GLOBAL QUANTUM OPTIMIZATION ALGORITHM FOR COMBINATORIAL OPTIMIZATION PROBLEMS IN NISQ DEVICES

A method, system and computer program product for employing quantum optimization algorithms for combinatorial optimization problems in noisy intermediate-scale quantum (NISQ) devices. An objective function of a combinatorial optimization problem to be minimized is defined. The input to the objective function corresponds to the circuit parameters for the ansatz of the Gauss-Newton based quantum algorithm (GNQA). The output of the objective function corresponds to the error-robust indicator value indicating whether the result of GNQA (solution of the combinatorial optimization problem) is a legitimate or illegitimate return value. After initializing the circuit parameters (θ) of the objective function, GNQA is employed for local optimization. Furthermore, a Bayesian optimization is employed for global optimization in response to the solution of the combinatorial optimization problem not reaching a correct solution, where the Bayesian optimization updates the circuit parameters to minimize the indicator value. Once a correct solution is reached, it is outputted.

TECHNICAL FIELD

The present disclosure relates generally to quantum optimization algorithms, and more particularly to implementing a global quantum optimization algorithm for combinatorial optimization problems that works effectively in noisy intermediate-scale quantum (NISQ) devices.

BACKGROUND

Quantum optimization algorithms are quantum algorithms that are used to solve optimization problems. Mathematical optimization deals with finding the best solution to a problem (according to some criteria) from a set of possible solutions. Mostly, the optimization problem is formulated as a minimization problem, where one tries to minimize an error which depends on the solution: the optimal solution has the minimal error. The power of quantum computing may allow problems which are not practically feasible on classical computers to be solved, or suggest a considerable speed up with respect to the best known classical algorithm.

SUMMARY

In one embodiment of the present disclosure, a method for employing quantum optimization algorithms for combinatorial optimization problems in noisy intermediate-scale quantum (NISQ) devices comprises defining an objective function of a combinatorial optimization problem to be minimized, where an input to the objective function corresponds to circuit parameters, and where an output of the objective function corresponds to an indicator value. The method further comprises initializing the circuit parameters of the objective function. The method additionally comprises employing a Gauss-Newton based quantum algorithm for local optimization to output the indicator value of the objective function based on the circuit parameters and to output a solution of the combinatorial optimization problem based on the circuit parameters. Furthermore, the method comprises employing Bayesian optimization for global optimization in response to the solution of the combinatorial optimization problem not reaching a correct solution, where the Bayesian optimization updates the circuit parameters to minimize the indicator value.

The foregoing has outlined rather generally the features and technical advantages of one or more embodiments of the present disclosure in order that the detailed description of the present disclosure that follows may be better understood. Additional features and advantages of the present disclosure will be described hereinafter which may form the subject of the claims of the present disclosure.

DETAILED DESCRIPTION

As stated in the Background section, quantum optimization algorithms are quantum algorithms that are used to solve optimization problems. Mathematical optimization deals with finding the best solution to a problem (according to some criteria) from a set of possible solutions. Mostly, the optimization problem is formulated as a minimization problem, where one tries to minimize an error which depends on the solution: the optimal solution has the minimal error. The power of quantum computing may allow problems which are not practically feasible on classical computers to be solved, or suggest a considerable speed up with respect to the best known classical algorithm.

One such optimization problem to be solved using a quantum optimization algorithm is a combinatorial optimization problem. The combinatorial optimization problem is the act of trying to find out the value (combination) of variables that optimizes an index (value) from among many options under various constraints. Examples of combinatorial optimization problems include the 2-Satisfiability (2-Sat) problem and the quadratic unconstrained binary optimization problem (QUBO).

The 2-Sat problem is a computational problem of assigning values to variables, each of which has two possible values, in order to satisfy a system of constraints on pairs of variables. It is a special case of the general Boolean satisfiability problem, which can involve constraints on more than two variables, and of constraint satisfaction problems, which can allow more than two choices for the value of each variable.

QUBO is a combinatorial optimization problem with a wide range of applications from finance and economics to machine learning. QUBO is a non-deterministic polynomial-time hardness (NP hard) problem, and for many classical problems from theoretical computer science, such as maximum cut, graph coloring and the partition problem, embeddings into QUBO have been formulated. Embeddings for machine learning models include support-vector machines, clustering and probabilistic graphical models. Moreover, due to its close connection to Ising models, QUBO constitutes a central problem class for adiabatic quantum computation, where it is solved through a physical process called quantum annealing.

In connection with solving combinatorial optimization problems, such as QUBO, a Gauss-Newton based quantum algorithm may be utilized to solve such a combinatorial optimization problem. Gauss-Newton based quantum algorithm (GNQA) is a quantum optimization algorithm that employs a parameter p, which bridges a gap between a gradient-method-based variational quantum eigensolver (VQE) and an exact search algorithm. The parameter p may correspond to the positive number for the Hamiltonian transformation, f(H)=(1−H)p.

When the parameter p is large enough, GNQA, as a global optimization method, rapidly converges towards one of the optimal solutions without being trapped in local minima or plateaus. A global optimization method or algorithm is utilized to locate the global minima or maxima of a function or a set of functions on a given set. In function minimization (the task of finding the input to a mathematical function which produces the smallest output), the symptom of being too exploitative is getting stuck in a so called “local minima” where an early non-optimal result leads you down a path of no return from which you cannot improve. Plateaus refer to the situation in which whatever action is taken there is very little change in the result, with an apparent noisy or random component to all the observations.

In order for GNQA to perform as a global optimization algorithm, a deep quantum circuit (i.e., the depth or longest path of the quantum circuit is large) is required in order to realize the necessary Hamiltonian transformations in solving the combinatorial optimization problem, such as QUBO.

However, there is a limit to the depth of quantum circuits to execute stably since the computation on current noisy intermediate-scale quantum (NISQ) devices includes noise in the results.

Furthermore, when the parameter p is small, GNQA can be executed with shallow circuits (i.e., the depth or longest path of the quantum circuit is small) and is able to converge to the solution faster than standard VQEs. Unfortunately, when the parameter p is small, GNQA has the possibility of being trapped in local minima.

As a result, GNQA cannot currently be used as a valid global optimization algorithm to solve combinatorial optimization problems in the case of the parameter p being small, especially considering the usage of NISQ devices. That is, there is not currently a practical and stable global quantum optimization algorithm for combination optimization problems which works effectively in NISQ devices.

The embodiments of the present disclosure provide the means for implementing a global quantum optimization algorithm for combinatorial optimization problems that works effectively in NISQ devices. In one embodiment of the present disclosure, such a global quantum optimization algorithm for combinatorial optimization problems (e.g., QUBO) works effectively in NISQ devices by combining GNQA using a small value for the parameter p (e.g., p=5), which is executed with shallow circuits, along with the framework of a Bayesian optimization. In such an embodiment, GNQA is utilized for local optimization (locate the local minima for an objective function of a combinatorial optimization problem) and the Bayesian optimization is utilized for global optimization (locate the global minima for the objective function of the combinatorial optimization problem). In this manner, the Bayesian optimization framework is built on top of GNQA to realize the global optimization algorithm with slightly shallow circuits. As a result, a global quantum optimization algorithm for combinatorial optimization problems (e.g., QUBO), even in the case when the value of the parameter p is small (e.g., p<10) which is desirable in NISQ devices, can work effectively in NISQ devices. These and other features will be discussed in greater detail below.

In some embodiments of the present disclosure, the present disclosure comprises a method, system and computer program product for employing quantum optimization algorithms for combinatorial optimization problems in noisy intermediate-scale quantum (NISQ) devices. In one embodiment of the present disclosure, an objective function of a combinatorial optimization problem (e.g., quadratic unconstrained binary optimization problem (QUBO)) to be minimized is defined. In one embodiment, the input to the objective function corresponds to circuit parameters

for the ansatz of the Gauss-Newton based quantum algorithm (GNQA). An “ansantz,” as used herein, refers to the initial guess to solve the combinatorial optimization problem. “N,” as used herein, refers to the problem size of a target combinatorial optimization problem (e.g., QUBO), which is equal to the number of qubits for formulating the problem (e.g., QUBO) in quantum systems. In one embodiment, the output of the objective function corresponds to the error-robust indicator value defined by g(x)=xTQx, where g(x) is the error-robust indicator function which corresponds to a value indicating whether the result of GNQA (solution of the combinatorial optimization problem) is a legitimate return value or an illegitimate value that indicates an error, where Q is an upper triangular matrix for formulating a target combinatorial optimization problem (e.g., QUBO problem),

and where x is a binary vector given by

and θresultdenotes the parameters obtained by GNQA. After initializing the circuit parameters (θ) of the objective function, a Gauss-Newton based quantum algorithm (GNQA) is employed for local optimization (locating a local minima for the objective function) to output an error-robust estimation result (indicator value) of the objective function based on the circuit parameters as well as to output a solution of the combinatorial optimization problem (e.g., the ground state energy level of the Hamiltonian) based on the circuit parameters. Furthermore, a Bayesian optimization is employed for global optimization (locating a global minima for the objective function) in response to the solution of the combinatorial optimization problem not reaching a correct solution, where the Bayesian optimization updates the circuit parameters to minimize the error-robust estimation result (indicator value). Once a correct solution is reached, it is outputted. In this manner, the Bayesian optimization framework is built on top of GNQA to realize the global optimization algorithm with slightly shallow circuits. As a result, a global quantum optimization algorithm for combinatorial optimization problems (e.g., QUBO), even in the case when the value of the parameter p is small (e.g., p<10) which is desirable in NISQ devices, can work effectively in NISQ devices.

In the following description, numerous specific details are set forth to provide a thorough understanding of the present disclosure. However, it will be apparent to those skilled in the art that the present disclosure may be practiced without such specific details. In other instances, well-known circuits have been shown in block diagram form in order not to obscure the present disclosure in unnecessary detail. For the most part, details considering timing considerations and the like have been omitted inasmuch as such details are not necessary to obtain a complete understanding of the present disclosure and are within the skills of persons of ordinary skill the relevant art.

Referring now to the Figures in detail,FIG.1illustrates an embodiment of the present disclosure of a communication system100for practicing the principles of the present disclosure. Communication system100includes a quantum computer101configured to perform quantum computations, such as the types of computations that harness the collective properties of quantum states, such as superposition, interference and entanglement, as well as a classical computer102in which information is stored in bits that are represented logically by either a 0 (off) or a 1 (on). Examples of classical computer102include, but not limited to, a portable computing unit, a Personal Digital Assistant (PDA), a laptop computer, a mobile device, a tablet personal computer, a smartphone, a mobile phone, a navigation device, a gaming unit, a desktop computer system, a workstation, and the like configured with the capability of connecting to network113(discussed below).

In one embodiment, classical computer102is used to setup the state of quantum bits in quantum computer101and then quantum computer101starts the quantum process. Furthermore, in one embodiment, classical computer102in conjunction with quantum computer101are configured to employ quantum optimization algorithms for combinatorial optimization problems (e.g., 2-Sat problem, QUBO) in NISQ devices as discussed further below.

In one embodiment, a hardware structure103of quantum computer101includes a quantum data plane104, a control and measurement plane105, a control processor plane106, a quantum controller107and a quantum processor108.

Quantum data plane104includes the physical qubits or quantum bits (basic unit of quantum information in which a qubit is a two-state (or two-level) quantum-mechanical system) and the structures needed to hold them in place. In one embodiment, quantum data plane104contains any support circuitry needed to measure the qubits' state and perform gate operations on the physical qubits for a gate-based system or control the Hamiltonian for an analog computer. In one embodiment, control signals routed to the selected qubit(s) set a state of the Hamiltonian. For gate-based systems, since some qubit operations require two qubits, quantum data plane104provides a programmable “wiring” network that enables two or more qubits to interact.

Control and measurement plane105converts the digital signals of quantum controller107, which indicates what quantum operations are to be performed, to the analog control signals needed to perform the operations on the qubits in quantum data plane104. In one embodiment, control and measurement plane105converts the analog output of the measurements of qubits in quantum data plane104to classical binary data that quantum controller107can handle.

Control processor plane106identifies and triggers the sequence of quantum gate operations and measurements (which are subsequently carried out by control and measurement plane105on quantum data plane104). These sequences execute the program, provided by quantum processor108, for implementing a quantum algorithm.

In one embodiment, control processor plane106runs the quantum error correction algorithm (if quantum computer101is error corrected).

In one embodiment, quantum processor108uses qubits to perform computational tasks. In the particular realms where quantum mechanics operate, particles of matter can exist in multiple states, such as an “on” state, an “off” state and both “on” and “off” states simultaneously. Quantum processor108harnesses these quantum states of matter to output signals that are usable in data computing.

In one embodiment, quantum processor108performs algorithms which conventional processors are incapable of performing efficiently.

In one embodiment, quantum processor108includes one or more quantum circuits109. Quantum circuits109may collectively or individually be referred to as quantum circuits109or quantum circuit109, respectively. A “quantum circuit109,” as used herein, refers to a model for quantum computation in which a computation is a sequence of quantum logic gates, measurements, initializations of qubits to known values and possibly other actions. A “quantum logic gate,” as used herein, is a reversible unitary transformation on at least one qubit. Quantum logic gates, in contrast to classical logic gate, are all reversible. Examples of quantum logic gates include RX (performs eiθX, which corresponds to a rotation of the qubit state around the X-axis by the given angle theta θ on the Bloch sphere), RY (performs eiθY, which corresponds to a rotation of the qubit state around the Y-axis by the given angle theta θ on the Bloch sphere), RXX (performs the operation e(−iθ/2X⊕X)on the input qubit), RZZ (takes in one input, an angle theta θ expressed in radians, and it acts on two qubits), etc. In one embodiment, quantum circuits109are written such that the horizontal axis is time, starting at the left hand side and ending at the right hand side.

Furthermore, in one embodiment, quantum circuit109corresponds to a command structure provided to control processor plane106on how to operate control and measurement plane105to run the algorithm on quantum data plane104/quantum processor108.

Furthermore, quantum computer101includes memory110, which may correspond to quantum memory. In one embodiment, memory110is a set of quantum bits that store quantum states for later retrieval. The state stored in quantum memory110can retain quantum superposition.

In one embodiment, memory110stores an application111that may be configured to implement one or more of the methods described herein in accordance with one or more embodiments. For example, application111may implement a program for employing quantum optimization algorithms for combinatorial optimization problems in NISQ devices as discussed below in connection withFIGS.2-3and4A-4B. Examples of memory110include light quantum memory, solid quantum memory, gradient echo memory, electromagnetically induced transparency, etc.

Furthermore, in one embodiment, classical computer102includes a “transpiler112,” which as used herein, is configured to rewrite an abstract quantum circuit109into a functionally equivalent one that matches the constraints and characteristics of a specific target quantum device. In one embodiment, transpiler112(e.g., qiskit.transpiler, where Qiskit® is an open-source software development kit for working with quantum computers at the level of circuits, pulses and algorithms) converts the trained machine learning model upon execution on quantum hardware103to its elementary instructions and maps it to physical qubits.

In one embodiment, quantum machine learning models are based on variational quantum circuits109. Such models consist of data encoding, processing parameterized with trainable parameters and measurement/post-processing.

In one embodiment, the number of qubits (basic unit of quantum information in which a qubit is a two-state (or two-level) quantum-mechanical system) is determined by the number of features in the data. This processing stage may include multiple layers of parameterized gates. As a result, in one embodiment, the number of trainable parameters is (number of features)*(number of layers).

Furthermore, as shown inFIG.1, classical computer102, which is used to setup the state of quantum bits in quantum computer101, may be connected to quantum computer101via a network113.

Network113may be, for example, a quantum network, a local area network, a wide area network, a wireless wide area network, a circuit-switched telephone network, a Global System for Mobile Communications (GSM) network, a Wireless Application Protocol (WAP) network, a WiFi network, an IEEE 802.11 standards network and various combinations thereof. Other networks, whose descriptions are omitted here for brevity, may also be used in conjunction with system100ofFIG.1without departing from the scope of the present disclosure.

Furthermore, classical computer102in conjunction with quantum computer101are configured to employ quantum optimization algorithms for combinatorial optimization problems (e.g., 2-Sat problem, QUBO) in NISQ devices as discussed further below in connection withFIGS.2-3and4A-4B. A description of the software components of classical computer102is provided below in connection withFIG.2and a description of the hardware configuration of classical computer102is provided further below in connection withFIG.3.

System100is not to be limited in scope to any one particular network architecture. System100may include any number of quantum computers101, classical computers102and networks113.

A discussion regarding the software components used by classical computer102for employing quantum optimization algorithms for combinatorial optimization problems (e.g., 2-Sat problem, QUBO) in NISQ devices is provided below in connection withFIG.2.

FIG.2is a diagram of the software components of classical system102(FIG.1) for employing quantum optimization algorithms for combinatorial optimization problems (e.g., 2-Sat problem, QUBO) in NISQ devices in accordance with an embodiment of the present disclosure.

Referring toFIG.2, in conjunction withFIG.1, classical computer102includes an initialization engine201configured to define an objective function of a combinatorial optimization problem (e.g., 2-Sat problem, QUBO) to be minimized.

In one embodiment, the objective function corresponds to finding the minimum value among a set of possible values calculated in solving the combinatorial optimization problem. In one embodiment, the objective function of the combinatorial optimization problem to be minimized is defined as having the following features.

In one embodiment, the input to the objective function corresponds to circuit parameters

for the ansatz of GNQA. An “ansantz,” as used herein, refers to the initial guess to solve the combinatorial optimization problem. “N,” as used herein, refers to the problem size of a target combinatorial optimization problem (e.g., QUBO), which is equal to the number of qubits for formulating the problem (e.g., QUBO) in quantum systems.

In one embodiment, the output of the objective function corresponds to the error-robust indicator value defined by g(x)=xTQx, where g(x) is the error-robust indicator function which corresponds to a value indicating whether the result of GNQA (solution of combinatorial optimization problem) is a legitimate return value or an illegitimate value that indicates an error, where Q is an upper triangular matrix for formulating a target combinatorial optimization problem (e.g., QUBO problem),

and where x is a binary vector given by

and where θresultdenotes the parameters obtained by GNQA.

In one embodiment, initialization engine201initializes the circuit parameters (θ) of the objective function. In one embodiment, initialization engine201initializes the circuit parameters (θ) of the objective function by randomly selecting such circuit parameters (θ) of the objective function.

Once initialized, initialization engine201prepares the state |φ(θ)using the initialized circuit parameters (θ) in the first trial (discussed further below) or the circuit parameters (θ) that were updated based on Bayesian optimization (discussed further below) in subsequent trials.

In one embodiment, such a state is prepared in connection with solving a combinatorial optimization problem, such as QUBO. QUBO is related and computationally equivalent to the Ising model. In particular, the generic formulation of QUBO corresponds to the Ising spin-glass Hamiltonian. As a result, in one embodiment, the prepared state |φ(θ)is utilized in the transformation of the Hamiltonian (discussed below). In one embodiment, given a guess or ansatz, quantum processor108calculates the expectation value of the system with respect to an observable, such as the Hamiltonian. The Hamiltonian of a system specifies its total energy—i.e., the sum of its kinetic energy (that of motion) and its potential energy (that of position)—in terms of the Lagrangian function and of the position and momentum of each of the particles. In one embodiment, quantum processor108executes quantum circuit109to perform the transformation of the Hamiltonian to realize:

where H is the Hamiltonian converted from a target combinatorial optimization problem (e.g., QUBO), f(H) is the Hamiltonian transformation and U is the unitary operator acting on |φ.

In one embodiment, quantum processor108estimates all the inner products (ψ|φ) upon execution of quantum circuit109to perform the transformation of the Hamiltonian. In one embodiment, quantum processor108performs the SWAP test to determine the inner products (ψ|φ). The SWAP test is a procedure in quantum computation that is used to check how much two quantum states differ. In one embodiment, the SWAP test takes two input states |ϕand |ψ∞ and outputs a Bernoulli random variable that is 1 with probability

(where the expressions here use bra-ket notation). This allows one to, for example, estimate the squared inner product between the two states, |ψ|ϕ|2to ε additive error by taking the average over

runs of the SWAP test. This requires

copies of the input states. The squared inner product roughly measures “overlap” between the two states.

Classical computer102further includes local optimization engine202configured to perform local optimization. Local optimization, as used herein, refers to locating a local minima for the objective function. A local minima corresponds to the point in the domain of the function (e.g., objective function of a combinatorial optimization problem), which has the minimum value. In one embodiment, local optimization involves the steps discussed above (preparing the state |φ(θ)∞ and having quantum computer101execute quantum circuit109to perform the transformation of the Hamiltonian to realize:

and to estimate all the inner products (ψ|ϕ)) as well as the steps discussed below performed by local optimization engine202.

In one embodiment, local optimization engine202utilizes GNQA for local optimization to locate the local minima for an objective function of a combinatorial optimization problem (e.g., QUBO).

In one embodiment, local optimization engine202receives the inner products (ψ|φ) of the transformation of the Hamiltonian. Upon receiving such inner products, local optimization engine202updates the circuit parameters (θ) based on the received inner products using GNQA. As discussed above, GNQA is a quantum optimization algorithm that employs a parameter p, which bridges a gap between a gradient-method-based variational quantum eigensolver (VQE) and an exact search algorithm. The parameter p may correspond to the positive number for the Hamiltonian transformation, f(H)=(1−H)P. In another embodiment, for NISQ devices, the following Hamiltonian transformation, f(H)=exp(−p2H2), is utilized.

In one embodiment, the solution of the combinatorial optimization problem, such as QUBO, is to solve the problem:

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where |ε*is the ground state of H. The optimal solution may then be estimated in terms of parameters, such as circuit parameters (θ). In one embodiment, such circuit parameters (θ) are updated using the received inner products in connection with finding a solution to the combinatorial optimization problem.

In one embodiment, local optimization engine202employs GNQA for local optimization to output the solution of the combinatorial optimization problem (e.g., QUBO) as discussed above. In one embodiment, the solution of the combinatorial optimization problem corresponds to the ground state energy level of the Hamiltonian.

Furthermore, in addition to updating the circuit parameters (θ) using the estimated inner products of the transformation of the Hamiltonian based on GNQA, in one embodiment, local optimization engine202employs GNQA for local optimization to output an error-robust estimation result (indicator value) of the objective function based on the updated circuit parameters. As previously discussed, the output of the objective function corresponds to the error-robust indicator value defined by g(x)=xTQx, where g(x) is the error-robust indicator function which corresponds to a value indicating whether the result of GNQA (solution of combinatorial optimization problem) is a legitimate return value or an illegitimate value that indicates an error, where Q is an upper triangular matrix for formulating a target combinatorial optimization problem (e.g., QUBO problem),

and where x is a binary vector given by

and where θresultdenotes the parameters obtained by GNQA.

In one embodiment, the steps discussed above in connection with performing local optimization may be repeated based on whether the maximum number of iterations have been reached. If the maximum number of iterations has not yet been reached, then the process discussed above in performing local optimization is repeated.

Once the maximum number of iterations has been reached, a “trial” is said to occur. After each trial has been completed, including the first trial, a determination is made as to whether the correct solution has been reached. If not, a subsequent trial is performed involving updating the circuit parameters (θ) based on Bayesian optimization (discussed further below) in subsequent trials during a global optimization.

Global optimization, as used herein, refers to locating a global minima for the objective function. A global minima corresponds to the smallest overall value of a function (e.g., objective function of a combinatorial optimization problem) over its entire range.

Referring again toFIG.2, classical computer102includes a global optimization engine203to determine if the correct solution has been reached. For example, global optimization engine203determines if the initial trial reaches the correct solution. That is, global optimization engine203determines if the solution of the combinatorial optimization problem provided by GNQA converges to a desired solution. For example, the solution of the combinatorial optimization problem provided by GNQA may correspond to the ground state energy level of the Hamiltonian. Such a solution may be said to reach a correct solution, when, as the iterations proceed, the output (ground state energy level of the Hamiltonian) gets closer and closer to a specific ground state energy level.

If a correct solution has not been reached, then global optimization engine203employs Bayesian optimization for global optimization, where the Bayesian optimization updates the circuit parameters (θ) to minimize the error-robust estimation result (indicator value corresponding to the output of the objective function of the combinatorial optimization problem). Bayesian optimization, as used herein, refers to a sequential design strategy for global optimization of black-box functions, such as the objective function of the combinatorial optimization problem, that does not assume any functional forms.

In one embodiment, the Bayesian optimization involves treating the objective function as a random function and place a prior (a prior probability distribution or “prior” of an uncertain quantity is the probability distribution, such as a probability distribution of circuit parameters (θ), that would express one's beliefs about this quantity before some evidence is taken into account over it) over it. The prior captures beliefs about the behavior of the function. After gathering the function evaluations, which are treated as data, the prior is updated to form the posterior distribution (distribution of circuit parameters (θ)) over the objective function. The posterior distribution, in turn, is used to construct an acquisition function (often also referred to as infill sampling criteria) that determines the next query point.

In one embodiment, the method of kriging, which uses Gaussian processes, is used to define the prior/posterior distribution (distribution of circuit parameters (θ)) over the objective function. Circuit parameters (θ) are then updated to minimize the variance among such a distribution. In one embodiment, the method Parzen-Tree Estimator is used to define the prior/posterior distribution (distribution of circuit parameters (θ)) over the objective function. In one embodiment, the Parzen-Tree Estimator constructs two distributions for “high” and “low” points and then finds the location that maximizes the expected improvement, which corresponds to updating the circuit parameters (θ) that minimizes the error-robust estimation result.

If, on the other hand, a correct solution has been reached, then global optimization engine203selects the solution of the combinatorial optimization problem outputted by the GNQA as the final solution.

In this manner, the Bayesian optimization framework is built on top of GNQA to realize the global optimization algorithm with slightly shallow circuits. As a result, a global quantum optimization algorithm for combinatorial optimization problems (e.g., QUBO), even in the case when the value of the parameter p is small (e.g., p<10) which is desirable in NISQ devices, can work effectively in NISQ devices.

A further description of these and other functions is provided below in connection with the discussion of the method for employing quantum optimization algorithms for combinatorial optimization problems in NISQ devices.

Prior to the discussion of the method for employing quantum optimization algorithms for combinatorial optimization problems in NISQ devices, a description of the hardware configuration of classical computer102(FIG.1) is provided below in connection withFIG.3.

Referring now toFIG.3, in conjunction withFIG.1,FIG.3illustrates an embodiment of the present disclosure of the hardware configuration of classical computer102which is representative of a hardware environment for practicing the present disclosure.

Computing environment300contains an example of an environment for the execution of at least some of the computer code (stored in block301) involved in performing the inventive methods, such as employing quantum optimization algorithms for combinatorial optimization problems in NISQ devices. In addition to block301, computing environment300includes, for example, classical computer102, network113, such as a wide area network (WAN), end user device (EUD)302, remote server303, public cloud304, and private cloud305. In this embodiment, classical computer102includes processor set306(including processing circuitry307and cache308), communication fabric309, volatile memory310, persistent storage311(including operating system312and block301, as identified above), peripheral device set313(including user interface (UI) device set314, storage315, and Internet of Things (IoT) sensor set316), and network module317. Remote server303includes remote database318. Public cloud304includes gateway319, cloud orchestration module320, host physical machine set321, virtual machine set322, and container set323.

Processor set306includes one, or more, computer processors of any type now known or to be developed in the future. Processing circuitry307may be distributed over multiple packages, for example, multiple, coordinated integrated circuit chips. Processing circuitry307may implement multiple processor threads and/or multiple processor cores. Cache308is memory that is located in the processor chip package(s) and is typically used for data or code that should be available for rapid access by the threads or cores running on processor set306. Cache memories are typically organized into multiple levels depending upon relative proximity to the processing circuitry. Alternatively, some, or all, of the cache for the processor set may be located “off chip.” In some computing environments, processor set306may be designed for working with qubits and performing quantum computing.

Volatile memory310is any type of volatile memory now known or to be developed in the future. Examples include dynamic type random access memory (RAM) or static type RAM. Typically, the volatile memory is characterized by random access, but this is not required unless affirmatively indicated. In classical computer102, the volatile memory310is located in a single package and is internal to classical computer102, but, alternatively or additionally, the volatile memory may be distributed over multiple packages and/or located externally with respect to classical computer102.

End user device (EUD)302is any computer system that is used and controlled by an end user (for example, a customer of an enterprise that operates classical computer102), and may take any of the forms discussed above in connection with classical computer102. EUD302typically receives helpful and useful data from the operations of classical computer102. For example, in a hypothetical case where classical computer102is designed to provide a recommendation to an end user, this recommendation would typically be communicated from network module317of classical computer102through WAN113to EUD302. In this way, EUD302can display, or otherwise present, the recommendation to an end user. In some embodiments, EUD302may be a client device, such as thin client, heavy client, mainframe computer, desktop computer and so on.

Remote server303is any computer system that serves at least some data and/or functionality to classical computer102. Remote server303may be controlled and used by the same entity that operates classical computer102. Remote server303represents the machine(s) that collect and store helpful and useful data for use by other computers, such as classical computer102. For example, in a hypothetical case where classical computer102is designed and programmed to provide a recommendation based on historical data, then this historical data may be provided to classical computer102from remote database318of remote server303.

Private cloud305is similar to public cloud304, except that the computing resources are only available for use by a single enterprise. While private cloud305is depicted as being in communication with WAN113in other embodiments a private cloud may be disconnected from the internet entirely and only accessible through a local/private network. A hybrid cloud is a composition of multiple clouds of different types (for example, private, community or public cloud types), often respectively implemented by different vendors. Each of the multiple clouds remains a separate and discrete entity, but the larger hybrid cloud architecture is bound together by standardized or proprietary technology that enables orchestration, management, and/or data/application portability between the multiple constituent clouds. In this embodiment, public cloud304and private cloud305are both part of a larger hybrid cloud.

Block301further includes the software components discussed above in connection withFIG.2to employ quantum optimization algorithms for combinatorial optimization problems in NISQ devices. In one embodiment, such components may be implemented in hardware. The functions discussed above performed by such components are not generic computer functions. As a result, classical computer102is a particular machine that is the result of implementing specific, non-generic computer functions.

In one embodiment, the functionality of such software components of classical computer102, including the functionality for employing quantum optimization algorithms for combinatorial optimization problems in NISQ devices, may be embodied in an application specific integrated circuit.

As stated above, in connection with solving combinatorial optimization problems, such as QUBO, a Gauss-Newton based quantum algorithm may be utilized to solve such a combinatorial optimization problem. Gauss-Newton based quantum algorithm (GNQA) is a quantum optimization algorithm that employs a parameter p, which bridges a gap between a gradient-method-based variational quantum eigensolver (VQE) and an exact search algorithm. The parameter p may correspond to the positive number for the Hamiltonian transformation, f(H)=(1−H)p. When the parameter p is large enough, GNQA, as a global optimization method, rapidly converges towards one of the optimal solutions without being trapped in local minima or plateaus. A global optimization method or algorithm is utilized to locate the global minima or maxima of a function or a set of functions on a given set. In function minimization (the task of finding the input to a mathematical function which produces the smallest output), the symptom of being too exploitative is getting stuck in a so called “local minima” where an early non-optimal result leads you down a path of no return from which you cannot improve. Plateaus refer to the situation in which whatever action is taken there is very little change in the result, with an apparent noisy or random component to all the observations. In order for GNQA to perform as a global optimization algorithm, a deep quantum circuit (i.e., the depth or longest path of the quantum circuit is large) is required in order to realize the necessary Hamiltonian transformations in solving the combinatorial optimization problem, such as QUBO. However, there is a limit to the depth of quantum circuits to execute stably since the computation on current noisy intermediate-scale quantum (NISQ) devices includes noise in the results. Furthermore, when the parameter p is small, GNQA can be executed with shallow circuits (i.e., the depth or longest path of the quantum circuit is small) and is able to converge to the solution faster than standard VQEs. Unfortunately, when the parameter p is small, GNQA has the possibility of being trapped in local minima. As a result, GNQA cannot currently be used as a valid global optimization algorithm to solve combinatorial optimization problems in the case of the parameter p being small, especially considering the usage of NISQ devices. That is, there is not currently a practical and stable global quantum optimization algorithm for combination optimization problems which works effectively in NISQ devices.

The embodiments of the present disclosure provide the means for implementing a global quantum optimization algorithm for combinatorial optimization problems that works effectively in NISQ devices by combining GNQA using a small value for the parameter p (e.g., p=5), which is executed with shallow circuits, along with the framework of Bayesian optimization as discussed below in connection withFIGS.4A-4B.

FIGS.4A-4Bare a flowchart of a method400for employing quantum optimization algorithms for combinatorial optimization problems in NISQ devices in accordance with an embodiment of the present disclosure.

Referring toFIG.4A, in conjunction withFIGS.1-3, in step401, initialization engine201of classical computer102defines an objective function of a combinatorial optimization problem (e.g., 2-Sat problem, QUBO) to be minimized.

As discussed above, in one embodiment, the objective function corresponds to finding the minimum value among a set of possible values calculated in solving the combinatorial optimization problem. In one embodiment, the objective function of the combinatorial optimization problem to be minimized is defined as having the following features.

In one embodiment, the input to the objective function corresponds to circuit parameters

for the ansatz of GNQA. An “ansantz,” as used herein, refers to the initial guess to solve the combinatorial optimization problem. “N,” as used herein, refers to the problem size of a target combinatorial optimization problem (e.g., QUBO), which is equal to the number of qubits for formulating the problem (e.g., QUBO) in quantum systems.

In one embodiment, the output of the objective function corresponds to the error-robust indicator value defined by g(x)=xTQx, where g(x) is the error-robust indicator function which corresponds to a value indicating whether the result of GNQA (solution of combinatorial optimization problem) is a legitimate return value or an illegitimate value that indicates an error, where Q is an upper triangular matrix for formulating a target combinatorial optimization problem (e.g., QUBO problem),

and where x is a binary vector given by

and where θresultdenotes the parameters obtained by GNQA.

In step402, initialization engine201of classical computer102initializes the circuit parameters (θ) of the objective function.

As stated above, in one embodiment, initialization engine201initializes the circuit parameters (θ) of the objective function by randomly selecting such circuit parameters (θ) of the objective function.

In step403, initialization engine201of classical computer102prepares the state |φ(θ)using the initialized circuit parameters (θ) in the first trial or the circuit parameters (θ) that were updated based on Bayesian optimization in subsequent trials.

As discussed above, in one embodiment, such a state is prepared in connection with solving a combinatorial optimization problem, such as QUBO. QUBO is related and computationally equivalent to the Ising model. In particular, the generic formulation of QUBO corresponds to the Ising spin-glass Hamiltonian. As a result, in one embodiment, the prepared state |φ(θ)is utilized in the transformation of the Hamiltonian. In one embodiment, given a guess or ansatz, quantum processor108calculates the expectation value of the system with respect to an observable, such as the Hamiltonian. The Hamiltonian of a system specifies its total energy—i.e., the sum of its kinetic energy (that of motion) and its potential energy (that of position)—in terms of the Lagrangian function and of the position and momentum of each of the particles.

In step404, quantum processor108of quantum computer101executes quantum circuit109to perform the transformation of the Hamiltonian to realize:

where H is the Hamiltonian converted from a target combinatorial optimization problem (e.g., QUBO), f(H) is the Hamiltonian transformation and U is the unitary operator acting on |φ.

In step405, quantum processor108of quantum computer101estimates all the inner products (ψ|φ) upon execution of quantum circuit109to perform the transformation of the Hamiltonian.

As stated above, in one embodiment, quantum processor108performs the SWAP test to determine the inner products (ψ|φ). The SWAP test is a procedure in quantum computation that is used to check how much two quantum states differ. In one embodiment, the SWAP test takes two input states |ϕand |ψ! and outputs a Bernoulli random variable that is 1 with probability

(where the expressions here use bra-ket notation). This allows one to, for example, estimate the squared inner product between the two states, |ψ|ϕ|2to ε additive error by taking the average over

runs of the SWAP test. This requires

copies of the input states. The squared inner product roughly measures “overlap” between the two states.

In step406, local optimization engine202of classical computer102receives the estimated inner products (ψ|φ) of the transformation of the Hamiltonian (obtained in step405).

In step407, local optimization engine202of classical computer102updates the circuit parameters (θ) based on the received inner products using GNQA.

As discussed above, GNQA is a quantum optimization algorithm that employs a parameter p, which bridges a gap between a gradient-method-based variational quantum eigensolver (VQE) and an exact search algorithm. The parameter p may correspond to the positive number for the Hamiltonian transformation, f(H)=(1−H)P. In another embodiment, for NISQ devices, the following Hamiltonian transformation, f(H) exp(−p2H2), is utilized.

In one embodiment, the solution of the combinatorial optimization problem, such as QUBO, is to solve the problem:

?12⁢❘"\[LeftBracketingBar]"φ⁡(θ)〉-❘"\[LeftBracketingBar]"ξ*〉2,?indicates text missing or illegible when filed

where |ε*is the ground state of H. The optimal solution may then be estimated in terms of parameters, such as circuit parameters (θ). In one embodiment, such circuit parameters (θ) are updated using the received inner products in connection with finding a solution to the combinatorial optimization problem.

In step408, local optimization engine202of classical computer102employs GNQA for local optimization to output the solution the combinatorial problem as discussed above.

As stated above, in one embodiment, the solution of the combinatorial optimization problem corresponds to the ground state energy level of the Hamiltonian.

In step409, local optimization engine202of classical computer102employs GNQA for local optimization to output an error-robust estimation result (indicator value) of the objective function based on the updated circuit parameters. As previously discussed, the output of the objective function corresponds to the error-robust indicator value defined by g(x)=xTQx, where g(x) is the error-robust indicator function which corresponds to a value indicating whether the result of GNQA (solution of combinatorial optimization problem) is a legitimate return value or an illegitimate value that indicates an error, where Q is an upper triangular matrix for formulating a target combinatorial optimization problem (e.g., QUBO problem),

and where x is a binary vector given by

and where θresultdenotes the parameters obtained by GNQA.

In step410, local optimization engine202of classical computer102determines whether a maximum number of iterations has been reached.

In one embodiment, the steps discussed above in connection with performing local optimization (e.g., steps403-409) may be repeated based on whether the maximum number of iterations have been reached. If the maximum number of iterations has not yet been reached, then initialization engine201of classical computer102prepares the state |φ(θ)using the circuitparameters (θ) previously utilized in step403.

Referring toFIG.4B, in conjunction withFIGS.1-3, once the maximum number of iterations has been reached, a “trial” is said to occur. After each trial has been completed, including the first trial, in step411, global optimization engine203of classical computer102determines whether the correct solution has been reached. If not, a subsequent trial is performed involving updating the circuit parameters (θ) based on the Bayesian optimization in subsequent trials during a global optimization.

For example, global optimization engine203determines if the initial trial reaches the correct solution. That is, global optimization engine203determines if the solution of the combinatorial optimization problem provided by GNQA converges to a desired solution. For example, the solution of the combinatorial optimization problem provided by GNQA may correspond to the ground state energy level of the Hamiltonian. Such a solution may be said to reach a correct solution, when, as the iterations proceed, the output (ground state energy level of the Hamiltonian) gets closer and closer to a specific ground state energy level.

If a correct solution has not been reached, then, in step412, global optimization engine203of classical computer102employs Bayesian optimization for global optimization, where the Bayesian optimization updates the circuit parameters (θ) to minimize the error-robust estimation result (indicator value corresponding to the output of the objective function of the combinatorial optimization problem).

As discussed above, Bayesian optimization, as used herein, refers to a sequential design strategy for global optimization of black-box functions, such as the objective function of the combinatorial optimization problem, that does not assume any functional forms.

In one embodiment, the Bayesian optimization involves treating the objective function as a random function and place a prior (a prior probability distribution or “prior” of an uncertain quantity is the probability distribution, such as a probability distribution of circuit parameters (θ), that would express one's beliefs about this quantity before some evidence is taken into account over it) over it. The prior captures beliefs about the behavior of the function. After gathering the function evaluations, which are treated as data, the prior is updated to form the posterior distribution (distribution of circuit parameters (θ)) over the objective function. The posterior distribution, in turn, is used to construct an acquisition function (often also referred to as infill sampling criteria) that determines the next query point.

In one embodiment, the method of kriging, which uses Gaussian processes, is used to define the prior/posterior distribution (distribution of circuit parameters (θ)) over the objective function. Circuit parameters (θ) are then updated to minimize the variance among such a distribution. In one embodiment, the method Parzen-Tree Estimator is used to define the prior/posterior distribution (distribution of circuit parameters (θ)) over the objective function. In one embodiment, the Parzen-Tree Estimator constructs two distributions for “high” and “low” points and then finds the location that maximizes the expected improvement, which corresponds to updating the circuit parameters (θ) that minimizes the error-robust estimation result.

Upon updating the circuit parameters (θ), initialization engine201of classical computer102prepares the state |φ(θ)using the updated circuit parameters (θ) in a subsequent trial in step403.

Referring to step411, if, however, a correct solution has been reached, then, in step413, global optimization engine203of classical computer102selects the solution of the combinatorial optimization problem outputted by the GNQA as the final solution.

In this manner, the Bayesian optimization framework is built on top of GNQA to realize the global optimization algorithm with slightly shallow circuits. As a result, a global quantum optimization algorithm for combinatorial optimization problems (e.g., QUBO), even in the case when the value of the parameter p is small (e.g., p<10) which is desirable in NISQ devices, can work effectively in NISQ devices.

As a result of the foregoing, the principles of the present disclosure provide a means for enabling GNQA to be used as a valid global optimization algorithm to solve combinatorial optimization problems in the case of the parameter p being small (e.g., p<10) in NISQ devices. In one embodiment, such an optimization algorithm is implemented using a classical-quantum hybrid global optimization method. Iterative trials are implemented in the classical-quantum hybrid system. In one trial, the process is executed by the Gauss-Newton-based quantum optimization, where the circuit parameters are input and the error-robust estimation result of the objective function is output. If the initial trial does not reach the correct solution, then a subsequent trial will be performed by replacing the initial parameters with the ones searched by the Bayesian optimization. Once the correct solution is reached, it is outputted. In this manner, the Bayesian optimization framework is built on top of GNQA to realize the global optimization algorithm with slightly shallow circuits.

Furthermore, the principles of the present disclosure improve the technology or technical field involving quantum optimization algorithms.

As discussed above, in connection with solving combinatorial optimization problems, such as QUBO, a Gauss-Newton based quantum algorithm may be utilized to solve such a combinatorial optimization problem. Gauss-Newton based quantum algorithm (GNQA) is a quantum optimization algorithm that employs a parameter p, which bridges a gap between a gradient-method-based variational quantum eigensolver (VQE) and an exact search algorithm. The parameter p may correspond to the positive number for the Hamiltonian transformation, f(H)=(1−H)p. When the parameter p is large enough, GNQA, as a global optimization method, rapidly converges towards one of the optimal solutions without being trapped in local minima or plateaus. A global optimization method or algorithm is utilized to locate the global minima or maxima of a function or a set of functions on a given set. In function minimization (the task of finding the input to a mathematical function which produces the smallest output), the symptom of being too exploitative is getting stuck in a so called “local minima” where an early non-optimal result leads you down a path of no return from which you cannot improve. Plateaus refer to the situation in which whatever action is taken there is very little change in the result, with an apparent noisy or random component to all the observations. In order for GNQA to perform as a global optimization algorithm, a deep quantum circuit (i.e., the depth or longest path of the quantum circuit is large) is required in order to realize the necessary Hamiltonian transformations in solving the combinatorial optimization problem, such as QUBO. However, there is a limit to the depth of quantum circuits to execute stably since the computation on current noisy intermediate-scale quantum (NISQ) devices includes noise in the results. Furthermore, when the parameter p is small, GNQA can be executed with shallow circuits (i.e., the depth or longest path of the quantum circuit is small) and is able to converge to the solution faster than standard VQEs. Unfortunately, when the parameter p is small, GNQA has the possibility of being trapped in local minima. As a result, GNQA cannot currently be used as a valid global optimization algorithm to solve combinatorial optimization problems in the case of the parameter p being small, especially considering the usage of NISQ devices. That is, there is not currently a practical and stable global quantum optimization algorithm for combination optimization problems which works effectively in NISQ devices.

Embodiments of the present disclosure improve such technology by defining an objective function of a combinatorial optimization problem (e.g., quadratic unconstrained binary optimization problem (QUBO)) to be minimized. In one embodiment, the input to the objective function corresponds to circuit parameters

for the ansatz of the Gauss-Newton based quantum algorithm (GNQA). An “ansantz,” as used herein, refers to the initial guess to solve the combinatorial optimization problem. “N,” as used herein, refers to the problem size of a target combinatorial optimization problem (e.g., QUBO), which is equal to the number of qubits for formulating the problem (e.g., QUBO) in quantum systems. In one embodiment, the output of the objective function corresponds to the error-robust indicator value defined by g(x)=xTQx, where g(x) is the error-robust indicator function which corresponds to a value indicating whether the result of GNQA (solution of the combinatorial optimization problem) is a legitimate return value or an illegitimate value that indicates an error, where Q is an upper triangular matrix for formulating a target combinatorial optimization problem (e.g., QUBO problem),

and where x is a binary vector given by

and θresultdenotes the parameters obtained by GNQA. After initializing the circuit parameters (θ) of the objective function, a Gauss-Newton based quantum algorithm (GNQA) is employed for local optimization (locating a local minima for the objective function) to output an error-robust estimation result (indicator value) of the objective function based on the circuit parameters as well as to output a solution of the combinatorial optimization problem (e.g., the ground state energy level of the Hamiltonian) based on the circuit parameters. Furthermore, a Bayesian optimization is employed for global optimization (locating a global minima for the objective function) in response to the solution of the combinatorial optimization problem not reaching a correct solution, where the Bayesian optimization updates the circuit parameters to minimize the error-robust estimation result (indicator value). Once a correct solution is reached, it is outputted. In this manner, the Bayesian optimization framework is built on top of GNQA to realize the global optimization algorithm with slightly shallow circuits. As a result, a global quantum optimization algorithm for combinatorial optimization problems (e.g., QUBO), even in the case when the value of the parameter p is small (e.g., p<10) which is desirable in NISQ devices, can work effectively in NISQ devices. Furthermore, in this manner, there is an improvement in the technical field involving quantum optimization algorithms.

The technical solution provided by the present disclosure cannot be performed in the human mind or by a human using a pen and paper. That is, the technical solution provided by the present disclosure could not be accomplished in the human mind or by a human using a pen and paper in any reasonable amount of time and with any reasonable expectation of accuracy without the use of a computer.