Continuous-time Quantum Annealing (QA) is a strategy for preparing the ground state of nontrivial many-body systems. In its standard form, the dynamics is generated by a time-dependent interpolation between a simple driving Hamiltonian and the target problem Hamiltonian, usually implemented through a linear schedule. This approach faces the crucial bottleneck of small spectral gaps, which may require exponentially long annealing times to ensure adiabaticity. Here, we show how to implement quantum control over the annealing schedule in a frustrated Ising ring, one of the simplest models exhibiting an exponentially small bottleneck gap. By optimizing smooth continuous-time annealing schedules with a dressed-CRAB approach, and using a digitized representation of the dynamics to efficiently evaluate gradients, we construct protocols that strongly outperform standard fixed schedules. The optimized dynamics bypasses the bottleneck through a strongly nonadiabatic mechanism, leading to efficient ground-state preparation despite the exponentially small minimum gap. In particular, the annealing time required to reach a fixed residual-energy threshold is found to grow linearly with system size rather than exponentially. We further examine a lowest-order variational counter-diabatic correction and find that, once schedule optimization is allowed, it does not lead to any improvement.
Continuous-time Quantum Annealing (QA) is a strategy for preparing the ground state of nontrivial many-body systems. In its standard form, the dynamics is generated by a time-dependent interpolation between a simple driving Hamiltonian and the target problem Hamiltonian, usually implemented through a linear schedule. This approach faces the crucial bottleneck of small spectral gaps, which may require exponentially long annealing times to ensure adiabaticity. Here, we show how to implement quantum control over the annealing schedule in a frustrated Ising ring, one of the simplest models exhibiting an exponentially small bottleneck gap. By optimizing smooth continuous-time annealing schedules with a dressed-CRAB approach, and using a digitized representation of the dynamics to efficiently evaluate gradients, we construct protocols that strongly outperform standard fixed schedules. The optimized dynamics bypasses the bottleneck through a strongly nonadiabatic mechanism, leading to efficient ground-state preparation despite the exponentially small minimum gap. In particular, the annealing time required to reach a fixed residual-energy threshold is found to grow linearly with system size rather than exponentially. We further examine a lowest-order variational counter-diabatic correction and find that, once schedule optimization is allowed, it does not lead to any improvement.
Abstract Exponentially small spectral gaps are known to be the crucial bottleneck for traditional Quantum Annealing (QA) based on interpolating between two Hamiltonians, a simple driving term and the complex problem to be solved, with a linear schedule in time. One of the simplest models exhibiting exponentially small spectral gaps is a ferromagnetic Ising ring with a single antiferromagnetic bond introducing frustration. Previous studies of this model have explored continuous-time diabatic QA, where optimized non-adiabatic annealing schedules provided good solutions, avoiding exponentially large annealing times. In our work, we move to a digital framework of Variational Quantum Algorithms, and present two main results: (1) we show that the model is digitally controllable with a scaling of resources that grows quadratically with the system size, achieving the exact solution using the Quantum Approximate Optimization Algorithm; (2) We combine a technique of quantum control—the Chopped RAndom Basis method—and digitized quantum annealing to construct smooth digital schedules yielding optimal solutions with very high accuracy.
We investigate the role of local counter-diabatic (CD) terms in enhancing the performance of discrete-time digital protocols for a frustrated Ising ring, a system with an exponentially small spectral gap that acts as a bottleneck for conventional quantum annealing. The techniques investigated range from a digitised version of a fixed-schedule protocol, including lowest-order analytical CD terms, all the way to a full-fledged Quantum Approximate Optimisation Algorithm (QAOA), including variational CD terms (DC-QAOA). By analyzing the resulting residual energy, we show that DC-QAOA combined with Chopped RAndom Basis (CRAB) quantum control techniques for optimizing the circuit parameters, can outperform all the other strategies. By monitoring the ground-state population during the dynamics we learn that DC-QAOA finds effective shortcuts towards the target state through intermediate excited states, a process that is unexpectedly enhanced by the inclusion of local CD unitaries. Our results highlight the importance of flexible variational control of CD dynamics and demonstrate that digital optimization can explore operator dynamics that remain inaccessible to standard analytical CD constructions.
Abstract Quantum annealing (QA) is a continuous-time heuristic quantum algorithm for solving or approximately solving classical optimization problems. The algorithm uses a schedule to interpolate between a driver Hamiltonian with an easy-to-prepare ground state and a problem Hamiltonian whose ground state encodes solutions to an optimization problem. The standard implementation relies on the evolution being adiabatic: keeping the system in the instantaneous ground state with high probability and requiring a time scale inversely related to the minimum energy gap between the instantaneous ground and excited states. However, adiabatic evolution can lead to evolution times that scale exponentially with the system size, even for computationally simple problems. Here, we study whether non-adiabatic evolutions with optimized annealing schedules can bypass this exponential slowdown for one such class of problems called the frustrated ring model. For sufficiently optimized annealing schedules and system sizes of up to 39 qubits, we provide numerical evidence that we can avoid the exponential slowdown. Our work highlights the potential of highly-controllable QA to circumvent bottlenecks associated with the standard implementation of QA.
The frustrated Ising ring is one of the simplest models exhibiting exponential closing spectral gaps, making it a paradigmatic and challenging benchmark for quantum annealing (QA). Ground-state preparation for this model has therefore been studied extensively in both continuous-time QA and digitized protocols such as the Quantum Approximate Optimization Algorithm (QAOA). Here, we use the frustrated Ising ring to investigate how the parity mapping affects the performance of both QA and QAOA. For QA, finite-size calculations show that the parity mapping increases the minimum spectral gap under the energy normalization used in this work, thereby enabling faster continuous-time ground state preparation protocols. An ideal implementation of Parity-QA, with a single global constraint, shows no evidence of exponential gap closing over the accessible system sizes, whereas a hardware-motivated decomposition into local constraints restores the exponential decrease, albeit with a smaller fitted exponent than conventional QA. For the digitized protocol, we find that the number of Parity-QAOA layers required to prepare the exact ground state remains constant over the simulated sizes, improving upon the quadratic scaling required by conventional QAOA. To investigate the role of constraints in Parity-QAOA, we further consider a modified Ising ring instance in which the constraint term is essential for preparing the target ground state. We then compare the corresponding resource requirements with those of conventional QAOA.
Recent advances in quantum technologies have intensified the search for algo rithms that can use quantum resources effectively. Quantum annealing and hybrid quantum–classical variational algorithms are two prominent approaches in this set ting. Conventional quantum annealing follows a prescribed Hamiltonian interpo lation and may therefore be limited by small spectral gaps along that path. This Thesis investigates how algorithmic performance changes when three elements are varied: the annealing schedule (Chapter 2), the independent parameters of a digi tized evolution (Chapter 3), and the encoding of the problem (Chapter 4). For weighted MaxCut instances with anomalously small annealing gaps, opti mized smooth schedules outperform linear digitized annealing at equal circuit depth. Rather than restoring adiabaticity or enlarging the gap, the dynamics deliberately leaves the instantaneous ground state and later returns the population to the target state. Smoothness allows these digital solutions to retain their quality as continuous time controls and when transferred between related instances. The schedule restriction is then removed for a frustrated Ising ring whose anneal ing gap closes exponentially with system size. Optimizing all circuit angles inde pendently yields numerically exact ground-state preparation at the critical depth Pcr 1 =(N2−1)/4. This quadratic threshold is explained by the dimension and sym metries of the reachable free-fermion manifold, demonstrating that controllability, rather than the gap of a particular interpolation, determines the required number of digital controls. Finally, the same ring family is reformulated through the parity mapping. For the original frustrated ring, Parity-QAOA reaches the exact ground state using only single-qubit rotations, without applying the constraint unitary. The number of layers equals the number of distinct coupling values and does not grow with system size. To clarify the role of constraints, an equal-magnitude signed ring is also studied, for which the constraint unitary is essential for preparing the target ground state. Together, these results identify the shape, controllability, and encoding of a quan tum protocol as resources for overcoming limitations imposed by more restricted routes to ground-state preparation.
Adiabatic control is a fundamental technique for manipulating quantum systems, guided by the quantum adiabatic theorem, which ensures suppressed nonadiabatic transitions under slow parameter variations. Quantum annealing, a heuristic algorithm leveraging adiabatic control, seeks the ground states of Ising spin glass models and has drawn attention for addressing combinatorial optimization problems. However, exponentially small energy gaps in such models often necessitate impractically long runtime to satisfy the adiabatic condition. Despite this limitation, improving the quality of approximate solutions remains crucial for practical applications. The quantum adiabatic brachistochrone provides a method to enhance adiabaticity by minimizing an action representing nonadiabaticity via the variational principle. While effective, its implementation requires detailed energy spectra, complicating its use in quantum annealing. Shortcuts to adiabaticity by counterdiabatic driving offer alternative approaches for accelerating adiabatic processes. However, the theory of shortcuts to adiabaticity often faces challenges such as nonlocal control requirements, high computational cost, and trade-offs between speed and energy efficiency. In this work, we propose a novel quantum adiabatic brachistochrone protocol tailored for quantum annealing that eliminates the need for energy spectrum information. Our approach builds on advancements in counterdiabatic driving to design efficient parameter schedules. We demonstrate the effectiveness of our method through numerical simulations on the transverse-field Ising chain and axial next-nearest neighbor Ising models.