Executive Summary

The specialization of quantum annealing to optimization problems may at first seem a practical limitation to the adoption of this quantum technology. But, the applications referenced in this research paper suggest that the niche filled by quantum annealers is not insignificant! 

Optimization problems are all around us – from the order in which we pay our bills to when we retire, from the design of the gardens in our back yards to when our trash gets picked up, from our work schedules to the flow of traffic, from where we park when we do get to work to the energy of the electrical grid lighting our way home at night – modern civilization runs on optimization.

What this research paper makes clear is that the specialization of quantum annealing has actually supported the pragmatic development of this technology and led to well-articulated sales verticals for quantum annealers.

Introduction

This year, in an industry first, D-Wave’s quantum annealing methods demonstrated quantum superiority [163, 201]. An international collaboration of scientists led by D-Wave performed simulations of quantum dynamics in programmable spin glasses on both D-Wave’s Advantage2 annealing quantum computer and the Frontier supercomputer at the Department of Energy’s (DOE) Oak Ridge National Laboratory [201]. As stated by the research team [163]:

“We demonstrate area-law scaling of entanglement in the model quench in two-, three- and infinite-dimensional spin glasses, supporting the observed stretched-exponential scaling of effort for classical approaches… and conclude that no known approach can achieve the same accuracy as the quantum annealer within a reasonable timeframe. Thus quantum annealers can answer questions of practical importance that classical computers cannot”.

Amazingly, D-Wave’s quantum computer performed the most complex simulation in minutes and with a level of accuracy that would take nearly one million years and require more than the entire world’s annual electricity consumption if using the DOE’s Frontier supercomputer [201].

Today, let’s take a closer look at quantum annealing: what is it, how does it work, and what are the applications?

What Is Quantum Annealing (QA)?

Tadashi Kadowaki and Hidetoshi Nishimori were first to propose quantum annealing (QA), a metaheuristic, quantum optimization algorithm that can be used to solve combinatorial optimization problems with problem-size-independent time complexity [2, 6, 8, 9, 10, 11, 14, 15, 16, 17, 18, 20, 23, 25, 28, 29, 30, 31, 32, 33, 34, 40, 42, 44, 45, 46, 47, 52, 136, 137, 138, 139, 152, 154, 157]. 

Related to adiabatic quantum computation and inspired by simulated annealing, but with thermal fluctuations replaced by adjustable quantum fluctuations [8, 12, 13, 20, 23, 25, 26, 42, 47, 167], QA is a non-Turing, restricted model of computation unable to execute Shor’s algorithm [7] and analogous to programmable RAM [1].

What Is The QA Problem Solving Process?

The fundamental concept in QA is transforming a combinatorial optimization problem into an energy function [167]. Quantum annealing uses the quantum tunnelling effect generated by quantum fluctuations to free the algorithm from the local optimal solution so that it can efficiently explore the energy landscape defined by the cost function – and achieve the global optimum quickly and accurately. This global optimum is the lowest energy state, or ground state, of the Hamiltonian operator (Ê), and corresponds to the problem’s best solution [2, 6, 8, 14, 15, 16, 23, 25, 28, 34, 42, 48, 56, 135, 144, 152, 167].

Importantly, the quantum annealing problem solving process requires a quantum annealer, a specialized quantum device [42, 43, 47, 57, 58, 119, 152]. The steps below are those required to solve industry-relevant problems using D-Wave’s quantum annealers [52].

1. QUBO Formulation

By design, quantum annealers are especially suited for quadratic unconstrained binary optimization (QUBO) problems [2, 8, 14, 15, 16, 28, 59, 60, 121, 132, 133, 134], and QUBO has become the standard input format to which the real-world application problems are converted [52]. An Ising model is formulated to represent a particular QUBO problem [19, 20, 138, 152].

2. Minor-Embedding

After QUBO formulation, the logical graph (where each node represents a variable and each edge denotes the interaction term between a pair of variables) is extracted from the Ising model [9, 52, 135]. The graph essentially shows the wiring of the connections, but not their strength – it maps the problem to the underlying hardware. This mapping to the hardware topology is called embedding, and the standard technique utilized is referred to as “minor-embedding” [23, 27]. Although well established algorithms exist, the minor-embedding process is, by itself, a non-trivial problem – it is the dominant costly step in terms of time and the largest detractor on performance for QA, as well [52].

3. Programming

Programming the quantum annealer requires setting the parameters that define the problem to be solved, also called final Hamiltonian. This involves setting the weights for each qubit bias (to control the magnetic field acting on the qubit) and coupler strength (to control the interaction between qubits) [52].

4. Initialization

After programming, the spin configuration of the QPU is initialized as the lowest energy configuration of an easy-to-implement, time-dependent, initial Hamiltonian, where the qubits are placed in an equal superposition of all possible states [19, 20, 52, 138, 152].

5. Annealing

In this step, the Ising model is solved. The system transitions from driver Hamiltonian (which governs the initial Hamiltonian) to the problem Hamiltonian gradually, taking the form of the Ising model and presenting the low-energy solution at the end of annealing [2, 8, 19, 20, 23, 24, 52, 135, 138, 152]. This step can also be the core of a hybrid approach where the inner loop of the calculation is performed by a quantum processor. Annealing can also be combined with a reverse annealing phase, which initializes the quantum annealer with a known (classical) solution and searches the state space around this local optimum [52].

6. Readout

At the end of the annealing phase, the qubits are in an eigenstate, or superposition of eigenstates, in the computational basis, where each eigenstate represents a possible minimum of the final Hamiltonian. The individual spin values of the qubits are read out and stored externally representing a candidate solution to the original problem [52].

7. Resampling

Quantum annealing is a heuristic, which means that there is only ever a non-zero probability that the computation results in a ground state of the system. Therefore, the anneal-readout cycle is repeated many times per input to acquire multiple candidate solutions [52].

Visualization: The Quantum Algorithm Workflow On An Annealer

Visualization of a typical quantum algorithm workflow on a D-Wave quantum annealer as presented by Yarkoni et al. [52]. The image is an adapted and extended version of the workflow presented by Goodrich et al. [200].

What Are The Applications Of Quantum Annealing?

The ability to efficiently explore high-dimensional combinational spaces makes quantum annealing capable of handling a wide range of optimization tasks [2, 8, 10, 11, 16, 21, 22, 142], and quantum annealing has been tested both for numerous applications and across multiple domains [28, 42, 58, 135, 152, 157]. The rest of this section is dedicated to those applications and industries finding utility in quantum annealing.

Aerospace

• Agile Earth Observation Scheduling – [55]
• Compressive Sensing [147, 148]
• Space Exploration – [53, 166]
• Trajectory Optimization – [51, 54]

Artificial Intelligence

• AI Training – [150, 151, 197]
• Associative Adversarial Networks (AANs) – [64]
• Autoencoders – [172]
• Machine Learning – [37, 61, 62, 65, 68]
• Reinforcement Learning – [47, 171]
• Restricted Boltzmann Machines (RBMs) – [66]
• Support Vector Machines (SNMs) – [199]
• Variational Neural Annealing (VNA) – [63]

Cybersecurity

• RSA-2048 – [167]

Energy

• Electrical Grid Optimization – [184, 186]
• EV Charging Stations – [187]
• Flow Optimization – [98, 159]
• Model Predictive Control (MPC) – [67]
• Refinery Scheduling – [102, 183]

Environmental Science

• Hydrologic Analysis – [160] 
• Mining-Induced Surface Change – [94]
• Seismic Inversion – [161, 162]
• Waste Collection Optimization – [185]
• Wave Propagation – [157]

Finance

• Banking & Financial Services – [193, 195]
• Holding Periods – [123]
• Monte Carlo Simulations – [153]
• Optimal Trading – [122]
• Portfolio Optimization – [2, 5, 49, 50, 194]

Life Science

• Computational Biology – [12]
• Drug Discovery – [188, 189]
• Genome Assembly – [116]
• Life Sciences – [190]
• Protein Design – [35, 164, 191, 192]
• Quantum Chemistry – [38]

Logistics

• Autonomous Vehicles – [149, 179]
• Cargo Loading – [177]
• Garden Optimization – [101]
• Fault Diagnosis – [146] 
• Logistics Routing – [156]
• Manufacturing – [168, 169, 170]
• Neighbourhood Descent – [118]
• Production Scheduling – [99, 103, 104, 106, 165, 178]
• Traffic Flow & Parking Management – [4, 105, 108, 111, 144]
• Vehicle & Rail Routing – [3, 110, 196]
• Workforce Scheduling – [44, 109, 140, 141, 174, 175, 176]

Marketing

• Election Forecasting – [112]
• Search Engine Ranking – [36] 

Materials Science

• Lennard–Jones Clusters – [115]
• Material Thermodynamics – [114, 155]
• Metamaterial Design – [16]
• Molecules – [113, 117, 158]
• Nuclear Shells – [56]
• Particle Tracking – [100]
• Qubit Lattice – [198]

Mathematics

• Big Data – [39, 46, 124, 125]
• Boolean Satisfiability – [143]
• Geometric Phase – [86]
• Graph Theory – [69, 70, 71, 72, 73, 74, 75, 76, 78, 79, 81, 82]
• Hilbert’s Tenth – [87]
• Integers, Polynomials, Prime Factorization – [85, 90, 92, 93, 96, 97]
• Jarzynski Equality – [88]
• Matrices – [84, 95, 145]
• Measurement Distributions – [91]
• NP-Complete – [40]
• SAT, MaxSAT, MAX-CUT, Max k-SAT – [80, 83, 89]
• Subgradient Approaches – [77] 

Telecommunications

• Graph-Based System Fault Detection – [131]
• MIMO – [41]
• Telecoms – [181, 182]
• Wireless Networks – [107, 120, 126, 127, 128, 129, 130, 173, 180]

Final Thoughts

For all of our optimization moments, quantum annealing stands to demonstrate its quantum supremacy – the right tool for the job, the quantum annealing revolution has begun.

In the end, though, we will likely witness the emergence of a hybrid computational future, a “best of all, worst of none approach” in which gate-based quantum computers and quantum annealers serve complementary roles – with the former tackling algorithmic problems requiring deep circuits and error correction, and the latter providing immediate utility for the optimization challenges that pervade industrial operations. 

After all, and as said by T. Harv Eker, author of Secrets of the Millionaire Mind, when faced with an “either/or” situation, we should ask, “How can I have both?”

Thanks for reading!

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