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 or see a list of the most important research papers here.

**2024**

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#### Optimization Applications as Quantum Performance Benchmarks

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#### Quantum Annealing-Based Algorithm for Efficient Coalition Formation Among LEO Satellites

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#### Real World Application of Quantum-Classical Optimization for Production Scheduling

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#### Solving the resource constrained project scheduling problem with quantum annealing

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#### Quantum Annealing-Infused Microgrids Formation: Distribution System Restoration and Resilience Enhancement

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#### Short-depth QAOA circuits and quantum annealing on higher-order ising models

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#### Using quantum annealing to design lattice proteins

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#### Scaling Advantage in Approximate Optimization with Quantum Annealing

**2023**

Publication

#### Inter-generational comparison of quantum annealers in solving hard scheduling problems

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#### Effectiveness of quantum annealing for continuous-variable optimization

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#### Efficient low temperature Monte Carlo sampling using quantum annealing

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#### Comparing Three Generations of D-Wave Quantum Annealers for Minor Embedded Combinatorial Optimization Problems

**2022**

Publication

#### On the Emerging Potential of Quantum Annealing Hardware for Combinatorial Optimization

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#### On constant-time quantum annealing and guaranteed approximations for graph optimization problems

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#### Quantum-Enhanced Selection Operators for Evolutionary Algorithms

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#### Quantum annealing algorithms for Boolean tensor networks

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#### High quality thermal Gibbs sampling with quantum annealing hardware

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#### Benchmarking Advantage and D-Wave 2000Q quantum annealers with exact cover problems

Publication

#### Parallel quantum annealing

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#### A Method for Reducing Time-to-Solution in Quantum Annealing Through Pausing

**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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#### SU(2) lattice gauge theory on a quantum annealer

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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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#### Two Quantum Ising Algorithms for the Shortest Vector Problem: One for Now and One for Later

**2020**

Publication

#### 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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#### Hybrid Quantum Annealing via Molecular Dynamics

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#### Accuracy and Minor Embedding in Subqubo Decomposition with Fully Connected Large Problems

Publication

#### Traffic Signal Optimization on a Square Lattice using the D-Wave Quantum Annealer

Publication

#### Designing Metamaterials with Quantum Annealing and Factorization Machines

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#### Reinforcement Quantum Annealing: A Quantum-Assisted Learning Automata Approach

**2019**

Publication

#### Unconventional Machine Learning of Genome-Wide Human Cancer Data

**2018**

Publication

#### Flight Gate Assignment with a Quantum Annealer

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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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#### Quantum Annealing Versus Classical Machine Learning Applied to a Simplified Computational Biology Problem

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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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#### 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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#### 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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#### A Study of Complex Deep Learning Networks on High Performance, Neuromorphic, and Quantum Computers

**2016**

Publication

#### Not Magic…Quantum

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#### Spanning Tree Calculations on D-Wave 2 Machines

**2015**

Publication

#### Adiabatic Quantum Computing Challenges

**2014**

Publication

#### 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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#### Consistency Tests of Classical and Quantum Models for a Quantum Annealer

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

**2013**

Publication

#### Experimental Determination of Ramsey Numbers

Publication

#### 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

Publication

#### Experimental Signature of Programmable Quantum Annealing

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#### Quantum Annealing with More Than One Hundred Qubits

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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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#### Robust Classification with Adiabatic Quantum Optimization

Publication

#### A Near-Term Quantum Computing Approach for Hard Computational Problems in Space Exploration

Publication

#### Quantum Speedup by Quantum Annealing

**2009**

Publication

#### Training a Large Scale Classifier with the Quantum Adiabatic Algorithm

**2008**

Publication

#### Training a Binary Classifier with the Quantum Adiabatic Algorithm

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