Digital-analog counterdiabatic quantum algorithm (DACQA)

Abstract

The invention pertains to a computer-implemented method for solving a QUBO or HUBO optimization problem in a trapped ion quantum computer using a Digital-Analog Counterdiabatic Quantum Optimization operation in a multi-qubit system. According to the invention, the method comprises, by using a central processing unit (CPU): receiving a problem Hamiltonian being defined based on the multi-qubit system and indicative of at least one first pair of interconnected qubits in the multi-qubit system, the problem Hamiltonian encodes the optimization problem and is indicative of at least one second pair of interconnected qubits in the multi-qubit system based on the problem Hamiltonian, realizing the multi-local terms using analog blocks and obtaining an optimal quantum circuit for performing the DAQC operation, the optimal quantum circuit comprising the optimal configuration of analog and digital blocks based on hardware restrictions, and providing an optimal quantum circuit to a quantum processing unit (QPU).

Description

[0001] Kipu Quantum GmbH -1- TITLE: DIGITAL-ANALOG COUNTERDIABATIC QUANTUM ALGORITHM (DACQA) Cross-relation to other applications This application claims priority of European patent application EP24158568.6 filed on 20 February 2024 and of European patent application EP24172875.7 filed on 26 April 2024. The entire disclosure of European patent application EP24158568.6 and of European patent application EP24172875.7 is incorporated herein by reference. Field of the invention The invention relates to quantum computing. Certain aspects of the invention are defined by the appended independent and dependent claims. The invention also relates to a computer-implemented method to perform digital-analog quantum computing (DAQC) in trapped ion quantum computers for solving optimization problems via counterdiabatic driving. In particular aspects and embodiments, the method is based on Digital-Analog quantum computing (DAQC) along with counterdiabatic driving (CD). Background of the invention This summary is provided to introduce a selection of concepts in a simplified form that are further described below in the detailed description. This summary is not intended to identify key features of the invention, nor is it intended to be used to limit the scope of the invention. Quantum computing: It is a field of computation, which aims at outperforming 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: The qubit may be considered as 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 |1), as well as in a superposition of those two states, e.g., This plays an important role in the development of quantum algorithms outperforming classical algorithms. One example of the physical device that may be used as a qubit is an electron spin. Kipu Quantum GmbH -2- 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 the quantum computing. In general, these gates are generated via some Hamiltonian that makes a qubit system to evolve in time. 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 consists of 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. 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. Hamiltonian and k-local terms: A Hamiltonian is an operator representing the total energy of a quantum system, essential 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 crucial 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. Quantum gate: A quantum gate is a fixed unitary evolution. An example is a multi-qubit operation: Kipu Quantum GmbH -3- Analog block: 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 Digital block: A digital block is a fixed unitary evolution up to a set of local rotations up to 2-qubits. The examples are parameter-fixed entangling quantum gates and single qubit rotations with arbitrary angles as In some aspects and embodiments of this invention, the analog block can be used as the global MS gate available in trapped ion quantum computers and defined as ^^^^^^^^ ^^^, ^^ ^, and the digitalblock as the one-parameter dependent 2-qubit MS gates ^^2^^^^ ^^^, 0 ^ and the single qubitrotations. 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. Global Mølmer-Sørensen (MS) gate: It is a quantum gate incorporating both XX and YY interactions across multiple qubits and is a powerful and sophisticated tool in the field of quantum computing, particularly within trapped ion quantum computers. This gate is defined by the unitary operator number of ions. The key feature of this gate is the summation over all qubits, leading to a complex entanglement pattern as it induces not just individual XX and YY interactions, but also cross terms involving different qubits. The squaring of the summed terms significantly enhances the gate's capability to create elaborate multipartite entangled states, crucial for advanced quantum algorithms and simulations. Kipu Quantum GmbH -4- Trotterization: It is a technique used in quantum computing to simulate the evolution of quantum systems governed by Hamiltonians that are sums of non-commuting terms. It breaks down the exponential of a sum of operators into a product of exponentials of these operators, ^^^^ ^ ^^ ^^^^^+^^^^^as ^^^ ≈ ൬^^^^^^^^^ allowing for an approximate simulation of the quantum system's dynamics. Here, ^^ and ^^ are, generally, non-commuting operators, ^^ represents time, and ^^ is the number of subdivisions or steps in the approximation. As ^^ 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. 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. 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. Noisy intermediated 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. 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. Kipu Quantum GmbH -5- 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. HUBO problem: Higher order unconstrained binary optimization (HUBO) problems involve optimizing polynomial functions of binary variables without explicit constraints. These problems are crucial 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. Ising spin glass Hamiltonian: The Ising spin glass model describes system of spins that can interact with each other in a complex manner. In this model, spins are 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. 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. 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 adiabatic evolution. 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, Kipu Quantum GmbH -6- 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. Counterdiabatic quantum computing: Counterdiabatic quantum computing (CDQC) 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). Counterdiabatic (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 counterdiabatic term) to the original Hamiltonian to cancel out the non-adiabatic transitions. Digitized counterdiabatic quantum optimization (DCQO): This algorithm aims to find the ground state of a problem Hamiltonian rapidly by making use of counterdiabatic (CD) protocols. This is achieved by adding a CD term to the adiabatic Hamiltonian to speed up the evolution of the quantum system, meanwhile suppressing any transitions between the instantaneous eigenstates. However, this often incurs large circuit depth since the quantum gates need to be decomposed into more elementary gates implementable by the hardware. DCQO is a purely digital approach to solve optimization problems rapidly. But it incurs large circuit depth since the quantum gates need to be decomposed into more elementary gates implementable by the hardware. This prevents solving industry use case problems in the current NISQ devices. A proposed solution is to perform CD optimization in the framework of DAQC. However, DAQC has not yet been designed to adapt to a specific hardware based on their specific analog blocks to solve a particular problem of interest because to achieve this, it requires to achieve an optimal configuration of digital and analog blocks available in the given hardware. Summary of the invention This invention relates to a method to perform CD optimization in the framework of DAQC algorithm in the ion trap quantum computers. Therefore, the method of the invention may be referred to as digital-analog counterdiabatic quantum optimization (DACQO). Kipu Quantum GmbH -7- The object of the invention is to provide a hardware specific solution for trapped ion quantum computers to solve an optimization problem. This object is achieved, for example, by the features of the independent claims in the appended claims. Further aspects, embodiments and examples are apparent from the dependent claims, the detailed description, and the accompanying drawings of the figures. In a first aspect, the invention pertains to a computer-implemented method for solving an optimization problem with a trapped ion quantum computer using digital-analog counterdiabatic quantum optimization. The method comprises by using a central processing unit (CPU): - providing a problem Hamiltonian being defined based on the multi-qubit system and indicative of at least one first pair of interconnected qubits in the multi-qubit system, the problem Hamiltonian being defined based on the DAQC operation encodes the optimization problem and is indicative of at least one second pair of interconnected qubits in the multi-qubit system; - based on the problem Hamiltonian, selecting a set of analog and digital blocks from the trapped ion quantum computer, - designing a quantum circuit using the set of analog and digital blocks, - obtaining an optimal set of digital and analog operations to construct an algorithm and using the algorithm to optimize a cost function encoded in the problem (target) Hamiltonian, (in particular, obtaining an optimal set of digital and analog operations by following steps 1 to 5 of Algorithm 1 and points 1 to 7 described in Example 1 for a homogeneous problem (see Fig.1) and complement it with the Example 2 (see Fig.2) for a non-homogeneous problem and use the algorithm to optimize a cost function encoded in the problem (target) Hamiltonian,) and providing an optimal quantum circuit from the the central processing unit (CPU) to a quantum processing unit (QPU) of the trapped ion quantum computer, (in particular, providing an optimal quantum circuit by following the steps 1 to 5 of the algorithm 1 and points 1 to 7 described in Example 1 for a Homogeneous Hamiltonian (see Fig.1) and following Example 2 for a Non- Homogeneous problem (see Fig. 2) to the optimal set of digital and analog operations to Kipu Quantum GmbH -8- optimize a cost function encoded in the problem Hamiltonian is used to provide the optimal quantum circuit), and obtaining the solution to the optimization problem by measuring the ground state of the target (final) Hamiltonian in the quantum processing unit (QPU) of the trapped ion quantum computer. According to one aspect of the present invention, the present technology can be configured to control a plurality of qubits, preferably of trapped ions, on a quantum computer, according to a set of circuit parameters ^^, to prepare at least one quantum state |ψ(^^)ۧ. According to one aspect of the present invention, the present technology relates to a computer- implemented method for solving an optimization problem using a computing system comprising at least one classical processing unit and at least one quantum processing unit, the quantum processing unit comprising at least one plurality of trapped ions, the method being configured to be executed by at least the computing system, the method comprising at least: • providing, using a classical processing unit, an initial Hamiltonian, preferably whose ground state corresponds to the ground state of a set of trapped ions; • providing, using the classical processing unit, a problem Hamiltonian that is: o defined based on a set of trapped ions taken among a plurality of trapped ions of a quantum processing unit, o indicative of at least one first pair of interconnected trapped ions in the set of trapped ions, and o indicative of an at least one second pair of interconnected trapped ions in the set of trapped ions, and wherein the problem Hamiltonian encodes the optimization problem; • providing, using a classical processing unit, a counter-diabatic Hamiltonian constructed based on the initial Hamiltonian and the problem Hamiltonian using a set of nested commutators; • Building a target Hamiltonian, using a classical processing unit, by summing the initial Hamiltonian, the problem Hamiltonian and the counter-diabatic Hamiltonian; • Based on the target Hamiltonian, selecting, using a classical processing unit, a set of analogue and digital blocks comprised by the quantum processing unit, wherein the analogue blocks comprise quantum gates greater than two qubits and their mathematical form corresponds to global Mølmer–Sørensen gates, also called MS gates, and wherein the digital blocks comprises quantum gates of two qubits; Kipu Quantum GmbH -9- • designing a quantum circuit using the set of analog and digital blocks, the quantum circuit comprising a set of layers of quantum gates where each layer comprises the maximum number of gates that can be applied in parallel. The designing step comprises preferably encoding the target Hamiltonian using this set of analogue and digital quantum gates in a sequence which is configured to produce a quantum circuit having a number of operations lower than a predetermined threshold. According to one aspect of the present invention, the designing step may comprise: o In a first layer, entangling each ^^ set of nearest neighbor ions using k-qubit analogue blocks and wherein ^^ number qubits implies ^^ / ^^ (^^ > ^^ and ^^ being a multiple of ^^) number of analogueblocks, k being an integer; ^^ ^^ o In a second layer, eliminating parasitic ^^^^^^^^terms due to the analogue blocks ^^† (^^ sin2 ^^ ,^^ applied in the first layer, using digital blocks ^^^^^^2 ), and wherein ^^ number of qubits implies ^^ / ^^ number of digital blocks; o In a third layer, entangling as many leftover as possible nearest neighbor ^^ ions using ^^ − 1 analogue blocks;^^ ^^ o In a fourth layer, eliminating parasitic ^^^^^^^^terms due to the analogue blocks applied in the third layer, using digital blocks ^^^^† (^^ sin2 o In a fifth layer, eliminating extra terms originating due to overlapping of analogue blocks, using digital blocks; ^^ ^^ o In a sixth layer, eliminate, preferably the left-over, parasitic ^^^^^^^^terms due to of MS gate using two-qubit MS, ^^2†^^^^ (^^ sin2 ^^^^ the previous layer,2 ). The gate ^^ count scales as ^^− 1.o In a seventh layer, entangling remaining pairs of ions using digital blocks; oObtaining the quantum circuit comprising 2^^^ − ^^^^^^ − ^^ + 1^ / ^^ layerscomprising an optimal set of analogue and digital operations; • constructing a quantum algorithm using the quantum circuit, this quantum algorithm being configured to perform at least one counter-diabatic optimization process; • providing the quantum circuit and the optimal set of digital and analog operations to the quantum processing unit to optimize a cost function encoded in the problem Hamiltonian using a counter-diabatic optimization process, the counter-diabatic optimization process comprises at least the following steps: o Counter-diabatically time evolving the quantum circuit under the encoded target Hamiltonian from the state of the initial Hamiltonian to a final state of a final Kipu Quantum GmbH -10- Hamiltonian up to a predetermined final time which is equal to or less than the coherence time of the qubit made by the set of trapped ions, this evolving step can comprise: ▪ Emitting a set of time-dependent electromagnetic pulses, using at least one electromagnetic pulse generation module, to drive a plurality of trapped ions under the total Hamiltonian, the set of time-dependent electromagnetic pulses being used to generate interactions among the trapped ions according to the total Hamiltonian; o Measuring, using a measurement module, the quantum state of each trapped ions of the set of trapped ions to obtain at least one state probability distribution, the state probability distribution containing a set of pairs of unique bitstring and its associated probability, the unique bitstring being the measured states of the trapped ions in at least one computational basis, the probability being the number of times the unique bitstring was measured relative to the total number of total measurements; o Post processing, using the classical processing unit, the result of the state probability distribution, the post processing comprising the following steps: ▪ Building an energy probability distribution as a set of triplets, each triplet containing one unique bitstring, one energy value, which is the expectation value of the unique bitstring with respect to the final Hamiltonian, and the probability associated to the bitstring and retrieved from the state probability distribution; ▪ Generating a post-processed energy probability distribution by performing a mapping of the probabilities of each triplet over a predetermined scale; o Calculating, using the classical processing unit, at least one expectation value from the post-processed energy probability distribution; o Extracting, using the classical processing unit, a ground state solution from the calculated expectation value; • obtaining the solution to the optimization problem by measuring, preferably using, the ground state of the final Hamiltonian in the quantum processing unit. A nested commutator is a mathematical operation involving repeated applications of the commutator, which measures the non-commutativity of two operators. Given two operators A and ^^, their commutator is defined as [A,B]=AB−BA. A nested commutator extends this by iteratively applying the commutator operation, such as [A,[A,B]] or deeper structures like Kipu Quantum GmbH -11- [A,[A,[A,B]]]. In quantum mechanics, nested commutators are particularly useful in constructing counter-diabatic Hamiltonians, where they systematically generate higher-order corrections to suppress non-adiabatic transitions and drive a system along an adiabatic path in finite time. According to one aspect of the present invention, the classical processing unit can also be called central processing unit and advantageously can belong to a classical computer, also called a classical computing subsystem. According to one aspect or an embodiment, to perform a global MS gate, a pair of counter- propagating laser beams or microwave fields is applied to the entire ion chain. These fields are bichromatically detuned symmetrically around a chosen vibrational mode of the trapped ions, typically the center-of-mass mode. The detuning ensures that the laser fields do not excite real electronic transitions but instead create a state-dependent force that couples the qubit states to the collective motion of the ions. Advantageously, this force induces a controlled oscillation in the ions, generating an effective XX and YY-type spin-spin interaction across all qubits, i.e. trapped ions. By carefully tuning the laser intensity, phase, and duration, the interaction strength can be precisely controlled to achieve a predetermined entangling operation. According to one aspect of the present invention or an embodiment, the present technology relates to a computing system for solving an optimization problem, comprising: • a classical processing unit configured to: o provide an initial Hamiltonian, wherein the ground state of said initial Hamiltonian corresponds to the ground state of a set of trapped ions taken among a plurality of trapped ions; o provide a problem Hamiltonian that is: ▪ defined based on a set of trapped ions taken among a plurality of trapped ions of a quantum processing unit, ▪ indicative of at least one first pair of interconnected trapped ions in the set of trapped ions, and ▪ indicative of an at least one second pair of interconnected trapped ions in the set of trapped ions, and wherein the problem Hamiltonian encodes the optimization problem o construct a counter-diabatic Hamiltonian based on said initial Hamiltonian and said problem Hamiltonian, preferably using a set of nested commutators; Kipu Quantum GmbH -12- o Build a target Hamiltonian by summing the initial Hamiltonian, the problem Hamiltonian, and the counter-diabatic Hamiltonian; o design a quantum circuit using a set of analogue and digital blocks selected based on said target Hamiltonian, wherein said analogue blocks comprise quantum gates for more than two qubits and correspond mathematically to global Mølmer–Sørensen gates, and said digital blocks comprise quantum gates for two qubits; o construct a quantum algorithm using the quantum circuit, configured to perform at least one counter-diabatic optimization process; o provide post-processing of a state probability distribution to generate an energy probability distribution, o map the probabilities of at least one triplet within said distribution over a predetermined scale, o calculate at least one expectation value from the post-processed energy probability distribution, and o extract a ground state solution from the calculated expectation value; • a quantum processing unit comprising the plurality of trapped ions, the quantum processing unit being configured to receive and execute the quantum circuit and the optimal set of digital and analogue operations to optimize a cost function encoded in the problem Hamiltonian using a counter-diabatic optimization process. Preferably, said counter-diabatic optimization process comprises emitting a set of time-dependent electromagnetic pulses to drive trapped ions of the set of trapped ions under the total Hamiltonian for the evolution from the state of the initial Hamiltonian to a final state of a final Hamiltonian up to a predetermined final time; • Optionally, an electromagnetic pulse generation module operatively connected to the quantum processing unit and configured to emit said set of time-dependent electromagnetic pulses for driving interactions among the trapped ions according to the total Hamiltonian; • a measurement module configured to measure the quantum state of trapped ions to obtain at least one state probability distribution, containing a set of pairs of unique bitstring and its associated probability; and • wherein the classical processing unit is further configured to: o post-process the result of the state probability distribution to build an energy probability distribution, o perform a mapping of probabilities, o calculate at least one expectation value, and Kipu Quantum GmbH -13- o extract the ground state solution. According to one optional aspect of the present invention, the quantum circuit comprises at least two layers comprising an optimal set of analogue and digital operations designed to minimize the number of operations below a predetermined threshold. According to one optional aspect of the present invention, the quantum algorithm is tailored for counter-diabatic optimization process to ensure rapid convergence to the ground state solution within the coherence time of the quantum processing unit. According to one optional aspect of the present invention, the quantum circuit is designed with a sequence of layers comprising alternating sets of analogue blocks for entangling nearest neighbor ions and digital blocks for eliminating parasitic and extra terms generated by overlapping of analogue blocks. According to one optional aspect of the present invention, the classical processing unit provides an initial Hamiltonian. This Hamiltonian is useful as its ground state mirrors the ground state of a specific set of trapped ions. The initial Hamiltonian serves as the starting point for the quantum counter-diabatic optimization process. According to one optional aspect of the present invention, the problem Hamiltonian is defined based on the set of trapped ions. Preferably, it comprises information about at least one first pair and one second pair of interconnected trapped ions, encoding the optimization problem into a quantum framework. This Hamiltonian transformation of the optimization problem is useful for the quantum algorithm's ability to find the solution. According to one optional aspect of the present invention, a counter-diabatic Hamiltonian is constructed from the initial and problem Hamiltonians, utilizing a series of nested commutators. This counter-diabatic Hamiltonian is preferably designed to facilitate a smooth transition from the initial to the problem Hamiltonian, mitigating non-adiabatic transitions and thereby enhancing the efficiency of the counter-diabatic optimization process. According to one optional aspect of the present invention, the target Hamiltonian is the sum of the initial, problem, and counter-diabatic Hamiltonians. It preferably represents the total Hamiltonian that governs the evolution of the set of trapped ions, guiding the set of ions trapped towards the solution of the optimization problem. Kipu Quantum GmbH -14- According to one optional aspect of the present invention, the classical processing unit designs a quantum circuit based on the target Hamiltonian, incorporating both analogue and digital blocks. Advantageously, analogue blocks include quantum gates for multi-qubit operations, akin to global Mølmer–Sørensen gates, while digital blocks involve two-qubit quantum gates. This hybrid approach allows for a versatile manipulation of quantum states. According to one optional aspect of the present invention, a quantum algorithm is constructed using the designed quantum circuit. Preferably, this quantum algorithm is specifically configured to execute at least one counter-diabatic optimization process, leveraging the dynamics dictated by the target Hamiltonian to find the ground state solution of the problem Hamiltonian. According to one optional aspect of the present invention, the quantum processing unit comprises a plurality of trapped ions and is tasked with receiving and executing the quantum circuit and operations designed by the classical processing unit. Preferably, it performs the counter-diabatic optimization process to optimize the cost function encoded in the problem Hamiltonian. According to one optional aspect of the present invention, the electromagnetic pulse generation module is connected to the quantum processing unit and emits time-dependent electromagnetic pulses. These pulses are preferably used for driving the interactions among the trapped ions according to the total Hamiltonian, facilitating the controlled evolution of the quantum state. According to one optional aspect of the present invention, the measurement module is used for measuring the quantum state of the trapped ions. It obtains a state probability distribution, which is a collection of pairs consisting of unique bitstrings and their associated probabilities. This data can be used for the post-processing tasks performed by the classical processing unit to extract the solution to the optimization problem. Indeed, according to one optional aspect of the present invention, the classical processing unit also handles post-processing of the quantum state probability distribution to generate an energy probability distribution. It can map probabilities of specific triplets within this distribution, calculate expectation values, and extract the ground state solution, which corresponds to the solution of the optimization problem. Kipu Quantum GmbH -15- In certain optional aspects and embodiments of the method of the invention, the problem Hamiltonian is an Ising spin glass Hamiltonian which encodes the solution to a QUBO or HUBO problem, in particular wherein the QUBO or HUBO problem comprises homogeneous (equal) and / or inhomogeneous (unequal) coefficients. In certain optional aspects and embodiments of the method of the invention, the total Hamiltonian consists of an adiabatic Hamiltonian and a counterdiabatic (CD) Hamiltonian and the adiabatic Hamiltonian consists of an initial Hamiltonian and a target Hamiltonian. In certain optional aspects and embodiments of the method of the invention, analog blocks from the set of analog and digital blocks are parametrized entangling unitary evolution with more than one parameter and a digital block is a fixed unitary evolution up to a set of local rotations up to 2-qubits. In certain optional aspects and embodiments of the method of the invention, the analog blocks are global Molmer Sorenson gates, and wherein the digital blocks are 2-qubit ^^^^^^^^gates or single qubit gates. In certain optional aspects and embodiments of the invention, the method comprises counterdiabatic driving of the problem Hamiltonian. In certain optional aspects and embodiments of the method of the invention, the multi-qubitsystem has ^^ qubits, and uses ^^ (^^ ≤ ^^) qubit analog blocks wherein n is an integer with ^^ ≥2. In certain optional aspects and embodiments, the trapped ion quantum computer allows implementing analog blocks on a non-nearest configuration of ions. In certain optional aspects and embodiments of the method of the invention, the selection of the set of analog and digital blocks from the trapped ion quantum computer comprises the steps of 3 to 5 of the Algorithm 1, namely: Step 3: Realizing the k-local (k>1) terms of the Hamiltonian using global MS gates (analog block) in trapped ions. Step 4: Realizing the 1-local terms using the single qubit rotations in trapped ions. Step 5: Obtaining a circuit with an optimal configuration of analog and digital blocks. Kipu Quantum GmbH -16- In certain optional aspects and embodiments of the method of the invention, designing the quantum circuit using the set of analog and digital blocks based on the problem Hamiltonian comprises the steps of 1 to 5 of the algorithm 1 and points 1 to 7 of the Example 1 for Homogeneous case (Fig.1) and following Example 2 for Non-Homogeneous case (Fig.2). In certain optional aspects and embodiments of the method of the invention, the optimal set of digital and analog operations to optimize the cost function encoded in the problem Hamiltonian is obtained by following the points 1 to 7 of the Example 1 for Homogeneous case (Fig.1) and following Example 2 for Non-Homogeneous case (Fig.2). In certain optional aspects of the invention or embodiments of the method of the invention, the optimal quantum circuit is obtained by following the points 1 to 7 of the Example 1 for a homogeneous case (Fig.1) and following Example 2 for a non-homogeneous case (Fig.2). In certain optional aspects of the invention or embodiments of the method of the invention, the set of operating parameters of the at least one analog block comprises rotation angles as an adjustable parameter. In certain optional aspects of the invention or embodiments of the method of the invention, the solution to the optimization problem is encoded in the ground state of the Ising spin glass Hamiltonian. In certain embodiments of the method of the invention, the ground state of the Ising spin glass Hamiltonian is found using counterdiabatic techniques of optimization. In certain optional aspects of the invention or embodiments of the method of the invention, each qubit in the intermediate quantum circuit is initialized, in particular in the |0ۧ state or another predefined state suitable for the given problem, generating an easy to prepare initial state for the CD driving. The invention also pertains to a data processing apparatus (or device or system) comprising means for carrying out the method described herein. The invention further pertains to a computer program [product] comprising a computer- readable storage medium, wherein the computer-readable storage medium stores a computer code which, when executed by at least one central processing unit (CPU) causes the CPU to perform the method described herein. Kipu Quantum GmbH -17- The invention further pertains to a computer program [product] comprising instructions which, when the program is executed by a computer, cause the computer to carry out the method described herein. The invention further pertains to the use of CD optimization algorithm to solve an optimization problem in trapped ion quantum computers by making us of a hardware specific digital-analog approach. Other features and advantages of the invention will be apparent upon reading the detailed description and reviewing the accompanying drawings of the figures. Various embodiments of the invention are further described in more detail with reference to the accompanying drawings. However, the invention may be embodied in many other forms and should not be constructed as limited to any certain structure or function discussed in the following description. According to the description, it will be apparent to the ones skilled in the art that the scope of the invention encompasses any embodiment thereof, which is disclosed herein, irrespective of whether this embodiment is implemented independently or in concert with any other embodiment of the invention. For example, the method disclosed herein may be implemented in practice by using any numbers of the embodiments provided herein. Furthermore, it will be understood that any embodiment of the invention may be implemented using one or more of the elements presented in the appended claims. The implementation of native operations on a quantum computer allows for more faithful execution of complex circuits. DAQC enables analog blocks provided by quantum hardware types such as trapped ions, superconducting circuits, neutral atoms, photonics, spin qubits, etc. to be combined with digital blocks. However, due to the limited hardware specifications and targeted problem, an optimal configuration of the digital and analog blocks is necessary to solve the problem. The present invention pertains to an algorithm capable of achieving the ground state solutions for QUBO of HUBO problems of arbitrary size using DAQC which involves less circuit depth quantum circuits as compared to purely digital methods. Kipu Quantum GmbH -18- A certain embodiment of the method of the invention (algorithm) is summarized in the steps given below (Algorithm 1). Step 1: In general, the method of the invention can be used for solving a QUBO or HUBO problem. Step 2: Encoding the solution of the optimization problem in the ground state of a many-body Hamiltonian. Step 3: Realizing the k-local (k>1) terms of the Hamiltonian using global MS gates (analog block) in trapped ions. Step 4: Realizing the 1-local terms using the single qubit rotations in trapped ions. Step 5: Obtaining a circuit with an optimal configuration of analog and digital blocks. Step 6: Using a CD optimization to find the ground state of the many-body Hamiltonian. Step 7: Using trotterization for the time evolution of the system to implement quantum gate- based optimization in trapped ion quantum computer. Step 8: To achieve Step 6, in certain embodiments, one starts with a n-qubit quantum circuit and a selected multi-qubit analog block (global MS gate), where each qubit is initialized in either the |0ۧ state or some predefined state suitable for the problem to be solved. For the trapped ion quantum computers, using the global n-qubit MS gate, one can achieve the optimal circuit configuration for the DACQO algorithm.

[0002] Kipu Quantum GmbH -19- Description of the figures Figure 1: Quantum circuit to solve an homogeneous QUBO or HUBO problem using CD ^^ ^2^^ optimization. Analog blocks are global MS gates, ^^^^^^ ^^, ^^ and digital blocks are ^^^^^^(^^, 2) and single qubit gates. The same structure is extendable to ^^ qubits with any ^^ qubit analog block according to the steps 1-8 and the algorithm shown in Example 1. Figure 2: Example of a 3-qubit quantum circuit to solve an inhomogeneous QUBO or HUBO problem with 3 qubit analog blocks single qubit rotations (Pauli Z gates) according to certain embodiments of the present invention. Figure 3: Scaling (circuit depth) as a function of the system size (number of qubits) for all cases. Figure 4: This panel illustrates the success probability achievable for varying problem sizes as the MS gate fidelity increases. The results are compared against the peak performance achievable through purely digital quantum simulations (run on IonQ noisy emulator). Additionally, the figure includes a benchmark indicating the minimum accuracy threshold set at 37 % of the ideal success probability (current benchmark for circuit success probability by IonQ, highlighting the efficacy of the algorithm in meeting or exceeding this benchmark across different fidelity levels. Figure 5: In this panel, we extrapolate the minimum gate fidelity required for the algorithm to achieve the stated minimum accuracy and to surpass the capabilities of digital simulation, up to a hypothetical 52-qubit system. The extrapolation is based on the observed fidelity trends for problem sizes up to 20 qubits (multiple measurements with 1024 shots and averaged overall), providing insights into the scalability of the algorithm in handling larger quantum systems with stringent accuracy requirements. The extrapolation employs an exponentialdecay model towards a limit, mathematically represented as ^^ + ^^^ − ^^^^^^^^^ , where ^^ =1 denotes the limiting fidelity value, ^^ represents the initial fidelity, ^^ = 0.1 is the decayconstant, and ^^ is the number of qubits. Figure 6: Enhancement factor for 16 node MIS problem using analog blocks of different qubit sizes. Kipu Quantum GmbH -20- Detailed description of the invention QUBO problem The method of the invention in certain embodiments was applied to a known optimization problem, namely the QUBO problem. QUBO problems involve finding binary values (0 or 1) for a set of variables to minimize a quadratic objective function. This can be defined as follows:^^^^^^^^^^^^^^^^: ^^^^^^ = σ^^,^^ ^^^^,^^^^^^^^^^ (4)where ^^ represents a vector of binary variables, with ^^^^taking binary values (0 or 1) for each variable and ^^^^,^^represents the coefficients of the quadratic terms that define the objective function. Solving QUBO problems is computationally challenging, particularly for large problem instances. The QUBO problem can be mapped to the Ising Hamiltonian of the form^^^^^^^^^^^^ = σ ^^,^^ J^^,^^ σ^^^^σ ^^ ^^ + σ ^^ ℎ^^^^^^ ^^(5) which encodes the solution in its ground state. To find the ground state of Equation (5), a CD technique is used which aims to provide rapid solutions to optimization problems by supplementing adiabatic driving (adiabatic quantum computing) by counterdiabatic terms. This is achieved by adding a CD term to the adiabatic Hamiltonian as^^ ^^^^ = ^^^^^^^^^^ + ^^ ^^^^^^^^^^ ^^^^ (6)where (^^^t^ ∈ [0, 1]) is a scheduling function that represents the interpolation from the initialHamiltonian to the final Hamiltonian. The adiabatic time dependent Hamiltonian comprises the initial and final (target) Hamiltonian as the transverse field and the Ising spin glass, respectively, given by Kipu Quantum GmbH -21- For the implementation, a native analog-block (MS gate) is used from trapped-ion quantum systems, given by where ^^^^ , ^^^^ = σ are a summation over the Pauli X and Y operators. The analog blockscan be applied to only nearest neighbor subset of qubits and the angles ^^, ^^ of each block ischosen according to the target Hamiltonian.Example 1: Homogeneous (equal coefficients, J^^,^^ = J, h^^,^^ = ℎ) HamiltonianIn certain embodiments of the method of the invention, digital and analog blocks are applied to realize a homogeneous Ising spin glass Hamiltonian whose ground state can be found by a CD optimization. After a basis transformation, the full Hamiltonian can be written as In this Hamiltonian, the local terms can be realized by digital gates and the 2-local terms using analog block, Equation (9). The quantum algorithm in the form of ^^ qubit quantum circuit using^^ (integer) qubit analog blocks (^^ > ^^ and ^^ is a multiple of ^^) can be designed as follows:1. Layer 1: Entangle each ^^ (integer) set of nearest neighbor ions using k-qubit MS gates ^^^^2 ^^^^ ^^^, ^^^^ ^^൯. ^^ number of qubits will require ^^ / ^^ (^^ > ^^ and ^^ isa multiple of ^^) number of MS gates. 2. Layer 2: Eliminate parasitic ^^ ^^ ^^ ^^^^^^terms due to MS gates applied in the previous layer (^^ 2^^ usingsin ^^ ,2) . ^^ number of qubits will require ^^ / ^^ number of these gates.3. Layer 3: Entangle as many leftovers as possible nearest neighbor ^^ ions. The gate ^^ count scales as ^^− 1. Kipu Quantum GmbH -22- 4. Layer 4: Eliminate parasitic ^^ ^^ ^^ ^^terms due to the previous layer of MS gate using (^^ sin2 ^^ ,^^ ^^ 2 ) as in step 2. The gate count scales as ^^− 1.5. Layer 5: Eliminate the extra terms originating due to overlapping MS gates using 2- qubit MS gates. The gate count scales as the number of MS gates in the second layer ^^ giving an estimate of ^^. 6. Eliminate parasitic ^^ ^^ ^^^^ ^^ ^^terms due to the previous layer of MS gate using two-qubit ^^ The gate count scales as ^^− 1.7. Finally, to entangle remaining pairs of ions, the 2-qubit MS gates are used. To estimate this, one can see that for ^^ = 1, there are 0 pairs, for ^^ = 2, there are 10 pairs, this willscale as ^^ − ^^ + ^^ − ^^^ − 1^ + .... + ^^ − ^^ + 1 = ^^^ − ^^^{^^ – ^^^ − 1^} / 2. But, asone needs an equal number of counter MS gates ^^2† (^^ sin2 the total would be = ^^^ − ^^^{^^ – ^^^ − 1^}. In this step ^^ / 2 number of gates can be donesimultaneously, giving the number of layers as 2^^^ − ^^^^^^ − ^^ + 1^ / ^^.The corresponding quantum circuit for a 8 qubit example is shown in Fig.1. For ^^-qubit system,the circuit depth of the algorithm is 6 + 2^^^ − ^^^^^^ − ^^ + 1^ / ^^. For a fully digital simulation thecircuit depth scales as 3(^^ − 1). For example, for a 8 qubit system and 4 qubit MS gate asanalog block the depth using DAQC will be 9 compared to Digital which is 21. The circuit depth scaling in time units for a ^^-qubit system is shown in Fig.2. Example 2: Non-homogeneous Hamiltonian In certain embodiments of the method of the invention, non-homogeneous Hamiltonians can be realized by introducing non-homogeneity using local rotations. As a simple example for a 3-qubit Hamiltonian realized using 3-qubit MS gates and local unitaries, it can be made non- homogeneous according to the Figure.2. The result of applying local rotations to the 3-qubit MS gate is to generate the following transformation Kipu Quantum GmbH -23- This creates inhomogeneity in the Hamiltonian and requires solving 12 equations to get theparameters ^^^^, ^^^^. In general, for a ^^ – qubit analog block, one needs (^^^^^ − 1^ / 2) gates and(4^^^^^ − 1^ / 2) number of equations to get the parameters. Therefore, it is also more reasonableto work with a smaller sized analog block. Furthermore, the circuit depth can be evaluated by ^^^^^−1^ ^^^^^^−^^+1^ + 2 ^^ means that each 4-qubit analog block in the Fig. 1 for the homogeneous case is replaced by 2 number of gates. The scaling of the algorithm using 4-qubit analog blocks is shown for both the Homogeneous and the Inhomogeneous cases as in Fig.3. It shows that the DACQO algorithm outperforms the purely digital simulation in both homogeneous and inhomogeneous cases. This is because the circuit depth plotted against the number of qubits scales less for the DACQO in general and can be used to solve a larger problem in the same device (trapped ions) with the given coherence time. Resultsthe number of ion pairs left to be entangled scales as ^^^^ = ^^^ − ^^^^^^ − ^^ + 1^ / 2 and thegate requirement scales as 2^^^^(including counterMS gates). In trapped ions, ideally N / 2 gates can be performed simultaneously, this leads to a circuit depth of 2(N −k)(N −k+1)N . The total circuit depth for a N qubit system can be expressed as the number of gate layers that can be executed sequentially, where each layer consists of gates that can be executed in parallel as follows 1^ ^^. we utilize a 4-qubit analog block to perform DACQA across systems comprising 4N qubits (since we use a 4-qubit analog block). This approach enables us to assess the fidelity thresholds necessary for analog blocks to outperform digital simulations and achieve satisfactory accuracy levels. The results of this analysis are presented in Fig. 4 and Fig.5. Additionally, we have extended our analysis to project the minimum fidelity thresholds necessary for surpassing the performance benchmarks set by digital simulations and meeting the established minimum accuracy standards. These projections, which estimate the fidelity requirements for systems scaling up to 52 qubits, leverage the observed trends in our current data to anticipate the evolving demands on quantum system performance as it expands. The findings indicate that for system sizes up to 20 qubits, MS gate fidelities within the 98% − 99% range are adequate to surpass purely digital simulations and fidelities between 99% − 99.5% are required to achieve minimum accuracy benchmark. The extrapolated data suggests that to meet or exceed both benchmarks, fidelities around approximately 99.5% are required, which are achievable with current state-of-the-art capabilities. Kipu Quantum GmbH -24- By accounting for both multi-qubit and single-qubit gate durations, we can calculate the overall circuit runtime in seconds. This quantity can be represented as the sum of the depths multiplied with their duration, with each depth’s duration determined by the execution times of respective layers, as shown below:^^^^^^^^^^^^^^ ^^^^^^^^^^^^^^ ^^^^^^^^ = ^^^^ · ^^^^^^^^ℎ[^^^^ ^^ ^^^, ^^^] + ^^^^ · ^^^^^^^^ℎ[^^^^,^^,^^ ]where ^^^^and ^^^^are the multi qubit and single qubit gate times, respectively and layers[^^^^ ^^ ^^^, ^^^], number of layers of multi-qubit and single qubit gates. Weconsider the gate times according to the current state of the art for 2-qubit MS gates and single qubit gates that are available in current hardwares (we consider the IonQ Forte system for our current analysis). For the 4-qubit MS gates, we can consider their anticipated gate times achievable with the current technology which is equivalent to the current 2-qubit gate times. In the current trapped ion systems, for example in the IonQ F orte system, the gate times are ^^^^= 930μs and ^^^^= 130μs. Based on these numbers achievable with the current state-of-the-art technology, we study the circuit runtime of the algorithm. for demonstrating the efficiency of the algorithm we consider different block sizes, and study the enhancement factor (^^^^^^^^^^^^^^^^ / ^^^^^^^^^^), where ^^ is the circuit runtime for a fixed problem size. We select the maximum independent set (MIS) problem for this purpose with three different instances involving weighted and non- weighted graphs and 16 nodes. To tackle this problem efficiently we can make use of the all-to-all connectivity of the trapped ion processors. We can represent the nodes with equal weights as the nearest neighbour qubits in trapped ion processors and apply the analog blocks only in Layer1 as shown the quantum circuit of Fig.1. This is because the problem does not possess an all-to- all connectivity. The result demonstrates that the algorithm shows about 1.6X−fold improvement in reducing the circuit runtime even with the smallest analog block of 2 qubits. Moreover, further reduction can be achieved with increasing block sizes and the algorithm is most effective for a non-weighted graph followed by a graph with mixed weights and a fully-non uniform graph. For a fully-non uniform graph, increasing the size of the block (beyond 4 qubits) leads to a disadvantage since the number of gates required for introducing non- homogeniety surpasses the number needed to realize all the many-body terms effectively. Therefore, the algorithm is problem dependent and demonstrates non-trivially that for a fully non-homogeneous problem (non-uniform MIS) thehighest reduction can be achieved with just a 2-qubit analog block [^^^^ ^^ ^^^, ^^^].

Claims

Kipu Quantum GmbH -25- Claims 1. A computer-implemented method for solving an optimization problem using a trapped ion quantum computer comprising a multi-qubit system comprising trapped ions, comprising: • providing, using a central processing unit of a classical computer, an initial Hamiltonian, preferably whose ground state corresponds to the ground state of a set of trapped ions of the multi-qubit system; • providing, using the central processing unit, a problem Hamiltonian that is defined based on the multi-qubit system and indicative of at least one first pair of interconnected qubits in the multi-qubit system and is indicative of an at least one second pair of interconnected qubits in the multi-qubit system, wherein the problem Hamiltonian encodes the optimization problem; • providing a counter-diabatic Hamiltonian constructed based on the initial Hamiltonian and the problem Hamiltonian, preferably using a set of nested commutators ; • summing the initial Hamiltonian, the problem Hamiltonian and the counter-diabatic Hamiltonian to build a target Hamiltonian; • based on the target Hamiltonian, selecting a set of analogue and digital blocks from the trapped ion quantum computer, wherein the analogue blocks comprise quantum gates greater than two qubits, and their mathematical form corresponds to global Mølmer– Sørensen gates, and wherein the digital blocks comprise quantum gates of two qubits; • designing a quantum circuit using the set of analog and digital blocks, the quantum circuit comprising a set of layers of quantum gates where each layer comprising the maximum number of gates that can be applied in parallel, this designing step comprising: o In a first layer, entangling each ^^ set of nearest neighbor ions using k-qubit analogue blocks ^^^^ ^^^, ^^ ^ =wherein ^^ number qubits implies ^^ / ^^ (^^ > ^^ and ^^ being a multiple of ^^) number of analogueblocks, k being an integer; ^^ ^^ o In a second layer, eliminating parasitic ^^^^^^^^terms due to the analogue blocks ng digital blocks ^^^^†^^^^ (^^ si 2^^ applied in the first layer, usin ^^ ,2 ), and wherein ^^ number of qubits implies ^^ / ^^ number of digital blocks; o In a third layer, entangling as many leftovers as possible nearest neighbor ^^ ions using ^^− 1 analogue blocks;Kipu Quantum GmbH -26- ^^ ^^ o In a fourth layer, eliminating parasitic ^^^^^^^^terms due to the analogue blocks applied in the third layer, using digital blocks ^^^^†^^^^ (^^ sin2; o In a fifth layer, eliminating extra terms originating due to overlapping of analogue blocks, using digital blocks; ^^ ^^ o In a sixth layer, eliminate parasitic ^^^^^^^^terms due to the previous layer of MS gate using two-qubit MS, ^^2†^^^^ (^^ sin2 ^^ ,^^ ^^ 2 ). The gate count scales as− 1.o In a seventh layer, entangling remaining pairs of ions using digital blocks; oObtaining the quantum circuit comprising 2^^^ − ^^^^^^ − ^^ + 1^ / ^^ layerscomprising an optimal set of analogue and digital operations; • constructing a quantum algorithm using the quantum circuit, this algorithm being configured to perform at least one counter-diabatic optimization; • providing the quantum circuit and the optimal set of digital and analog operations to a quantum processing unit of the trapped ion quantum computer to optimize a cost function encoded in the problem Hamiltonian using a counter-diabatic optimization process, obtaining the solution to the optimization problem by measuring the ground state of the final Hamiltonian in the quantum processing unit of the trapped ion quantum computer.

2. Method of claim 1, wherein the problem Hamiltonian is an Ising spin glass Hamiltonian which encodes the solution to a quadratic unconstrained binary optimization or higher- order unconstrained binary optimization problem, in particular wherein the quadratic unconstrained binary optimization or higher-order unconstrained binary optimization problem comprises homogeneous and / or inhomogeneous coefficients.

3. Method of claim 1 or 2, wherein the target Hamiltonian consists of an adiabatic Hamiltonian and a counter-diabatic Hamiltonian and the adiabatic Hamiltonian consists of an initial Hamiltonian and a target Hamiltonian.

4. Method of any of claims 1 to 3, wherein analogue blocks from the set of analogue and digital blocks are parametrized entangling unitary evolution with more than one parameter and a digital block is a fixed unitary evolution up to a set of local rotations up to 2-qubits.Kipu Quantum GmbH -27- 5. Method of any of claims 1 to 4, wherein the analogue blocks are global Molmer Sorenson gates, and wherein the digital blocks are 2-qubit ^^^^^^^^gates or single qubit gates.

6. Method of claim 1 to 5, comprising counter-diabatic driving of the problem Hamiltonian.

7. Method of any of claims 1 to 6, wherein the multi-qubit system has ^^ qubits, and uses ^^ (^^ ≤ ^^) qubit analogue blocks wherein n is an integer with ^^ ≥ 2.

8. Method of any of claims 1 to 7, wherein the trapped ion quantum computer is configured to allow implementing analog blocks on a non-nearest configuration of ions.

9. A computer product program for solving an optimization problem comprising instructions which, when executed by at least one processing unit, cause the processing unit to carry out the method of any one of claims 1 to 8.

10. A non-transitory computer readable medium comprising at least one computer program product according to the previous claim.

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