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.

## Our Resources

Publication

#### Quantum error mitigation in quantum annealing

Publication

#### Tutorial: calibration refinement in quantum annealing

White Paper

#### High-coherence fluxonium as a probe of D-Wave’s QPU environment

White Paper

#### Measuring Performance of the Leap Constrained Quadratic Model Solver: August 2023 Update

Application

#### Quantum computing for transport network design problems

Application

#### Optimization of Image Acquisition for Earth Observation Satellites via Quantum Computing

Application

#### Updated: Route Optimization for Multimodal Transport Systems

Application

#### Quantum in Production: Maximizing TV Commercial Reach

Publication

#### Milestones on the Quantum Utility Highway

Application

#### Efficient Earth Observation Satellites Mission Planning with Quantum Algorithm

Publication

#### Quantum Critical Dynamics in a 5,000-Qubit Programmable Spin Glass

Application

#### Quantum in Production: Optimizing eCommerce Logistics

Publication

#### Probing Flux and Charge Noise with Macroscopic Resonant Tunneling

Publication

#### Multilevel Resonant Tunneling in the Presence of Flux and Charge Noise

Publication

#### Coherent Quantum Annealing in a Programmable 2,000 Qubit Ising Chain

Publication

#### Annealing-based Quantum Computing for Combinatorial Optimal Power Flow

Application

#### Hybrid Quantum-Classical Algorithm for Solving Financial and Logistic Network

White Paper

#### Hybrid Solvers for Quadratic Optimization

Industry Brief

#### Quantum for Financial Services Overview

Industry Brief

#### Quantum for Life Sciences Overview

Industry Brief

#### Quantum for Manufacturing Overview

Application

#### Quantum Annealing Based Factory Layout Planning

Application

#### Quantum Annealing Algorithm for Expected Shortfall based Dynamic Asset Allocation

Application

#### Optimizing the Renewable Energy Grid with Quantum Computing

Publication

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

Application

#### Quantum Computing Approach to Railway Dispatching and Conflict Management Optimization on Single-Track Railway Lines

White Paper

#### The D-Wave Advantage System: An Overview

Application

#### Community Detection in Electrical Grids Using Quantum Annealing

Publication

#### Community Detection in Electrical Grids Using Quantum Annealing

Application

#### Paint Shop Optimization with Quantum Annealing

Application

#### Designing Peptide Therapeutics on a Quantum Computer

Application

#### Quadratic and Higher-Order Unconstrained Binary Optimization of Railway Dispatching Problem

Application

#### Sensor Placement Problems In Real Water Distribution Networks

Application

#### Traffic Flow Optimization using the D-Wave Quantum Annealer

Application

#### Reinforcement Learning using Quantum Boltzmann Machines on the D-Wave System

Application

#### Graph Clustering Approaches Using Quantum Annealing

Application

#### Particle Track Pattern Recognition via Content Addressable Memory and Adiabatic Quantum Optimization

Application

#### Hybrid Quantum Investment Optimization with Minimal Holding Period

Publication

#### Quantum Permutation Synchronization

Application

#### PayPal: Harnessing Quantum Computing in FinTech

Application

#### Mphasis: Business Applications of Quantum Computing

Application

#### Solving the Last Mile Resupply Problem

Application

#### Collision-free Route Planning of Urban Air Mobility

Application

#### Asset Liability Modelling in the Quantum Era

Application

#### Cryptocurrency Payments in Networks

Application

#### Quantum Molecular Unfolding

Application

#### Feature Selection for Recommender Systems

Application

#### Scaling Truly Farm to Table

Application

#### Quantum for Energy Exploration

Application

#### Energy Efficient Mobile Network Routing

Application

#### Practical Use of Quantum Computers in the Automobile Industry

Application

#### Optimization of Complex Systems

Application

#### Magnetic Monopole Kinetics in Qubit Spin Ice

Application

#### Lattice Qauge Theory

Application

#### Lossy Compression of Statistical Data

Application

#### Opportunities and Challenges in Quantum-Enhanced Machine Learning in Near-Term Quantum Computers

Application

#### Bayesian Networks Based Hybrid Quantum Classical Machine Learning

Application

#### C Code In, D-Wave QMI Out

Application

#### Quantum in Telecom

Publication

#### Multi-qubit Correction for Quantum Annealers

Application

#### Towards an Optimal Hybrid Algorithm for EV Charging Stations Placement

Application

#### GPS: Improvement in the formulation of the TSP

Publication

#### Essentially Exact Numerical Modelling of Flux Qubit Chains Subject to Charge and Flux Noise

Application

#### Optimising Rolling Stock Planning

Application

#### Framework for Implementing Quantum Applications in Financial Institutions

Application

#### Creating Sustainable Cities through Waste Collection Optimization

Application

#### Delivering Value with Quantum Computing: Kidney Exchange Network

Publication

#### Architectural Considerations in the Design of a Third-Generation Superconducting Quantum Annealing Processor

Publication

#### Qubit Spin Ice

Application

#### Applying a Quantum Annealing Based Restricted Boltzmann Machine for MNIST Handwritten Digit Classification

Publication

#### Improving Nonstoquastic Quantum Annealing with Spin-Reversal Transformations

Application

#### Statistical Classification of High-throughput Multi-nomics Cancer Data on Quantum Computing Architecture

Application

#### Optimal Control of Traffic Signals using Quantum Annealing

Application

#### mRNA Codon Optimization on Quantum Computers

Application

#### Save-On-Foods Quantum Computing in Grocery

Application

#### Display Advertising Optimization

Application

#### Quantum Annealing for Asset Sustainment

Application

#### Quantum Computing Algorithms for Optimized Planning and Scheduling

Application

#### Quantum Annealing Based Optimizations of Robotic Movement in Manufacturing

Application

#### Applications of Quantum Annealing in Computational Finance

Application

#### T-QARD and DENSO: A Great Collaboration for Factory Optimization

Application

#### Item Listing Optimization

Application

#### Computing Protein-Ligand Binding Free Energy using Quantum Annealing

Application

#### Telecommunications Network Optimisation

Application

#### Reconstructing Proton-Proton Collision Positions at the Large Hadron Collider with D-Wave

Application

#### Flight Gate Assignment with a Quantum Annealer

Application

#### Novel Machine Learning Algorithms for Quantum Annealing with Applications in High Energy Physics

Application

#### Quantum Annealing with Continuous Variables: An Application to Matrix Factorization

Application

#### Multilevel Quantum Annealing for Graph Partitioning

Application

#### Minimizing Polynomial Functions on a D-Wave Quantum Computer: An Overview

Application

#### Partition of Large Optimization Problems with One-Hot Constraint

Application

#### Allocating Railway Traffic with QUBO Formulated Models

Application

#### Demo: Antenna Selection

Application

#### Demo: Satellite Placement

Application

#### Our Experience Developing an Optimization Service Using D-Wave

Application

#### Quantitative Reverse Stress Testing using Simulated and Quantum Annealing Applied to XVA

Application

#### OpenJij: An Open-Source Project Towards a Unified Annealing Platform

Application

#### Demo: Sudoku

Application

#### Demo: Circuit Fault Diagnosis

Application

#### Demo: Use Mutual Information to Select Features in a Titanic Data Set

Application

#### Our Target Applications and Embedding Algorithm of Subproblem

Application

#### Demo: Solving a Maze

Application

#### Demo: Factoring

Application

#### Optimizing Quantum Annealing Performance via Quantum Control

Application

#### Demo: Analyzing the Structural Imbalance on a Signed Social Network.

Application

#### Demo: Graph Partitioning

Application

#### Quantum-Computational Hydrologic Inverse Analysis

Application

#### Demo: A Minimum Vertex Problem with Pipelines

Application

#### Programmation d'un D-Wave en Logique

Application

#### Quantum Programming Infrastructure

Application

#### Quantum Computation in a Topological Data Analysis Pipeline

Application

#### Forward-Reverse Error Mitigation Procedure for Quantum Annealers

Application

#### Simulations of the Ising Model on a Shastry-Sutherland Lattice by Quantum Annealing

Application

#### Quantum Chemistry on Quantum Annealers

Application

#### Simulating Electronic Structure on Noisy Quantum Hardware: from Gate to Annealing Models

Application

#### Demo: Map Coloring

Application

#### Leveraging Quantum Annealing for Large MIMO Processing in Cluster-Based Radio Access Networks

Application

#### Quantum Annealing for Factorization

Application

#### Demo: Job Shop Scheduling

Application

#### Observation of Topological Phenomena in a Programmable Lattice of 1,800 Qubits

Application

#### Solutions to Systems of Equations from Adiabatic Quantum Computing

Application

#### Efficiently Embedding QUBO Problems on Adiabatic Quantum Computers

Application

#### Challenging Collaborations with T-QARD

Application

#### Methods to Improve the Minimization of an Ising Objective Function

Application

#### Financial Portfolio Optimization

Application

#### Designing Metamaterials with the D-Wave 2000Q Quantum Annealer

Application

#### Case Studies in Machine Learning

Application

#### Hybrid Algorithm Development for Production Applications

Application

#### A Hybrid Quantum-Classical Recommendation System

Application

#### RNA Folding Using Quantum Computers

Application

#### Near-Term QC Applications in Industry

Application

#### Quantum Annealing Applied to Optimization Problems in Radiation Medicine

Application

#### A Quantum Macro Assembler

Application

#### A Quantum Annealing Approach to the Minimum Multicut Problem on General Graphs

Application

#### Quantum Annealing Approaches to Graph Partitioning on the D-Wave System

Application

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

Application

#### XACC: Enabling Quantum Acceleration in Scientific High-Performance Computing

Application

#### Nonnegative/Binary Matrix Factorization with a D-Wave Quantum Annealer

Application

#### Quantum Machine Learning for Election Modeling

Application

#### Phase Transitions in a Programmable Quantum Spin Glass Simulator

Application

#### Free Energy-Based Reinforcement Learning Using a Quantum Processor

Application

#### Material Simulation on D-Wave

Application

#### Job Shop Scheduling

Application

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

Application

#### Model Predictive Control for Finite Input Systems using the D-Wave Quantum Annealer

Application

#### Number Partitioning Problem

Application

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

Application

#### Griffiths-McCoy Singularity on the Diluted Chimera Graph: Monte Carlo Simulations and Experiments on the Quantum Hardware

Application

#### An Application of Quantum Annealing Computing to Seismic Inversion

Application

#### Image Acquisition Planning for Earth Observation Satellites with a Quantum Annealer

Application

#### A Hybrid Quantum Enabled RBM Advantage: Convolutional Autoencoders for Quantum Image Compression and Generative Learning

Application

#### On the Computational Viability of Quantum Optimization for PMU Placement

Application

#### Quantum-Assisted Greedy Algorithms

Application

#### Maximum-Likelihood Channel Decoding with Quantum Annealing Machine

Application

#### Discriminating Non-Isomorphic Graphs with an Experimental Quantum Annealer

Application

#### Dynamics of Longitudinal Magnetization in Transverse-Field Quantum Ising Model: From Symmetry-Breaking Gap to Kibble-Zurek Mechanism

Application

#### Dynamic Portfolio Optimization with Real Datasets Using Quantum Processors and Quantum-Inspired Tensor Networks

Application

#### Quantum Isomer Search

Application

#### Towards Analyzing Large Graphs with Quantum Annealing and Quantum Gate Computers

Application

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

Application

#### A Deceptive Step Towards Quantum Speedup Detection

Application

#### Not Magic…Quantum

Application

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

Application

#### Accuracy and Minor Embedding in Subqubo Decomposition with Fully Connected Large Problems: a Case Study About the Number Partitioning Problem

Application

#### Disorder-Induced Entropic Potential in a Flux Qubit Quantum Annealer

Application

#### Hybrid Quantum Annealing via Molecular Dynamics

Application

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

Application

#### Mathematical Methods for a Quantum Annealing Computer

Application

#### Quantum-Assisted Cluster Analysis on a Quantum Annealing Device

Application

#### Efficient Combinatorial Optimization Using Quantum Annealing

Application

#### Support Vector Machines on the D-Wave Quantum Annealer

Application

#### A Delay-Aware Trick for D-Wave to Solve Combinatorial Optimization in Business with Many Independent Subproblems in Bulk

Application

#### Multi-Agent Reinforcement Learning Using Simulated Quantum Annealing

Application

#### Boosting Quantum Computing Performance Through Services and Software

Application

#### Picking Efficient Portfolios from 3,171 US Common Stocks with New Quantum and Classical Solvers

Application

#### City as a Service: The Future of City Optimization by Quantum Computer and AI

Application

#### Development of Feature Selection Module in AutoML by Using D-Wave Advantage

Application

#### Portfolio Optimization Using the D-Wave Quantum Annealer

Application

#### Cooperative Control of Large Scale Traffic Signals

Application

#### Electronic Structure Calculations on Quantum Annealers

Application

#### Ferromagnetically Shifting the Power of Pausing

Application

#### Financial Portfolio Optimization Using Quantum Annealing

Application

#### Hamiltonian Path and Vehicle Routing Problems on Hybrid Solvers

Application

#### Scaling Asset Sustainment to Industrial Size: with Advantage

Application

#### Homomorphic Encryption for Quantum Annealing with Spin Reversal Transformations

Application

#### Hybrid-Quantum Approach for Optimal Lockdown

Application

#### Vehicle Routing Problems and Quantum Hybrid Systems

Application

#### Quantum Computing in Green Energy Production

Application

#### Multimodal Container Planning: a QUBO Formulation and Implementation on a Quantum Annealer

Application

#### Optimal Solution of the Nurse-Physician Scheduling Problem for Covid-19 Clinics Using Quantum Annealing

Application

#### Performance Evaluation of Domain Wall Encoding Scheme

Application

#### Optimization Problems for Benchmarking the Hybrid Solver Service Version 2 and the Advantage QPU

Application

#### Particle Track Pattern Recognition via Quantum Associative Memory

Application

#### QAR-Lab: Site Report

Application

#### Quantum and Classical Machine Learning for the Classification of Non‑Small‑Cell lung Cancer Patients

Application

#### Using Advantage to Solve Insurmountable Business Problems

Application

#### Quantum and Noisy: A Single-Qubit Study Reveals Interesting Properties of the D-Wave Annealer

Application

#### Quantum Annealing Approach to Molecule Unfolding

Application

#### Quantum Annealers as Quantum Laboratories

Application

#### Quantum Computing Applications for Financial Services

Application

#### Several Advanced Products by Using Quantum Annealing

Application

#### Simulating Quantum Configurational Tunneling Using Machine Learning Assisted Quantum Computation

Application

#### Solutions to Integer Programming from Quantum Annealing

Application

#### Solving Vehicle Routing Problem Using Quantum Annealing

Application

#### T-QARD Research Network Update

Application

#### The Effect of Reduced Noise on Annealing Search

Application

#### The Impact of Embedding on Quantum-Annealing Sampling: A Computer-Aided Design Perspective

Application

#### The Performance of Corrupted Biased Ferromagnets on D-Wave's 2000Q

Application

#### Towards Prediction of Financial Crashes with a D-Wave Quantum Computer

Application

#### Unfolding Measurement Distributions in High Energy Physics Experiments via Quantum Annealing

Application

#### Using Advanced Annealing Options of D-Wave 2000Q

Application

#### Benchmarking Quantum Annealing Controls with Portfolio Optimization

Application

#### Forecasting Financial Crashes with Quantum Computing

Application

#### Portfolio Optimization of 60 Stocks Using Classical and Quantum Algorithms

Application

#### Computing Molecular Excited States on a D-Wave Quantum Annealer

Publication

#### Computing Molecular Excited States on a D-Wave Quantum Annealer

Application

#### Quantum Portfolio Optimization with Investment Bands and Target Volatility

Publication

#### Quantum Computing in Green Energy Production

Publication

#### Scaling Advantage Over Path-Integral Monte Carlo in Quantum Simulation of Geometrically Frustrated Magnets

Publication

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

Publication

#### Quantum Annealing Simulation of Out-of-Equilibrium Magnetization in a Spin-Chain Compound

Publication

#### Simulating the Shastry-Sutherland Ising Model Using Quantum Annealing

Publication

#### Improving Performance of Logical Qubits by Parameter Tuning and Topology Compensation

Publication

#### Performance Benefits of Increased Qubit Connectivity in Quantum Annealing 3-Dimensional Spin Glasses

White Paper

#### D-Wave Hybrid Solver Service + Advantage: Technology Update

White Paper

#### Ocean Programs for Beginners

White Paper

#### Choosing Good Problems for Quantum Annealing

Publication

#### Coupler-Induced Errors for Flux-qubit Quantum Annealer: Measurements for One-Dimensional Ising Model with Twisted Boundary Fields

White Paper

#### Hybrid Solver for Discrete Quadratic Models

Publication

#### Theory Versus Practice in Annealing-Based Quantum Computing

White Paper

#### D-Wave Hybrid Solver Service: An Overview

White Paper

#### Programming the D-Wave QPU: Setting the Chain Strength

Publication

#### Hybrid Quantum Annealing via Molecular Dynamics

Publication

#### Probing Environmental Spin Polarization with Superconducting Flux Qubits

Publication

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

Publication

#### Demonstration of a Nonstoquastic Hamiltonian in Coupled Superconducting Flux Qubits

Publication

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

Publication

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

Publication

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

Publication

#### A Path Towards Quantum Advantage in Training Deep Generative Models with Quantum Annealers

White Paper

#### Next-Generation Topology of D-Wave Quantum Processors

Publication

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

Publication

#### PixelVAE++: Improved PixelVAE with Discrete Prior

White Paper

#### Practical Annealing-Based Quantum Computing

Publication

#### Quantum-Assisted Genetic Algorithm

Publication

#### The Mathematics of Quantum-Enabled Applications on the D-Wave Quantum Computer

White Paper

#### Probing Mid-Band and Broad-Band Noise in Lower-Noise D-Wave 2000Q Fabrication Stacks

Publication

#### Demonstration of Nonstoquastic Hamiltonian in Coupled Superconducting Flux Qubits

White Paper

#### Improved Coherence Leads to Gains in Quantum Annealing Performance

White Paper

#### Whitepaper

White Paper

#### A Head-to-Head Comparison of D-Wave and Rigetti QPUs

Publication

#### Flight Gate Assignment with a Quantum Annealer

Publication

#### Solving SAT and MaxSAT with a Quantum Annealer: Foundations, Encodings, and Preliminary Results

Publication

#### Computing Wasserstein Distance for Persistence Diagrams on a Quantum Computer

Publication

#### Theory of Open Quantum Dynamics with Hybrid Noise

White Paper

#### Performance Advantage in Quantum Boltzmann Sampling

Publication

#### Observation of Topological Phenomena in a Programmable Lattice of 1,800 Qubits

White Paper

#### Synopsis: Phase Transitions in a Programmable Quantum Spin-Glass Simulator

Publication

#### Phase Transitions in a Programmable Quantum Spin Glass Simulator

Publication

#### Mathematical Methods for a Quantum Annealing Computer

Publication

#### Quantum-Assisted Cluster Analysis on a Quantum Annealing Device

Publication

#### GumBolt: Extending Gumbel Trick to Boltzmann Priors

Publication

#### DVAE#: Discrete Variational Autoencoders with Relaxed Boltzmann Priors

Publication

#### Toward Robustness Against Label Noise in Training Deep Discriminative Neural Networks

Publication

#### Quantum Variational Autoencoder

Publication

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

Publication

#### DVAE++: Discrete Variational Autoencoders with Overlapping Transformations

Publication

#### Leveraging Adiabatic Quantum Computation for Election Forecasting

Publication

#### Efficient Combinatorial Optimization Using Quantum Annealing

Publication

#### Traffic Flow Optimization using a Quantum Annealer

Publication

#### From Near to Eternity: Spin-Glass Planting, Tiling Puzzles, and Constraint Satisfaction Problems

White Paper

#### Virtual Graphs for High-Performance Embedded Topologies

White Paper

#### Reverse Quantum Annealing for Local Refinement of Solutions

Publication

#### A Deceptive Step Towards Quantum Speedup Detection

White Paper

#### Quantum Performance Evaluation: A Short Reading List

Publication

#### Solving a Higgs Optimization Problem with Quantum Annealing for Machine Learning

White Paper

#### Optimization with Clause Problems

Publication

#### Experimental Demonstration of Perturbative Anticrossing Mitigation using Non-Uniform Driver Hamiltonians

Publication

#### Graph Partitioning using Quantum Annealing on the D-Wave System

Publication

#### Nonnegative/Binary Matrix Factorization with a D-Wave Quantum Annealer

Publication

#### Can Quantum Monte Carlo Simulate Quantum Annealing?

White Paper

#### Quantum Annealing Amid Local Ruggedness and Global Frustration

Publication

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

Publication

#### Quantum Eigenstate Tomography with Qubit Tunneling Spectroscopy

White Paper

#### Computational Power Consumption and Speedup

White Paper

#### Limits on Parallel Speedup for Classical Ising Model Solvers

White Paper

#### Boosting Integer Factoring Performance via Quantum Annealing Offsets

Publication

#### Benchmarking Quantum Hardware for Training of Fully Visible Boltzmann Machines

Publication

#### Discrete Variational Autoencoders

Publication

#### Not Magic…Quantum

Publication

#### Global Warming: Temperature Estimation in Annealers

Publication

#### Mapping Constrained Optimization Problems to Quantum Annealing with Application to Fault Diagnosis

Publication

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

Publication

#### Quantum Boltzmann Machine

Publication

#### A Frequency and Sensitivity Tunable Microresonator Array for High-Speed Quantum Processor Readout

Publication

#### Degeneracy, Degree, and Heavy Tails in Quantum Annealing

Publication

#### Fast Clique Minor Generation in Chimera Qubit Connectivity Graphs

Publication

#### Constructing SAT Filters with a Quantum Annealer

Publication

#### Performance of a Quantum Annealer on Range-Limited Constraint Satisfaction Problems

Publication

#### Benchmarking a Quantum Annealing Processor with the Time-to-Target Metric

Publication

#### Searching for Quantum Speedup in Quasistatic Quantum Annealers

Publication

#### Adiabatic Quantum Computing Challenges

Publication

#### Computational Role of Collective Tunneling in a Quantum Annealer

Publication

#### Discrete Optimization Using Quantum Annealing on Sparse Ising Models

Publication

#### Reexamining Classical and Quantum Models for the D-Wave One Processor

Publication

#### Quantum Annealing Correction for Random Ising Problems

Publication

#### A Quantum Annealing Approach for Fault Detection and Diagnosis of Graph-Based Systems

Publication

#### Quantum Optimization of Fully-Connected Spin Glasses

Publication

#### A Practical Heuristic for Finding Graph Minors

Publication

#### Entanglement in a Quantum Annealing Processor

Publication

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

Publication

#### Architectural Considerations in the Design of a Superconducting Quantum Annealing Processor

Publication