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arXiv:2406.16385 · Physical Review A

Validity condition for high-fidelity digitized quantum annealing

Alan C. Santos

Published Feb 18, 2025• 1 citation
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Abstract

Digitizing an adiabatic evolution is a strategy able to combine the good performance of gate-based quantum processors with the advantages of adiabatic algorithms, thus providing a hybrid model for efficient quantum information processing. In this work we develop validity conditions for high-fidelity digital adiabatic tasks. To this end, we assume a digitizing process based on the Suzuki-Trotter decomposition, which allows us to introduce a digitized adiabatic theorem. As a consequence of this theorem, we show that the performance of such a hybrid model is limited by the fundamental constraints on the adiabatic theorem validity, even in ideal quantum processors. We argue how our approach predicts the existence of intrinsic nonadiabatic errors reported by Barends et al. [Nature (London) 534, 222 (2016)] through an empirical study of digital annealing. In addition, our approach allows us to explain the existence of a scaling of the number of Suzuki-Trotter blocks for the optimal digital circuit with respect to the optimal adiabatic total evolution time, as reported by Mbeng et al. [Phys. Rev. B 100, 224201 (2019)] through robust numerical analysis of digital annealing. We illustrate our results through two examples of digitized adiabatic algorithms, namely, the two-qubit exact-cover problem and the three-qubit adiabatic factorization of the number 21.

DOI
10.1103/physreva.111.022618

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Digitizing an adiabatic evolution is a strategy able to combine the good performance of gate-based quantum processors with the advantages of adiabatic algorithms, thus providing a hybrid model for efficient quantum information processing. In this work we develop validity conditions for high-fidelity digital adiabatic tasks. To this end, we assume a digitizing process based on the Suzuki-Trotter decomposition, which allows us to introduce a digitized adiabatic theorem. As a consequence of this theorem, we show that the performance of such a hybrid model is limited by the fundamental constraints on the adiabatic theorem validity, even in ideal quantum processors. We argue how our approach predicts the existence of intrinsic nonadiabatic errors reported by Barends et al. [Nature (London) 534, 222 (2016)] through an empirical study of digital annealing. In addition, our approach allows us to explain the existence of a scaling of the number of Suzuki-Trotter blocks for the optimal digital circuit with respect to the optimal adiabatic total evolution time, as reported by Mbeng et al. [Phys. Rev. B 100, 224201 (2019)] through robust numerical analysis of digital annealing. We illustrate our results through two examples of digitized adiabatic algorithms, namely, the two-qubit exact-cover problem and the three-qubit adiabatic factorization of the number 21.
The simulation of adiabatic evolution has deep connections with adiabatic quantum computation, the quantum approximate optimization algorithm, and adiabatic state preparation. Here we address the error analysis problem in quantum simulation of adiabatic process using Trotter formulas. We show that with additional conditions, the circuit depth can be linear in simulation time TTT. The improvement comes from the observation that the fidelity error here can't be estimated by the norm distance between evolution operators. This phenomenon is termed the robustness of discretization in digital adiabatic simulation. It can be explained in three steps, from analytical and numerical evidence: (1) The fidelity error should be estimated by applying adiabatic theorem on the effective Hamiltonian instead. (2) Because of the specialty of Riemann-Lebesgue lemma, most adiabatic process is naturally robust against discretization. (3) As the Trotter step gets larger, the spectral gap of effective Hamiltonian tends to close, which results in the failure of digital adiabatic simulation.
Digitized adiabatic quantum factorization is a hybrid algorithm that exploits the advantage of digitized quantum computers to implement efficient adiabatic algorithms for factorization through gate decompositions of analog evolutions. In this paper, we harness the flexibility of digitized computers to derive a digitized adiabatic algorithm able to reduce the gate-demanding costs of implementing factorization. To this end, we propose a new approach for adiabatic factorization by encoding the solution of the problem in the kernel subspace of the problem Hamiltonian, instead of using ground-state encoding considered in the standard adiabatic factorization proposed by Peng etetet alalal. [Phys. Rev. Lett. 101, 220405 (2008)]. Our encoding enables the design of adiabatic factorization algorithms belonging to the class of Quadratic Unconstrained Binary Optimization (QUBO) methods, instead the Polinomial Unconstrained Binary Optimization (PUBO) used by standard adiabatic factorization. We illustrate the performance of our QUBO algorithm by implementing the factorization of integers NNN up to 8 bits. The results demonstrate a substantial improvement over the PUBO formulation, both in terms of reduced circuit complexity and increased fidelity in identifying the correct solution.
The simulation of adiabatic evolution has deep connections with Adiabatic Quantum Computation, the Quantum Approximate Optimization Algorithm and adiabatic state preparation. Here we address the error analysis problem in quantum simulation of adiabatic process using Trotter formulas. We show that with additional conditions, the circuit depth can be linear in simulation time TTT. The improvement comes from the observation that the fidelity error here can't be estimated by the norm distance between evolution operators. This phenomenon is termed as the robustness of discretization in digital adiabatic simulation. It can be explained in three steps, from analytical and numerical evidence: 1. the fidelity error should be estimated by applying adiabatic theorem on the effective Hamiltonian instead. 2. because of the specialty of Riemann-Lebesgue lemma, most adiabatic process is naturally robust against discretization. 3. as the Trotter step gets larger, the spectral gap of effective Hamiltonian tends to close, which results in the failure of digital adiabatic simulation.
Trotter errors in digitized quantum dynamics arise from approximating time-ordered evolution under noncommuting Hamiltonian terms with a product formula. In the adiabatic regime, such errors are known to exhibit long-time self-healing [Phys. Rev. Lett. \textbf{131}, 060602 (2023)], where discretization effects are effectively suppressed. Here we show that self-healing persists at finite evolution times once nonadiabatic errors induced by finite-speed ramps are compensated. Using counterdiabatic driving to cancel diabatic transitions and isolate discretization effects, we study both noninteracting and interacting spin models and characterize the finite-time scaling with the Trotter steps and the total evolution time. In the instantaneous eigenbasis of the driven Hamiltonian, the leading digital error maps to an effective harmonic perturbation whose dominant Fourier component yields an analytic upper bound on the finite-time Trotter error and reveals the phase-cancellation mechanism underlying self-healing. Our results establish finite-time self-healing as a generic feature of digitized counterdiabatic protocols, clarify its mechanism beyond the long-time adiabatic limit, and provide practical guidance for high-fidelity state preparation on gate-based quantum processors.
Adiabatic time evolution can be used to prepare a complicated quantum many-body state from one that is easier to synthesize and Trotterization can be used to implement such an evolution digitally. The complex interplay between nonadiabaticity and digitization influences the infidelity of this process. We prove that the first-order Trotterization of a complete adiabatic evolution has a cumulative infidelity that scales as O(T^{-2}δt^{2}) instead of O(T^{2}δt^{2}) expected from general Trotter error bounds, where δt is the time step and T is the total time. This result suggests a self-healing mechanism and explains why, despite increasing T, infidelities for fixed-δt digitized evolutions still decrease for a wide variety of Hamiltonians. It also establishes a correspondence between the quantum approximate optimization algorithm and digitized quantum annealing.
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.