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.

**2022**

Publication

#### This is a test

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#### Coherent Quantum Annealing in a Programmable 2000-qubit Ising Chain

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#### Hybrid Quantum Annealing for Larger-than-QPU Lattice-Structured Problems

**2021**

Publication

#### 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**

Publication

#### 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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#### 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**

Publication

#### 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**

Publication

#### 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**

Publication

#### 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**

Publication

#### 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**

Publication

#### 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

Publication

#### Glassy Chimeras Could be Blind to Quantum Speedup: Designing Better Benchmarks for Quantum Annealing Machines

**2013**

Publication

#### Evidence for Temperature Dependent Spin-Diffusion as a Mechanism of Intrinsic Flux Noise in SQUIDs

Publication

#### 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**

Publication

#### 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**

Publication

#### 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**

Publication

#### 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**

Publication

#### 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**

Publication

#### 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**

Publication

#### Sign- and Magnitude-Tunable Coupler for Superconducting Flux Qubits

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#### A Characterization of Global Entanglement

**2006**

Publication

#### 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**

Publication

#### 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**

Publication

#### 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

Publication

#### 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

Publication

#### Dephasing-Insensitive Quantum Information Storage and Processing with Superconducting Qubits

**2003**

Publication

#### 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**

Publication

#### 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**

Publication

#### 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

Publication