Workflow for Developing Quantum Applications

This chapter provides some insight into the process used by D-Wave and other companies to develop successful quantum applications.

Application development typically advances through the following steps.

1. Problem Discovery identifies problems in your company’s processes that can benefit from quantum technology.

2. Problem Description describes a problem in a way that enables developers to model it.

3. Problem Formulation develops mathematical models of a problem.

4. Software Implementation implements a mathematical model in code.

5. Test and Iterate iteratively evaluates and improves the application.

Problem Discovery

The application-development process may start with your company looking to benefit from quantum technology to improve existing business applications, develop new applications, or add new features. How do you identify good candidate problems for quantum-classical hybrid solutions? The following steps can help.

1. Assess your applications.

   Search the company’s processes for “bottlenecks”, places where problem solving takes a lot of time or returns unsatisfactory results. These processes may be workhorses that have been in use for years, but over those years increases in input data, changing hardware (computers and new sensing technology) may have slowed processing or decreased efficacy.

   Bear in mind that a less obvious but possibly fruitful investigative route is to also question staff about unsolved problems for which the company may have never tried to implement processes.

2. Identify candidate problems.

   Typically, good candidates are optimization problems of the type described in the Characteristics of candidate problems section below.

   Shortlist only the most promising problems, which should have these attributes:

   • Important/valuable to the business; resist selecting lower-value problems that are tempting due to the ease of formulation.

   • Hard: for well-known classes of problems you might know that your problem is formally hard (e.g., NP-hard) but for most problems it’s sufficient that it lacks satisfactory solutions, contains many interacting choices that cannot be solved sequentially, would add value if solved more quickly, etc.

   Often these are problems that your company has previously attempted to solve, or has developed a solution for, but perhaps with less satisfactory results than desired, lower efficiency than needed, higher costs than acceptable, etc.

3. Select problems to solve.

   Successful development and introduction of an application as a company process might hinge on close collaboration with someone in the organization, a “problem owner”, who has responsibility for the problem and authority over any changes to its solutions. Consider the following steps when selecting a problem:

   1. Prepare an elevator pitch (see a possible format described below) for each candidate problem identified in the previous step.

   2. Identify a problem owner for each candidate problem; support from such an invested owner may be a condition for advancing a new solution to the problem.

   3. In collaboration with the problem owner, ensure you can access the problem’s data (inputs, current solution runtimes and quality, etc).

   Proceed with only this curated set of problems.

Examples of Problem Discovery

• Problem Discovery for a Large Retailer

  As an illustrative example, consider the following scenario of discovery: you work for a large retailer and are tasked with looking into applying new technologies to improve efficiency and cut costs of business operations. Following the steps in this section produces the following results.

  • Assessing Applications

    You talk to representatives of each department and make a list of operational processes, which, for a large retailer, might include replenishing stocks of existing products, ordering optimum quantities of new products, routing deliveries from suppliers and to consumers, and many additional processes that occur daily, weekly, quarterly, etc.

  • Identifying Candidate Problems

    Among these processes you note that your operations personnel are spending many hours per week scheduling shifts to staff the company’s outlets. Is this a good candidate for new solutions?

    You look into the current scheduling process and find that it is implemented with in-house software plus some manual tuning on Excel sheets. Two decades ago a single manager used to spend a couple of hours on Fridays scheduling shifts for the company’s single outlet but now, with additional outlets that employ many more workers, the task occupies the time of multiple managers on both generating an initial schedule and then on making adjustments during the week. Employees are often unsatisfied by the resulting schedules that fail to account for their preferences on shift times.

    You set up meetings with some of these managers, and they provide some very rough figures to help you estimate the business cost of remaining with the existing solution. You realize that over months and years, a better solution would yield significant savings for the company. It would also increase employee satisfaction, and thus retention. A quick internet search shows that scheduling can be a hard, discrete optimization problem. Such problems are good candidates for quantum-classical hybrid solutions.

    Perhaps you identify additional problems in a similar way.

  • Selecting Problems

    For the identified scheduling problem, one of the involved managers agrees to act as the problem owner. Your manager allocates a senior developer for a couple of months to help you develop a proof of concept.

    (For the simplest of the additional candidate problems you identified, you are given approval to spend a few days jerry-rigging a proof of concept, by which you hope to demonstrate improved solution quality and justify a project budget. Your manager also considers another candidate you identified but suspects that changes there will require broad support in the company. You, your manager, and a problem owner create a short presentation on that problem’s importance to the business, the difficulties and cost of the existing process, and the benefits of improving the process. Following this presentation, the company’s Vice President of Operations sets up an ad-hoc “steering committee” with representatives of departments that would be effected by a change to this process and makes a budget request for developing an improved solution.)

• Elevator pitch for scheduling candidate problem.

  As an illustration example, the elevator pitch worksheet was used to create the following pitch points for the large retailer’s employee-scheduling problem.

  Table 5 Scheduling: Business Value and Use Cases

  Question	Answer
  Why is this problem important to the business?	Given that meeting demand for staffing is imperative for our outlets to function, we currently over-allocate, at high cost; our current poor fit to employee requirements harms retainment of trained personnel; managers spend a lot of time scheduling; any changes due to same-day no-shows are disruptive.
  What improvements would most increase business value?	Speed of generating the schedule (reduce time managers spend scheduling), better fit to staff preferences, ability to scale for our expanding to more outlets next year.
  Who are the main users?	Outlet managers are responsible for weekly scheduling.
  What problem are they trying to solve?	Meet the weekly staffing demand while considering employee preferences, minimizing paid overtime, and meetings various hard and soft constraints.
  How do/will they interact with a solution?	Ideally the system reads in the staffing-demand and employee preference spreadsheets submitted by email/uploaded on website, reads employee schedule-data files in database, and at a preset weekly time generates up to half a dozen alternative schedules for managers to choose from, which can be presented online or emailed. If needed, the application can be manually updated and run.
  What is the overall process flow?	Outlet manager files the demand (requirements) for the week by Friday, staff file their preferred shifts for the week by noon Monday, the application runs automatically at 1:00 PM and presents schedules, managers select one or update parameters and run again, and by 4:00 PM the formal schedule is released.

  Table 6 Scheduling: Existing Solution

  Question	Answer
  How is the problem solved today?	In-house software plus manual tuning on Excel sheets. Tuning requires many hours and the results are a poor match to staff preferences. The schedule always meets demand but only because it includes an expensive 20% margin of over-staffing.
  What improvements are required for a new solution to displace the current one?	The main needs are reducing human work and improving the match to staff preferences. Any solution must be robust to the expected expansion of outlets set in the 5-year plan. An attempt was made to modernize the existing software but without success.
  Are there any known bottlenecks?	Scheduling is a known hard problem.
  What are the data inputs and outputs?	Inputs are staffing demand file, staff preference sheets, employee schedule-data files (exist in current process); output is the full schedule for each outlet.
  What are the required system and/or process integrations?	The new application must read the emailed/uploaded demand and preference sheets, and must have permission to access the database files of employee qualifications, training, hours, and pay rate. It should run automatically and allow managers with permission to run manually.
  How are results delivered/presented to users?	Ideally as an online schedule as shown in the attached PowerPoint.
  Is there historical data that can be used to test a new solution?	Yes, we can use the last year’s filings and schedules.

Resources for Problem Discovery

• Characteristics of candidate problems.

  What type of problems can benefit from quantum technology?

  Quantum computers can solve some hard problems more efficiently than any known algorithm for classical computers.1

  A strong category of problems for quantum technology is optimization problems with quadratic interactions between discrete variables.2

  Optimization problems are problems that require an assignment of variables that results in the best, or very good, solutions. For example, defining in what order a set of products be assembled to make the most efficient use of the manufacturing machines on a factory floor.

  Discrete variables include the following categories of variables:

  • Binary variables can be assigned two values, such as 0 and 1 or True and False.

    Problems with these variables can be recognized by the True or False judgments required from their solutions. For example,

    • Scheduling: Did a task meet its deadline? Did the crew make it to the flight?

    • Networks: Did a network node experience failure?

    • Finance: Did a loan go into default?

  • Integer variables can be assigned whole numbers, such as those between -5 to 10.

    Problems with these variables optimize the number of something. For example,

    • Delivery: How many 11” x 5” x 14”-sized boxes should be loaded onto the truck?

  • Categorical (one-hot or “discrete”) variables can be assigned a value from a set, such as green, red, blue.

    Problems with these variables have several distinct options. For example,

    • Scheduling: Which shift should employee \(X\) work?

    • Map Coloring: Should the state be colored red, blue, green or yellow?

  Quadratic interactions represent relationships and correlations between the inputs of a problem. For example,

  • Scheduling: A missed deadline affects other tasks, preventing gaps between consecutive machine usages saves costs.

  • Networks: A failed network node changes the load on other nodes.

  • Finance: Diverse stocks lowers risk, a defaulted loan affects the risk to other loans.

  1

  It can be helpful to have a little familiarity with the relevant terminology. Classical solution techniques often classify problems by their variable types and interactions:

  • ILP, integer linear programming, deals with problems that have integer variables that do not interact with each other.

  • MILP, mixed integer linear programming, includes real (continuous) variables.

  • MIQP, mixed integer quadratic programming, allows for quadratic interactions between the variables.

  More academically, complexity classes classify problems in terms of how solution times of algorithms scale with input size. For example, P, is a class of problems for which algorithms scale polynomially (considered efficient on classical computers) while for other classes there may not be classical algorithms that run in polynomial time; NP, nondeterministic polynomial-time, is a class of problems for which proposed solutions can be verified quickly but no known algorithms guarantee solutions in polynomial time.

  2

  D-Wave’s hybrid constrained quadratic model solver also performs well on problems with some real variables. Problems with real variables optimize over an uncountable set. For example, in a gaspipe-maintenance problem, you might ask, Where should the sensor be installed? And the answer might be 2.46 meters along some axis.

• Elevator pitch for a candidate problem.

  You might use tables such as the following to create elevator pitches for each candidate problem:

  Table 7 Business Value and Use Cases

  Question	Answer	Comments
  Why is this problem important to the business?
  What improvements would most increase business value?		Examples: Speed, quality, scalability, etc
  Who are the main users?
  What problem are they trying to solve?
  How do/will they interact with a solution?
  What is the overall process flow?

  Table 8 Existing Solution

  Question	Answer	Comments
  How is the problem solved today?		What is working well? What is not?
  What improvements are required for a new solution to displace the current one?		Have other approaches been tried?
  Are there any known bottlenecks?
  What are the data inputs and outputs?		Are the necessary data inputs already in place?
  What are the required system and/or process integrations?
  How are results delivered/presented to users?
  Is there historical data that can be used to test a new solution?

• The Stating the Problem chapter provides examples of good problems in many industries and verticals, as well as links to further examples.

Problem Description

For any selected problem, the first step in attempting to develop a new solution or improve an existing solution is a clear and comprehensive description.

A good description specifies the following elements3 of the problem:

3

The following subsections provide simple examples that should make the abstract definitions given here concrete even to users with no prior optimization experience.

• Inputs: the data needed to represent an instance of the problem.

• Outputs: the preferred presentation of solutions to the problem.

• Parameters: dependencies that configure problem instances and set preferences on solutions.

• Decision Variables: the constituents of the problem to which the process attempts to assign good values.4

• Objectives to be Optimized: the goals the process attempts to accomplish by minimizing or maximizing certain aspects of the problem to the extent possible.

• Constraints: aspects of the problem and/or process with limited or no flexibility, which must be satisfied for solutions to be considered feasible.5

4

This initial set of variables and their definitions often develops and changes during the Problem Formulation step.

5

Constraints are often categorized as either “hard” or “soft”. Any hard constraint must be satisfied for a solution of the model to qualify as feasible. Soft constraints may be violated to achieve an overall good solution.

The following steps can help guide you.

1. Write a plain-language description of the problem as you currently understand it.

   A good problem description has the following constituents for its inputs and outputs:

   • Entities

     For example, in an employee-scheduling problem entities might include employees, time slots, supervisors, jobs, hourly rates, staffing demands, etc.

   • Relations

     For example, relations in an employee-scheduling might include a requirement that one supervisor be present for every three new hires, that no more than 20% of staff in any time slot be new hires, that two senior staff should not staff a small department simultaneously, etc.

   • Quantity being optimized

     For example, minimizing the makespan of a scheduling problem or selection of some number \(k\) of features in a machine-learning problem.

2. In collaboration with the problem owner, revise your initial description.

   • Describe the problem using the business/domain language, explained for non-specialist developers; that means, the description is immediately recognizable to experts in the field, with all domain-specific terminology clarified for the application developers who may not be knowledgeable in the problem’s domain or the company’s business.

   • Clear up any ambiguities and inconsistencies.

   • Ensure that the problem owner approves of the revised description.

3. Attempt to acquire at least one complete instance of the problem, including input data, runtime, solutions, tuning parameters, etc.

Examples of Problem Description

Description of a Large Retailer’s Scheduling Problem

As an illustrative example, consider the following scenario of problem description.

• Candidate Problem

  You are tasked with developing a new solution to the scheduling problem of the Problem Discovery section.

• Plain-Language Description

  Based on your initial work of problem discovery you know enough about the employee-scheduling problem to write a draft description. It might look something like this:

  “Our employee-scheduling problem is to generate a weekly schedule for our Madrid employees that optimally matches demand and scheduled shifts, with preference given to senior and full-time employees. The schedule should not include back-to-back shifts or more than 48 weekly hours per employee…”

• Revised Description

  After a few iterations with the problem owner, your final description might look like this:

  “Our employee-scheduling problem is to release every Monday by 4:00PM to all employees of our five Madrid outlets a schedule of their allocated shifts for the following week. Because employees have until noon Monday to file their availability and preferences for the following week, the scheduling application must generate a schedule within 1 hour to allow staff time to make adjustments and rerun once or twice if needed. Preferably, the application generates more than a single schedule per run, allowing staff discretion to select one.

  The optimal schedule should, to the extent possible, achieve the following objectives:6

  • Minimize the differences between our anticipated demand for work hours and scheduled hours.

  • Maximally account for seniority in selecting employees for available hours.

  • Maximally account for employees’ stated schedule preferences.

  • Minimize the number of employees needed to meet the anticipated work.

  • Maximize the number of employees with full-time schedules.

  • Minimize overtime.

  • Minimize the variance in overtime across our employees.

  The schedule must also take into consideration the following constraints:

  • Overtime must be in 4-hour blocks.

  • Full-time employees should not be alloted more than 48 hours per week.

  • Employees must not work back-to-back shifts.

  • Full-time employees must have two consecutive rest days each week.

  • Week-end shifts can be allocated only to eligible employees.”

• Problem Instance

  You also collect one, or preferably a few, problem instances that developers can use when designing a replacement application and testing it. These might include artifacts such as the following:

  • File with shift start and end hours for May, July, and October.

  • Currently used input data files on employees with the following details: name, employee ID number, rank, minimum and maximum available weekly hours, flag specifying full or part time, training (what roles the employee is qualified to fill).

  • Excel sheet with anticipated demand per outlet per role per day from several weeks in the previous year.

  • Scanned copies of the sheets employees file with the weekly availability and preferred hours.

  • Schedules for the last 6 weeks as generated by the current process.

6

When formulating a problem with multiple desired objective, such as here, it can be helpful to reduce the final objective to a weighted sum of just two or three objectives and reformulate the remainder as (soft) constraints. Often these objectives represent costs to the business or opportunities to profit, and can be costed; that costing can be used as the weightings between these various objectives, combining them into a single objective of improving profits. For example, you might define the objective to minimize overtime as a constraint on overtime not exceeding some threshold.

Resources for Problem Description

• Domain-driven design

• Work Sheet for Problem Description

  This worksheet can aid you in writing a problem description. You can print it and, in collaboration with the problem owner, fill in the Description column with answers to the question asked for each section, plus any additional pertinent information.

  Table 9 Problem Description

  Section	Description	Questions Answered
  General statement of the problem in business/domain language		What are the decision variables? What is the objective? What are the constraints?
  Timing goals and restrictions		How frequently is the problem solved? What is the expected solution runtime? Are problem instances independent or a series where one instance needs the previous solution as an input? Do other business processes depend on the output of the optimization?
  Scale of production problems		Is the company already solving problem instances at the desired scale? What is the largest problem instance currently solved? Is solving large instances more important than quality solutions7?
  Quality of solutions		What are the requirements for solution quality? Is optimality a requirement? How important is solution quality? How important is optimality versus feasibility8? Is the company satisfied with the quality of current solutions?
  Existing model		Is there an existing mathematical formulation for the problem?

  7

  Does the problem owner prefer to find higher-quality solutions for a smaller-than-desired problem instance over good solutions for a larger problem instance?

  8

  Finding the global optimal (best possible solution, which corresponds to the solver returning the lowest possible energy) versus finding a good, feasible solution.

Problem Formulation

A comprehensive problem description is a prerequisite for developing your initial model.9

The standard approach to problem formulation is to translate the problem description generated in the previous section into mathematical equations, specifically an objective subject to constraints.

Doing the Math

Although users with high-school mathematics can understand these models, depending on your previous experience this step may be challenging. D-Wave’s training course can reduce learning time but if developing mathematical models is outside your company’s expertise, it can make solid business sense to have D-Wave’s Professional Services organization handle this step for you.

Some developers find it best to directly represent the equations in code rather than in writing. You can try and see what works best for you.

The model you develop in this stage typically has the following elements:

• Variables: can be binary10, integer, and real

• Objective: the quantity being optimized (formulated as a quadratic/linear model to be minimized)

• Constraints: linear and quadratic relationships between variables that must or should11 be satisfied

Such a model is called a constrained quadratic model or sometimes a binary quadratic model.

Performance is sensitive to the model. As you develop your model consider various formulations: developing a few different models can be beneficial in that some might significantly outperform others.

9

Typically there are additional prerequisites such as managerial approvals, budgets, possibly reviews in your company’s legal and human-resources departments, input from your IT department, etc. For some applications, where change is disruptive, it can be helpful to discuss the process with someone in your company that has experience in project and change management. Updating established processes can have wide-ranging implications; understanding these early on enables you to initiate any needed administrative work in parallel to your application development.

10

Including “one hot” discrete variables.

11

Constraints are often categorized as either “hard” or “soft”. Any hard constraint must be satisfied for a solution of the model to qualify as feasible. Soft constraints may be violated to achieve an overall good solution.

Examples of Problem Formulation

• The Employee Scheduling example in D-Wave’s collection of code examples is a pedagogic example of formulating a small employee-scheduling problem.

• Ocean software’s collection of code examples on GitHub contains many examples of formulation.

Resources for Problem Formulation

• The Getting Started with D-Wave Solvers guide walks you through some basic examples of mathematical formulation, using very simple objectives and constraints, which can be a gentle introduction to the concepts. Likewise, the Ocean documentation documentation provides a series of code examples, for different levels of experience, which include such formulations.

• The Stating the Problem chapter of this guide provides references to examples categorized by field and the Reformulating a Problem chapter describes various techniques to mathematically formulate parts of your problem.

• D-Wave’s corporate website provides links to user applications.

• Building a Quantum Hybrid Application is a video recording of an August 2023 D-Wave webinar.

• There is a vast literature on operational optimization.

• D-Wave’s Learn: D-Wave’s online, instructor-supported training.

• D-Wave Launch: D-Wave’s Professional Services organization, which works with customers to accelerate their progress from getting started through production implementation.

Software Implementation

Your application will likely have multiple parts, including the following:

• Core optimization code

  This is the part that implements your formulation of the problem and submits it to a D-Wave solver for solution. It is recommended that you implement the model representing your problem, formulated as described in the previous section, and manage the submission using the Ocean SDK.

• Handling inputs and presenting results

  Your application may have rigidly defined inputs and expected outputs or, as part of your work, you may have freedom to define input formats and how to present solutions. Close collaboration with the problem owner and intended users of the application on these parts can prevent mistaken presumptions that might risk the project’s ultimate success.

• Integration with other applications

  Try to identify any needed integrations early: some might have long procurement or development schedules.

It is recommended that you manage and schedule your application development over multiple iterations of learning and improvement, as described in the next section.

Test and Iterate

Getting it Right

Many users, even those experienced in operations research, find it challenging to finesse a working model into a performant one. If your results fall short, especially on your first attempt to develop a quantum application, professional help might be key to your project’s success.

A useful approach to implementing your formulated problem as a software application includes the following steps of iterative development:

1. Build an initial prototype

   For your initial prototype, start with small problem inputs and use symbolic math, simple loops, etc to build a model without worrying much about construction performance.

   This prototype, which may also be considered a proof of concept, might be the first point in your development process that provides feedback on its feasibility and likelihood of success. As such, you may be balancing multiple conflicting needs; for example:

   • Speed to show initial results versus sufficient research to enable success

   • Demonstrating advantage on large problem instances versus minimal software work on small instances

   • Clear presentation of results versus minimal coding work on outputs

   • Ease of troubleshooting versus performance

   It can be helpful to define in advance the minimal requirements for the initial prototype to ensure timely delivery and successful completion of this important step. Take into consideration the complexity of the model, the experience of the developers in the relevant fields (both the problem and the quantum programming model), and the company’s scheduling requirements, and aim for the achievable.

2. Increase scale, complexity, performance

   • Iteratively increase problem inputs up to the size expected in your production environment. Input size affects not just solution times but also the time and memory usage required to build the model. As your model’s size increases, so does the importance of optimally building the model.

   • Consider various decomposition techniques when dealing with extremely large problems.

   • Consider various techniques to reduce problem size: presolve techniques, dropping (or even adding) constraints, using native discrete variables for the hybrid CQM solver, etc.

3. Test outputs, update model, and repeat the previous step

   Model validation includes many aspects; for example:

   • Comparisons with the current solution, between constrained and unconstrained models (i.e. representing one or more constraints as penalty models in the objective), between different solvers, and for varying solver runtimes.

   • Various inputs, preferably inputs similar to those expected in your production environment.

   • Multiple runs: results from heuristic solvers vary over executions for the same input, so the performance of a single execution is inconclusive.

   Often, initial models do not perform well and, before you can begin increasing the problem scale, you may need to further simplify your model. It might be that at this point your immediate goal is to just to achieve some “base model” that produces results that are not wrong.

   There are many options for simplifying your model to attain a working base from which to then build up the needed, comprehensive model. For example,

   • Relax some of the constraints

   • Drop some components of the problem formulation

   • Explore some alternative formulations for at least some parts of the model

     Bear in mind that there are often multiple ways you can view a problem; for example, you could represent an input toggle switch as either a binary variable (True or False) or a one-hot variable (ON or OFF) and you might represent the satisfiability (SAT) problem \((x_1 \vee \overline{x}_2 ) \wedge (\overline{x}_1 \vee x_2)\) of the Getting Started with D-Wave Solvers guide as either an objective to be minimized, \(0.1 x_1 + 0.1 x_2 - 0.2 x_1 x_2\), or a constraint to be met, \(x_1=x_2\).

Resources for Test Iterations

• Model Validation and Scaling tutorial video from D-Wave’s Qubits 2023 conference.

• The Ocean SDK documentation’s Scaling guide.

• D-Wave provides Launch programs to accelerate enterprises’ path from problem discovery through production implementation.

• Leap hosts a community where developers help each other.