Quasiparticle Decoherence in D-Wave Superconducting Qubits
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Observation of Macroscopic Landau-Zener Tunneling in a Superconducting Device
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Wigner Distribution Function Formalism for Superconductors and Collisionless Dynamics of the Superconducting Order Parameter
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Dephasing-Insensitive Quantum Information Storage and Processing with Superconducting Qubits
2003
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Non-Equilibrium Quasiclassical Theory for Josephson Structures
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Josephson-Phase Qubit without Tunneling
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Dynamical Effects of an Unconventional Current-Phase Relation in YBCO dc SQUIDs
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Anomalous Current-Phase Relation as Basis for HTS Qubit
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Nonequilibrium Quasiclassical Theory for Josephson Structures
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Quasiclassical Calculations of Spontaneous Current in Restricted Geometries
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Continuous Monitoring of Rabi Oscillations in a Josephson Flux Qubit
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Theory of Weak Continuous Measurements in a Strongly Driven Quantum Bit
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Tunable Coupling of Superconducting Qubits
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Dynamical Effects of an Unconventional Current-Phase Relation in YBCO dc-SQUIDs
2002
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Low-Frequency Characterization of Quantum Tunneling in Flux Qubits
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Multi-Terminal Superconducting Phase Qubit
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High Temperature PI/2-SQUID
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Time Reversal Breaking States and Spontaneous Current Pattern in Josephson Junctions of D-Wave Superconductors
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Quasiclassical Theory of Spontaneous Currents at Surfaces and Interfaces of D-Wave Superconductors
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d+is Versus d+id Time Reversal Symmetry Breaking States in Finite Size Systems
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DC-SQUID Based on the Mesoscopic Multi-Terminal Josephson Junction
2001
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Mesoscopic Multi-Terminal Josephson Structures. I. Effects of Nonlocal Weak Coupling
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Degenerate Ground State in a Mesoscopic YBa2CuO Grain Boundary Josephson Junction
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