Digital adiabatic evolution is universally accurate

TL;DR

The paper demonstrates universal robustness of digital adiabatic evolution, showing that discretized adiabatic simulations using either Trotterization or GQSP do not inherently accumulate errors with the total evolution time. It develops a general error-cancellation framework and provides concrete bounds: for $k$-th order Trotterization, $\\mathcal{I}(1)=O(\\beta_{\\text{ad}}^2T^{-2}+\\beta_{\\text{sim}}^2\\delta t^{2k})$, and for GQSP, $\\mathcal{I}(1)=O(\\beta_{\\text{ad}}^2T^{-2})$, with additional exponential improvements achievable via a $Q$-th order path. Numerical simulations on molecular ground-state preparation and linear-system solvers validate the theory, showing dramatic tightening of Trotter bounds and favorable scaling for GQSP. The results offer a path to accurate, efficient digital adiabatic algorithms on fault-tolerant and potentially near-term quantum devices, with broad implications for quantum chemistry and linear-algebra tasks and a framework for optimizing algorithmic parameters under circuit-depth constraints.

Abstract

Adiabatic evolution is a central paradigm in quantum physics. Digital simulations of adiabatic processes are generally viewed as costly, since algorithmic errors typically accumulate over the long evolution time, requiring exceptionally deep circuits to maintain accuracy. This work demonstrates that digital adiabatic evolution is intrinsically accurate and robust to simulation errors. We analyze two Hamiltonian simulation methods -- Trotterization and generalized quantum signal processing -- and prove that the simulation error does not increase with time. Numerical simulations of molecular systems and linear equations confirm the theory, revealing that digital adiabatic evolution is substantially more efficient than previously assumed. Remarkably, our estimation for the first-order Trotterization error can be 10^6 times tighter than previous analyses for the transverse field Ising model even with less than 6 qubits. The findings establish fundamental robustness of digital adiabatic evolution and provide a basis for accurate, efficient implementations on fault-tolerant -- and potentially near-term -- quantum platforms.

Figures (3)

• Figure 1: Schematic of the error-cancellation mechanism and relevant applications. (a) Diagram of error cancellation in digital adiabatic evolution. The transition amplitude is bounded by a smooth envelope (black dash-dotted line) and oscillates frequently. As $T \to +\infty$, the integral of the transition amplitude cancels out within each period (grey line), leaving a residual part (orange line) independent of $T$. (b) Adiabatic time evolution from time $t$ to $t' = t + \delta t$. Errors from Hamiltonian simulation (pink dashed arrow) and non-adiabaticity (pink dotted arrow) excite the ground state $|\phi_0(t/T)\rangle$ to higher energy states $|\phi_i(t'/T)\rangle$. Here $|\psi(t/T)\rangle$ is the state at time $t$, and $|\phi_i(t/T)\rangle$ is the $i$-th instantaneous eigenstate of the Hamiltonian $H(t/T)$ with eigenvalue $E_i(t/T)$. $\widetilde{E}_i(t/T)$ is the effective energy of the $i$-th eigenstate under the Hamiltonian simulation algorithm. (c) Applications of our theory to adiabatic state preparation and classical problems.
• Figure 2: Comparison of different infidelities, query complexity, and auxiliary qubits scaling. Here, first-order Trotterization is studied. (a) Infidelity of digital adiabatic evolution with primary Trotterization (yellow), sub-Trotterization (red), and Trotter error for a general time-dependent Hamiltonian evolution, with $T=50$ (solid line), $T=200$ (dashed line), and $T=500$ (dotted line). (b) The improvement factor $\eta$, defined as the ratio of the former to our proposed Trotterization error bound, as a function of $g_{\text{min}}^{-1}$ for systems with $n=3$ to $6$ qubits. Here we set $H_i = -\sum_{i=1}^n X_i$ and $H_f = \sum_{i=1}^{n-1} (-2Z_iZ_{i+1} + g\cdot Z_n+X_i)$. By varying $g$ from $0.01$ to $0.1$ we can change the minimum gap $g_{\text{min}}$ within the range [0.02, 0.31]. $T$ and $\delta t$ are chosen as the critical values required to ensure the adiabatic infidelity $\mathcal{I}_{\text{ad}}$ and simulation infidelity $\mathcal{I}_{\text{sim}}$ (seriously defined in Eq. (\ref{['eq:infi']})) are both below the threhold of $0.001$. Here, we consider first-order Trotterization.(c) Comparison of query complexity scaling with respect to $\epsilon$ between previous works and our work. Here we illustrate the performance of Trotterization (previous work), QDrift and Trotterization (this work). (d) Comparison of auxiliary qubits scaling with respect to $\epsilon$ between previous works and our work. Here we illustrate the performance of truncated Dyson series, Qubitization and GQSP (This work). Here we use the relationship $T=O(\epsilon^{-1/2})$ to plot the curves.
• Figure 3: Numerical results for the $N_2$ molecule and systems of linear equations. (a) Infidelity of adiabatic state preparation (ASP) for the $N_2$ system with first-order sub-Trotterization at a fixed $\delta t=0.5$. The plot shows total infidelity $\mathcal{I}$, infidelity from non-adiabaticity $\mathcal{I}_{\text{ad}}$, and infidelity from Hamiltonian simulation $\mathcal{I}_{\text{sim}}$. (b) Infidelity of ASP for the $N_2$ system with primary Trotterization or first-order sub-Trotterization at a fixed $T=100$. (c) Parameter optimization in ASP for the $N_2$ system with first-order sub-Trotterization. The dashed curve represents a fit of the data using the model $\mathcal{I}=p(T/r)^2+bT^{-2}$. The minimum points $T_c$ of each fitting curve are shown with green crosses. The purple solid line is the linear fit of $T_c$ corresponding to $\sqrt{r}$. (d) Infidelity of ASP for the $N_2$ system at a fixed $\delta t=0.2$. Here we choose $K=\lfloor 4\log T \rfloor$. (e) Infidelity of adiabatic quantum computation (AQC) using GQSP for a linear equation system with orders $K=2$, $3$, $4$ and $5$ at a fixed $T=1000$. (f) Asymptotic behavior of AQC infidelity $\mathcal{I}$ using GQSP for a linear equation system at a fixed $\delta t=0.2$. Here we choose $K= 2\lfloor \log T \rfloor$. The scheduling function $u(x)$ is chosen as zeroth, first, second, and $\infty$ order paths, respectively. A scheduling function $u(x)$ is called a $Q$-th order path if $u^{(q)}(0)=u^{(q)}(1)=0$ for every $1\leq q\leq Q$. If $u^{(1)}(0)\neq 0$ or $u^{(1)}(1)\neq0$, it is a zeroth order path. An $\infty$ order path satisfies $\lim_{T\to+\infty} u^{(q)}(0),u^{(q)}(1)=0$ for every $q\in\mathbb{N}^*$.

Theorems & Definitions (20)

• Lemma A.1
• proof
• Lemma A.2: Theory of Trotter Error with Commutator Scaling childs2021theory
• Lemma A.3
• proof
• Lemma A.4: GQSP for Hamiltonian simulation motlagh2024generalized
• proof
• Lemma A.5
• proof
• Theorem A.6: Theorem 1 in the main text