Articles and methods for quantum scheduling

Abstract

Provided herein is a system for scheduling tasks in quantum computing. The system includes a front-end scheduling module; a mapping module; and a back-end device module; wherein the front-end scheduling module is decoupled from the back-end device module; and wherein the mapping module is arranged and disposed to map a job input at the front-end scheduling module to the back-end device module. Also provided herein is a method of scheduling jobs for quantum computing using the system described herein.

Description

[0001] Attorney Docket No. 047162-7525WO1 (02644)

[0002] ARTICLES AND METHODS FOR QUANTUM SCHEDULING

[0003] CROSS-REFERENCE TO RELATED APPLICATIONS

[0004] The present application claims priority under 35 U.S.C. § 119(e) to U.S. Provisional Patent Application No. 63 / 689,480, filed August 30, 2024, which application is incorporated herein by reference in its entirety.

[0005] STATEMENT REGARDING FEDERALLY SPONSORED RESEARCH OR DEVELOPMENT

[0006] This invention was made with government support under 2127309 awarded by National Science Foundation (NSF). The government has certain rights in the invention.

[0007] BACKGROUND OF THE INVENTION

[0008] Quantum applications (also called jobs / tasks) today are scheduled by users on cloud systems, which in turn execute them on quantum processors (QPUs), and return the results to users. Currently, users must specify the particular QPU that they want to run each of their tasks on, and the task then enters into a queue for that QPU. This is inefficient for a few reasons. First, the tasks stay in these queues without migration until execution. This inflexibility combined with the ad hoc job execution heuristics results in load imbalance across devices, and long wait times for users. Some QPUs might be idle while the others are over- sub scribed. Additionally, iterative jobs like variational solvers that include both classical and quantum execution, additionally suffer a long network round trip between quantum and classical devices for every iteration. Recently, some systems have allowed the classical execution to run on a cloud instance. However, this instance is still paired with a specific quantum unit, and the classical execution cannot create more work onto other quantum units. Thus, resource execution is still inefficient. Furthermore, current systems also fail to benefit from parallelism because when jobs spawn multiple parallel instances, they are all run serially on the same device.

[0009] Accordingly, there remains a need in the art for articles and methods that improve upon existing articles and methods for quantum computing scheduling. The present disclosure meets this need. Attorney Docket No. 047162-7525WO1 (02644)

[0010] SUMMARY

[0011] In one aspect, a system for scheduling tasks in quantum computing includes a front-end scheduling module, a mapping module, and a back-end device module, wherein the front-end scheduling module is decoupled from the back-end device module and the mapping module is arranged and disposed to map a job input at the front-end scheduling module to the back-end device module.

[0012] In some embodiments, the front-end scheduling module is arranged and disposed to read metadata about incoming jobs, process the metadata, identify the correct que to place the jobs based upon the metadata, and change the entries in the front-end queues. In some embodiments, the mapping module is arranged and disposed to route the job to the device queues using the metadata and device-queue information.

[0013] In some embodiments, the back-end device module comprises one or more quantum processing units (QPUs). In some embodiments, the back-end device module comprises one or more cloudlets. In some embodiments, each of the one or more cloudlets comprises one or more classic computing elements. In some embodiments, each device of the back-end device module is virtualized.

[0014] In some embodiments, the mapping module maps the job to any available device of the back-end device module.

[0015] In another aspect, a method of scheduling jobs for quantum computing includes providing the system according to any of the embodiments disclosed herein, inputting a job to the front-end scheduling module, and mapping the job, with the mapping module, to any available device of the back-end device module. In some embodiments, the job comprises a hybrid job. In some embodiments, the method further includes mapping one or more quantum computing tasks of the hybrid job to one or more quantum computing unit (QPUs) and mapping one or more classic computing tasks of the hybrid job to one or more cloudlets. In some embodiments, the method further includes autonomously mapping a new quantum computing task spawned from the one or more classic computing tasks to the one or more QPUs.

[0016] BRIEF DESCRIPTION OF THE DRAWINGS

[0017] For a fuller understanding of the nature and desired objects of the present invention, reference is made to the following detailed description taken in conjunction with the Attorney Docket No. 047162-7525WO1 (02644) accompanying drawing figures wherein like reference characters denote corresponding parts throughout the several views.

[0018] FIGS. 1A-D show images illustrating cognitive model prototypes for which quantum implementations are developed herein. (A) Decision Walk on a lattice for two choices. (B) MPMW for one infinitely tall well. Its lowest three eigenvectors are shown. (C) Predator-Prey screen grid to model cognitive control. (D) LCA for two inputs. fi(.), zQ are nonlinear functions.

[0019] FIGS. 2A-B show schematics for realizing an 8-state Decision Walk on gate-based systems. H is the Hadamard gate. (A) Reflecting. (B) Absorbing.

[0020] FIG. 3 shows a schematic for realizing Decision Walk on annealing systems.

[0021] FIG. 4 shows a schematic illustrating a Quantum RBM network for Predator-Prey .

[0022] FIGS. 5A-E show graphs illustrating reflecting boundary Decision Walk on IBM Perth. (A) Timestep 0. (B) Timestep 1. (C) Timestep 2. (D) Timestep 3. (E) Timestep 4.

[0023] FIGS. 6A-E show graphs illustrating reflecting boundary Decision Walk with statevector simulation. (A) Timestep 0. (B) Timestep 1. (C) Timestep 2. (D) Timestep 3. (E) Timestep 4.

[0024] FIGS. 7A-C show graphs illustrating (A) gate count, (B) circuit depth, and (C) execution time of Decision Walk.

[0025] FIGS. 8A-E show graphs illustrating absorbing boundary Decision Walk with statevector simulation (Total probability <1 due to projection). (A) Timestep 0. (B) Timestep 1. (C) Timestep 2. (D) Timestep 3. (E) Timestep 4.

[0026] FIGS. 9A-E show graphs illustrating absorbing boundary Decision Walk on IBM Perth (Total probability <1 due to projection). (A) Timestep 0. (B) Timestep 1. (C) Timestep 2. (D) Timestep 3. (E) Timestep 4.

[0027] FIGS. 10A-C show graphs illustrating timestep 4 of absorbing walk with depolarizing noise. (A) / ? = 0.01. (B) = 0.001. (C) = 0.0001.

[0028] FIGS. 11A-C show graphs illustrating annealing for a 4-state Decision Walk. (A) Accuracy. (B) Size. (C) Execution time.

[0029] FIG. 12 shows a graph illustrating ansatz selection with VQE (the exact value is shown).

[0030] FIGS. 13A-B show graphs illustrating (A) simulated and (B) measured SSVQE B2 for

[0031] MPMW. Attorney Docket No. 047162-7525WO1 (02644)

[0032] FIGS. 14A-C show graphs illustrating execution time of the SSVQE algorithm for MPMW. (A) SPSA. (B) COBYLA. (C) COBYLA, IBM Quito.

[0033] FIGS. 15A-B show graphs illustrating QITE and Quantum Lanczos fox MPMW. (A) Convergence. (B) Accuracy of eigenstates.

[0034] FIGS. 16A-C show images illustrating sequences of moves for Predator-Prey screen with different RBM strategies. (A) Multiple anneals (RBM). (B) Sample-efficient anneal (RBMcff). (B) Two-sample parallel anneal (RBMpar2).

[0035] FIGS. 17A-B show graphs illustrating Predator-Prey performance. (A) Timing. (B) Accuracy.

[0036] FIG. 18 shows a graph illustrating quantum annealing for the LCA.

[0037] FIGS. 19A-B show schematics illustrating reorganization of quantum cloud scheduling. (A) Existing. (B) Embodiment disclosed herein.

[0038] FIGS. 20A-C show graphs illustrating impact of a parallelism-aware quantum cloud. (A) MPMW SSVQE. (B) Parallel Predator-Prey (time). (C) Parallel Predator-Prey (accuracy).

[0039] DETAILED DESCRIPTION

[0040] Definitions

[0041] As used herein, each of the following terms has the meaning associated with it in this section. Unless defined otherwise, all technical and scientific terms used herein generally have the same meaning as commonly understood by one of ordinary skill in the art to which this disclosure belongs. Generally, the nomenclature used herein and the laboratory procedures in molecular biology, immunology, animal pharmacology, pharmaceutical science, peptide chemistry, and organic chemistry are those well-known and commonly employed in the art. It should be understood that the order of steps or order for performing certain actions is immaterial, so long as the present teachings remain operable. Any use of section headings is intended to aid reading of the document and is not to be interpreted as limiting; information that is relevant to a section heading may occur within or outside of that particular section. All publications, patents, and patent documents referred to in this document are incorporated by reference herein in their entirety, as though individually incorporated by reference.

[0042] In the application, where an element or component is said to be included in and / or selected from a list of recited elements or components, it should be understood that the element Attorney Docket No. 047162-7525WO1 (02644) or component can be any one of the recited elements or components and can be selected from a group consisting of two or more of the recited elements or components.

[0043] In the methods described herein, the acts can be carried out in any order, except when a temporal or operational sequence is explicitly recited. Furthermore, specified acts can be carried out concurrently unless explicit claim language recites that they be carried out separately. For example, a claimed act of doing X and a claimed act of doing Y can be conducted simultaneously within a single operation, and the resulting process will fall within the literal scope of the claimed process.

[0044] As used herein, the singular form “a,” “an,” and “the” include plural references unless the context clearly dictates otherwise.

[0045] Unless specifically stated or obvious from context, as used herein, the term “about” is understood as within a range of normal tolerance in the art, for example within 2 standard deviations of the mean. “About” can be understood as within 10%, 9%, 8%, 7%, 6%, 5%, 4%, 3%, 2%, 1%, 0.5%, 0.1%, 0.05%, or 0.01% of the stated value. Unless otherwise clear from context, all numerical values provided herein are modified by the term about.

[0046] As used herein, the terms “comprises,” “comprising,” “containing,” “having,” and the like can have the meaning ascribed to them in U.S. patent law and can mean “includes,” “including,” and the like.

[0047] Unless specifically stated or obvious from context, the term “or,” as used herein, is understood to be inclusive.

[0048] Ranges provided herein are understood to be shorthand for all of the values within the range. For example, a range of 1 to 50 is understood to include any number, combination of numbers, or sub-range from the group consisting 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, or 50 (as well as fractions thereof unless the context clearly dictates otherwise).

[0049] As used herein, the term “ratio” refers to a relationship between two numbers (e.g., scores, summations, and the like). Although, ratios can be expressed in a particular order (e.g., a to b or cr.b), one of ordinary skill in the art will recognize that the underlying relationship between the numbers can be expressed in any order without losing the significance of the underlying relationship, although observation and correlation of trends based on the ration may Attorney Docket No. 047162-7525WO1 (02644) need to be reversed. For example, if the values of a over time are (4, 10) and the values of b over time are (2, 4), the ratio a.b will equal (2, 2.5), while the ratio b a will be (0.5, 0.4). Although the values of a and b are the same in both ratios, the ratios a.b and b.a are inverse and increase and decrease, respectively, over the time period.

[0050] Detailed Description

[0051] Provided herein are systems for task scheduling in quantum computing. In some embodiments, the system is cloud based. In some embodiments, the system is physically / locally based (e. ., a desktop quantum system). In some embodiments, the system includes a front-end scheduling module and a back-end device module. The front-end scheduling module is decoupled, or separated, from the back-end device module. For example, in some embodiments, the elements of the back-end device module are virtualized for executing a task, the virtualization decoupling the back-end device module from the frond-end scheduling module. As such, in contrast to existing systems, where a user is required to select a specific device for executing a job, the present system is arranged and disposed to receive jobs at the front-end scheduling module without selection of a specific device for execution. Instead, after the user submits the job to the front-end scheduling module, the system autonomously maps the job to any available device in the back-end device module.

[0052] In some embodiments, the front-end scheduling module is arranged and disposed to receive any suitable quantum computing job. Additionally or alternatively, in some embodiments, the front-end scheduling module is arranged and disposed to receive any suitable hybrid job, where each hybrid job includes a quantum computing task and a classical computing task. The front-end scheduling module includes any suitable arrangement of elements for receiving a job input from a user and identifying the correct que to place the jobs. For example, in some embodiments, the front-end scheduling module includes one or more elements for reading metadata about incoming jobs, processing the metadata, identifying the correct que to place the jobs based upon the metadata, and / or changing the entries in the front-end queues. Suitable metadata includes, but is not limited to, the type of computations (c.g., batched (z.e., run iteratively, between classical and quantum units) or standalone quantum execution), desired resource needs (e.g., number of quantum processing units (QPUs) and / or classical processing units, number of parallel executions that can be launched), priorities (e.g., among individual sub- Attorney Docket No. 047162-7525WO1 (02644) tasks), preferences (e.g., for certain QPUs), expected device noise fidelity, other device parameters, or combinations thereof. Suitable ques include, but are not limited to, a batch que, a standalone que, any other suitable que, or a combination thereof.

[0053] In some embodiments, the system also includes a mapping module. The mapping module autonomously maps, or routes, each job from the front-end scheduling module to the back-end device module. For example, in some embodiments, the mapping module uses the metadata and device-queue information to route the job to the device queues. Suitable elements of the mapping module include, but are not limited to, any elements capable of reading metadata and devicequeue information, and processing them together using logic to route jobs from the front end to back end.

[0054] In some embodiments, the mapping module is arranged and disposed to autonomously map jobs to any available device in the back-end device module. In some embodiments, the mapping module is configured to consider user specified preferences and / or priorities when mapping to the back-end device module. In some embodiments, the mapping module maps jobs and / or tasks in parallel. Additionally or alternatively, in some embodiments, the mapping module autonomously maps new tasks created / spawned during execution of existing tasks, enabling dynamic paralellism. For example, in some embodiments, the mapping module autonomously maps new quantum tasks, spawned from classical computing elements, to be executed by one or more QPUs in parallel.

[0055] The system may further include an execution selection module. The execution selection module uses metadata to select a task from the device queue to run on a processing unit. In such embodiments, the execution selection module chooses the order of execution from the jobs in the device queues. For example, the execution selection module can select a higher priority job for execution ahead of a lower priority job, even when the higher priority job was submitted / created after the lower priority job. Suitable elements of the execution selection module include, but are not limited to, any elements capable of reading metadata and selecting the next task for execution based upon the metadata.

[0056] The back-end device module includes any suitable elements for executing the job(s) input to the front-end scheduling module. For example, in some embodiments, the back-end device module includes one or more quantum processing units (QPUs). In some embodiments, the back- end device module also includes classical computing elements, such as, but not limited to, Attorney Docket No. 047162-7525WO1 (02644) central processing units (CPUs), graphics processing units (GPUs), or any other suitable classical computing element. In some embodiments, the classical computing elements are arranged in one or more cloudlets within a quantum cloud architecture of the back-end device module.

[0057] Also provided herein are methods of task scheduling in quantum computing. In some embodiments, the method includes receiving and reading the metadata of a job input by a user to the front-end scheduling module (without selecting a specific device), identifying the correct queue for the job based upon the metadata, and mapping the job, with the mapping module, to any available device in the back-end device module. Additionally or alternatively, in some embodiment, the method includes selecting a task for execution by one or more of the back-end devices using the execution selection module. In some embodiments, the method includes virtualizing the devices of the back-end device module, such that the front-end scheduling module is decoupled from the back-end device module. In some embodiments, the method includes autonomously mapping quantum computing tasks to one or more QPUs and autonomously mapping classical computing tasks to one or more classical computing elements. In some embodiments, the one or more classical computing elements may be grouped into one or more cloudlets. Additionally or alternatively, in some embodiments, the method includes autonomously mapping any new tasks spawned during execution of a job to the corresponding computing element.

[0058] The systems and methods according to one or more of the embodiments disclosed herein provide a new quantum scheduling architecture that improves performance and usability, as compared to existing systems. Without wishing to be bound by theory, it is believed that the structuring of the front-end, mapping, and execution selection, and multiple stages queues is new over existing schedulers having only queue stage with the queues being device specific. Therefore, the metadata used in the systems and methods described herein e.g., priorities, parallelism, etc.) is also new. Consequently, the scheduling logic is also new (e.g., so as to consider device utilization as well as user preferences, priorities).

[0059] As discussed in detail above, the systems and methods according to one or more of the embodiments disclosed herein decouple user and device scheduling, which is a form of virtualization. This decoupling facilitates automation and provides better distribution of tasks, avoiding imbalance. Additionally, the systems and methods according to one or more of the embodiments disclosed herein include classical computing as a peer element (e.g., cloudlet(s) in Attorney Docket No. 047162-7525WO1 (02644) the cloud). This enables automatic distribution of execution between classical and quantum resources to exploit the various types of parallelism found in applications. An example of the system according to one or more of the embodiments is illustrated in FIG. 19B.

[0060] As will be appreciated by those skilled in the art, by automating scheduling and / or enabling parallel cloud-based computing, the systems and methods disclosed herein reduce imbalances and wait times, reduce long-latency network round-trips, improve resource utilization, and / or increase throughput in quantum cloud computing, as compared to existing systems. Additionally, in contrast to existing systems, which currently treat quantum tasks as circuits that a user wants to evaluate on a particular QPU, the present systems considers tasks as applications that are complex, dynamic, and with needs for communication and synchronization. Accordingly, the systems and methods described herein permit users to develop and run new types of quantum applications which benefit from parallel execution, and multi-tasking, which has not been feasible on the existing cloud systems.

[0061] Those skilled in the art will recognize, or be able to ascertain using no more than routine experimentation, numerous equivalents to the specific procedures, embodiments, claims, and examples described herein. Such equivalents were considered to be within the scope of this invention and covered by the claims appended hereto. For example, it should be understood, that the particular technology used to implement the QPU (e.g., photonics, superconducting, etc.) may be modified without changing the scheduler architecture, and as such the various art- recognized alternatives are within the scope of the present application. That is, while the specific logic may be modified by those skilled in the art based upon the particular QPU, and using no more than routine experimentation, the scheduler organization would remain the same.

[0062] It is to be understood that wherever values and ranges are provided herein, all values and ranges encompassed by these values and ranges, are meant to be encompassed within the scope of the present invention. Moreover, all values that fall within these ranges, as well as the upper or lower limits of a range of values, are also contemplated by the present application.

[0063] The following examples further illustrate aspects of the present invention. However, they are in no way a limitation of the teachings or disclosure of the present invention as set forth herein. Attorney Docket No. 047162-7525WO1 (02644)

[0064] EXAMPLES

[0065] EXAMPLE 1 - QUATRO’. Unlocking Quantum Computing for Models of Human Cognition

[0066] Abstact

[0067] Research progress in quantum computing has, thus far, focused on a narrow set of application domains. Expanding the suite of quantum application domains is vital for the discovery of robust new software toolchains and architectural abstractions. This Example unlocks a new class of applications ripe for quantum computing research — computational cognitive modeling. Cognitive models are critical to understand and replicate human intelligence. The work described herein connects them to quantum computer architectures for the first time. Conceived and released herein is QUATRO, a collection of quantum computing applications from cognitive models. The development and execution of QUA TRO sheds light on gaps in the quantum computing stack that need to be closed to ease programming and drive performance. Among several contributions, ideas pertaining to quantum cloud scheduling (using data from gate- and annealing- based quantum computers), parallelization, and more are proposed and studied in this Example. Similar to precedents in classical computing, QUATRO is believed to enable novel research toward versatile quantum systems.

[0068] I. Introduction

[0069] Expanding the utility of quantum computers is an immediate need. These systems have grown from early prototypes to mature cloud platforms. However, their applications remain limited to a few areas within select domains, such as physics, chemistry, and recently, machine learning (ML). As articulated in the recent quantum visioning report, supporting new domains is crucial to boost adoption, to deliver real-world impact, and to identify drivers that sustain near- term progress and potentially inspire new long-term directions.

[0070] The key to realizing more widely useful quantum systems is the design of new architectural abstractions and a flexible stack. But, this is challenging — guided by benchmark applications in a few domains, current systems have developed depth in select areas, but at the expense of limiting their breadth.

[0071] Historically, when new computing paradigms emerged, new application suites were instrumental in triggering the similar progress we seek. Prominent examples include CloudSuite Attorney Docket No. 047162-7525WO1 (02644) with scale-out applications, web search for mobiles, enterprise IT applications, and DaCapo for Java. Originally conceived to introduce new applications to computer architects, these suites led to many subsequent hardware and software innovations that addressed the limitations they exposed.

[0072] As noted by the community, quantum computer architecture also critically needs new applications, those that come from new domains, and which help spur new research into expanding the quantum stack and identifying new abstractions.

[0073] In response, provided herein is QUATRO, an application suite drawn from a new domain, computational cognitive modeling. Cognitive models explain how humans process information and make decisions. They have been foundational in many fields such as psychology, artificial intelligence (Al), ML, and even economics. They are computationally demanding, containing methods with simultaneous constraint satisfaction and planning, which scale poorly with model complexity. Furthermore, theoretical and rigorous human studies have shown that a class of cognitive models using quantum probability theory can formally explain features of human behavior — such as biases in decision-making, effects of question framing and order — that have been difficult to capture using classical methods alone. However, these models today are run entirely classically, limiting the scale of models that can be used. Quantum computing can help unlock vital progress.

[0074] As shown in this Example, quantum computer architecture also benefits from targeting cognitive models. The models have been mapped to quantum hardware for the first time, and packaged them into QUATRO, a suite of computationally hard classical and quantum-theoretic cognitive models, for architecture.

[0075] QUATRO is significant for several reasons. Foremost, it brings new real-world applications to guide computer architecture. This allows identifying limitations across the full stack, complementing existing suites that contain kernels for hardware benchmarking. For example, as shown in Section IV, mapping certain models has required circuit innovation, and has also exposed the lack of compiler, language, and runtime support. Second, QUATRO is designed to have little overlap with other existing suites. It includes new constructs not present in them such as multi-eigenstate problems, non-variational eigensolvers, and Hamiltonian walks, being truly complementary. Third, the constructs in QUATRO are carefully chosen so that optimizing for them can also help other domains, avoiding over-specialization. Finally, Attorney Docket No. 047162-7525WO1 (02644)

[0076] QUA TRO spans both digital and analog quantum systems, and is suitable for both today’s hardware and future fault-tolerant designs. This enables architects to explore promising but less- studied problems.

[0077] QUATRO is used to characterize existing quantum systems. The analysis presented herein identifies, for the first time, new parallelization opportunities in eigensolvers and certain neural networks like restricted Boltzmann machines (RBMs), but which cannot be exploited by current clouds. Following this, a new proof-of-concept scheduler that offers speedup and improved throughput is outlined. Furthermore, it is demonstrated that non-unitary methods, which are less- studied today in system design, hold significant potential in the near-term. Finally, early fault- tolerant applications among QUATRO are identified, with existing hardware close to achieving reliable execution. Key findings are highlighted in Section VI and analyzed in Section VII. The present work expands quantum computing into a new application domain with broad impact — from cognitive science and neuroscience to Al and economics — while at the same time setting the stage for versatile quantum systems that are easier and faster to use.

[0078] In summary, the contributions discussed herein include, but are not limited to,:

[0079] 1) Development of QUATRO, a suite of real-world cognitive models, presenting a new application for quantum computing. QUATRO is suitable for current and future hardware.

[0080] 2) Discovery of new circuit-level constructs on subspace projections, techniques that benefit domains beyond cognitive modeling, with a fraction of resources over prior work.

[0081] 3) Identification of shortcomings and opportunities for research innovation in the software runtime and programming stack for quantum computers. Unlike prior work, the present work transcends low-level hardware and addresses the higher-level stack.

[0082] 4) Proposal and analysis of new quantum cloud organization using data from real quantum computers, showing 2.4x- 10x speedup for co-designed applications.

[0083] II. Motivation

[0084] Virtuous cycles among applications, algorithms, and systems have been central to the success of computing. A prime example is the symbiotic progress in GPU systems, and the deep neural networks they helped bring to scale. Quantum computing is yet to experience such growth. Quantum applications remain concentrated in certain areas of physics, chemistry, finance, and machine learning. As the field is maturing, there is a crucial need to identify new Attorney Docket No. 047162-7525WO1 (02644) applications that allow the broader society to benefit from quantum computing, and also, to help drive further systems research.

[0085] The present inventors have identified that cognitive models are well poised to drive the much-needed synergy between quantum systems and applications. They are computationally hard, and benefit from quantum computing. Additionally, mapping and executing real-world cognitive models helps quantum computer design by revealing performance and programmability gaps across the full-stack. This would not have been possible with traditional kernels alone. Connecting cognitive science with quantum computing serves to broaden quantum applications, and lays the groundwork for future hardware and software innovation.

[0086] A. Quantum Computing for Cognitive Modeling

[0087] Computational cognitive models describe how humans make decisions and generate behavior. Some of these use well-known computationally hard methods such as large-scale simultaneous constraint satisfaction, graph isomorphism techniques, combinatorial optimization, and planning. Additionally, there is a new class of cognitive models that employs quantum probability as a theoretical tool to describe human decision-making. These models leverage the richer axioms of quantum probability such as projections, non-commutativity, and superposition, to describe complex behavior that has resisted explanation from existing classical alternatives, and in several instances, involve exponentially fewer parameters. Below are two illustrative results. As an example of the expressiveness of quantum cognitive models, consider a two-stage task where individuals choose to play or quit a gamble before each stage. When the outcome of the first stage was known — regardless of whether it was a win or a loss, real-world data showed individuals preferred to play the gamble in the second stage. However, when the outcome was unknown, most individuals switched and preferred to quit. This violates the classic law of total probability, making it difficult for classical models to explain such “irrational” behavior. In contrast, a quantum cognitive model that represents decision-making as a superposition over the outcomes predicts different results, because quantum interference effects can generate classical violations of total probability.

[0088] Quantum cognitive models can also explain human cognitive functions with exponentially fewer parameters over classical models. For example, consider a strategic game involving n players, each choosing one of K possible actions. Classically representing an individual’s beliefs about the actions of all players requires Knjoint probabilities, representing an Attorney Docket No. 047162-7525WO1 (02644) exponential growth in model dimension. Instead, the same could be modeled with quantum probability using a single K-dimensional vector space, evaluated with n different bases representing the player’s different view points of the game.

[0089] While there could be a future classical formalism that can match quantum cognition in expressiveness, model conciseness, and accuracy, quantum cognition provides some of the best empirically-validated tools today to explain complex behavior. Unfortunately, all these models today are run entirely on classical systems, limiting the complexity of models being studied, and gating important progress, including understanding how a quantum formulation is able to better explain cognition. With the growing availability and power of quantum computers, the timing is opportune to explore their use to unlock large-scale modeling of human-level cognition.

[0090] B. Cognitive Models for Computer Architecture

[0091] End-to-end applications have played an important role in grounding computer architecture research in emerging paradigms. Key examples are, CloudSuite, mobile websearch, enterprise IT, and DaCapo. With the exception of DaCapo, the other suites did not focus on benchmarking, and instead mainly aimed at introducing computer architects to new domains. Mapping and executing these applications have unearthed inefficiencies across the entire stack, and exposed opportunities missed by simple kernels. Consequently, they were followed up by a wave of hardware and software research in the community addressing those optimizations. While the applications came from specific domains, the subsequent impact has been broad. Quantum computing is also in critical need of such application suites to sustain its momentum.

[0092] Cognitive models are particularly suited to help advance computer architecture innovation and co-design. They are real-world applications, unlike kernels used for performance benchmarking. Since they originate from a new domain not traditionally associated with quantum computing, mapping them exposed the limitations of the full quantum stack in being broadly useful. Moreover, as shown herein, they contain constructs different from those studied in quantum computer architecture so far. These constructs are also relevant to other domains, e.g., in condensed matter physics, or dynamics of electron systems, but have not been studied by architects, thus presenting new widely-applicable targets for systems research. These research opportunities are summarized in Section VII. Attorney Docket No. 047162-7525WO1 (02644)

[0093] Finally, studying cognitive models enables computer architects to help advance understanding of human cognition. Possibly, it could inform novel quantum architectures, much- like brain-inspired computing in classical systems.

[0094] III. Relevant Quantum Techniques

[0095] The following is a brief review of quantum concepts relevant to QUATRO. The Dirac (bra-ket) notation (h = 1), where |x) denotes a column vector x used to represent states, (x| is a row vector, and block letters (e g., W) are operators is used. A quantum system is described by its Hamiltonian, H. If the system’s initial state is |i / iinit), then its state after time t is, | / >nevv) = where U = e~lHtis the evolution operator.

[0096] Quantum walks: These describe the evolution of a particle over some states, and are used in cognitive models for decision-making. They capture how individuals process inputs (stimuli) to navigate their internal preferences (states) under uncertainty to make decisions.

[0097] Quantum walks are typically specified by [7, and implemented on gate-based computers (i.e., digital quantum computers) by compiling U into gates. Many specifications are possible, with coin-walks on graphs being a common approach. Cognitive models require a different approach, with walks specified by H, and their execution on gate-based systems requires additional computation.

[0098] Variational algorithms: They are near-term constructs, which use both classical and quantum computation to minimize a cost function (Hamiltonian) and find the lowest eigenstate (ground state). They are useful to find eigenstates in certain cognitive models. Important examples of variational methods are Variational Quantum Eigensolvers (VQEs) and the Quantum Approximate Optimziation Algorithm (QAOA).

[0099] SSVQE (SubSpace VQE) is a recently proposed technique to identify eigenstates higher than the ground state. These higher states are required by cognitive models, unlike many applications that only need the ground state.

[0100] Similar to VQE, SSVQE uses a parameterized quantum circuit (or ansatz) to generate candidate solutions for the optimization. Then, a classical optimizer evaluates the cost function for these states, and changes the circuit parameters to explore better states. However, SSVQE differs from VQE in computing the cost function since it must find excited states. Attorney Docket No. 047162-7525WO1 (02644)

[0101] Two relevant SSVQE variants are used herein. One approach (SSVQE B) finds one excited state at a time, e.g., the A111state using a weighted cost function. At each iteration, the algorithm inputs k + 1 orthogonal vectors, { | i j _Q, into the ansatz (e.g., k + 1 vectors whose elements are all zeros except for a 1 in a different position each), and obtains states and their energy expectations Then, a classical optimizer minimizes the cost function with weight w, w(fc|H|fc) + 2;=o {^j \H \<Pj), to obtain the ktheigenstate.

[0102] The other method (SSVQE C) also uses a weighted cost function but finds all eigenstates up to the kth state simultaneously. The cost function is Sj=o where the weights Wj decrease in value i.e., Wj < wJ -1. As shown in this Example, the two methods have different implementation tradeoffs.

[0103] Quantum imaginary time evolution (QITE): This is another method to obtain eigenvalues. Variational algorithms require identifying a good ansatz and optimizer to work well, but this is hard. QITE avoids this approach.

[0104] QITE works by evolving a quantum state with the time parameter as t = —IT. Then, the state is given by which represents state decay. If T is long, all higher energy components vanish, and the system decays into the groundstate. To obtain excited states, the Quantum Lanczos method that uses several QITE states with different T values is used.

[0105] QITE belongs to the class of non-unitary algorithms that can provide quantum speedup, but requiring additional steps to run on quantum computers because the hardware is unitary. Such methods have not been studied in prior systems research.

[0106] Quantum annealing: Cognitive models using simultaneous constraint satisfaction and quantum walks can be cast as optimization problems solved by quantum annealing. In quantum annealing, the system begins in the ground state of a simple Hamiltonian, which is slowly changed to the Hamiltonian whose minimum is to be found. If the rate of change is slow enough, the system ends up in the ground state of the target Hamiltonian, giving the solution we seek.

[0107] Quantum annealers represent analog quantum computers, programmed directly with the Hamiltonian unlike digital versions. This is attractive for cognitive models, which are commonly specified with the Hamiltonian, but these have not received adequate attention in computer architecture. Attorney Docket No. 047162-7525WO1 (02644)

[0108] Existing quantum annealers process Ising Hamiltonians given by + i bid , where azlis the Pauli Z operator with values ±1, and a^, hLare scalars and i,j refer to the qubits. General Hamiltonians, including those in cognitive models, also contain Pauli X (crx), and Pauli Y (0^) operators. Annealing them requires manual reformulation. IV. QUATRQ-, Quantum Cognitive Applications

[0109] Important cognitive models that can benefit from quantum execution, and which help architecture research, are identified herein. Table I lists the models, the probability theory they use, significance to cognitive science, the techniques used to map them to quantum hardware, and their relevance for systems research. The present implementations and key findings are all analyzed in Section VII.

[0110] TABLE I: The QUATRO applications: Selecting cognitive models for quantum implementation

[0111] A. Decision Walk

[0112] Decision Walk models two-choice decision-making (e.g., yes / no tasks) using the biased random walk of a particle on a 1 -dimensional lattice of states (FIG. 1 A). The states represent an individual’s preference for the choices, and the model is given by the Hamiltonian H. Over classical methods, this model accurately captures the intrinsic uncertainty in decision-making, and is more compact. Attorney Docket No. 047162-7525WO1 (02644)

[0113] There are two variants of Decision Walk one where it evolves over all states including the boundary states (Y / N in FIG. 1 A) for fixed timesteps (called reflecting boundaries), and another where the walk evolves only until it does not reach a boundary (called absorbing boundaries). The former represents the decision-making response to a prompt, and the latter captures the first time an individual arrives at a decision.

[0114] Reflecting boundaries (gate-based): The probability of a decision state (e.g., \d) = |0 ... 0)) after A timesteps is given by is the measurement operator. Realizing this walk on a digital quantum computer is straightforward, using the circuit construction in FIG. 2A for an example 3-qubit (8-state lattice) model.

[0115] Absorbing boundaries (gate-based): This walk evolves only in the non-boundary states, and requires projection after each timestep to contain its evolution. Thus, the probability for a decision after A timesteps is given by where P is the projector for the non-boundary states.

[0116] Realizing this walk on existing hardware is difficult because the hardware does not support partial projections. The only approach is post-selection, i.e., running the system many times and discarding the trials where the system was found in the boundaries. This requires a non-disruptive, runtime mechanism to identify when the system touches the boundaries.

[0117] Unfortunately, prior state-detection / checking circuits are ill-suited for the current purpose. Projection-based methods cannot be used since they change the state. Among alternatives, one approach detects only certain states, such as those with an even number of ones. In the present case, a boundary state can have any number of ones. The other, swap-based assertions, can check for approximate state membership but doubles the number of qubits to swap and restore the state.

[0118] To overcome these problems, a new circuit was developed to detect the boundary states with only one ancilla qubit. FIG. 2B shows the circuit with the proposed state detector for an example 8-state walk. The ancilla qubit is shown at the bottom. For any input state E-=oaJ ® 11), the detector’s output is (ao|0) + a717)) ® |0) + (Sf=i 10) ® | 1) i.e., the ancilla is in 11) if and only if the main qubits are not in the boundary states. Therefore, postselecting on this condition is sufficient to measure the correct probabilities. The detector is easily extended to larger systems and other states by varying the CNOT and Toffoli gates. This is a generic absorbing boundary walk design on existing quantum computers. Attorney Docket No. 047162-7525WO1 (02644)

[0119] Annealing walk: Digital quantum computers are programmed with U, which must be derived from the H given by the cognitive model. This is expensive. Alternatively, Decision Walk is studied on analog computers like quantum annealers that directly accept H. Unfortunately, existing annealers accept only Ising Hamiltonians, which are diagonal and can be decomposed into pairwise Pauli-Z terms (crz). The Decision Walk Hamiltonian is not diagonal, resulting in additional Pauli-X (<JX) and Pauli-Y (<jy) terms, and can have more than two Pauli operator interactions. Therefore, a custom approach is used.

[0120] Decision Walks are mapped based upon prior work that mapped molecular Hamiltonians to annealers. The multi-stage translation used herein is shown in FIG. 3. It begins by decomposing the A-qubit Decision Walk Hamiltonian into a sum of Pauli interactions. Next, a larger rN -qubit system is created and these interactions are mapped to the larger system using new pseudo-operators (e.g., X) that only use Pauli-Z operations. Finally, the Pauli-Z operations in the new system are converted to binary variables and quadratized, i.e., expand the Hamiltonian to AT -qubits such that each term has at most two variables. The result, HB, can be annealed with existing machines, and the original ground state can be recovered from the annealer’s output. Increasing r results in more accurate estimates.

[0121] Mapping Decision Walk to current digital and analog quantum computers needs custom constructs (e.g., partial projections, Ising transforms for annealing walks), which can be integrated into hardware, programming languages and compilers.

[0122] B. Multi-Particle Multi-Well (MPMW)

[0123] MPMWis a quantum cognitive model for multi -choice decision-making, and is the first formal model to integrate many cognitive parameters. The model has a landscape of potential wells, one for each choice. The properties of the wells like their height and width relate to cognitive parameters (e.g., a single infinitely tall well is shown in FIG. IB). Particles with varying energy values, representing an individual’s cognitive arousal, are used to model the dynamics of decision-making. Each particle enters the landscape serially and its final position is measured. When a particle falls within a well, the preference for the choice is incremented by one bit. This process repeats until a definitive choice is made.

[0124] Determining the position of particles in the landscape requires computing the eigenstates of its Hamiltonian. To derive this Hamiltonian from the landscape parameters, the Schrodinger Attorney Docket No. 047162-7525WO1 (02644) equation of the particle is used. Then, the position is discretized, and finite differences are used for the derivative to obtain the Hamiltonian. Next, its eigenstates are solved for.

[0125] Two approaches for eigensolving are identified: SSVQE, which is variational, and QITE / Quantum Lanczos, which is not. These methods were selected because they demand different design and execution support on quantum hardware.

[0126] SSVQE: Two variants were selected, SSVQE B and SSVQE C (Section III) that present different tradeoffs. SSVQE B runs a simpler optimization to find one eigenstate, but must be run multiple times to find multiple eigenstates. Instead, SSVQE C runs a single but complex optimization to find all eigenstates. While SSVQE C is more algorithmically efficient, SSVQE B is embarrassingly parallel and each instance is faster than SSVQE C. This is a new axis of parallelism — between states, complementary to existing methods that parallelize the expectation calculation of Pauli operators within a variational step, or those that parallelize shots for averaging. Unfortunately, no native methods exist to specify this parallelism in current systems. This is analyzed further in Section VII.

[0127] QITE / Quantum Lanczos: Variational algorithms like SSVQE require selecting a good ansatz and optimizer, which is generally difficult. Hence, an alternative non-variational algorithm, QITE, was picked for eigensolving. QITE is in the class of non-unitary quantum algorithms, which have not been studied in architecture. Being non-unitary, QITE is unsupported natively on quantum computers, which are unitary. Therefore, prior work was followed to determine unitary operators that approximate the same state evolution as with QITE. At each timestep, these operators are obtained by solving a linear system of equations with Pauli-operator expectations, measured from a quantum computer.

[0128] To find the excited states, the Quantum Lanczos algorithm based on QITE was used. This approach obtains different state vectors evolved with QITE to construct a vector subspace. Then, it runs a classical Lanczos iteration on this subspace to orthogonalize the states into excited eigenstates.

[0129] The architecture and software stack of digital quantum computers misses opportunities for parallelism, and co-design. Non-unitary methods and related primitives have little support from software libraries and compilers. Attorney Docket No. 047162-7525WO1 (02644)

[0130] C. Predator-Prey

[0131] Predator-Prey is used to model cognitive control, which governs other mental processes including decision-making. The model describes an individual playing a game shown in FIG. 1C. The game has a screen grid with three entities: an agent, which is the player’s screen icon, a prey, and a predator. The player must move the agent to capture the prey and avoid being caught by the predator. The model captures how the player allocates attention to the screen icons to discern their position, and uses those perceived positions to make a move.

[0132] Identifying the optimal attention allotment and deciding movement is a simultaneous constraint satisfaction problem. It was mapped with a quantum restricted Boltzmann machine (RBM) designed by the present inventors, shown in FIG. 4. RBMs were chosen because their structure aligns with the simultaneous interactions found in the brain, and they have long been used in cognitive modeling. The present RBM has a visible layer with nodes for the true position of the on-screen icons, attention, perceived positions, and the movement direction.

[0133] The RBM was mapped to quantum annealers since they are a good fit for its Ising interactions. The standard approach to run the RBM needs 2 anneals for inference (visible to hidden, and back) and 3 for training (+1 to update hidden nodes). Two new methods that benefit from different types of parallelism were identified. One approach (RBMeff) leverages computeparallelism by combining updates to both hidden and visible layers. This method is resource efficient, requiring only 1 anneal for inference and 2 for training, but has more complexity. Another approach (RBMparK) leverages data-parallelism where T different samples are annealed for simultaneously. This is inspired by asynchronous ML training algorithms, where the network weights are updated once after K samples, instead of K sequential updates. Neither, however, are supported natively on existing annealers. Their impact using workarounds is analyzed in Section VII.

[0134] Neural networks like RBMs have compute and data parallelism, but which is hard to specify and exploit currently.

[0135] D. Leak)> Competing Accumulator (LCA)

[0136] The LCA is a biologically inspired model for multi-choice decision-making and cognitive control. FIG. ID shows the LCA for two choices. Each choice has an accumulator x(t), which integrates the values of the input i(t) corresponding to that choice at timestep t. The Attorney Docket No. 047162-7525WO1 (02644) accumulator loses or leaks its value (by a factor A), and is inhibited by the output of the other choice (by a factor nonlinear function such as a sigmoid that acts on the accumulator’s output.

[0137] The LCA was selected because it is widely used, and its mathematical behavior has been well-characterized. The LCA uses simultaneous constraint satisfaction like Ising models, with the additional complexity of stateful dynamics / memory from the accumulator, and nonlinearity from the activation function. These aspects are crucial in its utility as a model, but they also create challenges in running it on existing machines.

[0138] The LCA was mapped to quantum annealers. However, its equations were converted into Ising Hamiltonians (cost functions) that the hardware can run. This process was run manually. It began by linearizing ( ) into ax + b using Taylor expansion. To obtain the value of ( ) for a given %(t), the annealing cost function (fx— ax( ) + b)2) was created and minimized, yielding fxto be « Since x(t) depends on previous values, it was replaced with a function of x(t — 1), and the LCA input. The result is a cost function with fx, x(t — 1), and the inputs i(t). Next, each variable was expanded with multiple qubits to represent floating point numbers. Finally, the cost function was quadratized. Annealing this cost function with the current input and previously accumulated value produces the current output and accumulated values.

[0139] There are different alternatives to anneal LCA over multiple timesteps. One approach is to anneal for a single timestep at once, where the classical outputs read from the previous anneal are used as inputs for the next. Since existing annealers do not offer easy variable initialization (reverse annealing does not suffice since only some of the variables are initialized), additional terms were used in the present cost function to realize it. This method has the overhead of reading and re-initializing qubits each step.

[0140] Another approach is to unroll the LCA dynamics for K steps, so that one anneal would return the output for all K steps. This amortizes the reading / initialization overheads but results in a much larger Hamiltonian than what is reliably run on current systems. The last approach uses the technique of Feynman’s clock, where a new problem is formulated that includes a timestep register along with the LCA variables. This method introduces complex variables that are unsupported presently. As a balance, K steps were annealed for at once. Attorney Docket No. 047162-7525WO1 (02644)

[0141] Dynamical systems and nonlinearities need heavy tranforms to map to annealers, which can be abstracted with libraries.

[0142] V. Experimental Setup

[0143] Platforms: Publicly available hardware platforms with mature software were chosen to evalaute QUATRO. For the gate-based applications, IBM systems were used (IBM Perth / Quito, which have 5 superconducting transmon qubits), and for annealing, D-Wave’s Advantage system4.1 solver was used, which has 5, 627 qubits connected in a Pegasus graph.

[0144] Cognitive model parameters: QUATRO can scale to the large sizes targeted, but existing hardware cannot fit them. Therefore, smaller versions were run. Table II shows the minimum model sizes preferred for evaluation, the actual sizes that could fit on hardware, and the implementation parameters.

[0145] TABLE II: Model and implementation parameters.

[0146] , , , , Model parameters

[0147] ModelT, , . . . , Implementation parameters

[0148] Ideal minimum Actual

[0149] Decision 100 states, 1000 8 states, 1 walk Gates. Qubits: 3 (ref), 4 (abs)

[0150] Walk walks 4 states, 1 walk Annealing (r:2-9, 60% pausing)

[0151] 10 dynamic wells, SSVQE (COBYLA / SPSA) _

[0152] MPMW 1000 positions, ... . . / +• x

[0153] 10 eigenstates positions, 3 eigenstates QITE (step size = 0.2, steps = 135)

[0154] Predator- 6 x 6 grid, 12 levels, 3 RBM (112 node; data: 100 tram,

[0155] 100 levels, 10 . mn x A «no /

[0156] Prey icons icons 100 test), 60% pausing

[0157] T„ . 10 units, multi- „ . , , 5-step unroll, 6-bit floats, 60%

[0158] LCA , 2 units, 1 layer 5 layer pausing

[0159] A large Decision Walk uses >100 states, and multiple walks to model interacting decisions. The smaller version evaluated here uses a single walk with 8 states for gate systems and 4 states for annealing. Annealing has fewer states since the actual number of qubits needed is much higher (>rN; Section IV). Anneals are paused to mitigate thermalization and noisy outputs.

[0160] MPMW scales to multiple wells and well-sizes, several eigen- states, and fine spatial resolution in the horizontal dimension. On actual hardware, 3 eigenstates were targeted for the simplest MPMW model, which has one infinite-height well. In finding eigenvalues with SSVQE, the SPSA optimizer was used in simulations as it is more accurate, but use COBYLA on actual hardware since it requires fewer executions. Attorney Docket No. 047162-7525WO1 (02644)

[0161] For the smaller Predator-Prey, the RBM has 56 visible and 56 hidden nodes. The nodes take binary values, but their biases and weights can be real. A single set of nodes was used for the true and perceived positions and they were encoded with one-hot vectors. The RBM was trained with contrastive divergence. Training data was obtained by identifying the best decision from an exhaustive search without lookahead. Train was on 100 randomly chosen screens over 30 epochs, and another 100 were used as test. While this data may seem small, evaluating it consumed hours of anneals. One-hot encoding of the screen coordinates was used, with a valid move of one step in any of the 8 cardinal and ordinal directions.

[0162] A large LCA model would have multiple layers, each with several units. In the experiments, a single layer with 2 units was used. 6 qubits were used to represent float values. Accuracy was quantified using the mean local relative error, which is the average error of a method f relative to exact LCA ( / ), when they run on the same inputs i and previous values (x.i) from the method. It is given by, avg

[0163] VI. Evaluation

[0164] A. Decision Walk

[0165] Gate-based implementation: FIGS. 5A-E and 6A-E show the probabilities of the states at various timesteps in the reflecting boundary Decision Walk using statevector simulations, and as measured from quantum hardware (IBM Perth), respectively. The simulations align closely with analytical calculations, but the results from quantum hardware deviate significantly, especially from timestep 2. The absorbing boundaries walk, shown in FIGS. 8A-E (simulations) and 9A-E (measured) is worse affected. Much of the distribution is lost even by timestep 2.

[0166] The cause of poor hardware performance is noise from running large circuits. FIGS. 7A- C show the gate count, depth, and execution time of the walks. While these models use only 3 or 4 qubits, the gate depths are in the hundreds, resulting in large noise. The absorbing boundary walk was also simulated with 1- and 2-qubit gate depolarizing noise to study its sensitivity. FIGS. 10A-C show the results at timestep 4 for different noise probabilities. There is noticeable distortion (vs FIG. 8E) at =0.001, suggesting a need for fault tolerance, but near-accurate result for p=Q.0001, indicating promise.

[0167] Decision Walk models are early fault tolerant, and with co- design, reliable execution is within reach. Attorney Docket No. 047162-7525WO1 (02644)

[0168] Annealing implementation: FIGS. 11 A-C show the results for finding the ground state of a 4-state Decision Walk. The results are shown for both simulated annealing (SA) and quantum annealing with pausing (QA-P). FIG. 11 A shows the absolute error of the measured eigenvalue (the exact value is -7.22). With SA, the error decreases with increased repetition (r), as expected. With QA-P, however, the error increases after an initial decrease. This is because bigger r results in more qubits (FIG. 1 IB) and complex interaction, worsening noise.

[0169] FIG. 11C shows the execution time of SA and QA-P. SA’s execution time grows linearly with r while QA-P is relatively flat. Even though the error for QA-P is higher at large r, the trend for small r (where its accuracy is comparable to SA) suggests that QA’s execution time can scale better.

[0170] Decision Walk on QA presents an accuracy -time tradeoff that can be exploited in the near-term, against SA.

[0171] B. MPMW

[0172] SSVQE: Identifying a suitable ansatz for SSVQE was nontrivial. 24 choices were simulated from prior work to solve for MPM ground state using standard VQE, and FIG. 12 shows their performance. The best choice was (Ansatz 1), which uses RYand controlled Rxgates. Circuit search methods from quantum ML were also used, but they did not perform competitively.

[0173] For optimizers, both SPSA and COBYLA were evaluated in the simulations, and SPSA was found to be better. However, SPSA required 2-10x more circuit evaluations, which could not be run on real hardware in reasonable time for convergence in this Example. So, only COBYLA was used for quantum hardware.

[0174] The SSVQE B and SSVQE C algorithms were run to find the 0th, 1stand 2ndeigenstates of MPMW, with exact values as 1.72, 6.21, and 11.78, respectively. FIGS. 13A-B show the progression of SSVQE B2, which only finds the 2ndeigenstate. The measured value (10.1) has a steady state error (14%). Other runs (SSVQE B0 / B1 / C) are similar.

[0175] FIGS. 14A-C show the execution time of SSVQE. SSVQE uses many circuit evaluations to compute the Pauli expectations, repeated for each eigenstate, causing long execution times. The evaluations increase with the SPSA optimizer. Attorney Docket No. 047162-7525WO1 (02644)

[0176] FIGS. 14A-C also show that each SSVQE B instance takes lesser time than SSVQE C, which finds all eigenvalues at once. Thus, if these multiple instances could be run in parallel, the time to solution will be faster — but this is unsupported.

[0177] Existing clouds do not offer parallelization primitives (hardware / software), missing potential speedup. Ansatz and optimizer selection is manual and heuristic.

[0178] QITE / Quantum Lanczos: FIGS. 15A-B show their performance for MPMW. FIG. 15A shows convergence of the ground state but also significant hardware noise. FIG. 15B shows the error in the calculated eigenvalues. The ground state error is low and is better than what SSVQE achieved. The excited states have a steady state error due to the unitary approximation, but the error is only slightly worse than with SSVQE. Importantly, however, QITE is much faster than SSVQE, despite being nonunitary. It required just 164.8 s in simulations (vs 12,000 s for SSVQE C with SPSA for comparable accuracy). This is because it has fewer evaluations, and does not tune an ansatz. Moreover, QITE / Quantum Lanczos do not require an ansatz and optimizer search, which is hard. Unfortunately, these methods have not received adequate attention in architecture.

[0179] Non-unitary methods are competitive or better than variational methods for the near-term, but have little library, programming language and compiler support.

[0180] C. Predator-Prey

[0181] FIGS. 16A-C show a few sequences of moves generated by the RBM with the three different annealing strategies (RBM, RBMeff, and RBMpar2), and the corresponding movements from the predator and prey. At each step, the deviation of the RBM’s move from a no-lookahead exhaustive search is shown.

[0182] FIG. 17A shows the execution time of the present RBM implementations. Compared to RBM, RBMefftakes nearly 100* longer. This is because RBMeffpacks 2 passes into one anneal resulting in significantly complex interactions, and thus, noise. The number of physical qubits to realize this network is 100* higher. On the other hand, RBMpar2is «2x faster. Even though it packs the processing of 2 samples into one anneal, the two samples are processed independently and do not complicate qubit interactions. Thus, it only needs 2x more qubits.

[0183] FIG. 17B shows the accuracy of the RBMs with simulated (SA) and quantum annealing (QA). RBMeffhas poor accuracy due to its complex interactions, which is more pronounced with Attorney Docket No. 047162-7525WO1 (02644) noise-sensitive QA. On the other hand, RBMpar2has comparable accuracy to RBM. Note that in training, RBMpar2updates the weights after 2 data samples, deviating from the standard algorithm that updates weights after each sample. However, this does not affect the RBM’s accuracy, similar to what has been observed in some classical ML training algorithms. This is a new opportunity to accelerate quantum RBMs. But, existing quantum clouds do not support such parallelism. In Section VII, parallel execution is emulated with O-strength qubit couplings to study its impact.

[0184] Data-parallelism can be exploited in quantum RBMs, but software / hardware support to specify and execute it is lacking.

[0185] D. LCA

[0186] FIG. 18 shows the two outputs of LCA ( / j, and / 2) for 10 timesteps, computed with different methods: SA, QA-P, and QA-NP (QA without pausing). 5 timesteps were found in one anneal, requiring 60 logical qubits and 252 physical qubits.

[0187] LCA with SA is close to the exact values, with a mean local error (Section V) of (0.58%, 0.49%) for the two LCA outputs. Among the QA methods, QA-NP has larger deviations due to noise, with an error of (5.75%, 7.54%). QA-P has a better fit with an error of (5.18%, 5.82%). Note that the error metric does not consider the conditioning of the problem i.e., for ill- conditioned calculations, errors are amplified over time.

[0188] For execution time, SA took 84.26 ms for one anneal with 100 reads and 1000 sweeps. For the best fidelity, QA required 175 ms QPU (quantum processing unit) time for 3500 reads with 50 us anneals each. With simple noise mitigation (i.e., pausing) QA for the LCA model achieved comparable accuracy to SA. However, QA also needed 3.5x more reads, and consequently, 2* longer time. This could be overcome in the near term.

[0189] Exiting annealers are noisy to run LCA, but mitigation with co-design can offer superior performance in the near-term

[0190] VII. Systems Design Implications

[0191] While the nascence of the quantum stack is known, identifying concrete directions for progress is crucial. The design and execution of QUATRO required innovation, and more importantly, identified new research across the stack for creating abstractions, and architecture (co-)design. Table III organizes such highlights marked throughout the Example. Attorney Docket No. 047162-7525WO1 (02644)

[0192] TABLE TIT: Innovation and research identified by QUATRO.

[0193] Exposed opportunities for future research Our innovation

[0194] Algorithms- 1. Optimizing for parallelism in neural 1. Showing parallelism in

[0195] Architecture Co- networks, and eigensolvers multi-eigensolvers design 2. Quantum walks on analog systems 2. Presenting data and

[0196] 3. Mapping nonlinear dynamics, projections compute parallelism in RBMs

[0197] 4. Accuracy-time tradeoffs in annealers 3. Projection state detector

[0198] Languages, 5. Parallelization primitives 4. Mapping Hamiltonian

[0199] Compilers, and 6. Libraries / primitives for partial projections, quantum walks and nonlinear

[0200] Tools non-unitary methods, nonlinear dynamics dynamics on annealers

[0201] 7. Transforms for annealers / analog systems

[0202] Runtimes, 8. Abstractions and co-design for parallelism 5. Conceptualize a parallelism

[0203] Systems 9. Scheduling for throughput and easier use aware cloud

[0204] Devices 10. Co-design for noise mitigation in quantum walks and nonlinear dynamics

[0205] Algorithm-Architecture Co-design: The quantum algorithms being studied and codesigned with in architecture are only a few, such as VQE, QAOA, and error mitigation / correction. This work presents new high-impact targets. A new state detector addressing one particular need for partial projections is described, and new axes of parallelism in multi-eigensolvers and RBMs are identified herein. These set the stage for new research into broadly exploring: the data and compute parallelism in quantum neural networks (like classical nets), mapping nonlinear dynamics and realizing partial projections in digital and analog hardware, studying quantum walks on analog systems, and optimizing for the accuracy-time tradeoff in annealers to achieve quantum advantage. Such research would help many domains since these constructs are widely-used.

[0206] Languages, Compilers and Tools: Quantum software is specialized in a few domains, and for a few operations. For example, there are no primitives to express parallelism, which, as shown herein, offers large speedups. Primitives and libraries are also lacking for widely-used constructs like partial projections, non-unitary methods, and nonlinear dynamics. This issue is more acute for analog quantum systems and annealers. Often, there is a need for heavy custom transformations — as with the quantum walks and nonlinear dynamics, specifying low-level device parameters — like the schedules for pausing, and post-processing. This approach is tedious, error-prone since quantum programming and output analysis is not intuitive, and inefficient as it wastes qubits and execution time. The solution is to develop languages with broader abstractions, with more reusable libraries and tools. Attorney Docket No. 047162-7525WO1 (02644)

[0207] Devices: Hardware supporting certain features in QUATRO such as partial projections might take years to be deployed at scale. However, QUATRO also has at least two applications closely aligned with the current focus on device co-design for noise mitigation. This Example highlights (in Section VI- A) that, with improved noise rate (at 104), quantum walk models can already be explored accurately, representing meaningful NISQ (noisy intermediate-scale quantum) or early fault tolerance. The LCA is another such application for annealers. These show that while current NISQ devices may not outperform the best classical systems for QUATRO, they hold promise to do so in the near future.

[0208] Runtimes and Systems: Existing cloud infrastructure for quantum is geared towards prototyping rather than broad utility. Thus, it lacks several abstractions needed for usability, and performance, such as parallelization frameworks. Barring a few initial studies, there is little exploration into co-design of runtimes and architecture. This Example delves deeper, and outlines a solution following the analysis with QUATRO.

[0209] FIG. 19A shows the scheduling on quantum clouds today. Users select QPUs for their jobs, and place them into device-specific queues where they reside until execution. This is problematic since users are unaware of the global scheduling demand or the several ad hoc heuristics that clouds employ to select jobs for execution. The result is severe load imbalance across devices, and hours-long wait times for users. Furthermore, this execution model is also ill suited for parallelism. When a multi-eigenstate solver like SSVQE spawns multiple parallel tasks, they are all run serially, because they are attached to a single QPU.

[0210] Outlined herein is an alternative cloud that improves performance and usability, based on two principles. The first is to decouple user and device scheduling, which is a form of virtualization. This allows better distribution of tasks avoiding imbalance, and automation. The second is to include classical computing as a peer element in the cloud. This enables automatically distributing execution between classical and quantum resources, to exploit the various types of parallelism found in applications. For example, a multi-eigensolver can launch multiple tasks for each eigenstate, all of which are executed in parallel before being aggregated. Each of these executions can further be interleaved across classical and quantum devices to distribute the Pauli operators, or expectation computations, in a MapReduce manner. This cloud is shown in FIG. 19B. Attorney Docket No. 047162-7525WO1 (02644)

[0211] The present Example differs from commercial clouds since they force users to specify target devices, and also do not support parallelism. It is also complementary to proposals like EQC which target parallelization of Pauli operators within one variational task. The present inventors generalize to interleaving across multiple heterogeneous tasks, such as those needed by multi-eigensolvers.

[0212] The promise of a parallelism-aware cloud using MPMW and Predator-Prey from QUATRO is evaluated herein. FIG. 20A shows the normalized speedup for MPMW obtained by executing the 3 instances of SSVQE B in parallel, or one SSVQE C instance vs serially running SSVQE B (current option). Parallel execution is «2.4* faster. This does not include the additional speedups from the other forms of parallelism possible, and reducing the delay between classical and quantum devices.

[0213] For the RBM in the Predator-Prey model, FIG. 20B shows the training and testing times, and FIG. 20C shows the accuracy when processing varying number of samples in parallel. Also included is a design trained serially, but tested in parallel (Tmixed). The RBM’s accuracy remains relatively unchanged up to 10-parallel evaluations during training, but is 10* faster than serial execution. Furthermore, testing can always run in parallel without loss of accuracy.

[0214] These results are highly promising, and are only a sample of the impactful research that QUATRO can inspire.

[0215] VIII. Related Work

[0216] Quantum applications: Numerous advances have helped make quantum computing a reality. However, realizing new applications has been a challenge despite great interest from many domains in adopting quantum computing. They remain concentrated in only a few areas. Cognitive modeling is identified herein as a new domain that can benefit from quantum computing. It is also shown herein that targeting this domain is beneficial for architects. A suite of full applications unlike benchmarks that use kernels for performance comparison is presented. This work also spans analog and digital quantum systems, which is rare.

[0217] Quantum computer architecture: Significant strides have been made in quantum programming and compilation, circuit synthesis, noise mitigation and reliability, and microarchitecture design. Recent work on improving the fidelity of partial measurements, and improving variational algorithms for near-term machines can help the cognitive models in QUATRO. Attorney Docket No. 047162-7525WO1 (02644)

[0218] One limitation of prior research is that it is heavily device-centric, with only a few analyses at the higher, organization and system levels. Ravi et al. raise the issue of growing application demand in the cloud, and proposed fidelity- and queuing-aware scheduling. Our work contributes to expanding such research, exposing other aspects like parallelism.

[0219] Existing architecture and software research is also almost entirely focused on gate-based systems, which are only one type of quantum computers. Many applications, including some of QUATRO, are more naturally suited to Hamiltonian-based computers (e.g., QuEra). Unfortunately, the stack for these types of systems is under-explored.

[0220] Parallelism and Scheduling: Prior work explored microarchitecture and circuit-level parallelism with co-location on gate-based computers and annealers while mitigating reliability issues that arise. Studied herein is a complementary form of embarrassing parallelism exposed by QUATRO, that can benefit from, but does not need co-location. Capturing all these forms of parallelism could enable near-term machines to be competitive for real applications.

[0221] IX. Conclusion

[0222] Quantum computing can benefit many domains, but it has been a challenge to identify new applications, and stimulate architecture-application co-design. This work presented cognitive modeling as a new application area for quantum computing. Described herein is QUATRO, a suite of real -world cognitive models that can be run on existing quantum hardware. Developing QUATRO required innovation, and running them helped identify new research in the quantum stack, some of which was evaluate with real data. This work simultaneously advances the cognitive sciences, and quantum computer architecture.

[0223] EQUIVALENTS

[0224] Although preferred embodiments of the invention have been described using specific terms, such description is for illustrative purposes only, and it is to be understood that changes and variations may be made without departing from the spirit or scope of the following claims.

[0225] INCORPORATION BY REFERENCE

[0226] The entire contents of all patents, published patent applications, and other references cited herein are hereby expressly incorporated herein in their entireties by reference.

Claims

1. Attomey Docket No. 047162-7525WO1 (02644)CLAIMSWhat is claimed is:

1. A system for scheduling tasks in quantum computing, the system comprising: a front-end scheduling module; a mapping module; and a back-end device module; wherein the front-end scheduling module is decoupled from the back-end device module; and wherein the mapping module is arranged and disposed to map a job input at the front-end scheduling module to the back-end device module.

2. The system of claim 1, wherein the front-end scheduling module is arranged and disposed to read metadata about incoming jobs, process the metadata, identify the correct que to place the jobs based upon the metadata, and change the entries in the front-end queues.

3. The system of claim 2, wherein the mapping module is arranged and disposed to route the job to the device queues using the metadata and device-queue information.

4. The system of claim 1, wherein the back-end device module comprises one or more quantum processing units (QPUs).

5. The system of claim 4, wherein the back-end device module comprises one or more cloudlets.

6. The system of claim 5, wherein each of the one or more cloudlets comprises one or more classic computing elements.

7. The system of claim 1, wherein each device of the back-end device module is virtualized.

8. The system of claim 1, wherein the mapping module maps the job to any available device of the back-end device module.

9. A method of scheduling jobs for quantum computing, the method comprising:Attomey Docket No. 047162-7525WO1 (02644) providing the system of claim 1 ; inputting a job to the front-end scheduling module; and mapping the job, with the mapping module, to any available device of the back-end device module.

10. The method of claim 9, wherein the job comprises a hybrid job.

11. The method of claim 10, further comprising: mapping one or more quantum computing tasks of the hybrid job to one or more quantum computing unit (QPUs); and mapping one or more classic computing tasks of the hybrid job to one or more cloudlets.

12. The method of claim 11, further comprising autonomously mapping a new quantum computing task spawned from the one or more classic computing tasks to the one or more QPUs.

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