qDRIFT and SqDRIFT: Randomized Compilation

• qDRIFT and SqDRIFT are randomized compilation techniques that reduce circuit depth and gate count by stochastically sampling Hamiltonian terms.
• They employ power-series error analysis and Richardson extrapolation to achieve high-order accuracy and exponential improvements over traditional methods.
• SqDRIFT decouples gate cost from the number of Hamiltonian terms, making it effective for dense chemistry Hamiltonians and scalable on near-term quantum devices.

The qDRIFT Randomized Compilation protocol and its higher-order generalizations (“SqDRIFT”, including qFLO) represent a class of randomized circuit compilers for quantum simulation, delivering gate count and circuit depth reductions that are sharply favorable, especially for high-order accuracy and near-term hardware. These algorithms leverage stochastic sampling of Hamiltonian terms, power-series expansions of expectation values, and classical post-processing (notably, Richardson extrapolation), achieving asymptotically exponential improvements over traditional product formulas in simulation error scaling versus circuit depth. They stand out by decoupling gate cost from the number of Hamiltonian summands—a key advantage in chemistry and materials contexts—and are robust, ancilla-free, and compatible with high-noise devices (Watson, 6 Nov 2024).

1. Foundations: qDRIFT Randomized Compilation Protocol

The qDRIFT protocol targets unitary simulation $U=e^{-iHT}$ for time-independent Hamiltonians decomposed as $H = \sum_k h_k H_k$, where $\|H_k\|=1$ and $h_k>0$ (Campbell, 2018, Watson, 6 Nov 2024). The aggregate strength is $\lambda = \sum_k h_k$. Rather than deterministically implementing Trotter steps over all $L$ terms, qDRIFT executes $N$ rounds, each involving stochastic sampling:

• At each round, select index $j$ with probability $p_j = h_j / \lambda$.
• Apply unitary $V_j = \exp(-i\lambda H_j t)$, corresponding to time-step $t = T / N$.

This yields the channel $$\mathcal{E}(\rho) = \sum_j p_j V_j \rho V_j^\dagger = \rho - i t [H, \rho] + O(t^2)$$ per step, with error accumulating linearly over $N$ steps. The circuit depth required to achieve diamond-norm error $\epsilon$ is $N = O((\lambda T)^2 / \epsilon)$—crucially independent of $L$ (Chen et al., 2020).

2. Higher-Order SqDRIFT (qFLO): Series Expansion and Richardson Extrapolation

SqDRIFT (also called qFLO) builds upon qDRIFT’s power-series error structure by explicitly extrapolating to higher orders via Richardson techniques (Watson, 6 Nov 2024):

• For step-size $s=1/N$, the $N$-fold channel is $E^N = \exp(-iT(\mathrm{ad}_H + \Delta(s)))$ with $\Delta(s)=O(sT)$.
• Observable expectation $f(s) = \operatorname{Tr}[A E^N(\rho_0)]$ expands as $$f(s) = f(0) + \sum_{j=2}^\infty \alpha_j s^j + O(s^K)$$.

Richardson extrapolation evaluates $f(s/y_j)$ at $m$ tailored points and combines them $\hat{F}_m = \sum_j b_j f(s/y_j)$ to cancel all terms up to $s^{m-1}$, ensuring $\hat{F}_m=f(0)+O(s^m)$. Chebyshev node choices for $\{y_j\}$ and corresponding weights $\{b_j\}$ yield controlled norms $\|\mathbf{b}\|_1 = O(\log m)$, and at order $m\sim\log(1/\epsilon)$, circuit depth per run is reduced to $O((\lambda T)^2\log(1/\epsilon))$, exponentially better than the $O(1/\epsilon)$ depth for qDRIFT alone.

3. Error Analysis and Circuit Depth Scaling

The extrapolated observable error in SqDRIFT satisfies

$$|\hat{A} - \operatorname{Tr}[A e^{-iHT} \rho_0 e^{iHT}]| \leq \|A\| \|\mathbf{b}\|_1 \sum_{j\geq m} [s(8\lambda T)]^j \sum_{l=1}^K \frac{(8\lambda T)^l}{l!}$$

Selecting $s$ and $m$ so the geometric tail falls below $\epsilon$ provides the depth bound

$N_{max} = O((\lambda T)^2 \log(1/\epsilon))$

and total gate count

$$O\left( (\lambda T)^2 \log^2(1/\epsilon) / \epsilon^2 \right)$$

since $O(1/\epsilon^2)$ repeated runs per point are needed to estimate the sample mean to within $O(\epsilon/\|\mathbf{b}\|_1)$ (Watson, 6 Nov 2024).

4. Comparison to qDRIFT and Deterministic Product Formulas

Method	Circuit Depth (per run)	Total Gate Count	Scaling With Terms $L$
qDRIFT	$O((\lambda T)^2/\epsilon)$	$O((\lambda T)^2/\epsilon)$	Independent
SqDRIFT/qFLO (qFLO)	$O((\lambda T)^2 \log(1/\epsilon))$	$O((\lambda T)^2 \log^2(1/\epsilon)/\epsilon^2)$	Independent
Trotter/Suzuki	$O(L^x [...] / \epsilon^y)$	$O(L^x [...] / \epsilon^y)$	Explicit dependence

SqDRIFT trades repeated short-depth experiments plus classical post-processing (no ancillas or control gates beyond native $e^{-iH_k t}$ unitaries) for dramatically reduced coherent depth (Watson, 6 Nov 2024). This reduction is especially pronounced when simulating dense chemistry Hamiltonians with $L=O(n^4)$, where qDRIFT/SqDRIFT’s circuit cost remains solely a function of $\lambda$ and not $L$ (Campbell, 2018).

5. Extensions and Hybrid Randomized Compilers

Several generalizations fall within the SqDRIFT paradigm:

• Markov Chain Random Compilation: Extends qDRIFT to compile time-dependent $H(t)$ using continuous-time Markov chains to stochastically select Hamiltonian terms, allowing dwell times and jump rates to be tuned for optimal error scaling, with overall gate count $O(C^2T^2/\epsilon_0)$ (Dubus et al., 10 Nov 2024).
• Stochastic Hamiltonian Sparsification: Interpolates between pure qDRIFT and randomized Trotter by optimally sparsifying Hamiltonian terms; convex optimization of term inclusion probabilities quadratically suppresses simulation error for given gate budget (Ouyang et al., 2019).
• Importance Sampling and Composite Channels: Sampling from cost-optimized distributions $q_j$ (e.g., minimizing expected CNOT or $T$-count) further suppresses overall gate cost while maintaining rigorous error bounds; composite protocols partition the Hamiltonian into deterministic (Trotter) and stochastic (qDRIFT) blocks (Kiss et al., 2022).
• Adaptive Random Sampling: Fluctuation-guided compilers update term-sampling probabilities in response to real-time state fluctuations, yielding improved error scaling when $\sum_j \Delta H_j \ll \sum_j \|H_j\|$ (Wu et al., 12 Sep 2025).

6. Concentration Results and Cost Analysis

Rigorous martingale and concentration-inequality analyses establish that a single realization of the random product formula concentrates sharply around ideal evolution, with probability $1-\delta$ requiring only

$N = O((n+\log(1/\delta)) (t\lambda)^2/\epsilon^2)$

gates for diamond-norm error $\le \epsilon$ (Chen et al., 2020). For typical chemistry Hamiltonians, hundreds-to-thousands fold speedups are demonstrated compared to Trotter-Suzuki for relevant time regimes and precision targets, including phase estimation to chemical accuracy (Campbell, 2018). Cost-aware importance sampling further compresses gate counts near per-term hardware cost minima (Kiss et al., 2022).

7. Practical Considerations for Near-Term Quantum Devices

SqDRIFT methods require no ancilla qubits or advanced control operations. Circuit depth at each point is determined by $\lambda$, $T$, and $\epsilon$, not $L$, enabling application to electronic structure problems with massive Hamiltonian decompositions (Watson, 6 Nov 2024). Robustness to imperfect state preparation and measurement noise follows from the well-conditioned structure and incoherent error averaging inherent to randomized compilation. Classical post-processing, including Richardson extrapolation, is computationally light; many short-depth circuits may be executed in parallel to exploit device throughput (Kiss et al., 2022, Watson, 6 Nov 2024). These features collectively render SqDRIFT highly suitable for early fault-tolerant and NISQ quantum hardware, particularly in contexts demanding scalable, chemistry-relevant quantum simulation.