Method and system for solving optimization problems
Patent Information
- Application Number
- PCT/EP2026/058466
- Authority / Receiving Office
- WO · WO
- Patent Type
- Applications
- Current Assignee / Owner
- Priority Date
- 2025-03-25
- Filing Date
- 2026-03-25
- Publication Date
- 2026-10-01
Smart Images
Figure EP2026058466_01102026_PF_FP_ABST
Abstract
Description
[0001] Description
[0002] Title: Method and System for Solving Optimization Problems Introduction
[0003] The present invention relates to the technical field of quantum computing, particularly to methods and systems for solving optimization problems. This application claims priority of Luxembourg patent application LU 509917, filed on 25 March 2025. The entire disclosure of the Luxembourg patent application LU 509917 is hereby incorporated herein by reference.
[0004] Background
[0005] Prior art solutions for solving optimization problems, particularly those formulated as Quadratic Unconstrained Binary Optimization (QUBO) or Higher-order Unconstrained Binary Optimization (HUBO), primarily rely on classical algorithms. Techniques such as simulated annealing, genetic algorithms, and greedy algorithms have been employed to tackle these problems. While these methods can yield satisfactory results for smaller instances, they often struggle with scalability and efficiency when applied to larger, more complex problems. The combinatorial nature of these optimization tasks leads to exponential growth in computational time, rendering classical approaches inefficient for large-scale instances. Consequently, these methods may fail to provide optimal solutions within a reasonable timeframe, particularly in applications requiring high accuracy.
[0006] In the realm of quantum computing, existing quantum algorithms have shown promise in addressing optimization problems. However, many of these algorithms are constrained by hardware limitations, including high error rates and limited qubit connectivity. Quantum annealing, for instance, is an analog approach that seeks to find low-energy states of a system but often requires long evolution times to ensure adiabaticity. This reliance on slow evolution can lead to non-adiabatic transitions, resulting in suboptimal solutions. Furthermore, purely quantum algorithms are typically tailored to specific problem types, limiting their applicability across a broader range of optimization challenges. As a result, these quantum methods may not effectively leverage the advantages of quantum computing for general optimization tasks.
[0007] For example, SEBASTIIAN V ROMERO et Al: " Bias-Field Digitized Counterdiabatic Quantum Algorithm for Higher-Order Binary Optimization" discloses a variant of the Bias-Field Digitized Counterdiabatic Quantum Optimization (BF-DCQO) algorithm. And Gomez Cadavid Alejandro et Al: " Bias-field digitized counterdiabatic quantum optimization" discloses an iterative quantum algorithm that enhances combinatorial optimization on noisy quantum hardware by incorporating counterdiabatic terms and bias fields.
[0008] Another notable approach in the prior art is the use of hybrid quantum-classical algorithms. These methods attempt to combine the strengths of classical optimization techniques withquantum computing capabilities. However, they often face challenges in balancing the classical and quantum components, leading to inefficiencies in execution. For example, while classical processors can handle certain computations, the integration with quantum processors may introduce latency and complexity that diminish the overall performance. Additionally, the dynamic adjustment of parameters in these hybrid systems can be cumbersome, making it difficult to achieve the desired optimization outcomes. Thus, while prior art solutions have made strides in addressing optimization problems, they do not adequately resolve the technical challenges associated with scalability, efficiency, and accuracy that the present invention aims to overcome.
[0009] In light of these disadvantages, it is, therefore, an objective of the present invention to partially overcome the aforementioned challenges by proposing a novel approach that addresses the inefficiencies and impracticalities associated with the prior art solutions.
[0010] Summary
[0011] The present invention has been designed to overcome at least some drawbacks present in prior art solutions.
[0012] The problem is solved by providing a method according to claim 1, a computing system according to claim 16 or 18, a computer program product according to claim 19 and a non-transitory computer readable medium according to claim 20. Further advantageous aspects and / or embodiments of the invention are presented in the dependent claims. According to an aspect, the present invention relates to a computer-implemented method for solving at least one optimization problem, preferably using at least one quantum processing unit, advantageously comprising a plurality of qubits and configured to apply a plurality of quantum gates on said plurality of qubits, the method being configured to be executed by a computing system, the method comprising:
[0013] a. An initialization phase, the initialization phase comprising:
[0014] i. Encoding, using a classical processing unit, at least one optimization problem, comprising a plurality of functions representing a plurality of binary variables, into a problem Hamiltonian comprising a plurality of quantum operators;
[0015] ii. Initializing, using the classical processing unit, a time-dependent Hamiltonian comprising an initial Hamiltonian defining the starting configuration of a set of qubits taken among the plurality of qubits of the quantum processing unit;
[0016] iii. Applying, using a bias-field generation module, a predetermined bias-field to the initial Hamiltonian to get a bias-field Hamiltonian; iv. Adding, using the classical processing unit, the bias-field Hamiltonian to the initial Hamiltonian to get a modified initial Hamiltonian;v. Using the modified initial Hamiltonian and the problem Hamiltonian to build a Counter-diabatic Hamiltonian using a set of nested commutators and the classical processing unit;
[0017] vi. Adding the initial Hamiltonian with the counter-diabatic Hamiltonian and the problem Hamiltonian to get a total Hamiltonian, using the classical processing unit;
[0018] vii. Initializing, using quantum gates of the quantum processing unit, the set of qubits into an initial quantum state representing the ground state of the modified initial Hamiltonian;
[0019] b. An evolution phase, the evolution phase comprising:
[0020] i. Evolving, using the quantum processing unit, the quantum state of the set of qubits based on the total Hamiltonian to reach the ground state of the problem Hamiltonian;
[0021] ii. Measuring, using a measurement module, the quantum state the set of qubits to get a probability distribution;
[0022] iii. Computing, using the classical processing unit, expectation values of the probability distribution;
[0023] c. An indexation phase, the indexation phase comprising:
[0024] i. Attributing, using the classical processing unit, an index to each qubit of the set of qubits;
[0025] ii. Selecting, using the classical processing unit, a predetermined number n of qubits equal or smaller to the number N of qubits of the set of qubits;
[0026] d. A tree-based processing phase, the tree-based processing phase being configured to construct a tree comprising n branches, the tree-based processing phase comprising n repetitions of the following set of steps: i. Selecting, using the classical processing unit, the index of a qubit taken among the n qubits;
[0027] ii. A first branching step, the first branching step comprising:
[0028] ■ Applying, using the bias-field generation module, a first predetermined constraint to the selected qubit to get a first modified total Hamiltonian;
[0029] ■ Evolving, using the quantum processing unit, the quantum state of the set of qubits based on the first modified total Hamiltonian to reach the ground state of the problem Hamiltonian;
[0030] ■ Measuring, using the measurement module, the quantum state of the set of qubits to get a first modified probability distribution;
[0031] ■ Computing, using the classical processing unit, expectation values of the first modified probability distribution;
[0032] ■ Creating, using the classical processing unit, a first branch of a tree, the first branch of the tree being associated with the expectation values of the first modified probability distribution;
[0033] iii. A second branching step, the second branching step comprising:■ Applying, using the bias-field generation module, a second predetermined constraint to the selected qubit to get a second modified total Hamiltonian;
[0034] ■ Evolving, using the quantum processing unit, the quantum state of the set of qubits based on the second modified total Hamiltonian to reach the ground state of the problem Hamiltonian;
[0035] ■ Measuring, using the measurement module, the quantum state of the set of qubits to get a second modified probability distribution;
[0036] ■ Computing, using the classical processing unit, expectation values of the second modified probability distribution;
[0037] ■ Creating, using the classical processing unit, a second branch of the tree, the second branch of the tree being associated with the expectation values of the second modified probability distribution;
[0038] iv. Selecting, using the classical processing unit, at least one process taken among a cutting process or a bounding process based on a predetermined set of rules:
[0039] ■ The cutting process comprising:
[0040] Comparing the first expectation value with the second expectation value:
[0041] If the first expectation value is lower than the second expectation value, cutting the second branch, and continuing the treebased processing phase with the first branch;
[0042] if the second expectation value is lower than the first expectation value, cutting the first branch, and continuing the tree-based processing phase with the second branch; Obtaining a tree made of branches corresponding to the lowest expectation values;
[0043] Getting a configuration of the set of qubits corresponding to the lowest expectation values of the problem Hamiltonian;
[0044] ■ The bounding process comprising:
[0045] Storing the first branch and the second;
[0046] Continuing the tree-based processing phase to explore each possible configuration of the set of qubits to identify the lowest expectation value of the problem Hamiltonian;
[0047] e. A solution determination phase, the solution determination phase comprising:
[0048] i. Computing, using the classical processing unit, the lowest expectation values of the problem Hamiltonian to get a finalprobability distribution corresponding to the lowest energy of the set of qubits;
[0049] ii. Extracting, using the classical processing unit, at least one bitstring from the final probability distribution, the bitstring corresponding to the solution of the optimization problem.
[0050] According to another aspect, the present invention relates to a computer product program designed for solving at least one optimization problem which, when executed by at least one processing unit, executes the method according to the present invention.
[0051] According to another aspect, the present invention relates to a non-volatile memory comprising at least one computer program product according to the present invention. The non-volatile memory is designed to retain data even in the absence of power. The computer program product is encoded as instructions or code that can be executed by a processor when accessed from the non-volatile memory.
[0052] According to another aspect, the present invention relates to a computing system for solving at least one optimization problem, comprising at least one classical processing unit, at least one quantum processing unit, at least one bias field generation module and at least one measurement module, and wherein:
[0053] a. the classical processing unit is configured to:
[0054] i. Encode at least one optimization problem into a problem Hamiltonian;
[0055] ii. Initialize a time-dependent Hamiltonian comprising an initial Hamiltonian defining the starting configuration of a set of qubits; iii. Add a bias-field Hamiltonian to the initial Hamiltonian to get a modified initial Hamiltonian;
[0056] iv. Use the modified initial Hamiltonian and the problem Hamiltonian to build a Counter-diabatic Hamiltonian using a set of nested commutators;
[0057] v. Add the initial Hamiltonian with the Counter-diabatic Hamiltonian and the problem Hamiltonian to get a total Hamiltonian;
[0058] vi. Compute expectation values of a probability distribution;
[0059] vii. Attribute an index to each qubit of the set of qubits;
[0060] viii. Select a predetermined number of qubits for tree-based processing; ix. Compute the lowest expectation values of the problem Hamiltonian to get a final probability distribution corresponding to the lowest energy of the set of qubits;
[0061] x. Construct a tree comprising n branches based on the expectation values of the probability distributions,
[0062] xi. select at least one process taken among a cutting process or a bounding process based on a predetermined set of rules to obtain a tree made of branches corresponding to the lowest expectation values;
[0063] xii. get a configuration of the set of qubits corresponding to the lowest expectation values of the problem Hamiltonianextract at least one bitstring from the final probability distribution, corresponding to the solution of the optimization problem;
[0064] b. the quantum processing unit is configured to:
[0065] i. Initialize the set of qubits into an initial quantum state representing the ground state of the modified initial Hamiltonian;
[0066] ii. Evolve the quantum state of the set of qubits based on the total Hamiltonian to reach the ground state of the problem Hamiltonian; c. the bias-field generation module is configured to apply:
[0067] i. a predetermined bias-field to the initial Hamiltonian to get a biasfield Hamiltonian;
[0068] ii. first and second predetermined constraints to a selected qubit to get first and second modified total Hamiltonians, respectively, with the second bias-field having a polarity opposite to the first bias-field; d. A measurement module is configured to measure the quantum state the set of qubits to get a probability distribution and to get first and second modified probability distributions based on the first and second modified total Hamiltonians, respectively.
[0069] According to another aspect, the present invention relates to a computer-implemented method for solving at least one optimization problem using at least one quantum processing unit comprising a plurality of qubits and configured to apply a plurality of quantum gates on said plurality of qubits, the method being configured to be executed by a computing system, the method comprising:
[0070] a. An initialization phase, the initialization phase comprising:
[0071] i. Encoding, using a classical processing unit, at least one optimization problem, comprising a plurality of functions representing a plurality of binary variables, into a problem Hamiltonian comprising a plurality of quantum operators;
[0072] ii. Initializing, using the classical processing unit, a time-dependent Hamiltonian comprising an initial Hamiltonian defining the starting configuration of a set of qubits taken among the plurality of qubits of the quantum processing unit;
[0073] iii. Applying, using a bias-field generation module, a predetermined bias-field to the initial Hamiltonian to get a bias-field Hamiltonian; iv. Adding, using the classical processing unit, the bias-field Hamiltonian to the initial Hamiltonian to get a modified initial Hamiltonian;
[0074] v. Using the modified initial Hamiltonian and the problem Hamiltonian to build a counter-diabatic Hamiltonian using a set of nested commutators and the classical processing unit;
[0075] vi. Adding the initial Hamiltonian with the counter-diabatic Hamiltonian and the problem Hamiltonian to get a total Hamiltonian, using the classical processing unit;
[0076] vii. Initializing, using quantum gates of the quantum processing unit, the set of qubits into an initial quantum state representing the ground state of the modified initial Hamiltonian;b. An evolution phase, the evolution phase comprising:
[0077] i. Evolving, using the quantum processing unit, the quantum state of the set of qubits based on the total Hamiltonian to reach the ground state of the problem Hamiltonian;
[0078] ii. Measuring, using a measurement module, the quantum state of the set of qubits to get a probability distribution;
[0079] iii. Computing, using the classical processing unit, expectation values of the probability distribution for use in selecting qubits in the indexation phase;
[0080] c. An indexation phase, the indexation phase comprising:
[0081] i. Attributing, using the classical processing unit, an index to each qubit of the set of qubits;
[0082] ii. Selecting, using the classical processing unit, a predetermined number n of qubits equal or smaller to the number N of qubits of the set of qubits;
[0083] d. A tree-based processing phase, the tree-based processing phase being configured to construct a tree comprising n branches, the tree-based processing phase comprising n repetitions of the following set of steps: i. Selecting, using the classical processing unit, the index of a qubit taken among the n qubits;
[0084] ii. A first branching step, the first branching step comprising:
[0085] ■ Applying, using the bias-field generation module, a first predetermined constraint to the selected qubit to get a first modified total Hamiltonian;
[0086] ■ Evolving, using the quantum processing unit, the quantum state of the set of qubits based on the first modified total Hamiltonian to reach the ground state of the problem Hamiltonian;
[0087] ■ Measuring, using the measurement module, the quantum state of the set of qubits to get a first modified probability distribution;
[0088] ■ Computing, using the classical processing unit, first expectation values of the first modified probability distribution;
[0089] ■ Creating, using the classical processing unit, a first branch of a tree, the first branch of the tree being associated with the expectation values of the first modified probability distribution;
[0090] iii. A second branching step, the second branching step comprising:
[0091] ■ Applying, using the bias-field generation module, a second predetermined constraint to the selected qubit to get a second modified total Hamiltonian;
[0092] ■ Evolving, using the quantum processing unit, the quantum state of the set of qubits based on the second modified total Hamiltonian to reach the ground state of the problem Hamiltonian;■ Measuring, using the measurement module, the quantum state of the set of qubits to get a second modified probability distribution;
[0093] ■ Computing, using the classical processing unit, second expectation values of the second modified probability distribution;
[0094] ■ Creating, using the classical processing unit, a second branch of the tree, the second branch of the tree being associated with the expectation values of the second modified probability distribution;
[0095] iv. Selecting and executing, using the classical processing unit, at least one process taken among a cutting process or a bounding process based on a predetermined set of rules:
[0096] ■ The cutting process comprising:
[0097] Comparing the first expectation value with the second expectation value:
[0098] If the first expectation value is lower than the second expectation value, cutting the second branch, and continuing the treebased processing phase with the first branch;
[0099] if the second expectation value is lower than the first expectation value, cutting the first branch, and continuing the tree-based processing phase with the second branch; Obtaining a tree made of branches corresponding to the lowest expectation values;
[0100] Getting a configuration of the set of qubits corresponding to the lowest expectation values of the problem Hamiltonian;
[0101] ■ The bounding process comprising:
[0102] Storing the first branch and the second;
[0103] Continuing the tree-based processing phase to explore each possible configuration of the set of qubits to identify the lowest expectation value of the problem Hamiltonian;
[0104] e. A solution determination phase, the solution determination phase comprising:
[0105] i. Computing, using the classical processing unit, the lowest expectation values of the problem Hamiltonian to get a final probability distribution corresponding to the lowest energy of the set of qubits;
[0106] ii. Extracting, using the classical processing unit, at least one bitstring from the final probability distribution, the bitstring corresponding to the solution of the optimization problem.According to another aspect, the present invention relates to a computer-implemented method for solving at least one optimization problem using at least one quantum processing unit comprising a plurality of qubits and configured to apply a plurality of quantum gates on said plurality of qubits, the method being configured to be executed by a computing system, the method comprising:
[0107] a. An initialization phase, the initialization phase comprising:
[0108] i. Encoding, using a classical processing unit, at least one optimization problem into a problem Hamiltonian;
[0109] ii. Initializing, using the classical processing unit, a time-dependent Hamiltonian comprising an initial Hamiltonian defining the starting configuration of a set of qubits taken among the plurality of qubits; iii. Applying, using a bias-field generation module, a predetermined bias-field to the initial Hamiltonian to get a bias-field Hamiltonian; iv. Adding, using the classical processing unit, the bias-field Hamiltonian to the initial Hamiltonian to get a modified initial Hamiltonian;
[0110] v. Using the modified initial Hamiltonian and the problem Hamiltonian to build a counter-diabatic Hamiltonian;
[0111] vi. Adding the initial Hamiltonian with the counter-diabatic Hamiltonian and the problem Hamiltonian to get a total Hamiltonian, using the classical processing unit;
[0112] vii. Initializing, using quantum gates of the quantum processing unit, the set of qubits into an initial quantum state representing the ground state of the modified initial Hamiltonian;
[0113] b. An evolution phase, the evolution phase comprising:
[0114] i. Evolving, using the quantum processing unit, the quantum state of the set of qubits based on the total Hamiltonian to reach the ground state of the problem Hamiltonian;
[0115] ii. Measuring, using a measurement module, the quantum state of the set of qubits to get a probability distribution;
[0116] iii. Computing, using the classical processing unit, expectation values of the probability distribution for use in selecting qubits in the indexation phase;
[0117] c. An indexation phase, the indexation phase comprising:
[0118] i. Attributing, using the classical processing unit, an index to each qubit of the set of qubits;
[0119] ii. Selecting, using the classical processing unit, a predetermined number n of qubits equal or smaller to the number N of qubits of the set of qubits;
[0120] d. A tree-based processing phase, the tree-based processing phase being configured to construct a tree comprising n branches, the tree-based processing phase comprising n repetitions of the following set of steps: i. Selecting, using the classical processing unit, the index of a qubit taken among the n qubits;
[0121] ii. A first branching step, the first branching step comprising:■ Applying, using the bias-field generation module, a first predetermined constraint to the selected qubit to get a first modified total Hamiltonian;
[0122] ■ Evolving, using the quantum processing unit, the quantum state of the set of qubits based on the first modified total Hamiltonian to reach the ground state of the problem Hamiltonian;
[0123] ■ Measuring, using the measurement module, the quantum state of the set of qubits to get a first modified probability distribution;
[0124] ■ Computing, using the classical processing unit, first expectation values of the first modified probability distribution;
[0125] ■ Creating, using the classical processing unit, a first branch of a tree, the first branch of the tree being associated with the expectation values of the first modified probability distribution;
[0126] iii. A second branching step, the second branching step comprising:
[0127] ■ Applying, using the bias-field generation module, a second predetermined constraint to the selected qubit to get a second modified total Hamiltonian;
[0128] ■ Evolving, using the quantum processing unit, the quantum state of the set of qubits based on the second modified total Hamiltonian to reach the ground state of the problem Hamiltonian;
[0129] ■ Measuring, using the measurement module, the quantum state of the set of qubits to get a second modified probability distribution;
[0130] ■ Computing, using the classical processing unit, second expectation values of the second modified probability distribution;
[0131] ■ Creating, using the classical processing unit, a second branch of the tree, the second branch of the tree being associated with the expectation values of the second modified probability distribution;
[0132] iv. Selecting and executing, using the classical processing unit, at least one process taken among a cutting process or a bounding process based on a predetermined set of rules;
[0133] e. A solution determination phase, the solution determination phase comprising:
[0134] i. Computing, using the classical processing unit, the lowest expectation values of the problem Hamiltonian to get a final probability distribution corresponding to the lowest energy of the set of qubits;ii. Extracting, using the classical processing unit, at least one bitstring from the final probability distribution, the bitstring corresponding to the solution of the optimization problem.
[0135] According to another aspect, the present invention relates to a computing system for solving at least one optimization problem, comprising at least one classical processing unit, at least one quantum processing unit comprising a plurality of qubits, at least one bias field generation module and at least one measurement module, wherein:
[0136] a. the classical processing unit is configured to:
[0137] i. Encode at least one optimization problem into a problem Hamiltonian;
[0138] ii. Initialize a time-dependent Hamiltonian comprising an initial Hamiltonian defining the starting configuration of a set of qubits selected from the plurality of qubits of the quantum processing unit;
[0139] iii. Add a bias-field Hamiltonian generated by a bias-field generation module to the initial Hamiltonian to get a modified initial Hamiltonian;
[0140] iv. Use the modified initial Hamiltonian and the problem Hamiltonian to build a Counter-diabatic Hamiltonian using a set of nested commutators;
[0141] v. Add the initial Hamiltonian with the Counter-diabatic Hamiltonian and the problem Hamiltonian to get a total Hamiltonian;
[0142] vi. Attribute an index to each qubit of the set of qubits for use in selecting qubits during tree-based processing;
[0143] vii. Select a predetermined number of qubits for tree-based processing from the set of qubits, where n is equal to or smaller than the total number N of qubits in the set of qubits;
[0144] viii. Compute expectation values of:
[0145] ■ a probability distribution obtained from measurements following evolution under the total Hamiltonian;
[0146] ■ a modified probability distribution obtained from measurements following evolution under a first modified total Hamiltonian;
[0147] ■ a second modified probability distribution obtained from measurements following evolution under a second modified total Hamiltonian;;
[0148] ix. Construct a tree comprising n branches based on the expectation values of the probability distributions, wherein for each of n repetitions corresponding to the n branches of the tree, the processing unit is configured to:
[0149] ■ select an index of a qubit from among the n qubits;
[0150] ■ create a first branch associated with expectation values of the first modified probability distribution;■ create a second branch associated with expectation values of the second modified probability distribution;
[0151] x. Execute at least one process taken among:
[0152] ■ a cutting process, preferably configured to compare expectation values from different branches and retain branches corresponding to lower expectation values while discarding branches with higher expectation values; or ■ a bounding process, preferably configured to store multiple branches for comprehensive exploration of the solution space;
[0153] wherein the process is selected based on a predetermined set of rules to obtain a tree made of branches corresponding to the lowest expectation values;
[0154] xi. Compute the lowest expectation values from among the expectation values computed from the probability distributions to get a final probability distribution corresponding to the lowest energy of the set of qubits;
[0155] xii. Determine a configuration of the set of qubits corresponding to the lowest expectation values of the problem Hamiltonian among all branches of the tree;
[0156] xiii. extract at least one bitstring from the final probability distribution, the bitstring corresponding to the solution of the optimization problem;
[0157] b. the quantum processing unit, configured to apply a plurality of quantum gates on said plurality of qubits, is configured to:
[0158] i. Initialize the set of qubits into an initial quantum state representing the ground state of the modified initial Hamiltonian using quantum gates;
[0159] ii. Evolve the quantum state of the set of qubits based on the total Hamiltonian to reach the ground state of the problem Hamiltonian; iii. Evolve the quantum state of the set of qubits based on the first modified total Hamiltonian following application of first predetermined constraint;
[0160] iv. Evolve the quantum state of the set of qubits based on the second modified total Hamiltonian following application of second predetermined constraint
[0161] c. the bias-field generation module is configured to:
[0162] i. Generate a predetermined bias-field Hamiltonian by applying a predetermined bias-field to the initial Hamiltonian;
[0163] ii. Apply first and second predetermined constraints to a selected qubit to get first and second modified total Hamiltonians, respectively, with the second bias-field having a polarity opposite to the first bias-field;
[0164] d. The measurement module is configured to:i. measure the quantum state of the set of qubits to obtain a probability distribution; and
[0165] ii. measure the quantum state of the set of qubits following evolution based on the first modified total Hamiltonian to obtain a first modified probability distribution;
[0166] iii. measure the quantum state of the set of qubits following evolution based on the second modified total Hamiltonian to obtain a second modified probability distribution.
[0167] Before providing below a detailed review of embodiments of the invention, some optional characteristics that may be used in association or alternatively will be listed hereinafter: According to a further aspect, the problem Hamiltonian comprises a plurality of quantum operators.
[0168] According to a different aspect, the cutting process comprises:
[0169] a. Comparing the first expectation value with the second expectation value:
[0170] i. If the first expectation value is lower than the second expectation value, cutting the second branch, and continuing the tree-based processing phase with the first branch;
[0171] ii. if the second expectation value is lower than the first expectation value, cutting the first branch, and continuing the tree-based processing phase with the second branch;
[0172] b. Obtaining a tree made of branches corresponding to the lowest expectation values;
[0173] c. Getting a configuration of the set of qubits corresponding to the lowest expectation values of the problem Hamiltonian.
[0174] According to another aspect, the bounding process comprises:
[0175] a. Storing the first branch and the second;
[0176] b. Continuing the tree-based processing phase to explore each possible configuration of the set of qubits to identify the lowest expectation value of the problem Hamiltonian.
[0177] According to a different aspect, the first branching step comprises:
[0178] a. Applying, using the bias-field generation module, a first predetermined constraint to the selected qubit to get a first modified total Hamiltonian; b. Evolving, using the quantum processing unit, the quantum state of the set of qubits based on the first modified total Hamiltonian to reach the ground state of the problem Hamiltonian;
[0179] c. Measuring, using the measurement module, the quantum state of the set of qubits to get a first modified probability distribution;
[0180] d. Computing, using the classical processing unit, first expectation values of the first modified probability distribution;e. Creating, using the classical processing unit, a first branch of a tree, the first branch of the tree being associated with the expectation values of the first modified probability distribution.
[0181] According to a further aspect, the second branching step comprises:
[0182] a. Applying, using the bias-field generation module, a second predetermined constraint to the selected qubit to get a second modified total Hamiltonian; b. Evolving, using the quantum processing unit, the quantum state of the set of qubits based on the second modified total Hamiltonian to reach the ground state of the problem Hamiltonian;
[0183] c. Measuring, using the measurement module, the quantum state the set of qubits to get a second modified probability distribution;
[0184] d. Computing, using the classical processing unit, second expectation values of the second modified probability distribution;
[0185] e. Creating, using the classical processing unit, a second branch of the tree, the second branch of the tree being associated with the expectation values of the second modified probability distribution.
[0186] According to a different aspect, the encoding step comprise mapping, using the classical processing unit, the optimization problem to the problem Hamiltonian, by associating at least one quantum operator of the problem Hamiltonian to each function of the optimization problem.
[0187] According to a further aspect, the selected qubit taken among the set of qubits is the qubit having the nearest expectation value to zero.
[0188] According to a different aspect, the first predetermined constraint is a first predetermined bias field.
[0189] According to a further aspect, the second predetermined constraint is a second predetermined bias field having a polarity opposite to the first predetermined bias field. According to a different aspect, the problem Hamiltonian is derived from an optimization problem selected from a group comprising: protein folding, portfolio optimization, logistics optimization, and machine learning model training.
[0190] According to another aspect, the tree-based processing phase further comprises updating the bias-field Hamiltonian based on the computed expectation values of the modified probability distributions, thereby dynamically adjusting the search space for the optimization problem.
[0191] According to a further aspect, the counter-diabatic Hamiltonian is constructed using a set of nested commutators designed to minimize non-adiabatic transitions during the evolution phase, the constructing step comprising at least the following steps:
[0192] a. Providing the modified initial Hamiltonian Hoand the final Hamiltonian Hp;b. Obtaining an interpolated Hamiltonian by defining a parametric interpolation of the Hamiltonian Hsof the set of qubits along a continuous evolution parameter A(t), as:
[0193]
[0194] = (l -A(O>o+ A(t)Hp
[0195] where A(t) is a time-dependent schedule function satisfying A(0) = 0 and A(T) = 1.
[0196] c. Computing the derivative of the interpolated Hamiltonian with respect to A expressed as:
[0197] dM =Hp-H0
[0198] d. Constructing an approximate adiabatic gauge potential A for the Hamiltonian HS(A) of the set of qubits, wherein A is expressed as a finite series expansion of nested commutators, such as:
[0199] AA= ia1[d1 / / s(A), / / (A)] + ia [dM^, HW], HsW] + --- where anare predetermined or variationally optimized coefficients, and the series is truncated at a predetermined order based on an additional predetermined set of rules;
[0200] e. Determining the time derivative (t) of the continuous evolution parameter A(t); and
[0201] f. Generating the counter-diabatic Hamiltonian comprising the original interpolated Hamiltonian and an additional driving term dependent on the adiabatic gauge potential, expressed as:
[0202] H
[0203]
[0204] CD(t) = + A(t)XA
[0205] According to a different aspect, the quantum gates applied by the quantum processing unit comprise single-qubit gates and multi-qubit gates, and wherein the evolution of the quantum state is achieved through a sequence of single-qubit gates and of multi-qubit gates operations that approximate the time evolution under the total Hamiltonian.
[0206] According to a further aspect, the initialization phase further comprises selecting an initial Hamiltonian that has a known ground state.
[0207] According to a different aspect, the encoding step further comprises encoding the optimization problem into the problem Hamiltonian using an Ising model representation for Quadratic Unconstrained Binary Optimization problems or a generalized Ising modelwith higher-order interactions for Higher-order Unconstrained Binary Optimization problems.
[0208] According to another aspect, the predetermined set of rules for selecting between a cutting process or a bounding process comprises, when a predetermined time limit is selected:
[0209] • when the predetermined time limit is equal or higher than a predetermine time threshold, selecting the cutting process;
[0210] • when the predetermined time limit is lower than a predetermined time threshold, selecting the bounding process;
[0211] According to a further aspect, the predetermined set of rules for selecting between a cutting process or a bounding process comprises, when a predetermined level of accuracy is selected:
[0212] • When the predetermined level of accuracy is lower than a predetermined accuracy threshold, selecting the cutting process;
[0213] • When, a predetermined level of accuracy is equal or higher than a predetermined accuracy threshold, selecting the bounding process.
[0214] According to a different aspect, the present invention further comprises a post-processing phase wherein the bitstring extracted from the final probability distribution is decoded, using the processing unit, to retrieve the solution to the optimization problem in its native representation.
[0215] According to another aspect, the quantum processing unit comprises at least one among: superconducting quantum circuit, trapped ion quantum processor, and neutral atom quantum processor, each configured to implement quantum gate operations, preferably required by the method.
[0216] According to a different aspect, the present invention further comprises a feedback loop wherein the results from the solution determination phase are used to refine the encoding of the optimization problem or adjust parameters of the total Hamiltonian.
[0217] Based on the highly technical field of the present invention, some definitions and explanations will now be given about some technical terms and words used in this description:
[0218] • Quantum computing: It refers to a field of computation that aims to outperform classical computation by exploiting quantum mechanical phenomena. In this regard, it is necessary to introduce a qubit, which is a basic unit of quantum information.• Qubit: It refers to the quantum analog of a classical bit. It is representative of a physical system that may be in two different states, generally denoted by |0) and 11>, as well as in a superposition of those two states, e.g.,|0^1>. This plays an important role in the development of quantum algorithms outperforming classical algorithms. One example of a physical device that may be used as a qubit is an electron spin.
[0219] • Unitary operation: The time evolution of qubits is specified by a unitary operator acting on qubit states. The unitary operator plays the role of a gate in quantum computing. In general, these gates are generated via some Hamiltonian that makes a qubit system evolve in time.
[0220] • Quantum circuit: A quantum circuit is a model for quantum computation in which a computation is a sequence of quantum gates, which are reversible transformations on a quantum mechanical analog of an n-bit register. This register comprises n qubits. Circuit depth: It is defined as the number of parallel unitary operations that can be performed for a given algorithm to run on a given quantum hardware. • Coherence time: It is the duration over which a quantum system, such as a qubit in a quantum computer, maintains its quantum state without significant decoherence. Decoherence is the process by which a quantum system loses its quantum mechanical properties, typically due to interactions with its external environment, leading to the loss of superposition and entanglement. Coherence time sets a limit on the time available to perform quantum operations or computations before the quantum information is degraded.
[0221] • Gate: In quantum computing, a gate is a fundamental operation applied to one or more qubits that changes their quantum state. These gates operate according to the principles of quantum mechanics, such as superposition, entanglement, and interference. Mathematically, quantum gates are represented as unitary matrices, meaning their operations are reversible and preserve the total probability of the system. When applied to qubits, they transform the state vector in the complex Hilbert space, enabling quantum computations.
[0222] Quantum gates come in various types, each with specific roles. Single-qubit gates act on individual qubits to manipulate their states. For example, the Hadamard gate (H) creates superposition by evenly distributing the probability across all possible states, while the Pauli gates (X, Y, Z) perform rotations operations or rotations around specific axes. Other single-qubit gates like the Phase gates (S, T) introduce specific phase shifts to the quantum state.
[0223] Two-qubit gates, on the other hand, operate on pairs of qubits and are useful for generating entanglement. For instance, the Controlled-NOT gate (CNOT) flips the state of a target qubit depending on the state of a control qubit. Another exampleis the SWAP gate, which exchanges the quantum states of two qubits. Extending these principles, multi-qubit gates, such as the Toffoli gate (a generalization of CNOT with two control qubits) and the Fredkin gate (a controlled-SWAP gate), are used in more complex quantum operations.
[0224] Key features of quantum gates include their reversibility, as all quantum gates are unitary and inherently reversible, and their role in universality, where a combination of certain gates (like Hadamard, Phase, and CNOT) can be used to implement any quantum algorithm. Additionally, quantum gates enable non-classical behavior, such as creating superposition and entanglement, phenomena that have no direct equivalents in classical computing.
[0225] In a quantum circuit, quantum gates are applied sequentially to qubits, forming the building blocks of quantum algorithms. These operations allow quantum computers to solve problems that are computationally infeasible for classical systems.
[0226] • Gate error: It refers to inaccuracies that occur during the execution of quantum gates, which are fundamental operations on qubits. These errors can be caused by a variety of factors including environmental disturbances, imperfections in qubit fabrication, or inaccuracies in the control mechanisms that execute quantum operations. Gate errors are significant because they directly impact the fidelity of quantum operations, thereby affecting the accuracy and reliability of quantum computations. As such, mitigating gate errors is useful, involving strategies like improving qubit design to enhance stability, refining control techniques to reduce execution imperfections, and implementing quantum error correction protocols to detect and correct errors without disturbing the quantum state.
[0227] • Gate fidelity: It refers to the accuracy with which quantum gates — the basic operations that manipulate qubits — perform their intended tasks. It quantifies the degree of closeness between the theoretical or ideal quantum gate operation and the actual operation as implemented in a quantum computing system. High fidelity is useful for practical quantum computations as it ensures the reliability of quantum algorithms, particularly in systems that require a complex sequence of operations.
[0228] • Analog block (or gate): An analog block is a parametrized entangling unitary evolution with more than one parameter. An example is a two-parameter- dependent multiqubit operation in trapped ions (global MS gate) as
[0229] U^s(9, 0 ) = e“^cos< / > Sx + sin< / > sy^
[0230] W
[0231]
[0232] here SxM=• Digital block (or gate): A digital block is a fixed unitary evolution up to a set of local rotations upto 2-qubits. The examples are parameter-fixed entangling quantum gates and single qubit rotations with arbitrary angles as
[0233] 0 e^2cr / )et7r / 4cr1zcr / , and
[0234] ty(0) =
[0235] U
[0236]
[0237] ^e. O ^ e-^a^.
[0238] • State fidelity: It refers to a measure of the closeness between two quantum states.
[0239] It is often used to quantify the accuracy of a quantum operation or to compare the theoretical and experimental states of a quantum system. More specifically, the state fidelity between two quantum states, represented as p and o, is defined by the formula:
[0240] - \2
[0241] Trjjpfrjp]
[0242]
[0243] • Neutral atom quantum computers: Neutral atoms set-ups have been used for years now as a truly versatile experimental platform for quantum simulation. The shift in interest from quantum simulation to quantum computation with industrial applications in mind followed with the recent technological developments for trapping and manipulating arrays of single atoms led to the use of neutral atoms platforms to perform analog quantum computing. One of the most commonly used atomic species for quantum computation is rubidium-87 due to its well know electronic structure as well as the availability of lasers and other necessary hardware. Another important element to realize quantum computation with neutral atoms is the use of Rydberg states. Rydberg states are electronic states of an atom which has high quantum number, thus where the outer shell electron is promoted far away from the atomic core. This yields long range interactions with neighboring atoms. There are different ways to exploit Rydberg states that yield different spin state mapping schemes for neutral atoms platforms. Optical tweezers are far-off resonance laser beams used to trap single atoms in space and reorganize them into any 2-dimensional or 3-dimensional patterns. Cooled down atomic clouds, Rydberg states and optical tweezers constitute important parts of the most neutral atom quantum computation platforms. A two spin states mapping scheme is described here as an example of the invention.
[0244] • Ground-Rydberg qubits: An atom with an outer shell electron promoted to a Rydberg state will interact with a neighboring atom in its natural ground state through a Van der Waals interaction. This interaction creates the Rydberg blockade phenomenon.• Trapped ion quantum computers: Trapped ions are a leading technology in the field of quantum computing, distinguished by their high-fidelity operations and long coherence times. This technology utilizes ions-atoms that have been ionized by adding or removing electrons confined and suspended in free space using electromagnetic fields. These trapped ions serve as qubits, the fundamental units of quantum information, in a quantum computer. Qubits are stored in stable electronic states of each ion, and quantum information can be transferred through the collective quantized motion of the ions in a shared trap. For single qubit operations, lasers are applied to induce coupling between the qubit states or, for entanglement between qubits, coupling between the internal qubit states and the external motional states.
[0245] • Quantum Dot Quantum Computers: Quantum dot quantum computers use semiconductor nanocrystals (quantum dots) where electrons are confined in three spatial dimensions, creating discrete quantum states that can be controlled electrically. This approach hopes to leverage existing semiconductor manufacturing techniques to build quantum computers, potentially allowing easier integration with classical electronic infrastructure.
[0246] • Photonic Quantum Computers: Photonic quantum computers use particles of light (photons) to perform quantum computations. This type of quantum computing does not require cooling the system to near absolute zero temperatures and can potentially operate at room temperature. Photons are manipulated using optical elements like mirrors, beam splitters, and phase shifters to perform quantum operations. Companies like Xanadu are pioneering this technology, focusing on building quantum computers that can be integrated into existing fiber-optic infrastructure.
[0247] • Superconducting Quantum Computers: Superconducting quantum computers utilize materials that become superconductors at very low temperatures, allowing them to conduct electricity without resistance. This property is crucial for maintaining the coherence of qubits, which are the basic units of quantum information in these systems. Superconducting qubits are typically created using small circuits made from superconducting materials, such as aluminum or niobium, which are cooled to temperatures close to absolute zero (-273.15°C). At these extreme temperatures, the circuits exhibit quantum mechanical behaviors. The most common types of superconducting qubits include the transmon, flux, and phase qubits, each differing slightly in design and how they manipulate quantum states. Operations on qubits are performed using microwave pulses, which excite the qubits between their energy states, allowing for the implementation of quantum gates necessary for computation. The entire system requires sophisticatedcryogenic equipment to maintain the low temperatures and an environment that minimally disturbs the qubits.
[0248] • Hamiltonian and k-local terms: A Hamiltonian is an operator representing the total energy of a quantum system, useful for describing the system's evolution over time. Particularly relevant are k-local Hamiltonians, where each term in the Hamiltonian involves interactions among at most k qubits. This concept is important in the realm of quantum simulations and algorithms, as it realistically models physical systems which typically exhibit local interactions. For instance, a 2-local Hamiltonian includes terms that describe interactions between pairs of qubits but not three or more simultaneously.
[0249] • Problem Hamiltonian: The problem Hamiltonian (also referred to here as target Hamiltonian) is the Hamiltonian that needs to be mapped to the quantum computer or simulator and simulated using DACQO algorithm.
[0250] • Initial Hamiltonian: The term refers to the system Hamiltonian that is designed to have a simple and known ground state. It serves as the starting point of the annealing process. As the annealing progresses, the Hamiltonian of the system gradually transforms from this initial Hamiltonian to a problem Hamiltonian.
[0251] • Final Hamiltonian: The term refers to the second sub-Hamiltonian that encodes the solution to the problem being solved in its ground state. In quantum annealing, the system evolves towards this Hamiltonian such that the quantum state will end up in the ground state of this Hamiltonian, effectively solving the problem.
[0252] • Auxiliary Hamiltonian: The term refers to the third sub-Hamiltonian that comprises local fields over each qubit provided. While its specific purpose might vary depending on the problem and the annealing protocol, auxiliary Hamiltonians are typically introduced to assist the annealing process and ensure a smooth transition from the initial to the problem Hamiltonian.
[0253] • Instantaneous Hamiltonian: The term refers to the Hamiltonian at any given point in time during the annealing or evolution process. As the annealing progresses, the system Hamiltonian changes from the initial to the final Hamiltonian. The Hamiltonian at any particular moment during this transformation can be considered the instantaneous Hamiltonian.
[0254] • Expectation value: It is the average of the measurable quantity of a quantum system which can include the energy, momentum or other dynamical quantity. It is obtained by the following relation:
[0255]
[0256] |(V’(t)IV’(0>l2
[0257] wherein 0 (t) is the time dependent (or maybe independent) operator of interest and >( is the time dependent wavefunction of the quantum system that determines its state at a particular time t.
[0258] • Quantum processor: A quantum processor is a programable quantum devices composed of several informational units (qubits) that can be tuned in order to perform quantum algorithms.
[0259] • Trotterization: It is a technique used in quantum computing to simulate the evolution of quantum systems governed by Hamiltonians that are sums of noncommuting terms. It breaks down the exponential of a sum of operators into a product of exponentials of these operators, as e(A+B)t~ e e J allowing for an approximate simulation of the quantum system's dynamics. Here, A and B are, generally, non-commuting operators, t represents time, and n is the number of subdivisions or steps in the approximation. As n approaches infinity, this approximation becomes exact. This method is based on the Trotter-Suzuki formula, which provides a way to approximate the evolution operator of the entire system through a sequence of simpler operations that are easier to implement on a quantum computer.
[0260] • Methods of quantum computing: Among the approaches to do quantum computing, there are mainly two, analog and digital methods. The analog method aims to design the hardware to mimic a certain problem (Hamiltonian). While being high in accuracy, it has the disadvantage that it is specific to a given problem and lacks flexibility. In digital quantum computation (DQC), the digital quantum gates (single and two-qubit) are used to tune the hardware to reach a certain target Hamiltonian. Digital quantum computing (DQC): is based on the application of sequences of single-qubit and multi-qubit gates. These operations are termed “digital blocks”, and the protocol is known as a digital quantum algorithm. DQC is limited by the fact that it is resource-consuming and does not display robustness to errors in the computation. This makes DQC challenging to implement important and complex problems and reach quantum advantage. Analog quantum computing: Analog quantum computing involves configuring quantum systems to emulate other quantum systems directly, making it highly effective for simulating quantum physics phenomena. Unlike digital quantum computing, which uses quantum bits (qubits) to perform calculations in a manner analogous to binary bits in classical computing, analog quantum computing operates by leveraging the natural quantum mechanical properties of the system. This method allows for a more intuitive representation of quantum simulations, particularly useful in fieldssuch as material science and chemistry where the inherent complexities of quantum interactions are a central focus.
[0261] • Digital-Analog Quantum Computing: An alternative approach is Digital-Analog Quantum Computing (DAQC), which uses single and two qubit gates as digital blocks and multi-qubit gates as analog blocks aiming to solve the problem with fewer resources (reduced circuit depth) than is required by DQC. Moreover, due to reduced circuit depth it incurs less error in the simulation compared to the DQC paradigm and is more accurate.
[0262] • Quantum annealing: It is an analog computation method that may be used to find a low-energy state, preferably the ground state, of a system. Quantum annealing is used for problems where the search space is discrete (e.g. combinatorial optimization problems) with many local minima. Similar in concept to classical annealing, the method relies on the underlying principle that natural systems tend towards lower energy states because lower energy states are more stable. However, while classical annealing uses classical thermal fluctuations to guide a system to a low-energy state and ideally its global energy minimum, quantum annealing may use quantum effects, such as quantum tunneling, to reach a global energy minimum more accurately and / or more quickly than classical annealing. In quantum annealing thermal effects and other noise may be present to aid the annealing. However, the final low-energy state may not be the global energy minimum. Adiabatic quantum computation, therefore, may be considered a special case of quantum annealing for which the system, ideally, begins and remains in its ground state throughout an adiabatic evolution. Thus, those with skill in the art will appreciate that quantum annealing systems and methods may generally be implemented on an adiabatic quantum computer. Throughout this description, any reference to quantum annealing is related to adiabatic quantum computation unless the context requires otherwise. Quantum annealing uses quantum mechanics as a source of disorder during the annealing process. The considered problem (i.e. the problem to be solved) is encoded in a Hamiltonian HP, and the algorithm introduces quantum effects by adding a disordering Hamiltonian Hothat does not commute with HP. A quantum annealer is a programable quantum device that is capable of interpolating an initial Hamiltonian Hoat time t=0 with a final or problem Hamiltonian HPat a time t=T. An appropriate design of a quantum annealer for solving the considered problem configured to produce a desired final Hamiltonian (problem Hamiltonian) comprises a tunable scheduling function. The quantum annealer may comprise at least one scheduling function that is a timedependent tunable parameter of the used hardware. Quantum computers (quantum processors) may include quantum annealing processors, digitized quantum processors, gate-based processors, or adiabatic quantum computation.• Noisy-intermediate scale (NISQ) devices: It refers to the current generation of quantum computers that are characterized by their relatively small number of qubits and the presence of noise and errors in their operations.
[0263] • Optimization problem: Optimization problems are central to many scientific, engineering, and economic applications and fall under a broad category of mathematical problems where the goal is to find the optimal solution from a set of available alternatives. The optimal solution is determined according to specific criteria, typically defined by an objective function that needs to be maximized or minimized. The computational complexity of finding optimal solutions in optimization problems are categorized into polynomial time (P), where the solution time scales polynomially with input size, or NP hard, that exhibit a solution time that can grow exponentially with the problem size, making them computationally challenging, especially for large-scale instances. Quantum computing aims to address this complexity.
[0264] • Cost function: A cost function defines the optimization problem. It is translated into a Hamiltonian, where the ground state represents the optimal solution. This cost function shapes the energy landscape. Quantum annealing aims to find the ground state, which corresponds to the cost function's minimum. Annealing starts with a simple "initial Hamiltonian." As the process progresses, the system is increasingly governed by the cost function's energy landscape. At the end of annealing, the quantum system's state gives a solution to the optimization problem, evaluated using the cost function.
[0265] • Optimization algorithms: This may include simulated annealing, parallel tempering, Markov Chain Monte Carlo techniques, branch and bound / cut algorithms, and greedy algorithms, which may be performed by a classical computer. Optimization algorithms may also include algorithms performed by a quantum computer, such as quantum annealing, quantum approximate optimization algorithm (QAOA) or other noisy intermediate-scale quantum (NISQ) algorithms, quantum implemented fault-tolerant optimization methods, or other quantum optimization algorithms.
[0266] • Quantum optimization algorithms: Quantum optimization algorithms are quantum algorithms that are used to solve optimization problems. A combinatorial optimization problem refers to the problem of finding the most efficient solution from a finite set of possibilities that satisfies certain criteria and constraints, often involving discrete or combinatorial structures like graphs, sequences, or configurations. Variational quantum algorithms: Variational quantum algorithms (VQA) are a family of hybrid quantum-classical algorithms, that mix the power of quantum computing with subroutines that contain classical optimization algorithms. The variational quantum algorithms use the concept of the variational quantum ansatz. This is a parametrized quantum state, which can be produced bya parametrized quantum circuit in the case of digital quantum computers, or by a parametrized evolution in the case of analog quantum computers, ora combination of both in the case of digital-analog paradigm. Once the ansatz is built, then a cost or objective function is evaluated using the quantum processor. The objective function serves as the driving force of these algorithms, it encapsulates the problem to be solved and quantifies its outcome. Later, the expectation value of a cost function is the input for a classical optimization algorithm, which updates the parameters of the variational ansatz to obtain a quantum state that maximizes or minimizes the objective function according to the requirement of the problem. • Variational quantum algorithms: Variational quantum algorithms (VQA) are a family of hybrid quantum-classical algorithm, that mix the power of quantum computing with subroutines that contain classical optimization algorithms. The variational quantum algorithms use the concept of the variational quantum ansatz. This is a parametrized quantum state, which can be produced by a parametrized quantum circuit in the case of digital quantum computers, or by a parametrized evolution in the case of analog quantum computers, or a combination of both in the case of digital-analog paradigm. Once the ansatz is built, then a cost or objective function is evaluated using the quantum processor. The objective function serves as the driving force of these algorithms, it encapsulates the problem to be solved and quantifies its outcome. Later, the expectation value of a cost function is the input for a classical optimization algorithm, which updates the parameters of the variational ansatz to obtain a quantum state that maximizes or minimizes the objective function according to the requirement of the problem. • QUBO problem: QUBO problems are a type of optimization problem and have a broad range of applications in various fields like machine learning, finance, and logistics. In a QUBO problem, one seeks to minimize a quadratic objective function of binary variables (variables that can take values of 0 or 1). The objective function is a sum of terms, each involving either a single variable or a product of two variables. These problems are unconstrained, meaning there are no explicit constraints on the variables apart from their binary nature.
[0267] • HUBO problem: Higher order unconstrained binary optimization (HUBO) problems involve optimizing polynomial functions of binary variables without explicit constraints. These problems are important in various fields, including operations research and physics. Solving HUBO problems requires advanced algorithms, with quantum computing offering promising new solutions. Quantum techniques exploit unique quantum properties to solve HUBO problems more efficiently than classical methods.
[0268] • Ising spin glass Hamiltonian: The Ising spin glass model describes a system of spins that can interact with each other in a complex manner. In this model, spinsare represented by variables that can take values of +1 or -1, typically denoting their 'up' or 'down' states. The model is defined on a lattice (or graph) where spins are placed on vertices and interactions between spins are represented by edges. These interactions can be ferromagnetic or antiferromagnetic, leading to complex energy landscapes. The energy of a spin configuration in the Ising model is given by a Hamiltonian, which is a function of the spin states and their interactions. The goal often involves finding the ground state of this Hamiltonian, which corresponds to the minimum energy configuration.
[0269] • Problem encoding: Encoding a problem in quantum computing refers to the process of translating a specific computational problem or algorithm into a form that is suitable for a quantum computer to execute. For example, the QUBO problem can be encoded into the Ising spin glass Hamiltonian.
[0270] • Adiabatic Quantum Computing (AQC): is a quantum computing methodology that leverages the adiabatic theorem of quantum mechanics. It starts with a quantum system in an easily identifiable ground state and then slowly evolves this system according to a changing Hamiltonian, a process known as adiabaticevolution. The final Hamiltonian of this evolution is designed such that its ground state encodes the solution to the computational problem of interest. AQC's strength lies in its robustness against certain types of errors and decoherence, as the system ideally remains in its lowest energy state throughout the computation. This approach is particularly effective for solving optimization problems and is closely related to quantum annealing. However, maintaining adiabaticity throughout the computation, especially for complex problems, poses a significant challenge, as it often requires exceedingly slow evolution to prevent transitions to higher energy states. Shortcuts to Adiabaticity: It is a computational paradigm proposed to speed up Adiabatic evolution which can be infinitely slow for practical purposes using Counter-diabatic driving. The concept behind Counter-diabatic driving is to add an auxiliary Hamiltonian to the system such that the combined evolution is equivalent to the desired adiabatic process but achieved in a shorter time. This auxiliary Hamiltonian effectively counteracts the non-adiabatic transitions that would typically occur during a faster evolution.
[0271] • Counter-diabatic quantum computing: Digitized Counter-diabatic Quantum Computing (DCQC) is a concept in quantum computing that aims to improve the efficiency and fidelity of quantum computations, particularly in the context of adiabatic quantum computing (AQC). Counter-diabatic (CD) driving, also known as transitionless quantum driving, is a technique used to speed up adiabatic processes without causing non-adiabatic transitions. It involves adding an auxiliary Hamiltonian (a counter-diabatic term) to the original Hamiltonian to cancel out the non-adiabatic transitions.• Digitized Counter-diabatic Quantum Computing (DCQC): This method combines the principles of Counter-diabatic (CD) driving with digitized quantum computation to achieve efficient and accurate solutions for quantum optimization and simulation problems. CD driving, or shortcuts to adiabaticity, involves adding an auxiliary Counter-diabatic Hamiltonian HCDto the system’s evolution, ensuring the system remains in its instantaneous eigenstate and avoiding non-adiabatic transitions. In DCQC, the total Hamiltonian H(t) = (1 -
[0272]
[0273] + (t)Had+ HCD(t) is digitized, where Htis a simple initial driving Hamiltonian (e.g., transverse-field terms), Hadis the adiabatic Hamiltonian that encodes the problem to be solved, and HCD(t) suppresses transitions by introducing non-commuting terms such as o’ of. The time evolution under H(t) is approximated using a sequence of quantum gates, leveraging Trotter-Suzuki decomposition for discretization. This hybrid approach reduces the need for slow adiabatic evolution by enhancing system fidelity and precision through HCD(t), while digitization ensures compatibility with gate-based quantum devices. DCQC offers scalability, flexibility, and inherent error mitigation, making it a promising tool for high-fidelity quantum computation on current and near-term hardware.
[0274] • Bias-Field Digitized-Counter-diabatic Quantum Optimization (BF-DCQO): It is a hybrid quantum optimization technique that integrates digitized Counter-diabatic (CD) driving with dynamically adjusted bias fields to solve combinatorial optimization problems efficiently. The method combines a time-dependent Hamiltonian composed of the problem Hamiltonian Had, a Counter-diabatic Hamiltonian HCD(t), an initial driving Hamiltonian Hitand a bias-field Hamiltonian HB. The problem Hamiltonian encodes the optimization problem, while HCD(t) suppresses non-adiabatic transitions during evolution, ensuring higher accuracy and fidelity. The bias field Hamiltonian, HB=
[0275]
[0276] dynamically refines the solution space by guiding the system toward promising configurations. BF-DCQO operates by alternately performing quantum evolution using Trotterized approximations of the time-evolution operator and updating the bias fields based on measured outcomes to iteratively improve the solution. This combination allows BF-DCQO to benefit from the robustness of Counter-diabatic principles and the adaptive refinement provided by classical bias-field updates, making it suitable forimplementation on gate-based quantum hardware while addressing scalability and noise resilience.
[0277] • Adiabatic gauge potential: It is a concept in quantum mechanics that quantifies how the eigenstates of a system's Hamiltonian respond to changes in the Hamiltonian's parameters during an adiabatic process. It plays a useful role in ensuring that a quantum system remains in its ground state during adiabatic evolution, which can be useful for applications like adiabatic quantum computing and quantum annealing. This potential is mathematically represented asA(t) = i( »(t) I t I ’(O) where
[0278]
[0279] |ψ(t)⟩ is an eigenstate of the Hamiltonian H(t), and i is the imaginary unit. Understanding and managing this gauge potential is useful for minimizing non-adiabatic transitions and optimizing the performance of quantum computational processes that rely on adiabatic changes.
[0280] • Nested commutator: In quantum mechanics and mathematical physics, a nested commutator is a higher-order expression that involves multiple layers of commutation between operators. Commutators are a fundamental aspect of the algebraic structure of quantum mechanics and are defined for any two operators A and B as: [A, B] = AB - BA
[0281] A nested commutator involves commutators within commutators. For instance, a simple example of a nested commutator would be a second-order commutator like:
[0282] [[A, B], C] = (AB - BA)C - C(AB - BA) = ABC - BAC - CAB + CBA Obtaining the exact adiabatic gauge potential (A(t)) for a many-body system is challenging. Its implementation is not optimal because, in many cases, A(t) can contain exponentially many terms with non-local many-body interactions. An alternative approach is to consider the approximate form of the adiabatic gauge potential that can be obtained from the nested commutator method.
[0283] • Scheduling function: A scheduling function is defined as a function that controls the evolution of the Hamiltonian from an initial (in particular easily prepared) ground state (the initial Hamiltonian) to the final Hamiltonian that encodes the solution to the problem of interest over the course of the computation. The time dependent scheduling function comprises at least one one-body CD term for adding an effective gauge potential that counters excitations arising from a non- adiabatic time evolution. Graph: Here, we understand graph as a mathematical object that describes a structure. This structure comprises vertices (also called nodes) and edges (also called links or lines). A vertex represents an object. A relation between two vertices is represented by an edge connecting them.
[0284] • Analog Counter-Diabatic Quantum Computing (ACQC): It is a quantum computing protocol designed to enhance the efficiency and accuracy of adiabatic quantumprocesses in analog quantum computers by incorporating counter-diabatic driving techniques. It features a dynamically tailored scheduling function based on the hardware capabilities that efficiently guides the system from an initial prepared ground state to a final state of the problem Hamiltonian, ensuring precise adiabatic evolution while mitigating non-adiabatic errors. By adding a hardware-specific counter-diabatic term, which introduces an effective gauge potential, ACQC counteracts excitations and enhances robustness against unwanted transitions.
[0285] • Graph coloring problem: A mathematical problem consisting in coloring each vertex of a graph with the minimum number of colors with the constraint that two vertices that are connected by an edge cannot share the same color. This problem arises in many practical applications, such as scheduling, frequency assignment, and register allocation. The minimal number of colors needed to color the entire graph is called the chromatic number. Computing the chromatic number is a NP- hard problem. To map a scheduling problem to a graph coloring problem, jobs are represented by vertices. If two jobs cannot be executed simultaneously, they are connected by an edge. The chromatic number of the corresponding graph then corresponds to the minimal needed to execute all jobs and hence, the optimal scheduling. For this example, we assumed that each job requires the same amount of time.
[0286] • Recursive largest first algorithms: The recursive largest first algorithm (RFL) is a classical heuristic algorithm to solve the graph coloring problem. This algorithm works as follows: Finding a large (in literature also called maximal) independent set and assign to this set a single color Remove the independent set from step 1 from the graph resulting in a subgraph. Repeat step 1 with the subgraph The performance of this algorithm strongly depends on the “quality” of the found independent sets.
[0287] • Maximum independent set (MIS): A kind of a combinatorial optimization problem comprising a graph of vertices connected by edges and for which the solution is the graph containing the maximum number of colored vertices without having two colored vertices connected by an edge. A Maximum Independent Set is the largest possible independent set in a graph.
[0288] • Branch-and-Bound (BB): Branch-and-bound is a complete search method that systematically partitions the solution space (branching) and uses bounds to exclude suboptimal regions (bounding). The algorithm progresses as follows: o Branching: The solution space is recursively divided into smaller subsets, often by fixing variables (e.g., xi= 0, xi= 1).
[0289] o Bounding: For each subset, bounds on the objective function are computed.
[0290] These bounds may come from solving a version of the problem (e.g., dropping integrality constraints in integer programming).o Pruning: Subsets whose bounds indicate that they cannot improve upon the current best solution are excluded from further exploration.
[0291] o Termination: The process continues until all subsets have been explored or pruned, ensuring that the optimal solution has been found. Branch-and-bound explores the entire tree unless pruning eliminates significant portions of the search space. While it guarantees finding the optimal solution, the computational effort can grow exponentially with problem size.
[0292] Branch-and-Cut (BC): Branch-and-cut extends branch-and-bound by incorporating cutting planes to tighten the problem formulation and reduce the size of the search tree. Cutting planes are additional constraints derived from the problem’s structure that exclude infeasible or suboptimal regions without removing feasible solutions. The procedure modifies branch-and-bound as follows:
[0293] o Cutting Planes: At each node of the tree, cutting planes are added to the problem to tighten the feasible region.
[0294] o Branching: If the solution to the problem is not integral, the node is branched further by partitioning the solution space (e.g., fixing variables). o Pruning: Suboptimal branches are pruned based on the bounds and constraints.
[0295] o Iteration with Cuts: As the tree progresses, additional cuts may be applied to reduce unnecessary exploration.
[0296] Brief description of the drawings
[0297] For a better understanding of the present invention, as well as other aspects and further features thereof, reference is made to the following description which is to be used in conjunction with the accompanying drawings, where:
[0298] FIG. 1: Figure 1 illustrates a sequence of steps of the method as described in an embodiment of the present invention.
[0299] FIG. 2: Figure 2 illustrates a system according to an embodiment of the present invention. FIG. 3: Figure 3 illustrates a schematical representation of a method for solving a spin glass problem according to an embodiment of the present invention.
[0300] FIG. 4: Figure 4 illustrates the performances of the application of a method for solving a spin glass problem according to an embodiment of the present invention.
[0301] FIG. 5: Figure 5 illustrates some results of solving a Low-Autocorrelation Binary Sequences problem according to an embodiment of the present invention.
[0302] FIG. 6: Figure 6 illustrates some results of solving a Low-Autocorrelation Binary Sequences problem according to another embodiment of the present invention.Detailed description
[0303] In the context of the present specification, “device” or “unit” may refer to any computer hardware that is capable of running software appropriate to the relevant task at hand. Thus, some (non-limiting) examples of devices include personal computers (desktops, laptops, netbooks, etc.), smartphones, and tablets, as well as network equipment such as routers, switches, and gateways. It should be noted that a device acting as a device in the present context is not precluded from acting as a server to other devices. The use of the expression “a device” does not preclude multiple devices being used in receiving / sending, carrying out or causing to be carried out any task or request, or the consequences of any task or request, or steps of any method described herein.
[0304] In the context of the present specification, a “database” is any structured collection of data, irrespective of its particular structure, the database management software, or the computer hardware on which the data is stored, implemented or otherwise rendered available for use. A database may reside on the same hardware as the process that stores or makes use of the information stored in the database or it may reside on separate hardware, such as a dedicated server or plurality of servers. It can be said that a database is a logically ordered collection of structured data kept electronically in a computer system In the context of the present specification, the expression “information” includes information of any nature or kind whatsoever capable of being stored in a database. Thus information includes, but is not limited to audiovisual works (images, movies, sound records, presentations etc.), data (location data, numerical data, etc.), text (opinions, comments, questions, messages, etc.), documents, spreadsheets, lists of words, etc. In the context of the present specification, the expression “component” or “module” is meant to include software (appropriate to a particular hardware context) that is both necessary and sufficient to achieve the specific function(s) being referenced.
[0305] In the context of the present specification, the expression “computer readable medium” is intended to include media of any nature and kind whatsoever, including RAM, ROM, disks (CD-ROMs, DVDs, floppy disks, hard drivers, etc.), USB keys, solid state-drives, tape drives, etc.
[0306] In the context of the present specification, the words “first”, “second”, “third”, etc. have been used as adjectives only for the purpose of allowing for distinction between the nouns that they modify from one another, and not for the purpose of describing any particular relationship between those nouns. Thus, for example, it should be understood that, the use of the terms “first module” and “third module” is not intended to imply any particular order, type, chronology, hierarchy or ranking (for example) of / between the module, nor is their use (by itself) intended imply that any “second module” must necessarily exist in any given situation. Further, as is discussed herein in other contexts, reference to a “first” element and a “second” element does not preclude the two elements from being the same actual real-world element. Thus, for example, in some instances, a “first” module and a“second” module may be the same software and / or hardware, in other cases they may be different software and / or hardware.
[0307] The functions of the various elements shown in the figures, including any functional block labeled as a "processor" or a “processing module” or a “processing unit”, may be provided through the use of dedicated hardware as well as hardware capable of executing software in association with appropriate software. When provided by a processor, the functions may be provided by a single dedicated processor, by a single shared processor, or by a plurality of individual processors, some of which may be shared. In some embodiments of the present invention, the processor may be a general- purpose processor, such as a central processing unit (CPU) or a processor dedicated to a specific purpose, such as a digital signal processor (DSP). Moreover, explicit use of the term a "processor" should not be construed to refer exclusively to hardware capable of executing software, and may implicitly include, without limitation, application specific integrated circuit (ASIC), field programmable gate array (FPGA), read-only memory (ROM) for storing software, random access memory (RAM), and non-volatile storage. Other hardware, conventional and / or custom, may also be included. In some embodiments of the present invention, the processor can be a quantum processor comprising a plurality of qubits.
[0308] Software modules, or simply modules which are implied to be software, may be represented herein as any combination of flowchart elements or other elements indicating performance of process steps and / or textual description. Such modules may be executed by hardware that is expressly or implicitly shown. Moreover, it should be understood that module may include for example, but without being limitative, computer program logic, computer program instructions, software, stack, firmware, hardware circuitry or a combination thereof which provides the required capabilities.
[0309] The examples and conditional language recited herein are principally intended to aid the reader in understanding the principles of the present invention and not to limit its scope to such specifically recited examples and conditions. It will be appreciated that those skilled in the art may devise various arrangements which, although not explicitly described or shown herein, nonetheless embody the principles of the present invention and are included within its spirit and scope.
[0310] Furthermore, as an aid to understanding, the following description may describe relatively simplified implementations of the present invention. As persons skilled in the art would understand, various implementations of the present invention may be of a greater complexity.
[0311] In some cases, what are believed to be helpful examples of modifications to the present invention may also be set forth. This is done merely as an aid to understanding, and, again, not to define the scope or set forth the bounds of the present invention. These modifications are not an exhaustive list, and a person skilled in the art may make other modifications while nonetheless remaining within the scope of the present invention. Further, where no examples of modifications have been set forth, it should not beinterpreted that no modifications are possible and / or that what is described is the sole manner of implementing that element of the present invention.
[0312] Moreover, all statements herein reciting principles, aspects, and implementations of the present invention, as well as specific examples thereof, are intended to encompass both structural and functional equivalents thereof, whether they are currently known or developed in the future. Thus, for example, it will be appreciated by those skilled in the art that any block diagrams herein represent conceptual views of illustrative circuitry embodying the principles of the present invention. Similarly, it will be appreciated that any flowcharts, flow diagrams, state transition diagrams, pseudo-code, and the like represent various processes which may be substantially represented in computer-readable media and so executed by a computer or processor, whether or not such computer or processor is explicitly shown.
[0313] With these fundamentals in place, we will now consider some non-limiting examples to illustrate various implementations of aspects of the present invention.
[0314] According to an embodiment, the present invention relates to a computer-implemented method and a system for solving optimization problems, specifically those formulated as Quadratic Unconstrained Binary Optimization (QUBO) or Higher-order Unconstrained Binary Optimization (HUBO) problems.
[0315] Advantageously, the present invention leverages the capabilities of digital quantum hardware, integrating classical and quantum processing units to enhance computational efficiency and accuracy. The present invention aims to address the challenges posed by traditional optimization algorithms, particularly in handling large-scale instances that are computationally intensive and time-consuming.
[0316] According to an embodiment, the present invention utilizes a branch-and-bound / cut bias field Digitized Counter-diabatic Quantum Optimization (BF-DCQO) approach. This technique combines the robustness of quantum heuristic methods with the systematic exploration of the solution space afforded by branch-and-bound / cut algorithms. By dynamically adjusting bias fields based on the outcomes of quantum computations, the present invention can iteratively refine the search for optimal solutions, thereby improving the overall performance of the optimization process.
[0317] According to an embodiment, the invention can be applied to a wide range of optimization problems across various fields, including but not limited to finance, logistics, and machine learning. The method can effectively tackle complex combinatorial optimization challenges, such as protein folding and portfolio optimization, by encoding these problems into Hamiltonians suitable for quantum processing. This encoding allows for the efficient simulation of the optimization landscape, facilitating the identification of optimal configurations.According to an embodiment, the functionalities of the present invention comprises the initialization of a problem Hamiltonian, the evolution of quantum states through a series of quantum gate operations, and the measurement of expectation values to guide the optimization process. The method can also incorporate feedback mechanisms to refine the encoding of the optimization problem based on the results obtained, ensuring that the solution quality is continuously improved throughout the computational iterations.
[0318] According to an embodiment, the method may comprise a feedback loop that utilizes results obtained from the solution determination phase. This feedback mechanism can enhance the overall efficiency of the optimization process by allowing for real-time adjustments based on the outcomes of previous iterations. By analyzing the results, the method can identify areas for improvement, ensuring that the optimization problem is continuously refined.
[0319] According to an embodiment, the feedback loop may enable adjustments to the parameters of the total Hamiltonian. This flexibility can be useful in optimizing the performance of the quantum processing unit, as it allows for the tuning of the Hamiltonian to better align with the evolving state of the set of qubits. Such adjustments can lead to improved fidelity in reaching the ground state, thereby enhancing the quality of the solutions obtained.
[0320] According to an embodiment, the iterative nature of the feedback loop can significantly improve solution quality over time. By continuously integrating results and making necessary adjustments, the method can converge more rapidly to optimal solutions. This iterative improvement can be particularly advantageous in complex optimization problems, where traditional methods may struggle to find satisfactory solutions efficiently.
[0321] According to an embodiment, the invention may further enhance the performance of quantum optimization algorithms by integrating classical processing units that manage the encoding and branching processes. This hybrid approach can leverage the strengths of both classical and quantum computing, allowing for a more flexible and scalable solution to complex optimization problems. The invention can thus represent a significant advancement in the field of quantum computing, providing a powerful tool for researchers and practitioners seeking to solve challenging optimization tasks.
[0322] According to an embodiment, a computer-implemented method is provided for solving at least one optimization problem utilizing a quantum processing unit (QPU). This QPU comprises a plurality of qubits and is configured to apply a plurality of quantum gates on said plurality of qubits, enabling the execution of complex quantum algorithms. The method is particularly configured to employ a tree-based optimization approach, specifically a branch-and-bound or cut technique, integrated with a bias-field digitized counter-diabatic quantum optimization (BF-DCQO) algorithm. This configuration allows for efficient exploration of the solution space, leveraging the unique capabilities of quantum computing to address optimization challenges.According to an embodiment, the method comprises a plurality of phases:
[0323] • An initialization phase;
[0324] • An evolution phase;
[0325] • An indexation phase;
[0326] • A tree-based processing phase; and
[0327] • A solution determination phase.
[0328] According to an embodiment, the initialization phase is configured to encode the optimization problem into a problem Hamiltonian using a classical processing unit (CPU). The optimization problem comprises multiple functions that represent binary variables, which are advantageously translated into a Hamiltonian comprising quantum operators. The encoding can utilize representations such as QUBO or HUBO. This encoding step is useful as it transforms the optimization problem into a format suitable for quantum processing, facilitating the subsequent computational steps.
[0329] According to an embodiment, the initialization phase further comprises the creation of a time-dependent Hamiltonian. This Hamiltonian is initialized to define the starting configuration of a selected set of qubits from the QPU. A predetermined bias-field can then be applied using a bias-field generation module, resulting in a bias-field Hamiltonian. This step introduces external influences that can guide the quantum system, made of a set of qubits, towards optimal configurations, enhancing the efficiency of the optimization process.
[0330] According to an embodiment, the method continues by modifying the initial Hamiltonian through the addition of the bias-field Hamiltonian, resulting in a modified initial Hamiltonian. This modified Hamiltonian, in conjunction with the problem Hamiltonian, is utilized to construct a Counter-diabatic Hamiltonian. This construction employs a set of nested commutators, which are mathematical tools that facilitate the management of quantum states during evolution. The use of nested commutators is advantageous as it minimizes non-adiabatic transitions, thereby improving the fidelity of the quantum state evolution.
[0331] According to an embodiment, a total Hamiltonian is then established by combining the initial Hamiltonian, the Counter-diabatic Hamiltonian, and the problem Hamiltonian. This total Hamiltonian serves as the foundation for the evolution phase, where the quantum state of the set of qubits is evolved based on this comprehensive Hamiltonian. The evolution phase allows the quantum system to explore the solution space and converge towards the ground state of the problem Hamiltonian.
[0332] According to an embodiment, during the evolution phase, the quantum state of the set of qubits is measured using a measurement module. This measurement yields a probability distribution that reflects the quantum state of the set of qubits. The classical processing unit then computes expectation values from this distribution, providing insights into thestate of the set of qubits. These expectation values are used then for subsequent decisionmaking processes within an optimization framework.
[0333] According to an embodiment, the method comprises an indexation phase, wherein each qubit is attributed an index by the classical processing unit. This indexing facilitates the selection of a predetermined number of qubits for further processing. The ability to index qubits allows for organized manipulation and analysis of the quantum states, enhancing the method's efficiency in navigating the solution space.
[0334] According to an embodiment, the tree-based processing phase is initiated, which constructs a tree comprising multiple branches. This phase involves repeated selections of qubits based on their expectation values, particularly focusing on the qubit with the nearest expectation value to zero, for example. This strategic selection process is advantageous as it targets qubits that are likely to yield optimal configurations, thereby streamlining the search for solutions.
[0335] According to an embodiment, the first branching step involves applying a first predetermined constraint to the selected qubit, resulting in a modified total Hamiltonian. The quantum state of the qubits is then evolved based on this modified Hamiltonian, followed by measurement to obtain a modified probability distribution. The expectation values from this distribution are computed, and a corresponding branch of the tree is created. This branching mechanism allows for systematic exploration of potential solutions, enhancing the likelihood of identifying optimal configurations.
[0336] According to an embodiment, a second branching step is executed, applying a second predetermined constraint to the selected qubit, which preferably has an opposite polarity to the first constraint. This dual-branching approach enables the exploration of different configurations, further enriching the search process. The subsequent evolution and measurement steps mirror those of the first branching step, ensuring a comprehensive evaluation of the qubit's potential states.
[0337] According to an embodiment, the method incorporates a selection process between cutting and bounding strategies based on predetermined rules. The cutting process involves comparing expectation values from the two branches and retaining the branch with the lower expectation value. This selective pruning of branches is advantageous as it reduces computational overhead by eliminating less promising paths in the solution space. Conversely, the bounding process retains both branches for further exploration, allowing for a more exhaustive search for optimal solutions.
[0338] According to an embodiment, the solution determination phase is executed, wherein the classical processing unit computes the lowest expectation values of the problem Hamiltonian. This computation results in a final probability distribution that corresponds to the lowest energy configuration of the qubits. The extraction of at least one bitstring from this distribution yields the solution to the optimization problem. This final extraction step iscrucial as it translates the quantum results back into a format that can be interpreted and utilized in practical applications.
[0339] In summary, the present invention integrates classical and quantum processing techniques to effectively solve optimization problems. Each phase of the method is designed to leverage the strengths of quantum computing, while maintaining the robustness of classical algorithms. The combination of tree-based optimization strategies with counter-diabatic quantum techniques enhances the overall efficiency and accuracy of the solution process.
[0340] In a more detailed way, and according to an embodiment illustrated by figures 1 and 2, the present invention relates to a computer-implemented method 100 for solving at least one optimization problem using preferably at least one quantum processing unit 220 comprising a plurality of qubits and configured to apply a plurality of quantum gates on said plurality of qubits.
[0341] Advantageously, and as described hereafter, the method 100 is configured to be executed by a computing system 200.
[0342] According to an embodiment, the method 100 comprises:
[0343] 1. An initialization phase 110, the initialization phase 110 comprising:
[0344] a. Encoding, using a classical processing unit 210, at least one optimization problem, comprising a plurality of functions representing a plurality of binary variables, into a problem Hamiltonian comprising a plurality of quantum operators;
[0345] b. Initializing, using the classical processing unit 210, a time-dependent Hamiltonian comprising an initial Hamiltonian defining the starting configuration of a set of qubits taken among the plurality of qubits of the quantum processing unit;
[0346] c. Applying, using a bias-field generation module 230, a predetermined biasfield to the initial Hamiltonian to get a bias-field Hamiltonian; According to an embodiment, the step of applying the bias field is adapted to the nature of the qubits of the set of qubits. For example, the step of applying the bias field can comprise:
[0347] i. In superconducting qubits, tuning external magnetic flux through at least one superconducting loop and / or by adjusting current and / or voltage through at least one circuit control line.
[0348] ii. In trapped ions qubits, modulating at least one laser pulse applied to at least one trapped ion of a set of trapped ions forming the set of qubits at a predetermined frequency to generate the bias. iii. In neutral atoms qubits, applying at least one laser pulse and / or magnetic field gradient to at least one neutral atom of a set of neutral atoms forming the set of qubits, to control frequencies and / or detunings of atomic transitions.d. Adding, using the classical processing unit 210, the bias-field Hamiltonian to the initial Hamiltonian to get a modified initial Hamiltonian;
[0349] e. Using the modified initial Hamiltonian and the problem Hamiltonian to build a Counter-diabatic Hamiltonian using a set of nested commutators and the classical processing unit 210;
[0350] f. Adding the initial Hamiltonian with the Counter-diabatic Hamiltonian and the problem Hamiltonian to get a total Hamiltonian, using the classical processing unit 210;
[0351] g. Initializing, using quantum gates of the quantum processing unit 220, the set of qubits into an initial quantum state representing the ground state of the modified initial Hamiltonian;
[0352] An evolution phase 120, the evolution phase comprising:
[0353] a. Evolving, using the quantum processing unit 220, the quantum state of the set of qubits based on the total Hamiltonian to reach the ground state of the problem Hamiltonian; Preferably, the evolution step comprises adjusting, advantageously smoothly, parameters of at least one timedependent schedule function A(t) overtime;
[0354] b. Measuring, using a measurement module 240, the quantum state of the set of qubits to get a probability distribution;
[0355] c. Computing, using the classical processing unit 210, expectation values of the probability distribution, preferably of the sigma z operators from the probability distribution;
[0356] An indexation phase 130, the indexation phase comprising:
[0357] a. Attributing, using the classical processing unit, an index to each qubit of the set of qubits;
[0358] b. Selecting, using the classical processing unit, a predetermined number n of qubits equal or smaller to the number N of qubits of the set of qubits; A tree-based processing phase 140, the tree-based processing phase being configured to construct a tree comprising n branches, the tree-based processing phase comprising n repetitions of the following set of steps:
[0359] a. Selecting, using the classical processing unit 210, the index of a qubit taken among the n qubits, preferably the qubit having the nearest expectation value to zero; Advantageously, only one index is selected;
[0360] b. A first branching step, the first branching step comprising:
[0361] i. Applying, using the bias-field generation module 230, a first predetermined constraint to the selected qubit to get a first modified total Hamiltonian, preferably by applying a first bias field to the selected qubit to get a first modified total Hamiltonian; ii. Evolving, using the quantum processing unit 220, the quantum state of the set of qubits based on the first modified total Hamiltonian to reach the ground state of the problem Hamiltonian;
[0362] iii. Measuring, using the measurement module 240, the quantum state of the set of qubits to get a first modified probability distribution;iv. Computing, using the classical processing unit 210, expectation values of the first modified probability distribution, preferably of the sigma z operators from the first modified probability distribution; v. Creating, using the classical processing unit 210, a first branch of a tree, the first branch of the tree being associated with the expectation values of the first modified probability distribution; c. A second branching step, the second branching step comprising:
[0363] i. Applying, using the bias-field generation module 230, a second predetermined constraint to the selected qubit to get a second modified total Hamiltonian, preferably by applying a second bias field to this qubit to get a second modified total Hamiltonian, the second bias-field having a polarity opposite to the first bias-field; ii. Evolving, using the quantum processing unit 220, the quantum state of the set of qubits based on the second modified total Hamiltonian to reach the ground state of the problem Hamiltonian; iii. Measuring, using the measurement module 240, the quantum state of the set of qubits to get a second modified probability distribution; iv. Computing, using the classical processing unit 210, expectation values of the second modified probability, preferably of the sigma z operators from the second modified probability distribution; v. Creating, using the classical processing unit 210, a second branch of the tree, the second branch of the tree being associated with the expectation values of the second modified probability distribution; vi. Selecting, using the classical processing unit 210, at least one process taken among a cutting process or a bounding process based on a predetermined set of rules:
[0364] 1. The cutting process comprising:
[0365] a. Comparing the first expectation value with the second expectation value:
[0366] i. If the first expectation value is lower than the second expectation value, cutting the second branch, and continuing the tree-based processing phase with the first branch; ii. if the second expectation value is lower than the first expectation value, cutting the first branch, and continuing the tree-based processing phase with the second branch; b. Obtaining a tree made of branches corresponding to the lowest expectation values;
[0367] c. Getting a configuration of the set of qubits corresponding to the lowest expectation values of the problem Hamiltonian;
[0368] 2. The bounding process comprising:a. Storing the first branch and the second;
[0369] b. Continuing the tree-based processing phase to explore each possible configuration of the set of qubits to identify the lowest expectation value of the problem Hamiltonian;
[0370] 5. A solution determination phase 150, the solution determination phase comprising:
[0371] a. Computing, using the classical processing unit 210, the lowest expectation values of the problem Hamiltonian to get a final probability distribution corresponding to the lowest energy of the set of qubits;
[0372] b. Extracting, using the classical processing unit 210, at least one bitstring from the final probability distribution, the bitstring corresponding to the solution of the optimization problem.
[0373] According to an embodiment, the encoding step involves mapping the optimization problem to a problem Hamiltonian using a classical processing unit. This mapping is configured to translate the optimization problem into a form that can be processed by a quantum system, i.e. by the quantum processing unit. By associating at least one quantum operator of the problem Hamiltonian to each function of the optimization problem, the method ensures that the quantum representation accurately reflects the underlying mathematical structure of the problem.
[0374] According to an embodiment, when the optimization problem is encoded as a Quadratic Unconstrained Binary Optimization (QUBO), it results in a many-body Hamiltonian. This representation allows for efficient processing of the optimization problem by leveraging the capabilities of quantum computing. The many-body Hamiltonian can capture complex interactions among multiple qubits, facilitating the exploration of the solution space in a manner that is inherently suited to quantum algorithms. According to an embodiment, the present invention may use an Ising model representation specifically for Quadratic Unconstrained Binary Optimization (QUBO) problems. The Ising model is a well-established framework in statistical mechanics that can effectively represent binary variables and their interactions, facilitating the mapping of complex optimization problems into a quantum-compatible format.
[0375] According to an embodiment, if the optimization problem is encoded as a Higher-order Unconstrained Binary Optimization (HUBO), it results in a spin Hamiltonian with higher-order interactions. This capability enhances the method's flexibility, allowing it to address a broader range of optimization problems that may involve more intricate relationships among variables. The higher-order interactions can lead to more accurate representations of the problem, potentially improving the quality of the solutions obtained.
[0376] According to an embodiment, the technical advantages of this encoding approach include improved scalability and efficiency in solving complex optimization problems. By utilizing many-body Hamiltonians and higher-order interactions, the method can exploit quantumparallelism, enabling faster convergence to optimal solutions compared to classical approaches. This is particularly beneficial for large-scale instances of optimization problems, where traditional algorithms may struggle due to combinatorial complexity. According to an embodiment, the mapping process can also enhance the robustness of the quantum optimization algorithm. By accurately encoding the problem into a Hamiltonian that reflects its structure, the method can mitigate the effects of noise and errors inherent in current quantum devices. This robustness is useful for achieving reliable results in practical applications, making the method suitable for near-term quantum hardware.
[0377] According to an embodiment, the method involves selecting a qubit from a set of qubits based on its expectation value. Preferably, the qubit chosen is the one that has the nearest expectation value to zero. This selection criterion is significant as it targets qubits that are likely to be in a state of minimal energy.
[0378] According to an embodiment, this approach can improve the efficiency of the optimization algorithm. By focusing on qubits with expectation values close to zero, the method may reduce the likelihood of non-adiabatic transitions during the evolution phase. This can lead to a more reliable convergence towards the ground state of the problem Hamiltonian, ultimately resulting in higher accuracy in the solutions obtained.
[0379] According to an embodiment, the selection of the qubit with the nearest expectation value to zero can facilitate a more effective branching strategy in the tree-based processing phase. By prioritizing qubits that exhibit minimal energy, the method can streamline the exploration of the solution space, allowing for quicker identification of optimal configurations. This can enhance the overall computational efficiency of the algorithm. According to an embodiment, this feature can also contribute to the robustness of the method against noise and errors inherent in quantum computing. By concentrating on qubits that are less likely to be influenced by external perturbations, the method may achieve greater fidelity in the results, making it suitable for implementation on near-term quantum devices.
[0380] According to an embodiment, the first predetermined constraint is a first predetermined bias field. This bias field serves as a parameter in the optimization process, influencing the behavior of the quantum system during the execution of the algorithm. By applying a specific bias field, the method can effectively guide the quantum state evolution towards more promising configurations, thereby enhancing the likelihood of finding optimal solutions to the optimization problem.
[0381] According to an embodiment, the first predetermined bias field can be dynamically adjusted based on the results obtained during the optimization process. This adaptability allows the method to refine its search space iteratively, focusing on areas that yield better performance. The ability to modify the bias field in response to intermediate results cansignificantly improve the efficiency of the optimization algorithm, making it more robust against noise and errors inherent in quantum computing.
[0382] According to an embodiment, the use of a first predetermined bias field can facilitate the exploration of the solution space by introducing a directional influence on the qubits. This directional influence can help in overcoming local minima, which are common challenges in optimization problems. By strategically applying the bias field, the method can enhance the convergence speed towards the global optimum, thereby reducing the computational resources required for solving complex optimization tasks.
[0383] According to an embodiment, the implementation of the first predetermined bias field can lead to improved accuracy in the results obtained from the quantum processing unit. The bias field can be designed to align with the characteristics of the specific optimization problem being addressed, ensuring that the quantum system remains focused on relevant configurations. This tailored approach can yield higher fidelity in reaching the ground state of the problem Hamiltonian, ultimately resulting in more reliable solutions.
[0384] According to an embodiment, the first predetermined bias field can be integrated seamlessly into the overall framework of the optimization method. This integration allows for a coherent interaction between classical and quantum components of the present invention, enabling efficient communication and processing of information. The synergy between the bias field and the quantum algorithm can enhance the overall performance of the optimization process, making it suitable for a wide range of applications in fields such as finance, logistics, and machine learning.
[0385] According to an embodiment, the method can comprise a second predetermined constraint, which can be a second predetermined bias field. Preferably, this bias field is characterized by having a polarity that is opposite to that of the first predetermined bias field. The introduction of this opposing bias field can facilitate a more comprehensive exploration of the solution space during the optimization process.
[0386] According to an embodiment, the functionality of the second predetermined bias field allows for the manipulation of the quantum state of the qubits in a manner that enhances the algorithm's ability to escape local minima. By applying a bias field with an opposite polarity, the method can effectively alter the energy landscape, thereby promoting transitions between different states. This can lead to a more robust search for the global optimum of the optimization problem.
[0387] According to an embodiment, the technical advantage of utilizing a second predetermined bias field lies in its potential to improve the convergence rate of the optimization algorithm. By dynamically adjusting the bias fields, the method can adapt to the evolving state of the set of qubits, allowing fora more efficient traversal of the solution space. This adaptability can result in faster identification of optimal solutions, particularly in complex optimization scenarios.According to an embodiment, the method may utilize a problem Hamiltonian that is specifically designed for various optimization problems. This versatility allows the method to be applicable across multiple domains, including protein folding, portfolio optimization, logistics optimization, and machine learning model training. Each of these applications presents unique challenges that can benefit from the present invention.
[0388] According to an embodiment, by tailoring the Hamiltonian to reflect the specific characteristics and constraints of the optimization problem at hand, the method can achieve more accurate and efficient solutions. This customization is configured to address the complexities inherent in each type of optimization problem.
[0389] According to an embodiment, the step of evolving the quantum state of each qubit of the set of qubits comprises time-evolving, preferably counter-diabatically, the quantum state of the set of qubits from the state of the modified initial Hamiltonian to a final state of the problem Hamiltonian, advantageously up to a predetermined final time which is equal to or less than the coherence time of the qubits of the set of qubits.
[0390] For example, this evolving step can comprise:
[0391] • In the case of the set of qubits comprising at least one qubit made of at least one trapped ion, emitting a set of time-dependent electromagnetic pulses, using at least one electromagnetic pulse generation module, to drive a plurality of trapped ions, the set of time-dependent electromagnetic pulses being used to generate interactions among the trapped ions according to the problem Hamiltonian;
[0392] • In the case of the set of qubits comprising at least one qubit made of at least one superconducting qubit, emitting a set of time-dependent flux pulses, using at least one flux pulse generation module, to drive a plurality of superconducting qubits, the set of time-dependent flux pulses being used to generate interactions among the superconducting qubits according to the problem Hamiltonian;
[0393] • In the case of the set of qubits comprising at least one qubit made of at least one neutral atom, emitting a set of time-dependent electromagnetic pulses, using at least one electromagnetic pulse generation module, to drive a plurality of neutral atoms, the set of time-dependent electromagnetic pulses being used to generate interactions among the neutral atoms according to the problem Hamiltonian, and / or emitting a set of time-dependent magnetic pulses, using at least one magnetic pulse generation module, to drive a plurality of neutral atoms, the set of timedependent magnetic pulses being used to generate interactions among the neutral atoms according to the problem Hamiltonian.
[0394] According to an embodiment, the tree-based processing phase can include the functionality of updating the bias-field Hamiltonian. This update can be based on the computed expectation values derived from the modified probability distributions, preferably from the sigma z operators of the modified probability distributions.According to an embodiment, dynamically adjusting the bias-field Hamiltonian can provide significant technical advantages. This adjustment allows the algorithm to refine its search space in real-time, focusing computational resources on more promising areas of the solution landscape. As a result, the method may achieve higher accuracy and efficiency in finding optimal solutions.
[0395] According to an embodiment, the ability to update the bias-field Hamiltonian based on expectation values can lead to improved convergence rates. By continuously refining the bias fields, the method can effectively navigate the complexities of the optimization problem, potentially reducing the number of iterations required to reach a satisfactory solution.
[0396] According to an embodiment, the counter-diabatic Hamiltonian is constructed using a set of nested commutators. This construction allows for a systematic approach to managing the Hamiltonian's evolution, ensuring that the quantum system, i.e. the set of qubits, remains in its desired state throughout the computation. Preferably, this construction is based on the framework of Adiabatic Gauge Potential.
[0397] According to an embodiment, the constructing step of the counter-diabatic Hamiltonian comprises several sub-steps.
[0398] First, the constructing step can comprise a sub-step of providing the modified initial Hamiltonian Hoand the final Hamiltonian Hp, which define the initial and target configurations of the set of qubits.
[0399] Then, the constructing step can comprise a sub-step of obtaining an interpolated Hamiltonian by defining at least one parametric interpolation of the Hamiltonian Hsof the set of qubits along a continuous evolution parameter A(t), as:
[0400]
[0401] Hs(A(t)) = (1 - A(t))H0+ A(t)Hp, where A(t) is a time-dependent schedule function satisfying A(0) = 0 and A(T) = 1.
[0402] Then, the constructing step can comprise a sub-step of computing the derivative of the interpolated Hamiltonian with respect to A expressed as ∂λHs(λ) = Hp- H0
[0403] Then, the constructing step can comprise the sub-step of constructing an approximate adiabatic gauge potential AAfor the Hamiltonian HS(X) of the set of qubits, wherein A is expressed as a finite series expansion of nested commutators, such as being approximately equal to iα1[∂λHs(λ), H(λ)] + ia2[[d Hs(A), H( )], HS(A)] + ••• where anare predetermined or variationally optimized coefficients, and the series is truncated at apredetermined order based on an additional predetermined set of rules related to computational feasibility and / or a level of accuracy.
[0404] Then, the constructing step can further comprise a sub-step of determining the time derivative (t) of the continuous evolution parameter A(t).
[0405] Then, the constructing step can further comprise a sub-step of generating the counter-diabatic Hamiltonian comprising the interpolated Hamiltonian and at least an additional driving term dependent on the adiabatic gauge potential, expressed as: HCD(t) =
[0406]
[0407] + A(t)XA.
[0408] According to an embodiment, the design of the counter-diabatic Hamiltonian aims to minimize non-adiabatic transitions during the evolution phase. By effectively counteracting these transitions, the present invention can maintain the coherence and stability of the set of qubits.
[0409] According to an embodiment, enhancing the fidelity of reaching the ground state of the problem Hamiltonian is a significant advantage of this approach. Higher fidelity translates to a greater likelihood of obtaining the optimal solution to the optimization problem being addressed. This improvement in accuracy can lead to more efficient computations and better performance in practical applications, such as combinatorial optimization and machine learning.
[0410] According to an embodiment, the implementation of nested commutators in constructing the counter-diabatic Hamiltonian can also provide flexibility in adapting to various optimization problems. This adaptability allows the method to be applied across a range of scenarios, including those with different complexities and requirements. As a result, the present invention can be utilized in diverse fields, enhancing its applicability and relevance in quantum computing.
[0411] According to an embodiment, the quantum gates applied by the quantum processing unit may include both single-qubit and multi-qubit gates. This configuration allows for versatile manipulation of qubit states, enabling the implementation of complex quantum algorithms. Single-qubit gates can perform operations on individual qubits, while multi-qubit gates facilitate interactions between multiple qubits.
[0412] According to an embodiment, the evolution of the quantum state of the set of qubits can be achieved through a sequence of quantum gate operations. This sequence is configured to approximate the time evolution under the total Hamiltonian, ensuring that the quantumsystem transitions smoothly between states. By utilizing a series of gate operations, the present invention can effectively simulate the dynamics dictated by the Hamiltonian. According to an embodiment, the integration of single-qubit and multi-qubit gates within the quantum processing unit can enhance the overall computational power. This capability allows for the execution of a broader range of quantum algorithms, making the present invention adaptable to various optimization challenges. The combination of gate types can also facilitate error correction and noise resilience, which can be important for practical implementations of quantum computing.
[0413] According to an embodiment, the initialization phase of the method may include the selection of an initial Hamiltonian that possesses a known ground state. This can provides a reliable starting point for the present invention. By utilizing an initial Hamiltonian with a well-defined ground state, the present invention can ensure that the set of qubits is initialized into a quantum state that accurately represents this ground state. This facilitates a more efficient evolution of the quantum system towards the solution of the optimization problem.
[0414] According to an embodiment, the selection of an initial Hamiltonian with a known ground state can enhance the overall fidelity of the quantum computation. This is because starting from a known ground state minimizes the risk of the system being initialized in a superposition that does not correspond to the desired configuration. As a result, the method may achieve a more stable and predictable evolution of the quantum state, leading to improved accuracy in the final solution.
[0415] According to an embodiment, the predetermined set of rules for selecting between a cutting process or a bounding process can enhance the efficiency of the optimization method.
[0416] For example, the set of rules can comprise the following criteria:
[0417] a. When a predetermined time limit is selected:
[0418] i. if the predetermined time limit is equal or higher than a predetermine time threshold, selecting the cutting process;
[0419] ii. if the predetermined time limit is lower than a predetermined time threshold, selecting the bounding process;
[0420] and / or
[0421] b. When a predetermined level of accuracy is selected:
[0422] i. if the predetermined level of accuracy is lower than a predetermined accuracy threshold, selecting the cutting process;ii. if, the predetermined level of accuracy is equal or higher than a predetermined accuracy threshold, selecting the bounding process.
[0423] By incorporating criteria based on the improvement of the objective function, the present invention can dynamically adapt its strategy to focus on the most promising branches of the solution space. This adaptability may lead to faster convergence towards optimal solutions, thereby reducing computational time and resource consumption.
[0424] According to an embodiment, the selection criteria may also consider the depth of the tree. By evaluating the depth, the present invention can prioritize branches that are more likely to yield better results based on their position within the search space. This hierarchical approach can facilitate a more structured exploration of potential solutions, allowing for a more efficient pruning of suboptimal branches.
[0425] According to an embodiment, the combination of these criteria can provide a robust framework for decision-making during the optimization process. By leveraging both the improvement of the objective function and the tree depth, the method can achieve a balanced exploration and exploitation strategy. This dual focus can enhance the overall performance of the optimization algorithm, making it more effective in solving complex problems.
[0426] According to an embodiment, the technical advantages of this approach include improved scalability and flexibility in handling various optimization problems. The ability to switch between cutting and bounding processes based on real-time evaluations can lead to a more responsive optimization strategy. This responsiveness can be particularly beneficial in scenarios where the optimization landscape is dynamic or where the problem size varies significantly.
[0427] According to an embodiment, the method can comprise a post-processing phase that enhances the overall functionality of the optimization process. This phase is configured to translate the results obtained from the quantum processing unit into a format that is understandable and usable for practical applications. By decoding the bitstring extracted from the final probability distribution, the present invention ensures that the solution aligns with the original optimization problem's requirements.
[0428] According to an embodiment, the decoding process can facilitate the retrieval of the solution in its native representation. This capability can be useful for applications where the optimization problem is defined in specific terms, such as binary variables or other structured formats. By providing the solution in its original context, the method can improve the usability and applicability of the results in real-world scenarios.
[0429] Overall, the described method leverages the unique capabilities of quantum processing units, combined with classical computational techniques, to efficiently solve complexoptimization problems. Each phase of the method is designed to enhance the accuracy and speed of the optimization process, making it suitable for a wide range of applications in fields such as finance, logistics, and machine learning.
[0430] According to an embodiment, the present invention also relates to a computing system configured to solve at least one optimization problem. This system comprises several components: at least one classical processing unit, at least one quantum processing unit, at least one bias field generation module, and at least one measurement module. Each of these components plays a role in the overall functionality of the system, enabling efficient problem-solving capabilities.
[0431] According to an embodiment, the classical processing unit is configured to perform multiple tasks. It can encode at least one optimization problem into a problem Hamiltonian, which serves as a mathematical representation of the optimization problem. This encoding process allows the system to translate complex optimization problems into a format suitable for quantum processing. The classical processing unit can also be configured to initialize a time-dependent Hamiltonian, which includes an initial Hamiltonian that defines the starting configuration of a set of qubits. This initialization is configured to establish the baseline state from which the quantum computation will evolve. Furthermore, the classical processing unit adds a bias-field Hamiltonian to the initial Hamiltonian, resulting in a modified initial Hamiltonian. This modification is advantageous as it allows for the incorporation of external influences that can guide the optimization process. According to an embodiment, the classical processing unit utilizes the modified initial Hamiltonian and the problem Hamiltonian to construct a Counter-diabatic Hamiltonian through a set of nested commutators. This construction enhances the fidelity of the quantum evolution, minimizing non-adiabatic transitions during the computation. The classical processing unit is then configured to combine the initial Hamiltonian, the Counter-diabatic Hamiltonian, and the problem Hamiltonian to derive a total Hamiltonian. This total Hamiltonian encapsulates the entire system's dynamics and is used for guiding the quantum state evolution.
[0432] According to an embodiment, the classical processing unit computes expectation values derived from a probability distribution, preferably of the sigma z operators derived from a probability distribution. This computation provides insights into the quantum state of the system, allowing for the evaluation of the optimization problem's progress. Additionally, the classical processing unit is configured to attribute an index to each qubit in the set, facilitating the management of qubit states during processing. It also selects a predetermined number of qubits for tree-based processing, which allows for structuring the optimization search space effectively.
[0433] According to an embodiment, the classical processing unit computes the lowest expectation values of the problem Hamiltonian, resulting in a final probability distribution that corresponds to the lowest energy configuration of the qubits. This step allow to identify the optimal solution to the optimization problem. The classical processing unit is alsoconfigured to construct a tree comprising at least n branches based on the expectation values of the probability distributions. This tree structure allows for systematic exploration of potential solutions. The classical processing unit then is configured to select at least one process, either a cutting process ora bounding process, based on the predetermined set of rules, mentioned previously. This selection process is advantageous as it enables the system to focus on the most promising branches, for example, enhancing computational efficiency.
[0434] According to an embodiment, the quantum processing unit comprises a plurality of qubits and is configured to apply a plurality of quantum gates on said plurality of qubits. It is configured to initialize the set of qubits into an initial quantum state that represents the ground state of the modified initial Hamiltonian. This initialization is configured to ensure that the quantum computation begins from a well-defined state. The quantum processing unit is then configured to evolve the quantum state of the set of qubits based on the total Hamiltonian, aiming to reach the ground state of the problem Hamiltonian. This evolution process leverages quantum mechanics to explore the solution space more effectively than classical methods.
[0435] According to an embodiment, the quantum processing unit may comprise various types of quantum processors, including superconducting quantum circuits, trapped ion quantum processors, and neutral atom quantum processors. Each of these processors is designed to leverage the unique properties of quantum mechanics to perform computations that are infeasible for classical systems.
[0436] According to an embodiment, superconducting quantum circuits can provide high-speed operations and scalability. They utilize Josephson junctions to create qubits, which can be manipulated using microwave pulses. This technology can achieve low error rates and is suitable for implementing complex quantum algorithms, making it advantageous for solving optimization problems efficiently.
[0437] According to an embodiment, trapped ion quantum processors can offer exceptional coherence times and high-fidelity gate operations. They utilize ions confined in electromagnetic fields, where quantum states are manipulated using laser beams. This method can facilitate precise control over qubit interactions, enhancing the reliability of quantum computations and enabling the execution of intricate algorithms.
[0438] According to an embodiment, neutral atom quantum processors can utilize optical lattices to trap and manipulate neutral atoms as qubits. This approach can allow for flexible qubit connectivity and scalability. The ability to control interactions between atoms with high precision can lead to improved performance in quantum algorithms, particularly in optimization tasks.
[0439] According to an embodiment, the integration of one of several of these diverse quantum processing technologies can enhance the overall robustness and versatility of the quantum computing system. By employing different types of quantum processors, themethod can adapt to various computational requirements and leverage the strengths of each technology, ultimately improving the efficiency and accuracy of solving complex optimization problems.
[0440] According to an embodiment, the bias-field generation module is configured to apply a predetermined bias-field to the initial Hamiltonian, resulting in a bias-field Hamiltonian. This application is configured to allow for the dynamic adjustment of the Hamiltonian, guiding the optimization process. The bias-field generation module also imposes first and second predetermined constraints on a selected qubit, generating first and second modified total Hamiltonians. As mentioned hereabove, the second bias-field is preferably designed to have a polarity opposite to the first bias-field, providing a mechanism for exploring different configurations of the qubit states.
[0441] According to an embodiment, the bias-field generation module is configured according to the nature of the qubits of the set of qubits. For example, the bias-field generation module can comprise at least one laser to generate pulse when the set of qubits comprises at least one qubit made from a trapped ion or a neutral atom. For example, the bias-field generation module can comprise at least one superconducting loop when the set of qubits comprises at least one superconducting qubit. Advantageously, the bias-field generation module comprises the devices needed to apply a constraint and / or a stimulus on at least one qubit of the set of qubits.
[0442] According to an embodiment, the measurement module is configured to measure the quantum state of the set of qubits, yielding a probability distribution. This measurement allows for extracting information about the quantum state and determining the outcomes of the optimization process. The measurement module also generates first and second modified probability distributions based on the first and second modified total Hamiltonians, respectively.
[0443] In summary, the computing system integrates classical and quantum processing capabilities, enabling efficient solutions to complex optimization problems. Each component contributes to the overall functionality, enhancing the system's ability to navigate the optimization landscape effectively.
[0444] We will know describe some examples of implementation of the present invention.
[0445] Example 1: Performance on Randomized Spin Glass Problems
[0446] This example illustrates, using figure 3, the application of the present invention to randomized fully connected spin glass problems. Spin glass systems are characterized by complex interactions among spins, which can lead to a multitude of local minima in the energy landscape. The optimization of such systems is crucial in various fields, including statistical mechanics and combinatorial optimization.
[0447] In this example, the set of qubits sizes varied from 12 to 30 qubits, allowing for a comprehensive analysis of the algorithm's performance across different scales. For eachset of qubits size, 10 random instances were generated, ensuring a robust evaluation of the method's effectiveness. The random instances were designed to reflect the inherent complexity and variability of spin glass problems, providing a realistic benchmark for the bf-DCQO approach.
[0448] The bf-DCQO algorithm utilized in this example was constructed using two Trotter steps, which serve to approximate the time evolution operator necessary for the quantum computation. The use of Trotter steps is configured to manage the complexity of the Hamiltonian evolution, particularly in the context of quantum optimization problems. By employing this method, the algorithm effectively navigates the energy landscape of the spin glass system, iteratively refining the bias fields and exploring the solution space. Overall, the performance results demonstrate the efficacy of the branch-and-cut bf-DCQO method in addressing randomized spin glass problems. The ability to achieve high success probabilities in finding optimal solutions, even in the presence of significant complexity, underscores the potential of this approach for solving challenging combinatorial optimization problems in quantum computing.
[0449] Example 2: Branch and Cut Procedure for Spin Glass Problem
[0450] This example illustrates the implementation of the present invention. The method, previously described, is applied to a fully connected spin-glass problem characterized by the following Ising Hamiltonian:
[0451]
[0452] W = £ i<;7iX^z+ Xi,
[0453] where Jij denotes the coupling strength between spins i and j, hi represents the local bias fields, and σiz is the Pauli-Z operator.
[0454] According to an embodiment, the size of the set of qubits considered in this example can range from 12 to 30 spins, with the coupling strengths Jj and local bias fields hi randomly generated from a uniform distribution within the interval [-1,1].
[0455] To facilitate the optimization process, bias-field digitized counter-diabatic quantum optimization algorithms were constructed utilizing two Trotter steps. This approach approximates the time-evolution operator e~iHt, allowing for the simulation of the quantum dynamics of the set of qubits. Initially, the root problem was addressed using the digitized counter-diabatic quantum optimization algorithm, which provided an initial configuration of bias fields and corresponding spin states, denoted as bj = ±1.
[0456] For the branching process, constraints were introduced by fixing certain spin variables bj. The bias-field digitized counter-diabatic quantum optimization algorithm was then employed to solve the dual problem for both possible spin directions, effectively exploring the solution space. The branch-and-cut procedure systematically pruned branches thatexhibited higher Hamiltonian values, thereby concentrating computational resources on the more promising branches that are likely to yield optimal solutions.
[0457] As the algorithm progressed, additional constraints bk= ±1 were incrementally imposed on the set of qubits. This iterative process continued until the depth of the tree reached the number of variables in the problem, ensuring a comprehensive exploration of the solution space while maintaining computational efficiency. According to an embodiment, the number n of selected qubits, i.e. of branches, is equal to the number of variable in the problem.
[0458] The results of the experiments are depicted in Figure 4, which presents the average success probability of finding optimal solutions as a function of the number of branches performed. The x-axis represents the number of branches, with a maximum value corresponding to the system size. This configuration allows for a direct comparison of the algorithm's performance as the branching process progresses. Error bars are included in the figure, represented by shadow plots, to indicate the standard deviation of the success probabilities across the 10 random instances. This statistical representation provides insight into the variability and reliability of the results.
[0459] In summary, this example illustrates the effectiveness of the branch-and-cut technique when applied to a spin-glass problem using BF-DCQO. By leveraging the quantum optimization capabilities of the DCQO algorithm and systematically pruning less promising branches, the method enhances the likelihood of identifying optimal solutions within the defined constraints of the spin-glass system.
[0460] Example 3: Results for the low autocorrelation binary sequences problem using bf-DCQO
[0461] This example focuses on the application of the present invention to a Higher-order Unconstrained Binary Optimization problem, specifically the Low-Autocorrelation Binary Sequences (LABS) problem. LABS are sequences that exhibit low autocorrelation properties, making them valuable in various applications, including communications and signal processing.
[0462] The experimental setup involved simulations of the present invention on quantum systems comprising 15 to 24 qubits. Each simulation was designed to evaluate the effectiveness of the present invention in identifying optimal LABS configurations. For each branch created during the tree-construction phase, a single bias-field iteration was performed, allowing the algorithm to refine its search for optimal solutions iteratively.
[0463] The bf-DCQO circuits utilized in this example were constructed using two Trotter steps, which facilitated the approximation of the time evolution operator necessary for the quantum computations. This approach ensured that the quantum state evolution remained accurate while minimizing the computational resources required.The results, as depicted in Figure 5, clearly demonstrate a positive correlation between the number of branches executed and the success probability of discovering optimal sequences. As the number of branches increased, the algorithm's ability to explore the solution space expanded, leading to a higher likelihood of identifying sequences that met the low autocorrelation criteria.
[0464] In summary, this example illustrates the effectiveness of the present invention in solving the LABS problem. The findings indicate that the branch-and-cut strategy significantly enhances the optimization process, yielding improved success rates in finding optimal solutions as the branching depth increases.
[0465] Example 4: Branching Strategies and Their Effectiveness
[0466] This example elucidates the comparative effectiveness of two distinct branching strategies: branch-and-bound and branch-and-cut processes, within the context of solving optimization problems using quantum computing techniques.
[0467] In the branch-and-bound approach, the present invention systematically explores the entire solution space by generating branches until a predetermined stopping condition is satisfied. This results in the formation of a complete binary tree of possibilities, where each node represents a potential solution. The exhaustive nature of this method ensures that all possible configurations are considered, thereby guaranteeing the identification of the optimal solution. However, this comprehensive exploration can lead to significant computational overhead, particularly as the size of the problem increases. The exponential growth of the search space can render this method inefficient for large-scale optimization problems.
[0468] Conversely, the branch-and-cut technique introduces a more strategic approach to the exploration of the solution space. Instead of generating all possible branches, this method evaluates the performance of each branch based on the expectation value of the problem Hamiltonian. The present invention selectively retains branches that demonstrate promising results while discarding those that yield poorer outcomes. This targeted pruning of the search space allows for a more efficient exploration, as it focuses computational resources on the most promising configurations. Consequently, the branch-and-cut methodology is designed to converge more rapidly towards optimal solutions, as it avoids unnecessary computations associated with less favorable branches.
[0469] During the development of the present invention, the data demonstrated that the branch-and-cut technique consistently achieves faster convergence compared to the branch-and-bound approach. By concentrating on branches that exhibit better performance, the branch-and-cut strategy enhances computational efficiency, thereby reducing the time required to arrive at a solution. This efficiency is particularly advantageous in the context of quantum computing, where the inherent limitations of current quantum hardware necessitate the optimization of algorithmic performance.In summary, the comparison of branch-and-bound and branch-and-cut processes highlights the advantages of the latter in terms of computational efficiency and speed of convergence. The ability to focus on promising branches while discarding less favorable ones positions the branch-and-cut technique as a superior choice for solving complex optimization problems in quantum computing environments.
[0470] Figure 6 illustrate computational experiments for two distinct 100-qubit HUBO instances: the first instance relates to sub-figures a, b and c, and the second instance relates to the sub-figures d, e and f.
[0471] According to this example, figures a) and d) depict, for each instance, the final energy distributions obtained by Branch & Bound (B& B) bf-DCQO, alongside the distributions resulting from a greedy algorithm and the pure greedy-pass algorithm for both instances, constrained to the same number of function evaluations.
[0472] The reference solution, Eref, corresponds to the minimum energy achieved by simulated annealing with 5 times 108function evaluations. The bar plots in panels (b) and (e) illustrate the approximation ratio, Emin over Eref, as a function of the number of function evaluations for both B& B bf-DCQO and simulated annealing. Here, function evaluations represent the number of energy measurements, which, for simulated annealing, corresponds to the number of spin-flips performed, while for B& B bf-DCQO, it accounts for both the number of quantum measurements and spin-flips applied in a greedy-pass correction. Additionally, figures (c) and (f) illustrate the how B& B bf-DCQO converges to Eref within a binary tree structure. The coloured nodes represent the moment where Emin over Eref is equal to 1.
[0473] As the skilled person in the art knows, a greedy algorithm (GA) and simulated annealing (SA) are both optimization techniques. A Greedy algorithm makes the best immediate choice at each step, hoping to find the global optimum through locally optimal decisions. It is simple, fast, and efficient for certain problems but often gets stuck in local optima because it does not explore alternatives once a choice is made. On the other hand, Simulated Annealing is an optimization method inspired by the annealing process. It starts by allowing worse solutions to be accepted with a certain probability, which helps the algorithm escape local optima. As the algorithm progresses, this probability decreases (controlled by a cooling schedule), making the algorithm increasingly greedy over time. Essentially, Simulated Annealing combines aspects of randomness and greediness, where the level of exploration reduces gradually. In fact, if the temperature in Simulated Annealing is set to zero from the beginning, it behaves exactly like a Greedy algorithm, only accepting better solutions. While Greedy algorithms are suitable for simple problems where local decisions lead to a global solution, Simulated Annealing is better suited for complex, non-convex problems where escaping local optima is necessary to find a near-global solution.In the present description, the words “data”, “piece of information” and “information” can be used for the same purpose, i.e. one data can be or comprise information or a piece of information, and a piece of information can be or comprise a data.
[0474] In the present description, one module can comprise several modules, one module can be formed of several modules. For example, the measurement module can comprise several other modules like the bias field generation module, for example.
[0475] According to an embodiment, the present invention can be configured to cooperate with at least one input module including one or more user interface devices such as a keyboard, pointer, number pad, or touch screen.
[0476] According to an embodiment the classical information processing module comprises at least one processor. In the present description, a processor may be any logic processing unit, such as one or more digital processors, microprocessors, central processing units, graphics processing units, application-specific integrated circuits, programmable gate arrays, programmed logic units, digital signal processors, network processors, and the like.
[0477] In the present description, a storage device is at least one non-transitory or tangible storage device. A storage device can, for example, include one or more volatile storage devices, for instance random access memory, and one or more non-volatile storage devices, for instance read only memory, flash memory, magnetic hard disk, optical disk, solid state disk, and the like.
[0478] In the present description, a storage device can include or store processor-executable instructions and / or processor-readable data associated with the operation of the present invention and / or with the execution of the method of the present invention. Execution of processor-executable instructions and / or data causes the at least one processor, and / or control modules or units, to carry out various processes and actions.
[0479] Classical data can include processor-executable instructions that, when executed by a processor, cause the processor to control, initialize, write to, manipulate, read out, and / or otherwise send data to / from a quantum module.
[0480] Classical data can include data used or obtained by the operation of the present invention. Classical data can include data associated with, e.g., created by, referred to, changed by, a processor executing processor-executable instructions, such as, control instructions. Quantum data or quantum information can comprise one or more qubits. A qubit or quantum bit is a logical building block of a quantum computer comparable to a binary digit in a classical digital computer. A qubit conventionally is a defined physical system having two or more discrete states called computational states or basis states. Basis states logically are analogous to binary states. These states may be labeled |0> and |1>.As well known by the skilled person in the relevant art (see for example “Quantum entanglement”, Ryszard Horodecki et al., Rev. Mod. Phys. 81, 865), an entangled state corresponds to at least two systems, for example two particles, sharing one wave function, and where the determination of a property of one of the particles determines a property of the other particle.
[0481] Unless otherwise specified herein, or unless the context clearly dictates otherwise the term about modifying a numerical quantity means plus or minus ten percent. Unless otherwise specified, or unless the context dictates otherwise, between two numerical values is to be read as between and including the two numerical values.
[0482] In the present description, some specific details are included to provide an understanding of various disclosed implementations. The skilled person in the relevant art, however, will recognize that implementations may be practiced without one or more of these specific details, parts of a method, components, materials, etc.
[0483] In the present description and appended claims "a", "an", "one", or "another" applied to "embodiment", "example", or "implementation" is used in the sense that a particular referent feature, structure, or characteristic described in connection with the embodiment, example, or implementation is included in at least one embodiment, example, or implementation. Thus, phrases like "in one embodiment", "in an embodiment", or "another embodiment" are not necessarily all referring to the same embodiment. Furthermore, the particular features, structures, or characteristics may be combined in any suitable manner in one or more embodiments, examples, or implementations.
[0484] As used in this description and the appended claims, the singular forms of articles, such as "a", "an", and "the", may include plural referents unless the context mandates otherwise. Unless the context requires otherwise, throughout this description and appended claims, the word "comprise" and variations thereof, such as, "comprises" and "comprising" are to be interpreted in an open, inclusive sense, that is, as "including, but not limited to".
[0485] Modifications and improvements to the above-described implementations of the present invention may become apparent to those skilled in the art. The foregoing description is intended to be exemplary rather than limiting. The scope of the present invention is, therefore, intended to be limited solely by the scope of the appended claims.
Claims
Claims1. A computer-implemented method (100) for solving at least one optimization problem using at least one quantum processing unit (220) comprising a plurality of qubits and configured to apply a plurality of quantum gates on said plurality of qubits, the method (100) being configured to be executed by a computing system (200), the method (100) comprising:a. An initialization phase, the initialization phase comprising:i. Encoding, using a classical processing unit (210), at least one optimization problem, comprising a plurality of functions representing a plurality of binary variables, into a problem Hamiltonian comprising a plurality of quantum operators;ii. Initializing, using the classical processing unit (210), a timedependent Hamiltonian comprising an initial Hamiltonian defining the starting configuration of a set of qubits taken among the plurality of qubits of the quantum processing unit (220);iii. Applying, using a bias-field generation module (230), a predetermined bias-field to the initial Hamiltonian to get a bias-field Hamiltonian;iv. Adding, using the classical processing unit (210), the bias-field Hamiltonian to the initial Hamiltonian to get a modified initial Hamiltonian;v. Using the modified initial Hamiltonian and the problem Hamiltonian to build a counter-diabatic Hamiltonian using a set of nested commutators and the classical processing unit (210);vi. Adding the initial Hamiltonian with the counter-diabatic Hamiltonian and the problem Hamiltonian to get a total Hamiltonian, using the classical processing unit (210);57vii. Initializing, using quantum gates of the quantum processing unit (220), the set of qubits into an initial quantum state representing the ground state of the modified initial Hamiltonian;b. An evolution phase (120), the evolution phase (120) comprising:i. Evolving, using the quantum processing unit (210), the quantum state of the set of qubits based on the total Hamiltonian to reach the ground state of the problem Hamiltonian;ii. Measuring, using a measurement module (230), the quantum state of the set of qubits to get a probability distribution;iii. Computing, using the classical processing unit (210), expectation values of predetermined observables for at least some qubits from the set of qubits based on the probability distribution;c. A tree-based processing phase (140), the tree-based processing phase (140) being configured to construct a tree comprising at least two branches, the tree-based processing phase (140) comprising one or more repeated applications of:i. A selection step of selecting, using the classical processing unit, a qubit from the set of qubits, the selected qubit being associated to a selected root in the tree, the selected qubit being the qubit with an expectation value of the predetermined observables closest to a predefined value for each predetermined observable, wherein for each repeated selection step a qubit is selected which has not been selected previously;ii. A first branching step, the first branching step comprising:I. Applying, using the bias-field generation module (230), a first predetermined constraint to the selected qubit to get a first modified total Hamiltonian;II. Evolving, using the quantum processing unit (220), the quantum state of the set of qubits based on the first modified total Hamiltonian to reach the ground state of the problem Hamiltonian;58III. Measuring, using the measurement module (240), the quantum state of the set of qubits to get a first modified probability distribution;IV. Computing, using the classical processing unit (210), first expectation values based on the first modified probability distribution;V. Creating, using the classical processing unit (210), a first branch of a tree from the selected root, the first branch of the tree being associated with the first expectation values, the first expectation values comprising a first expectation value of the problem Hamiltonian;iii. A second branching step, the second branching step comprising:I. Applying, using the bias-field generation module (230), a second predetermined constraint to the selected qubit to get a second modified total Hamiltonian;II. Evolving, using the quantum processing unit (210), the quantum state of the set of qubits based on the second modified total Hamiltonian to reach the ground state of the problem Hamiltonian;III. Measuring, using the measurement module (240), the quantum state of the set of qubits to get a second modified probability distribution;IV. Computing, using the classical processing unit (210), second expectation values of the second modified probability distribution, the second expectation values comprising a second expectation value of the problem Hamiltonian;V. Creating, using the classical processing unit (210), a second branch of the tree from the selected root, the second branch of the tree being associated with the second expectation values;d. A solution determination phase (150), the solution determination phase (150) comprising:59i. Computing, using the classical processing unit (210), the lowest expectation values of the problem Hamiltonian based on the constructed tree to get a final probability distribution corresponding to the lowest energy of the set of qubits;ii. Extracting, using the classical processing unit (210), at least one bitstring from the final probability distribution, the bitstring corresponding to the solution of the optimization problem.
2. Computer-implemented method (100) according to claim 1, wherein the tree-based processing phase (140) comprises:a. Selecting and executing during the tree-based processing phase (140), after each selection step, first branching step, and second branching step, using the classical processing unit (210), at least one process taken among a cutting process or a bounding process based on a predetermined set of rules:i. The cutting process comprising:I. Comparing the first expectation value of a preselected observable with the second expectation value of the preselected observable, the preselected observable preferably being the problem Hamiltonian, and:i. if the compared first expectation value is lower than the compared second expectation value, cutting the second branch, and continuing the tree-based processing phase with the first branch;ii. if the compared second expectation value is lower than the compared first expectation value, cutting the first branch, and continuing the tree-based processing phase with the second branch;II. Obtaining a tree made of branches corresponding to the lowest expectation values of the preselected observable;60III. Getting a configuration of the set of qubits corresponding to the lowest expectation values of the preselected observable;ii. The bounding process comprising:I. Storing the first branch and the second branch;II. Continuing the tree-based processing phase to explore each possible configuration of the selected qubits to identify the lowest expectation value of the preselected observable.
3. Computer-implemented method (100) according to claim 2, wherein the bounding process further comprises checking if the first expectation value of a preselected observable exceeds a preset bound and if true discarding the first branch during further processing and / or wherein the bounding process further comprises checking if the second expectation value of a preselected observable exceeds a preset bound and if true discarding the second branch during further processing.
4. Computer-implemented method (100) according to any one of the preceding claims, wherein the predetermined observables comprise sigma z for each qubit and / or the predefined value is zero for each predetermined observable.
5. Method (100) according to any one of the preceding claims, wherein the encoding step comprise mapping, using the classical processing unit (210), the optimization problem to the problem Hamiltonian, by associating at least one quantum operator of the problem Hamiltonian to each function of the optimization problem.
6. Method (100) according to any one of the preceding claims, wherein the selected qubit taken among the set of qubits is the qubit having the expectation value of sigma z nearest to zero.
7. Method (100) according to any one of the preceding claims, wherein the first predetermined constraint is a first predetermined bias field, and wherein the second predetermined constraint is a second predetermined bias field having a polarity opposite to the first predetermined bias field.
618. Method (100) according to any one of the preceding claims, wherein the problem Hamiltonian is derived from an optimization problem selected from a group comprising: protein folding, portfolio optimization, logistics optimization, and machine learning model training.
9. Method (100) according to any one of the preceding claims, wherein the treebased processing phase further comprises updating the bias-field Hamiltonian based on the computed expectation values of the modified probability distributions, thereby dynamically adjusting the search space for the optimization problem.
10. Method (100) according to any one of the preceding claims, wherein the quantum gates of the quantum processing unit (220) comprise single-qubit gates and multiqubit gates, and wherein the evolution of the quantum state is achieved through a sequence of single-qubit gates and of multi-qubit gates operations that approximate the time evolution under the total Hamiltonian.
11. Method (100) according to any one of the preceding claims, wherein the initialization phase further comprises selecting an initial Hamiltonian that has a known ground state.
12. Method (100) according to any one of the preceding claims, wherein the encoding step further comprises encoding the optimization problem into the problem Hamiltonian using an Ising model representation for Quadratic Unconstrained Binary Optimization problems or a generalized Ising model with higher-order interactions for Higher-order Unconstrained Binary Optimization problems.
13. Method (100) according to any one of the preceding claims, further comprising a post-processing phase wherein the bitstring extracted from the final probability distribution is decoded, using the processing unit (210), to retrieve the solution to the optimization problem in its native representation.
14. Method (100) according to any one of the preceding claims, wherein the quantum processing unit (220) comprises at least one among: superconducting quantumcircuit, trapped ion quantum processor, and neutral atom quantum processor, each configured to implement quantum gate operations.
15. Method (100) according to any one of the preceding claims, further comprising a feedback loop wherein the results from the solution determination phase are used to refine the encoding of the optimization problem or adjust parameters of the total Hamiltonian.
16. Computing system (200) for solving at least one optimization problem, comprising at least one classical processing unit (210), at least one quantum processing unit (220) comprising a plurality of qubits and configured to apply a plurality of quantum gates on said plurality of qubits, at least one bias-field generation module (230) and at least one measurement module (240), and wherein:a. the classical processing unit (210) is configured to:i. Encode at least one optimization problem into a problem Hamiltonian;ii. Initialize a time-dependent Hamiltonian comprising an initial Hamiltonian defining the starting configuration of a set of qubits of the plurality of qubits;iii. Add a bias-field Hamiltonian generated by the bias-field generation module (230) to the initial Hamiltonian to get a modified initial Hamiltonian;iv. Use the modified initial Hamiltonian and the problem Hamiltonian to build a counter-diabatic Hamiltonian using a set of nested commutators;v. Add the initial Hamiltonian with the counter-diabatic Hamiltonian and the problem Hamiltonian to get a total Hamiltonian;vi. Compute expectation values of predetermined observables for at least some qubits from the set of qubits based on a probability distribution obtained from measuring a quantum state of the set of qubits by the measurement module (240);vii. Assist in performing a tree-based processing on one or more qubits from the set of qubits, the tree-based processing comprising one or more repeated applications ofI. A selection step of selecting, using the classical processing unit, a qubit from the set of qubits, the selected qubit being associated to a selected root in the tree, the selected qubit being the qubit with an expectation value of the predetermined observables closest to a predefined value for each predetermined observable, wherein for each repeated selection step a qubit is selected which has not been selected previously;II. A first branching step, the first branching step comprising:A. Applying, using the bias-field generation module (230), a first predetermined constraint to the selected qubit to get a first modified total Hamiltonian;B. Evolving, using the quantum processing unit (220), the quantum state of the set of qubits based on the first modified total Hamiltonian to reach the ground state of the problem Hamiltonian;C. Measuring, using the measurement module (240), the quantum state of the set of qubits to get a first modified probability distribution;D. Computing, using the classical processing unit (210), first expectation values based on the first modified probability distribution;E. Creating, using the classical processing unit (210), a first branch of a tree from the selected root, the first branch of the tree being associated with the first expectation values, the first expectation values comprising a first expectation value of the problem Hamiltonian;III. A second branching step, the second branching step comprising:A. Applying, using the bias-field generation module (230), a second predetermined constraint to theselected qubit to get a second modified total Hamiltonian;B. Evolving, using the quantum processing unit (210), the quantum state of the set of qubits based on the second modified total Hamiltonian to reach the ground state of the problem Hamiltonian;C. Measuring, using the measurement module (240), the quantum state of the set of qubits to get a second modified probability distribution;D. Computing, using the classical processing unit (210), second expectation values of the second modified probability distribution, the second expectation values comprising a second expectation value of the problem Hamiltonian;viii. Compute the lowest expectation values of the problem Hamiltonian based on the constructed tree to get a final probability distribution corresponding to the lowest energy of the set of qubits; ix. obtain a configuration of the set of qubits corresponding to the lowest expectation values of the problem Hamiltonian and extract at least one bitstring based on the final probability distribution, the at least one bitstring corresponding to the solution of the optimization problem;b. the quantum processing unit (220) is configured to:i. Initialize the set of qubits into an initial quantum state representing the ground state of the modified initial Hamiltonian;ii. Evolve the quantum state of the set of qubits based on the total Hamiltonian to reach the ground state of the problem Hamiltonian; c. the bias-field generation module (230) is configured to apply:i. a predetermined bias-field to the initial Hamiltonian to generate the bias-field Hamiltonian;ii. first and second predetermined constraints to a selected qubit to get first and second modified total Hamiltonians, respectively, with the second bias-field having a polarity opposite to the first bias-field;65d. a measurement module (240) configured to measure the quantum state of the set of qubits to get a probability distribution and to get first and second modified probability distributions based on the first and second modified total Hamiltonians, respectively.
17. Computing system (200) according to claim 16, configured to select and execute, during tree-based processing, after each selection step, first branching step, and second branching step, using the classical processing unit (210), based on a predetermined set of rules, at least one process taken amonga. a cutting process configured to compare expectation values of a preselected observable from the first and second branch and to retain the branch corresponding to lower expectation values while discarding branches with higher expectation values; orb. a bounding process configured to store multiple branches for a more exhaustive search for optimal solutions.
18. Computing system (200) for solving at least one optimization problem, comprising at least one classical processing unit (210), at least one quantum processing unit (220) comprising a plurality of qubits and configured to apply a plurality of quantum gates on said plurality of qubits, at least one bias-field generation module (230), and at least one measurement module (240), the computing system (200) configured to execute a method according to any one of claims 1 to 15.
19. Computer product program for solving at least one optimization problem which, when executed by a computing system according to any one of claims 16 to 18, executes the method according to any one of claims 1 to 15.
20. Non-transitory computer readable medium comprising at least one computer program product according to claim 19.