Tradeoffs between quantum and classical resources in linear combination of unitaries

TL;DR

This work addresses the resource trade-offs in quantum algorithms built on linear combinations of unitaries (LCU) by introducing a hybrid LCU framework that partition-unitaries into groups, applying coherent LCU within groups while sampling randomly across groups. A central concept is the reduction factor $R$, which bounds the sampling overhead and interpolates between fully coherent $R=\mathcal{P}$ and fully randomized $R=1$ implementations, enabling substantial circuit-depth and ancilla reductions with only modest sampling overhead changes. The authors provide a decomposition theorem, estimator design, and multi-round generalization, then demonstrate the framework across non-Hermitian Hamiltonian simulation, quantum linear system solvers, ground-state preparation, and quantum error detection, showing concrete resource benefits such as 32x depth reduction in LCHS and favorable scaling in QLSS. This approach offers a practical path toward hardware-efficient, error-tolerant quantum algorithms for the early fault-tolerant era, balancing quantum and classical resources with tunable performance depending on the problem and hardware constraints.

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.

Figures (7)

• Figure 1: Linear combination of unitaries (LCU) methods. The target operation is given by $K_{\rm LCU}=\sum_{i=1}^m p_i U_i$ for a probability distribution $\{p_i\}$ and unitary operators $\{U_i\}$. The unitary ${\rm PRE}_p$ in (a) encodes $\{p_i\}$, and $K_{\rm LCU}\rho K^\dagger_{\rm LCU}$ is realized (up to normalization) when the final measurement after ${\rm PRE}^\dagger_p$ is all zero. In (b), the indexes $i,j$ are independently sampled from the probability distribution $\{p_i\}$. By measuring $X$ in the top single qubit, we can recover the property of $K_{\rm LCU}\rho K_{\rm LCU}^\dagger$ without explicitly preparing it.
• Figure 2: Idea of our decomposition for LCU-based CP map $\tilde{\Lambda}_{p,\mathcal{U}}$ or $\ket{\psi}\mapsto \sum_ip_i U_i\ket{\psi}$. To simulate the target state $\sum_{i} p_i U_i\ket{\psi}$, our method randomly uses the controlled version of the coherent LCU method for the partial superpositions $\{K_k|\psi\rangle\}$ and recovers the full state by measuring Pauli X in the top register as well as the virtual LCU method. The probability $q_kq_{k'}$ for virtually mixing $K_k\ket{\psi}$ with $K_{k'}|\psi\rangle$ is determined by $K_k$.
• Figure 3: Quantum circuit in our decomposition of $\tilde{\Lambda}_{p,\mathcal{U}}$. Similar to the virtual LCU, the final effective state in the third line is simulated by taking the expectation of $X_{\rm B}\otimes \Pi_{\rm A}$; see Eq. \ref{['eq:main_equality']}. The unitary $L_k$ is defined as Eq. \ref{['eq:LCUop']} for the LCU operator $K_k=\sum_{i\in S_k}(p_i/q_k)U_i$. Although $\Pi_{\rm A}$ denotes the initial state of the $\lceil\log_2 m\rceil$-qubit system A, $L_k$ acts only on a subset of the qubits in A. Note that if $k=k'$, then we can omit the 1-qubit control for $L_k$.
• Figure 4: A series of partitions of $[m]$ satisfying the assumption of Theorem \ref{['thm:2']}.
• Figure 5: The $M$-dependence of the $R-\mathcal{P}$ for Eq. \ref{['eq:lchs_target_lcu']}. The corresponding partition is described below Eq. \ref{['eq:lchs_target_lcu']}. The parameters are chosen as $\|L\|=2$ and $T=1$. We plot the final expression of Eq. \ref{['eq:lchs_rp_eval']} as the dotted lines.

Theorems & Definitions (17)

• Theorem 1: Decomposition of LCU-based CP maps
• proof : Sketch of the proof of Theorem \ref{['thm:1']}
• Theorem 2
• Theorem 3
• Lemma 1
• Lemma 2
• proof
• Lemma 3: Slutsky's lemma shao2008mathematical
• Lemma 4: Delta method shao2008mathematical
• Lemma 5