Analog counterdiabatic quantum computing - Nature

IntroductionMany scientific, technological, and industrial applications involve finding the best configuration from a vast set of discrete possibilities, such as optimizing logistics routes, scheduling tasks, designing efficient networks, or selecting the best portfolio of financial assets. Such challenging problems can be mapped mathematically to combinatorial optimization problems where the goal is to identify the best solution from an exponentially large search space while satisfying specific constraints1.Therefore, solving combinatorial optimization problems is highly relevant across multiple domains, as it enables improvements in efficiency, cost reduction, and performance enhancement in fields such as artificial intelligence, telecommunications, finance, healthcare, and energy systems1,2,3. However, the computational complexity of these problems often makes them computationally challenging on classical computers4,5. With the recent developments in quantum computing, speeding up the computation of industry-relevant problems is within the reach of noisy quantum processors6,7. Typically, the Hamiltonian of an analog quantum system is used to encode the problem’s cost function2. A promising approach to solve the problem is to use an analog quantum computing device, where an initial quantum state is evolved into the ground state of the problem Hamiltonian via an adiabatic evolution8,9.Recently, arrays of neutral atoms trapped in optical tweezers have emerged as a promising processor platform for analog quantum computing10,11,12, besides the already available quantum annealing processor7,13. These analog quantum computers make use of hundreds of atoms, where each atom serves as a qubit and harness the strongly interacting atomic Rydberg state to generate multipartite entanglement14. Moreover, the atomic arrays can be configured so that the quantum many-body ground state natively encodes the solution of optimization problems such as the maximum independent set (MIS) via adiabatic evolution6,15,16. This approach can be used to tackle industrially relevant optimization problems3. A recent proposal demonstrates how to solve non-native combinatorial optimization problems on this processor17. However, in this finite-time adiabatic evolution, non-adiabatic errors are not avoidable due to the limited coherence time of the processor. The errors result in reduced computation fidelity. One way to address this challenge is by finding optimal scheduling functions to describe the adiabatic evolution18, although this can be resource-demanding and would require multiple iterations on the processor. An alternative way to circumvent the non-adiabatic excitations is by using counterdiabatic (CD) protocol as introduced in19,20,21,22. The main idea behind CD protocols is to introduce an additional term to the fast-evolving adiabatic Hamiltonian to suppress the transition between eigenstates. However, the application of these initial proposals for CD protocols suffered from the difficulty in calculating the exact CD terms for large systems. Moreover, the required knowledge of instantaneous eigenstates to obtain the CD terms hindered its applications in adiabatic quantum computing (AQC). There have been several attempts to overcome this challenge23. Notably, a proposal for a variational CD protocol24,25 represents significant progress in this direction. This approach offers a method to construct approximate CD terms variationally, without requiring knowledge of the Hamiltonian spectra. In this regard, several theoretical advancements have been made to improve this protocol26,27, alongside experimental realizations on both digital and analog quantum processors28,29,30,31,32,33. Additionally, digital-analog methods have recently been proposed34, where sequential applications of digital and analog operations lead to targetting a wide range of Hamiltonians and use cases in a short depth circuit.In this work, we introduce a tailored method to enhance the performance of current analog quantum processors by applying analog counterdiabatic quantum computing (ACQC) techniques, specifically designed for direct implementation on the neutral atom quantum computing platform. Our method focuses on minimizing non-adiabatic errors through the introduction of CD terms, realized through the use of analytically calculated scheduling functions that control the amplitude, detuning, and phase of the driving laser used in neutral atom quantum computing experiments. This approach significantly improves the fidelity of the computation in comparison to standard adiabatic protocols. Recognizing the limitations of current processor i.e., short coherence time, noise, and the lack of flexibility in the control variables, we tailor the CD protocols to accommodate these constraints. To demonstrate the effectiveness of our proposed CD protocols, we tackle an industrially relevant combinatorial optimization problem, the maximum independent set (MIS) problem featuring up to 100 nodes across several instances and benchmark our results against conventional finite-time adiabatic quantum optimization protocols executed on actual processor. The MIS problem maps to use cases such as network resilience, where identifying the largest set of non-adjacent nodes helps in designing robust communication networks. It also has applications in resource allocation, scheduling, and computational biology, where finding optimal, non-conflicting subsets is critical. This makes the applicability of the method relevant for industry. Additionally, we discuss the implementation of more advanced CD protocols on next-generation programmable neutral atom quantum processor, equipped with individual addressing capabilities.In contrast to gate-based (digital) quantum processors, where CD protocols are typically implemented through sequences of discrete gates, our approach leverages the fully analog dynamics of neutral atom platforms. In a digital processor, CD terms can be engineered through gate sequences that approximate a wide variety of Hamiltonians, which provides flexibility and universality but typically at the cost of deeper circuits and reduced solution quality on current noisy devices28,30. By contrast, in an analog setup, such as neutral atom platforms, the control is continuous. This allows one to target specific classes of problems with higher-quality solutions within short coherence times, though with less generality than digital schemes. In this sense, the two approaches are complementary: digital implementations offer flexibility, while analog implementations, such as ACQC, provide efficient and high-fidelity performance for tailored problem classes.ResultsWe demonstrate how to enhance the capability of a neutral atom processor with ground-Rydberg qubits35,36 by deploying the ACQC protocol to solve the MIS problem. For this purpose, we calculated the counterdiabatic potential analytically, taking into account the processor’s controllability, which includes one-body Pauli terms. This tailored approach compensates for the non-adiabatic transitions of the driving part of the ground-Rydberg qubit system. We then show a way to directly implement the CD protocol on the neutral atom processor through well-designed scheduling functions, including the CD coefficients.Hardware implementation of ACQCThe ground-Rydberg qubits are described by the following Hamiltonian$$\frac{{H}_{\mathrm{Ryd}}(t)}{\hslash }=\mathop{\underbrace{\frac{\Omega (t)}{2}\left[\cos \varphi (t)\mathop{\sum }\limits_{i=1}^{N}{\sigma }_{i}^{x}-\sin \varphi (t)\mathop{\sum }\limits_{i=1}^{N}{\sigma }_{i}^{y}\right]-\Delta (t)\mathop{\sum }\limits_{i=1}^{N}{\sum }_{i=1}^{N}{n}_{i}}}\limits_{{H}_{\mathrm{drive}}}+\mathop{\underbrace{\mathop{\sum }\limits_{i B to ensure that the penalty for violating the independent set condition is higher than the reward for including additional vertices. The objective is to find a configuration of xv that minimizes H(x). This minimization problem with qudratic terms can be mapped to finding the ground state of an Ising Hamiltonian. This can be tackled using AQC methods.Adiabatic quantum computing is a well-known approach for solving combinatorial optimization problems, especially when using analog quantum computing processor. In this method, one begins by selecting an initial Hamiltonian Hi, whose ground state is both known and easy to prepare. The system is then adiabatically evolved towards the problem Hamiltonian Hp by slowly changing the driving terms as defined by a time-dependent Hamiltonian H(t) = f(t)Hi + g(t)Hp. For sufficiently slow evolution, the adiabatic theorem ensures that the system remains close to its ground state throughout the process. In this case, the wave function of the system follows the instantaneous eigenstates of the Hamiltonian while the optimization solutions are encoded to be the ground state of the final Hamiltonian, which is the target state. However, the adiabatic evolution requires long computation times, which are limited by the experiment, for example, the coherence time of the neutral atom system. A nonadiabatic evolution or the noise of the system can lead to excitations in the energy spectrum and can reduce the target-state fidelity, in other words, the success probability. Therefore, we propose analog counterdiabatic quantum computing method.The ground-Rydberg HamiltonianConsider the Hamiltonian of neutral atom platform using ground-Rydberg qubits in Eq. (1). Since the initial state of this processor is \({\left|0\right\rangle }^{\otimes N}\), the initial conditions read Ω(0) = 0 and Δ(0) is negative. At the final evolution time t = T, the Rabi frequency is back to zero and the ground state of the final Hamiltonian HRyd(T) encodes the solution of the combinatorial optimization problem, for example minimizing the cost function in Eq. (10). Combining the initial and target constraints, one obtains the boundary conditions of the control functions as follows$$\Omega (0)=0,\,\Delta (0)=-{\Delta }_{1},$$ (11) $$\Omega (T)=0,\,\Delta (T)={\Delta }_{2},$$ (12) where − Δ1 and Δ2 are the minimal (negative) and maximal (positive) experimental limitations of the detuning. Analytically, the minimal eigenvalue of the initial Hamiltonian is 0, and the second minimal eigenvalue is Δ1 > 0. Therefore, choosing a larger value of Δ1 can enlarge the gap between the initial ground state and the initial first excited state. For the final constraint, Δ2 can influence the minimal eigenvalue and the energy gap between the ground state and the first excited state at the final time. Depending on the specific graph and its structure, Δ2 should be set based on the number of vertices and edges to satisfy the condition that the ground state encodes the solution of the MIS problems.Note that no boundary condition applies to φ since its effect vanishes at initial and final evolution times due to Ω. In all standard protocols, the choice is to choose a constant phase: φ = 0.ACQC protocolHereafter, we show a way to design the scheduling functions Ω(t), Δ(t), and φ(t) to not only fulfill the boundary conditions, but also improve the success probability for a shorter computational time.The idea of counterdiabaticity20,21 is to add an auxiliary Hamiltonian HCD to the adiabatic Hamiltonian. This helps to guide the system more reliably to the desired state by preventing non-adiabatic transitions. Therefore, the Hamiltonian becomes$${H}_{{\rm{tot}}}(t)={H}_{{\rm{ad}}}(t)+{H}_{{\rm{CD}}}(t).$$ (13) There are different ways to add CD terms. Considering a current ground-Rydberg quantum computing platform, the Ising mode interactions exist when two neighboring atoms i and j are in the Rydberg states, ni and nj terms in Eq. (1). In the case of many-body systems, the exact adiabatic gauge potential of the dynamic system cannot be found or the energy spectrum is too expensive to calculate, obviously a nested commutator CD terms protocol24 could be a possible solution which is a variational method to search for an approximation of adiabatic gauge potential. However, it requires additional many-body terms, which are currently not available to be added to analog neutral atom quantum computing processor.To find a way around, we develop an ACQC method which does not require additional many-body interaction terms added to the quantum computing system. This can be directly implemented on the current neutral atom quantum processors without optimization or post-processing on processor.To ensure that the system follows the desired adiabatic path and reaches the ground state of a Hamiltonian Had(t) at the final time t = T, the constraint for HCD(t) to be the solution of the adiabatic gauge potential24 of Had(t) is$$\left[{\partial }_{t}{H}_{{\rm{ad}}}-i[{H}_{{\rm{ad}}},{H}_{{\rm{CD}}}],{H}_{{\rm{ad}}}\right]=0.$$ (14) To avoid introducing extra many-body terms beyond \({\sigma }_{i}^{z}{\sigma }_{j}^{z}\) terms of the Rydberg system, an efficient solution is to search for the counterdiabaticity of the independent spins under the control field where the Hamiltonian is the driving part of Rydberg Hamiltonian in Eq. (1), Had(t) = Hdrive(t). Then, the adiabatic gauge potential of Had(t) in the limit of zero interactions can be easily solved by choosing the following CD ansatz$${\left.{H}_{{\rm{CD}}}(t)\right|}_{J\to 0}={f}_{x}(t){\sum }_{i}{\sigma }_{i}^{x}+{f}_{y}(t){\sum }_{i}{\sigma }_{i}^{y}+{f}_{z}(t){\sum }_{i}{n}_{i},$$ (15) where the general solution of the CD coefficients fx,y,z in Eq. (15) can be analytically calculated directly through Eq. (14) as$${f}_{x}(t)=-\frac{\Omega \mathop{\Delta }\limits^{^\circ }-\Delta \mathop{\Omega }\limits^{^\circ }}{2({\Omega }^{2}+{\Delta }^{2})}\sin \varphi +\frac{\Omega \Delta \mathop{\varphi }\limits^{^\circ }}{2({\Omega }^{2}+{\Delta }^{2})}\cos \varphi ,$$ (16) $${f}_{y}(t)=-\frac{\Omega \mathop{\Delta }\limits^{^\circ }-\Delta \mathop{\Omega }\limits^{^\circ }}{2({\Omega }^{2}+{\Delta }^{2})}\cos \varphi -\frac{\Omega \Delta \mathop{\varphi }\limits^{^\circ }}{2({\Omega }^{2}+{\Delta }^{2})}\sin \varphi ,$$ (17) $${f}_{z}(t)=\frac{{\Omega }^{2}\mathop{\varphi }\limits^{^\circ }}{{\Omega }^{2}+{\Delta }^{2}}.$$ (18) Finally, the total Hamiltonian with CD terms in Eq. (13) should be implemented through the Rydberg Hamiltonian \({\widetilde{H}}_{{\rm{Ryd}}}(t)={H}_{{\rm{tot}}}(t)\) with the updated scheduling functions as follows:$${\widetilde{H}}_{{\rm{Ryd}}}(t)=\frac{\widetilde{\Omega }(t)}{2}\left[\cos \widetilde{\varphi }(t){\sum }_{i}{\sigma }_{i}^{x}-\sin \widetilde{\varphi }(t){\sum }_{i}{\sigma }_{i}^{y}\right]-\widetilde{\Delta }(t){\sum }_{i}{n}_{i}+{\sum }_{i < j}{J}_{i,j}{n}_{i}{n}_{j}.$$ (19) Therefore, the counterdiabatic scheduling functions are calculated as$$\widetilde{\Omega }(t)=\sqrt{{g}_{1}^{2}+{g}_{2}^{2}},$$ (20) $$\widetilde{\Delta }(t)=\Delta -\frac{{\Omega }^{2}\mathop{\varphi }\limits^{^\circ }}{{\Omega }^{2}+{\Delta }^{2}},$$ (21) $$\widetilde{\varphi }(t)=\varphi -\phi ,$$ (22) with$${g}_{1}(t)=\Omega \left(1+\frac{\Delta \mathop{\varphi }\limits^{^\circ }}{{\Omega }^{2}+{\Delta }^{2}}\right),$$ (23) $${g}_{2}(t)=-\frac{\Omega \mathop{\Delta }\limits^{^\circ }-\Delta \mathop{\Omega }\limits^{^\circ }}{{\Omega }^{2}+{\Delta }^{2}},$$ (24) $$\phi (t)={\rm{atan2}}({g}_{2},{g}_{1}),$$ (25) where atan2(y, x) returns the four-quadrant inverse tangent of y and x. Obviously, the scheduling functions Ω(t), Δ(t), and φ(t) are free to be chosen with respect to the boundary conditions in Eqs. (11 and 12) and the experimental limitations. Once the scheduling functions are set, the ACQC control protocol in Eqs. (20–25) are designed and implemented on commercial neutral atom processor.