Constant Depth Digital-Analog Counterdiabatic Quantum Computing

TL;DR

This work introduces digital-analog counterdiabatic quantum computing (DACQC) to implement nested commutator CD terms at constant depth for a fixed truncation order, using commutator product formulas with augmented analog blocks and single-qubit rotations. By leveraging native multi-qubit interactions and dressing them with digital controls, higher-order NC CD Hamiltonians become feasible on near-term hardware with system-size independent depth. The authors demonstrate the approach on two-dimensional Ising spin glasses and the XXZ model, showing favorable error scaling and significant circuit-depth reductions that translate into improved ground-state preparation fidelity. The results imply a qualitatively new resource scaling for CD protocols, enabling faster and more scalable quantum state preparation with practical implications for quantum simulation and optimization on current devices, while outlining hardware challenges and the path toward robust implementations.

Abstract

We introduce a digital-analog quantum computing framework that enables counterdiabatic protocols to be implemented at constant circuit depth, allowing fast and resource-efficient quantum state preparation on current quantum hardware. Counterdiabatic protocols suppress diabatic excitations in finite-time adiabatic evolution, but their practical application is limited by the non-local structure of the required Hamiltonians and the resource overhead of fully digital implementations. Counterdiabatic terms can be expressed as truncated expansions of nested commutators of the adiabatic Hamiltonian and its parametric derivative. Here, we show how this algebraic structure can be efficiently realized in a digital-analog setting using commutator product formulas. Using native multi-qubit analog interactions augmented by local single-qubit rotations, this approach enables higher-order counterdiabatic protocols whose implementation requires a constant number of analog blocks for any fixed truncation order, independent of system size. We demonstrate the method for two-dimensional spin models and analyze the associated approximation errors. These results show that digital-analog quantum computing enables a qualitatively new resource scaling for counterdiabatic protocols and related quantum control primitives, with direct implications for quantum simulation, optimization, and algorithmic state preparation on current quantum devices.

Figures (9)

• Figure 1: Digital-analog counterdiabatic quantum computing (DACQC) circuit construction. A native multi-qubit interaction implementing the hardware connectivity graph $S$ is used to define an analog block (AB), corresponding to continuous-time evolution under the native Hamiltonian. By applying appropriate single-qubit rotations $R_{\alpha}(\theta)$, where $\alpha \in \{x, y, z\}$, before and after the interaction, one obtains augmented analog blocks (AABs) that implement the adiabatic Hamiltonian $H$ and its parametric derivative $\partial H$. Sequences of these AABs are then composed according to product-formula constructions to generate the commutators appearing in the CD Hamiltonian. Applying product formula decompositions recursively allows the implementation of higher-order NCs. For illustration, the figure shows the first-order ($l=1$) product formula for a single time step $m$. Repeating this construction for $M$ time steps realizes the full time evolution.
• Figure 2: The errors in the PF decompositions $U_{\text{GC}}$ of the unitary operators $U_{\text{exact}}^{(l)}(\delta t) = e^{\delta t C^{(l)}}$ as functions of the time step $\delta t$. The error is defined in terms of the 2-norm $\lVert. \lVert$. We fit the functions $f(\delta t) = b \, \delta t^{\nu}$ to the error, shown as solid and dashed lines, and extract the fitted exponents $\nu$. Here, we consider the two-dimensional Ising model of size $3\times 3$ with random coefficients and set $\lambda=0.5$ in Eq. \ref{['eq:spin-glass-time-dependent']}. The exponents $\nu$ are consistent with the upper-bound scaling (lower bounds on $\nu$) presented in Sec. \ref{['sec:dacqc_trotter']}: $\nu_1 = 3/2$ for Eqs. \ref{['eq:1nc-gc-def-integrated']} and \ref{['eq:2nc-product-formula']} and $\nu_2 = 3/4$ for Eq. \ref{['eq:nested_gc_decomposition']}.
• Figure 3: Target-state fidelity Eq. \ref{['eq:fidelity-def']} as a function of instantaneous time for the Ising spin-glass of Eq. \ref{['eq:spin-glass-NN-ham']} defined on a $3\times3$ square lattice. (a) We use Eq. \ref{['eq:1nc-gc-def-integrated']} for the first-order NC unitary ($l=1$) and (b) Eq. \ref{['eq:nested_gc_decomposition']} for the second-order one ($l=2$). The different colors indicate different numbers of Trotter steps $M$. The black reference line is obtained by direct exponentiation of the Hamiltonian with a small time step for which the line is converged. The couplings are random, $J_{a,b}\in (-J, J]$ and $h_{a}\in (-h, h]$, with $J=h=1$.
• Figure 4: Target-state fidelity Eq. \ref{['eq:fidelity-def']} for the spin-glass Hamiltonian Eq. \ref{['eq:spin-glass-NN-ham']} on a $3\times 3$ lattice. As a benchmark, we include the fidelity obtained by evolving the initial state using exact exponentiation in adiabatic quantum computing (AQC). We also show the corresponding DCQC-DCQC(Reference)-results using the same time step $\delta t$ as in the DACQC simulation. The time step is set to $\delta t = 0.02/J$ for DACQC and $\delta t = 0.08/J$ for DCQC. All the three results are obtained using a circuit sampler with $20{,}000$ shots.
• Figure 5: The errors in the PF decompositions $U_{\text{PF}}$ of the unitary operators $U_{\text{exact}}^{(l)}(\delta t) = e^{\delta t C^{(l)}}$ as functions of the time step $\delta t$, similar to Fig. \ref{['fig:error-scaling-ising']}. Here, we consider the two-dimensional XXZ model with $\Delta = 0.5$, system size $3\times 3$, and set $\lambda=0.5$ in Eq. \ref{['eq:time-dependent-xxz-hamiltonian']}. The fitted exponents $\nu$ are consistent with the upper-bound scaling of Eq. \ref{['eq:upper_bound_scaling']}.