In adiabatic quantum computing the aim is to track an eigenstate as the Hamiltonian changes. In the usual setup this is achieved using the natural time-dependent Hamiltonian evolution of the system and the main technical tool is the adiabatic theorem. We propose several alternative processes that achieve the same goal, but can easily be implemented on a gate-based quantum computer without the overhead of simulating time-dependent Hamiltonian evolution. We give a general framework for deriving `adiabatic' theorems for these processes.
As an application, we give various algorithms for solving the Quantum Linear Systems Problem (QLSP) with optimal scaling in the condition number. One of these algorithms was previously developed in [Cunningham, Roland 2024] and another can be seen as a randomised version of the discrete adiabatic algorithm of [Costa et al. 2022].
We also describe versions of Trotterisation in our framework, which allows several results from [An et al. 2025] to be reproduced in a randomised setting. In particular, bounds on the Trotter error in terms of the fidelity are obtained that are asymptotically better than the standard bounds.
In adiabatic quantum computing the aim is to track an eigenstate as the Hamiltonian changes. In the usual setup this is achieved using the natural time-dependent Hamiltonian evolution of the system and the main technical tool is the adiabatic theorem. We propose several alternative processes that achieve the same goal, but can easily be implemented on a gate-based quantum computer without the overhead of simulating time-dependent Hamiltonian evolution. We give a general framework for deriving `adiabatic' theorems for these processes.
As an application, we give various algorithms for solving the Quantum Linear Systems Problem (QLSP) with optimal scaling in the condition number. One of these algorithms was previously developed in [Cunningham, Roland 2024] and another can be seen as a randomised version of the discrete adiabatic algorithm of [Costa et al. 2022].
We also describe versions of Trotterisation in our framework, which allows several results from [An et al. 2025] to be reproduced in a randomised setting. In particular, bounds on the Trotter error in terms of the fidelity are obtained that are asymptotically better than the standard bounds.
Adiabatic Quantum Computing (AQC) is a powerful framework for designing quantum algorithms, yet its practical implementation is often hampered by the difficulty of engineering precise, continuous time-dependent Hamiltonians or the substantial computational overhead of time discretisation. This thesis develops alternative adiabatic frameworks based on dissipative and stochastic quantum evolution to circumvent these challenges.The core methodology introduces novel dynamics where discrete quantum operations, such as time-independent Hamiltonian evolution with Poisson-distributed phase randomisation or randomised unitary applications (qubitization), replace continuous evolution. This stochastic approach allows the system’s average behaviour to be rigorously analysed using new, generalised adiabatic theorems, including the development of novel quantitative bounds for unbounded Hamiltonians and specialised dissipative generators.Algorithmically, this work demonstrates that these discrete, stochastic procedures can achieve the optimal asymptotic time complexity of continuous AQC while mitigating discretisation costs. Specifically, adapted-schedule implementations are shown to yield optimal scaling for the Quantum Linear Systems Problem (QLSP), achieving complexity proportional to the condition number O(κ), and O(N/M) scaling for unstructured search. By establishing highly efficient, implementable alternatives to continuous AQC, this research offers a significant pathway toward robust and scalable quantum computation. The study also investigates the general family of diagonal Hamiltonians for optimisation problems, revealing that finding the optimal scheduling required for peak performance is, in general, an NP-hard problem.
Recently, several approaches to solving linear systems on a quantum computer have been formulated in terms of the quantum adiabatic theorem for a continuously varying Hamiltonian.Such approaches have enabled near-linear scaling in the condition number κ of the linear system, without requiring a complicated variable-time amplitude amplification procedure.However, the most efficient of those procedures is still asymptotically suboptimal by a factor of log(κ).Here, we prove a rigorous form of the adiabatic theorem that bounds the error in terms of the spectral gap for intrinsically discrete-time evolutions.In combination with the qubitized quantum walk, our discrete adiabatic theorem gives a speed-up for all adiabatic algorithms.Here, we use this combination to develop a quantum algorithm for solving linear systems that is asymptotically optimal, in the sense that the complexity is strictly linear in κ, matching a known lower bound on the complexity.Our O[κ log(1/ )] complexity is also optimal in terms of the combined scaling in κ and the precision .Compared to existing suboptimal methods, our algorithm is simpler and easier to implement.Moreover, we determine the constant factors in the algorithm, which would be suitable for determining the complexity in terms of gate counts for specific applications.
Adiabatic quantum computing is a general framework for preparing eigenstates of Hamiltonians on quantum devices. However, its digital implementation requires an efficient Hamiltonian simulation subroutine, which may introduce extra computational overhead or complicated quantum control logic. In this work, we show that the time-step sizes in time discretization can be much larger than expected, and the overall complexity is greatly reduced. Remarkably, regardless of the general convergence order of the numerical method, we can choose a uniform time-step size independent of tolerated error and evolution time for sufficiently accurate simulation. Furthermore, with the boundary cancellation condition where the continuous diabatic errors are exponentially suppressed, we provide strong evidence for a superpolynomial convergence of first-order Trotter with uniform time-step size. We apply our analysis to the example of adiabatic unstructured search and show several preferable features of the Trotterized adiabatic approach: It can match the Grover lower bound, it does not require knowledge of the number of marked states, and its performance can be asymptotically comparable with that of the quantum approximate optimization algorithm.
Different types and approaches for using quantum mechanical events for computation have developed quickly with the quantum technology revolution. One of these, adiabatic quantum computing, uses the adiabatic theorem as a paradigm for computation. According to the adiabatic theorem, if conditions gradually change and a system is permitted to adjust its configuration (changing the probability density), the associated eigenstate will be of the final Hamiltonian if the system starts in an eigenstate of its initial Hamiltonian. Simply put, a quantum mechanical system that is subjected to rapidly changing and varying conditions, there isn’t enough time for a function to adapt which leaves the spatial probability density unchanged. A closely related optimization process that utilizes the adiabatic theorem known as quantum annealing, finds a global minimum over a candidate set of solutions by processing quantum fluctuations. Quantum annealing is used primarily in combinatorial optimization problems where the search space is large and discrete, containing multiple local minima.
Solving linear systems of equations is a fundamental problem with a wide variety of applications across many fields of science, and there is increasing effort to develop quantum linear solver algorithms. Subaşı . [Phys. Rev. Lett. 122, 060504 (2019)] proposed a randomized algorithm inspired by adiabatic quantum computing, based on a sequence of random Hamiltonian simulation steps, with suboptimal scaling in the condition number κ of the linear system and the target error ϵ . Here we go beyond these results in several ways. Firstly, using filtering [Lin and Tong, Quantum 4, 361 (2020)] and Poissonization techniques [Cunningham and Roland, ArXiv:2406.03972 (2024)], the algorithm complexity is improved to the optimal scaling O ( κ log ( 1 / ϵ ) ) —an exponential improvement in ϵ , and a shaving of a log κ scaling factor in κ . Secondly, the algorithm is further modified to achieve constant factor improvements, which are vital as we progress towards hardware implementations on fault-tolerant devices. We introduce a cheaper randomized walk operator method replacing Hamiltonian simulation—which also removes the need for potentially challenging classical precomputations; randomized routines are sampled over optimized random variables; circuit constructions are improved. We obtain a closed formula rigorously upper bounding the expected number of times one needs to apply a block-encoding of the linear system matrix to output a quantum state encoding the solution to the linear system. The upper bound is 837 κ at ϵ = 10 − 10 for Hermitian matrices.
The adiabatic theorem is a powerful result concerning the evolution of quantum mechanical systems. Intuitively, it states the following: suppose we want to study the evolution of a quantum mechanical system governed by a given Hamiltonian H (Def. 6.1), and the initial state of the system is the eigenstate of H corresponding to its lowest eigenvalue. The laws of quantum physics tell us that the evolution of the system over time follows the Schrüdinger equation, Eq. (6.1). Suppose now that the Hamiltonian is not fixed, but rather, it changes very slowly over time, so that the evolution of the system is described by the differential equation idtd∣ψ(t)⟩=H(t)∣ψ(t)⟩, and H(t) is slowly changing. If, for any value of the time parameter t, there is always a gap between the lowest eigenvalue of H(t) and all other eigenvalues, then the adiabatic theorem asserts that the state of the system always remains in an eigenstate with the lowest eigenvalue throughout the evolution. One specific case of interest in this chapter occurs when the time-dependent Hamiltonian H(t) varies from an initial (time-independent) Hamiltonian HI to a final Hamiltonian HF. If the change happens slowly, and the initial state of the system is an eigenstate of HI with minimum eigenvalue, the adiabatic theorem asserts that the evolution eventually leads to the eigenstate with minimum eigenvalue of the final Hamiltonian HF. This result can be used for optimization: imagine that the initial Hamiltonian encodes an easy problem for which we can construct the eigenstate with minimum eigenvalue, and the final Hamiltonian encodes a difficult optimization problem whose solution is given by the eigenstate with minimum eigenvalue. By the adiabatic theorem, evolving the eigenstate of the initial Hamiltonian while slowly changing the Hamiltonian into the final one leads to the solution of the difficult optimization problem. Of course we need to know how slowly HI should be changed into HF. The adiabatic theorem is at the heart of the quantum approximate optimization algorithm (QAOA), a framework that has been widely used to implement optimization algorithms on existing quantum hardware, because it has low requirements of quantum resources–although it does not guarantee improvements over classical algorithms in general. In this chapter we first discuss the adiabatic theorem, and then give an overview of QAOA.