Tradeoffs between quantum and classical resources in linear combination of unitaries

Abstract: The linear combination of unitaries (LCU) algorithm is a building block of many quantum algorithms. However, because LCU generally requires an ancillary system and complex controlled unitary operators, it is not regarded as a hardware-efficient routine. Recently, a randomized LCU implementation with many applications to early FTQC algorithms has been proposed that computes the same expectation values as the original LCU algorithm using a shallower quantum circuit with a single ancilla qubit, at the cost of a quadratically larger sampling overhead. In this work, we propose a quantum algorithm intermediate between the original and randomized LCU that manages the tradeoff between sampling cost and the circuit size. Our algorithm divides the set of unitary operators into several groups and then randomly samples LCU circuits from these groups to evaluate the target expectation value. Notably, we analytically prove an underlying monotonicity: larger group sizes entail smaller sampling overhead, by introducing a quantity called the reduction factor, which determines the sampling overhead across all grouping strategies. Our hybrid algorithm not only enables substantial reductions in circuit depth and ancilla-qubit usage while nearly maintaining the sampling overhead of LCU-based non-Hermitian dynamics simulators, but also achieves intermediate scaling between virtual and coherent quantum linear system solvers. It further provides a virtual ground-state preparation scheme that requires only a resettable single-ancilla qubit and asymptotically shows advantages in both virtual and coherent LCU methods. Finally, by viewing quantum error detection as an LCU process, our approach clarifies when conventional and virtual detection should be applied selectively, thereby balancing sampling and hardware overhead.

• The paper introduces a hybrid LCU algorithm that reduces quantum circuit depth and ancilla qubits by partitioning unitaries.
• It characterizes the tradeoff using the reduction factor R, which interpolates smoothly between coherent and randomized LCU schemes.
• Applications in simulation, linear system solving, and error detection demonstrate significant resource savings with only modest increases in sampling overhead.

Tradeoffs between Quantum and Classical Resources in Linear Combination of Unitaries

Introduction

The linear combination of unitaries (LCU) is a foundational primitive in quantum algorithms for simulation, linear system solving, ground state preparation, and error detection. While conventional LCU protocols realize operator linear combinations using coherent control, they generally require significant hardware resources—multiple ancilla qubits and circuit depth scaling with the number of terms. In contrast, randomized or "virtual" LCU methods minimize quantum resources, but incur significant classical sampling overhead, with an unfavorable quadratic dependence on the inverse projection probability.

This work introduces a hybrid quantum algorithm that interpolates between these extremes. By partitioning the unitary set into groups and selectively applying coherent LCU on subsets while randomizing over groups, the scheme enables fine-grained negotiation between sampling complexity and quantum circuit resource requirements. The critical insight is the reduction factor $R$, which quantifies sampling overhead and encapsulates the trade-off governed by partition granularity. The algorithm analytically bridges the scaling regimes of fully-coherent and fully-randomized LCUs and is validated across representative algorithmic tasks.

LCU Methods: Coherent, Randomized, and Hybrid

The fully coherent LCU approach realizes the target Kraus operator $K = \sum_{i=1}^m c_i U_i$ by block-encoding and projective measurement. Success probability $\mathcal{P}$ depends on the operator norm structure, with sampling overhead scaling as $\mathcal{O}(1/\mathcal{P})$. This method requires $\lceil \log_2 m \rceil$ ancilla qubits and $\mathcal{O}(m\log m)$ gates excluding the controlled-$U_i$ gates.

Randomized or "virtual" LCU, in contrast, achieves hardware efficiency by stochastically sampling pairs $(i,j)$ according to $p_i p_j$ and classically reweighting measurement outcomes. The associated sampling cost is quadratically larger, scaling as $\mathcal{O}(1/\mathcal{P}^2)$, although only a single ancilla qubit and shallower circuits are needed.

The hybrid protocol partitions the unitaries into $G$ non-empty groupings $\{S_k\}$, each associated with weight $q_k = \sum_{i\in S_k}p_i$ and normalized operators $K_k = \sum_{i\in S_k} (p_i/q_k) U_i$. Within each group $S_k$, coherent LCU is performed, while inter-group combinations are handled randomly. This yields an effective CP map

Figure 1: Idea of our decomposition for LCU-based CP map $\tilde{\Lambda}_{p, \mathcal{U}}$; coherent LCU is applied within groups $K_k$, and the full operator is synthesized by random mixing.

and allows explicit control over the quantum/classical balance. The reduction factor $R$ is defined as

$$R[\{S_k\}; \rho] = \sum_{k=1}^G q_k \operatorname{tr}[K_k^\dagger K_k \rho]$$

and governs the estimation variance of expectation values.

Analysis of the Reduction Factor and Sampling Complexity

The reduction factor $R$ is monotonic in the refinement of partitions:

Figure 2: A series of partitions of $[m]$ illustrating the monotonicity of $R$ with respect to partition refinement.

• For the trivial partition $S_1 = [m]$ (fully coherent LCU), $R = \mathcal{P}$.
• For maximal fragmentation $S_k = \{k\}$ (fully randomized LCU), $R = 1$.

For arbitrary intermediate partitions, $R$ interpolates smoothly between these limits. Explicit bounds are established:

$\mathcal{P} \leq R[\{S_k\}; \rho] \leq 1$

and finer partitioning always increases $R$ (and, correspondingly, reduces quantum resource needs per circuit at the expense of more samples).

The sample complexity for estimating $$\operatorname{Tr}[OK \rho K^\dagger]/\operatorname{Tr}[K \rho K^\dagger]$$ within additive error $\epsilon$ is

$$N = \mathcal{O}\left( \frac{R}{\mathcal{P}^2} \cdot \frac{\|O\|^2 \log(1/\delta)}{\epsilon^2} \right)$$

where $O$ is the observable. Adjusting partition granularity enables controlled movement between the scaling of randomized and coherent LCU.

Figure 3: The dependence of $R - \mathcal{P}$ (sampling overhead penalty) on the number of coherent terms $M$ used in the hybrid scheme; increasing $M$ reduces $R$ close to $\mathcal{P}$, bridging the two regimes.

Applications

Non-Hermitian Dynamics Simulation

The hybrid algorithm is instantiated for non-Hermitian dynamics, targeting evolution operators of the form $\exp(-At)$ with $A$ non-Hermitian. LCU decomposition via numerical quadrature can accumulate terms with widely varying weights. Experimentally, partitioning terms such that "tail" elements utilize randomized LCU while the main body is handled coherently yields up to 32-fold reduction in depth and a savings of 4 ancilla qubits, with negligible degradation in sampling cost.

Quantum Linear System Solver

For algorithms solving $M |x\rangle = |b\rangle$, using the Fourier-based LCU decomposition of $M^{-1}$, the hybrid LCU strategy groups time-evolution elements and interpolates between the classical sample costs of randomized ($\mathcal{O}(1)$) and coherent ($\mathcal{O}(\log^{-1}(\kappa/\varepsilon)$)) LCU, while allowing the number of ancilla qubits to be tuned in $\mathcal{O}(\log K)$ increments.

Ground State Preparation

Hybridization is especially beneficial for ground-state property estimation, enabling ground-state filters (e.g., using repeated cosine transforms) with only a single resettable ancilla qubit, yet achieving sampling overhead and total-time complexity that matches coherent methods asymptotically:

$$\mathcal{O}\left( p_0^{-1} \Delta^{-1} \varepsilon^{-2} \log(1/\varepsilon) \right)$$

where $p_0$ is ground state overlap and $\Delta$ the spectral gap.

Quantum Error Detection

By viewing error detection as LCU over stabilizer groups, the reduction factor precisely characterizes the relative sample cost for hybrid strategies. For rare errors, randomization (virtual QED) incurs little extra sampling cost; for frequent errors, coherent QED is favorable. Simulations with the Steane code under biased noise corroborate this effect.

Figure 4: The projection probabilities for various bit-flip/phase-flip ratios and the reduction factor $R$ under the hybrid quantum error detection for the Steane code, showing favorable sampling complexity as noise becomes more biased.

Theoretical and Practical Implications

The explicit monotonic dependence of $R$ on resource allocation offers precise guidance for algorithmic deployment in early fault-tolerant quantum computing. Hybrid LCU enables significant reductions in ancilla usage and gate depth for only modest increases in the number of samples, especially in regimes where the success probability $\mathcal{P}$ is not excessively small.

From a theoretical standpoint, this establishes a controlled interpolation between virtual and coherent LCU frameworks, sets the stage for information-theoretic analysis of quantum/classical trade-offs (e.g., in terms of correlation or entropic measures), and suggests new routes for resource-efficient filter design in quantum simulation, potentially in combination with QSVT-based techniques.

Conclusion

The presented hybrid LCU algorithm offers a systematic framework for managing the trade-off between quantum and classical resources in expectation-value estimation tasks. By partitioning the LCU operator and controlling the balance between quantum circuit complexity and classical sampling, the method bridges coherent and randomized LCU, with rigorous monotonicity of the sampling penalty. Theoretical guarantees and practical case studies, ranging from simulation and linear solvers to error detection, demonstrate the approach's flexibility. Future work may focus on integrating these concepts with amplitude amplification and advanced block-encodings, and on a deeper information-theoretic understanding of quantum resource utilization.