---
title: Sampling-Based Linear Combination of Unitaries
url: https://www.emergentmind.com/topics/sampling-based-linear-combination-of-unitaries
type: topic
---

# Sampling-Based Linear Combination of Unitaries

Sampling-based linear combination of unitaries (LCU) algorithms provide a set of NISQ- and fault-tolerance–compatible techniques for realizing non-trivial (often non-unitary) operators as expectation values by leveraging classical sampling and minimal quantum-coherent resources. These frameworks enable the simulation of nonunitary processes, spectral filtering, dynamical observables, and quantum measurement protocols by expanding the target operator as a (possibly integral) sum where each term is proportional to a unitary, estimated by simple quantum circuits, and aggregated via classical post-processing. Recent advances systematically clarify the trade-offs between quantum hardware requirements and classical sampling overhead, and demonstrate regimes where sampling-based (randomized or hybrid) LCU schemes are optimal or nearly so for ground-state preparation, quantum linear system solvers, non-Hermitian dynamics, error detection, and observable estimation.

## 1. Operator Representation in Sampling-based LCU

A general target operator can be expressed as a linear (or integral) combination of unitaries,
\[
F(A) = \int_V f(\mathbf{t})\, G(A,\mathbf{t})\, d\mathbf{t}
\]
or, in the discrete case, as
\[
K = \sum_i c_i U_i
\]
where $f(\mathbf{t})$ is a probability density over a $d$-dimensional domain $V\subset\mathbb{R}^d$ and $G(A,\mathbf{t}) = c(\mathbf{t}) U(A,\mathbf{t})$ with $\|U(A,\mathbf{t})\|=1$, or $c_i > 0$. This representation supports the realization of filtered projectors, Green's function inverses, and general nonunitary processes after suitable truncation and normalization [2509.14451].

When targeting state transformations or CP maps on density operators, the induced channel is
\[
\Lambda(\rho) = K \rho K^\dagger
\]
with normalization given by $\|c\|_1 = \sum_i |c_i|$ and probabilistic weights $p_i = c_i/\|c\|_1$ [2512.06260]. In many algorithmic settings, composite LCUs (products of LCUs associated to time steps or filter segments) appear, denoted $
A^{(\nu)} = \prod_{k=1}^\nu A_k
$ [2506.15658].

## 2. Sampling-based LCU Protocols: Randomized, Hybrid, and Classical Post-Processing

Sampling-based implementations circumvent the need for a large coherent ancilla and complex multi-controlled unitaries by randomizing the selection of LCU terms and estimating the expectation values via quantum measurements. The main protocols include:

- **Randomized (Virtual) LCU:** Sample index pairs $(i, j)$ with probability $p_ip_j$ and estimate $\operatorname{Re}\operatorname{tr}[O U_i \rho U_j^\dagger]$ using a Hadamard-test-style circuit with a single ancilla. This approach computes the same expectation as the original coherent LCU algorithm but with quadratic sampling overhead: the sample complexity for error $\epsilon$ scales as $O(1/(P^2\epsilon^2))$, where $P$ is the postselection probability in the coherent approach [2512.06260, 2506.15658].

- **Hybrid Grouping Strategy:** Interpolate between fully coherent and randomized LCU by partitioning the $m$-term sum into $G$ groups $S_k$. Each group forms a smaller coherent LCU, and sampling proceeds on a coarse-grained index set. The key resource parameter is the “reduction factor” $R$, which determines sampling overhead: $R$ strictly decreases as group sizes increase, interpolating between $R=1$ (randomized LCU) and $R=P$ (fully coherent LCU) [2512.06260].

- **Classical Post-Processing (LCU-CPP):** For continuous or large discrete LCUs, one samples a set $\{\mathbf{t}_k\}_{k=1}^K$ (using e.g., Monte Carlo, trapezoid, or quasi-Monte Carlo sequences) and estimates $\operatorname{Re}\operatorname{Tr}(G(A,\mathbf{t}_k)\rho)$ at each point. These values are aggregated classically to form the final estimator of $\operatorname{Tr}[F(A)\rho]$. No complex ancilla operations or block encodings are involved [2509.14451].

## 3. Quasi-Monte Carlo and Classical Integration in LCU-CPP Frameworks

Quasi-Monte Carlo (QMC) methods provide deterministic low–discrepancy sampling sequences (e.g., Halton, Sobol) to accelerate convergence over standard Monte Carlo. QMC replaces i.i.d. random samples by a sequence $\{\mathbf{x}_k\}_{k=1}^K$ on $[0,1]^d$, transformed into $\mathbf{t}_k$ via the inverse CDF of $f$. By the Koksma–Hlawka inequality,
\[
\left|\int_V f(\mathbf{t}) h(\mathbf{t}) d\mathbf{t} - \frac{1}{K}\sum_{k=1}^K h(\mathbf{t}_k)\right| \leq V_{\rm HK}(h) D^*(\{\mathbf{t}_k\})
\]
yielding bias scaling as $O((\ln K)^d/K)$, superior to MC ($O(K^{-1/2})$) and the trapezoid rule ($O(K^{-2/d})$) [2509.14451]. In LCU-CPP, QMC achieves the lowest end-to-end error for moderate shot counts $M=10^2$–$10^3$ and minimal circuit depth, which is significant for NISQ-era implementations.

At each QMC point, the Hadamard test is performed $M$ times to estimate $\nu_k$, and the samples are averaged; statistical shot-noise decreases as $O((KM)^{-1/2})$. Practical benchmarks (ground-state and Green's function estimation on Heisenberg chains) demonstrate lowest errors for QMC across all relevant $K$ and $M$ [2509.14451].

## 4. Circuit Structures, Resource Scaling, and Algorithmic Trade-offs

Resource requirements are determined by LCU protocol choice and grouping:

| Method            | Ancilla Qubits         | Quantum Circuit Cost                  | Sampling Overhead                 |
|-------------------|-----------------------|---------------------------------------|-----------------------------------|
| Coherent LCU      | $O(\log m)$           | $O(m\log m)$, $m$ controlled-$U_i$    | $O(P^{-1}\epsilon^{-2})$          |
| Randomized LCU    | $1$                   | $O(1)$, two controlled-$U_i$/shot     | $O(P^{-2}\epsilon^{-2})$          |
| Hybrid LCU        | $O(\max_k \log |S_k|)$| $O(\max_k |S_k|\log |S_k|)$           | $O(R/(P^2\epsilon^2))$            |
| LCU-CPP (QMC)     | $1$                   | controlled-$G$ + $2$ H-gates/shot     | $O((KM)^{-1/2})$; bias $\sim$ QMC |

For LCU-CPP and randomized LCU, each run requires only a single ancilla qubit, and circuit depth effectively reduces to the cost of implementing $G(A, \mathbf{t})$ or $U_i$, plus two Hadamard gates. Coherent LCU demands logarithmic ancilla scaling (in $m$), high gate count, and exponential cost for large $m$. Hybrid algorithms achieve intermediate hardware and sampling cost by adjusting group size $|S_k|$ [2512.06260].

Sampling overhead dominates for randomized LCU when $P$ is small, i.e., for nonselective target operators or low-probability subspaces. Hybrid grouping reduces this penalty by gathering high-weight terms into coherent clusters, lowering $R$ toward $P$ [2512.06260].

## 5. Observable Estimation and Shadow Tomography Integration

Estimating expectation values $\operatorname{Tr}[O K \rho K^\dagger]$ in sampling-based LCU can be performed directly for a single observable through repeated runs. For simultaneous estimation of many observables, the connection between randomized LCU and classical shadow tomography is exploited. By instrumenting each shot with a random Clifford rotation and computational basis measurement, the resulting classical shadows encode observables' values; an unbiased estimator for $U \rho V^\dagger$ is formed by
\[
\hat v_{a,b}(R,z) = 2 i^b (-1)^a\, \hat \sigma_{a,b}(R,z)
\]
where $a, b$ are ancilla measurement outcomes and $R, z$ label the classical shadow [2506.15658].

For $M$ observables, the shot complexity is $O(\epsilon^{-2}\max_m \|O_m\|_{\rm shadow}^2 \log(M/\delta))$, leveraging shadow norm bounds to ensure efficient estimation [2506.15658]. This framework enables concurrent multi-observable estimation in quantum simulation and phase estimation routines, at sample cost competitive with the best postselected and randomized approaches.

## 6. Applications and Case Studies

Sampling-based LCU schemes have been instantiated in a diverse range of quantum algorithms:

- **High-precision Hamiltonian simulation:** Decomposition of Trotter steps into Pauli LCUs, with randomized or grouped samplings yielding sample complexities of $O(\epsilon^{-2}\|O\|_{\rm shadow}^2\log(M/\delta))$, and depth per shot scaling as $O(T^{1+1/(4k)}\log(1/\epsilon))$ for $k$th-order methods [2506.15658].
- **Ground-state preparation and spectral filtering:** Gaussian-filtered projectors realized by QMC-integrated LCU-CPP or by hybrid virtual+coherent schemes that balance ancilla usage and sampling cost, achieving total time complexity $O(p_0^{-1}\Delta^{-1}\epsilon^{-2}\log(1/\epsilon))$ while minimizing circuit resources [2509.14451, 2512.06260].
- **Quantum linear system solvers:** Discretized Fourier-integral LCUs over $O(JK)$ terms, with coherent, randomized, and hybrid groupings yielding respective scalings in ancilla count and sample overhead. Hybrid approach recovers the best scaling of both extremes while minimizing hardware needs [2512.06260].
- **Non-Hermitian and open-system dynamics:** Dynamical maps expanded as integrals or sums over parameterized unitaries, sampled according to $f$, with grouping techniques (large cluster/coherent for dominant regions, singleton/virtual for tails) enabling substantial circuit depth reduction at negligible sampling cost increase [2512.06260].
- **Quantum error detection and syndrome extraction:** Projectors onto error-free subspaces represented as uniform LCUs over stabilizers, allowing randomized sampling (virtual detection), coherent detection, or hybrid strategies targeting biased error models. Resource trade-offs are custom-fitted to code structure and noise [2512.06260].

## 7. Caveats, Regimes, and Hardware Relevance

Sampling-based LCU methods are not universally optimal: for large $M$ (shots per QMC point) or low $d$ (integration dimensions), trapezoidal and Simpson rules can outperform QMC. When $M=1$, shot-noise dominates, and MC and QMC are both better than deterministic grid-based rules. The hybrid grouping strategy enables near-optimal trade-offs only when one can partition the operator expansion sensibly; poor groupings incur unnecessary sampling or circuit costs [2509.14451, 2512.06260].

Randomized and hybrid LCU approaches are especially valuable in NISQ and early fault-tolerant regimes due to their minimal circuit depth and flexible ancilla requirements; however, the inherent variance and sometimes large sample overhead (scaling as $P^{-2}$) remain their main limitations. All estimators for “virtual” effective states $U\rho V^\dagger$ are classical constructs—these states are not physically prepared but must be estimated via carefully designed quantum instruments and classical post-processing [2506.15658, 2512.06260].

Sampling-based LCU, and in particular QMC-LCU-CPP, delivers practical, provably faster convergence for nonunitary operator estimation, avoids large high-dimensional prefactors, and achieves minimal end-to-end error for practical shot counts at the lowest possible circuit depth for quantum hardware [2509.14451].

Source: https://www.emergentmind.com/topics/sampling-based-linear-combination-of-unitaries