Quantum Leverage Score Approximation

• Quantum leverage score approximation is a technique that estimates the importance of matrix rows or columns using quantum state preparation and dequantized sampling methods.
• It employs block-encoding, quantum singular-value transformation, and amplitude estimation to construct efficiency gains in low-rank approximation, regression, and optimization.
• The approach achieves quadratic to exponential speedups under specific conditions while maintaining rigorous error bounds and computational complexity guarantees.

Quantum leverage score approximation studies how to estimate, approximate, or sample the statistical leverage structure of a matrix by exploiting either gate-based quantum linear-algebra primitives or classical dequantized sampling schemes. For a matrix $A=UDV^\intercal$ of numerical rank $r$, the row leverage scores are $\ell_{R,i}(A)=\|U_{i,\bullet}\|_2^2$ and the column leverage scores are $\ell_{C,j}(A)=\|V_{j,\bullet}\|_2^2$; in the full-column-rank case, they are equivalently $\ell_i=a_i^\top(A^\top A)^{-1}a_i$. These quantities define importance-sampling distributions, can be encoded into quantum states, and appear as algorithmic primitives in low-rank approximation, regression, John ellipsoid computation, and QAOA parameter estimation (Shao, 2023, Li et al., 2024).

1. Definitions and core geometric quantities

Let $A\in\mathbb R^{m\times n}$ have singular value decomposition $A=U_A\Sigma_A V_A^T$, where $U_A\in\mathbb R^{m\times r}$ and $V_A\in\mathbb R^{n\times r}$ are orthonormal and $\Sigma_A\in\mathbb R^{r\times r}$ is diagonal with singular values $r$0. The $r$1th statistical leverage score is

$r$2

and the matrix coherence is

$r$3

The coherence measures how “spiky” the row-space basis $r$4 is (Zuo et al., 2021).

Equivalent formulations are central in both randomized linear algebra and quantum algorithms. For a full-column-rank matrix $r$5 with row vectors $r$6, the leverage scores satisfy

$r$7

with $r$8 and $r$9 (Li et al., 2024). In truncated form, if

$\ell_{R,i}(A)=\|U_{i,\bullet}\|_2^2$0

then the rank-$\ell_{R,i}(A)=\|U_{i,\bullet}\|_2^2$1 leverage scores are

$\ell_{R,i}(A)=\|U_{i,\bullet}\|_2^2$2

This rank-restricted variant is the form used in subspace embeddings and in the Hamiltonian-Guided Leverage Embedding pipeline (Mukherjee et al., 5 Jun 2026).

Quantum formulations repackage the leverage distribution as a state. The quantum leverage-score states are

$\ell_{R,i}(A)=\|U_{i,\bullet}\|_2^2$3

Measuring $\ell_{R,i}(A)=\|U_{i,\bullet}\|_2^2$4 or $\ell_{R,i}(A)=\|U_{i,\bullet}\|_2^2$5 yields an index according to the corresponding leverage-score distribution (Shao, 2023).

2. Access models and algorithmic primitives

The computational model is decisive. In the quantum-inspired classical setting, the matrix is not accessed entry-by-entry in the usual RAM model. Instead, one assumes query and sampling access to $\ell_{R,i}(A)=\|U_{i,\bullet}\|_2^2$6 through data structures supporting column-norm sampling,

$\ell_{R,i}(A)=\|U_{i,\bullet}\|_2^2$7

and entrywise sampling inside a chosen column,

$\ell_{R,i}(A)=\|U_{i,\bullet}\|_2^2$8

Under a binary-tree data structure, any entry $\ell_{R,i}(A)=\|U_{i,\bullet}\|_2^2$9 can be queried or updated in $\ell_{C,j}(A)=\|V_{j,\bullet}\|_2^2$0 time, any row or column norm can be computed, and sampling from any row or column is available in polylogarithmic time (Zuo et al., 2021).

Gate-based quantum algorithms usually assume oracle or QRAM access and then build block-encodings. One formulation uses an oracle

$\ell_{C,j}(A)=\|V_{j,\bullet}\|_2^2$1

from which one can construct, in $\ell_{C,j}(A)=\|V_{j,\bullet}\|_2^2$2 queries plus $\ell_{C,j}(A)=\|V_{j,\bullet}\|_2^2$3 overhead, a unitary $\ell_{C,j}(A)=\|V_{j,\bullet}\|_2^2$4 that is an $\ell_{C,j}(A)=\|V_{j,\bullet}\|_2^2$5 block-encoding of $\ell_{C,j}(A)=\|V_{j,\bullet}\|_2^2$6 when $\ell_{C,j}(A)=\|V_{j,\bullet}\|_2^2$7 (Li et al., 2024). A more abstract version assumes an $\ell_{C,j}(A)=\|V_{j,\bullet}\|_2^2$8 block-encoding of $\ell_{C,j}(A)=\|V_{j,\bullet}\|_2^2$9 available at cost $\ell_i=a_i^\top(A^\top A)^{-1}a_i$0, so that the top-left corner of the unitary approximates $\ell_i=a_i^\top(A^\top A)^{-1}a_i$1 (Shao, 2023).

Three primitives recur across the literature. Quantum Singular-Value Transformation is used to implement matrix polynomials and, in particular, sign- or inverse-like spectral filters. Amplitude amplification raises the success probability of post-selection steps, and amplitude estimation converts amplitudes into additive numerical estimates. In inversion-based constructions, QRAM is also used to prepare normalized row states

$\ell_i=a_i^\top(A^\top A)^{-1}a_i$2

in $\ell_i=a_i^\top(A^\top A)^{-1}a_i$3 time (Li et al., 2024).

3. Quantum-inspired classical approximation

A prominent dequantized approach is the two-stage procedure of "A quantum-inspired algorithm for approximating statistical leverage scores" (Zuo et al., 2021). The method separates subspace recovery from score evaluation. Algorithm 1, QiSVD, builds a succinct low-rank representation $\ell_i=a_i^\top(A^\top A)^{-1}a_i$4 approximating $\ell_i=a_i^\top(A^\top A)^{-1}a_i$5. Algorithm 2, QiSLS, estimates $\ell_i=a_i^\top(A^\top A)^{-1}a_i$6 by first estimating $\ell_i=a_i^\top(A^\top A)^{-1}a_i$7 and then taking its squared norm.

QiSVD samples $\ell_i=a_i^\top(A^\top A)^{-1}a_i$8 columns according to $\ell_i=a_i^\top(A^\top A)^{-1}a_i$9, forms a sketch matrix $A\in\mathbb R^{m\times n}$0 with rescaled sampled columns, then samples $A\in\mathbb R^{m\times n}$1 rows from the mixture distribution

$A\in\mathbb R^{m\times n}$2

and forms a second sketch $A\in\mathbb R^{m\times n}$3. After computing the top $A\in\mathbb R^{m\times n}$4 singular values $A\in\mathbb R^{m\times n}$5 of $A\in\mathbb R^{m\times n}$6 and the corresponding right singular vectors $A\in\mathbb R^{m\times n}$7, the algorithm outputs

$A\in\mathbb R^{m\times n}$8

where $A\in\mathbb R^{m\times n}$9 and $A=U_A\Sigma_A V_A^T$0. QiSLS then estimates the inner products $A=U_A\Sigma_A V_A^T$1 by inner-product sampling, assembles an approximation to $A=U_A\Sigma_A V_A^T$2, and returns

$A=U_A\Sigma_A V_A^T$3

The analysis is organized around concentration and perturbation bounds. Column sampling ensures $A=U_A\Sigma_A V_A^T$4, and with $A=U_A\Sigma_A V_A^T$5 one has

$A=U_A\Sigma_A V_A^T$6

with probability at least $A=U_A\Sigma_A V_A^T$7. The analogous row-sampling step gives

$A=U_A\Sigma_A V_A^T$8

Because $A=U_A\Sigma_A V_A^T$9, the top $U_A\in\mathbb R^{m\times r}$0 right singular vectors of $U_A\in\mathbb R^{m\times r}$1 approximate those of $U_A\in\mathbb R^{m\times r}$2. The resulting projector estimate satisfies

$U_A\in\mathbb R^{m\times r}$3

and the final leverage-score estimator satisfies

$U_A\in\mathbb R^{m\times r}$4

with probability at least $U_A\in\mathbb R^{m\times r}$5 (Zuo et al., 2021).

The same work gives explicit complexity bounds. Step 5, the SVD of $U_A\in\mathbb R^{m\times r}$6, costs $U_A\in\mathbb R^{m\times r}$7, and the overall running time is

$U_A\in\mathbb R^{m\times r}$8

The theoretical analysis therefore yields dependence polynomial in $U_A\in\mathbb R^{m\times r}$9, the condition number $V_A\in\mathbb R^{n\times r}$0, and logarithmic in matrix size. Numerical experiments on synthetic low-rank matrices up to thousands-by-hundreds showed that even for moderate sampling size $V_A\in\mathbb R^{n\times r}$1 theory, the leverage-score estimates track the true $V_A\in\mathbb R^{n\times r}$2 closely, the errors stabilize below prescribed $V_A\in\mathbb R^{n\times r}$3 in repeated trials, and practical $V_A\in\mathbb R^{n\times r}$4 can be much smaller than worst-case bounds (Zuo et al., 2021).

A frequent misconception is that this line of work requires quantum hardware. It does not: the approach is fully classical, and its claimed advantage is obtained under the sample-and-query data-structure model rather than a gate-based model (Zuo et al., 2021).

4. Gate-based quantum algorithms

Gate-based quantum leverage-score algorithms follow two main patterns: direct preparation of a leverage-score state, and direct estimation of quadratic forms involving $V_A\in\mathbb R^{n\times r}$5.

The state-preparation approach constructs an approximation to the partial isometry $V_A\in\mathbb R^{n\times r}$6 by QSVT. Given an $V_A\in\mathbb R^{n\times r}$7 block-encoding of $V_A\in\mathbb R^{n\times r}$8, one can construct in time

$V_A\in\mathbb R^{n\times r}$9

a unitary whose top-left block is an approximation $\Sigma_A\in\mathbb R^{r\times r}$0. Starting from the uniform superposition over column indices, post-selecting on the success ancilla, and then applying amplitude amplification for $\Sigma_A\in\mathbb R^{r\times r}$1 repetitions yields a state close to

$\Sigma_A\in\mathbb R^{r\times r}$2

The main theorem gives preparation time

$\Sigma_A\in\mathbb R^{r\times r}$3

and for any fixed row or column index $\Sigma_A\in\mathbb R^{r\times r}$4, single-score estimation is available in

$\Sigma_A\in\mathbb R^{r\times r}$5

time with error

$\Sigma_A\in\mathbb R^{r\times r}$6

The same work proves lower bounds showing that the speedup is at least quadratic and that the dependence on $\Sigma_A\in\mathbb R^{r\times r}$7 and $\Sigma_A\in\mathbb R^{r\times r}$8 is nearly tight (Shao, 2023).

The inversion-based approach starts from a block-encoding of $\Sigma_A\in\mathbb R^{r\times r}$9 and uses QSVT to approximate the inverse map $r$00 on the spectrum of $r$01. For condition number $r$02, an even polynomial of degree

$r$03

approximates $r$04 on $r$05, yielding a block-encoding of $r$06. Combined with row-state preparation, a Hadamard test estimates

$r$07

and amplitude estimation reduces the statistical error. In the all-scores setting, a theorem cited in the John-ellipsoid work gives multiplicative approximation

$r$08

using

$r$09

queries to $r$10 and

$r$11

total time, with success probability $r$12 (Li et al., 2024).

These two patterns target different outputs. State-preparation algorithms directly sample according to leverage scores without explicitly materializing all of them, whereas inversion-based algorithms produce numerical estimates of individual or all scores. This suggests a division between sampling-centric and estimation-centric quantum subroutines, although both rely on the same block-encoding and QSVT toolkit.

5. Complexity, lower bounds, and scope of the speedups

The complexity landscape is best understood by keeping the access model explicit. In the sample-and-query classical setting, the dequantized algorithm achieves additive error $r$13 in time polynomial in $r$14, $r$15, $r$16, $r$17, and $r$18, with only polylogarithmic dependence on $r$19 and $r$20 (Zuo et al., 2021). In the block-encoding model, quantum state preparation scales as

$r$21

while full multiplicative approximation of all leverage scores can scale as

$r$22

in total time (Shao, 2023, Li et al., 2024).

The strongest positive claims are regime-dependent. For the dequantized algorithm, the summary narrative states an exponential speedup in $r$23 compared with classical $r$24 SVD when $r$25 are large, the target rank satisfies $r$26, and $r$27 is well conditioned (Zuo et al., 2021). For leverage-score sampling on a quantum computer, the claimed speedup is at least quadratic and could be exponential for well-conditioned matrices, while for all-score approximation in the tall dense regime $r$28, the quantum approach yields a quadratic speedup $r$29 provided $r$30 (Shao, 2023, Li et al., 2024).

Lower bounds materially constrain further improvement. In the entry-oracle model, any quantum algorithm that prepares $r$31 must make

$r$32

queries. In the block-encoding model, any such algorithm must invoke the block-encoding

$r$33

times. Even estimating the largest leverage score up to constant relative error requires $r$34 entry-oracle queries when $r$35 and $r$36 block-encoding calls. These bounds are obtained by reductions to Grover search and are used to argue that the known quantum algorithms are close to optimal (Shao, 2023).

A second misconception is that leverage-score approximation automatically confers an end-to-end advantage. The papers are more specific. The condition number and precision dependence can be substantial: the classical quantum-inspired algorithm has polynomial overhead in $r$37 and $r$38, which can be large in ill-conditioned or high-precision regimes, and its hidden constants in $r$39 can be significant (Zuo et al., 2021). On the quantum side, the dependence on $r$40 is stated to be unavoidable in QSVT-based constructions, and access assumptions such as efficient block-encodings or QRAM enter linearly through the cost parameter $r$41 (Shao, 2023).

6. Applications and later developments

Leverage-score approximation is used as a subroutine rather than an endpoint in several later constructions. One application is quantum rigid regression with vector solution outputs. The quantum algorithm in (Shao, 2023) uses leverage-score sampling on the “rigid” scores of

$r$42

to form a sketch and then solve a smaller regression problem; the paper states that this achieves polynomial speedups over the best classical algorithm known.

Another application is the quantum algorithm for approximating the John ellipsoid. That work presents the first quantum algorithm for the problem, running in

$r$43

time by using recent advances in quantum algorithms for spectral approximation and leverage score approximation. In the tall matrix regime, the result is a quadratic speedup over the best classical algorithms cited there (Li et al., 2024).

A hybrid variational-quantum application appears in "Hamiltonian-Guided Leverage Embedding: Robust Subspace Compression for Efficient QAOA Parameter Estimation" (Mukherjee et al., 5 Jun 2026). At each candidate $r$44, the method draws $r$45 QAOA measurement shots, computes sample energies, chooses either uniform weights

$r$46

or Boltzmann-type weights

$r$47

builds a weighted Ising feature matrix

$r$48

computes an approximate dominant $r$49-dimensional column space in $r$50 time, and samples

$r$51

rows according to leverage-score probabilities. The formal guarantees include $r$52-subspace embedding, rank preservation, a master inequality controlling energy approximation error through the residual fraction

$r$53

and an argmin-shift bound

$r$54

Empirically, benchmarks on HamLib Max-Cut and Maximum Independent Set report that, without HGLE, optimizers vary widely, whereas with HGLE warm start all achieve $r$55 on both problems; the cross-optimizer variance on MIS collapses from $r$56 to $r$57 (Mukherjee et al., 5 Jun 2026).

This application also clarifies the relation between “quantum” and “quantum-inspired” leverage methods. HGLE is a hybrid pipeline in which the quantum device generates measurement samples, but the leverage-score computation itself is classical randomized linear algebra on the induced feature matrix. A plausible implication is that leverage-score approximation has become a reusable compression primitive across both quantum linear-algebra algorithms and variational-quantum workflows.

The current literature also records several extension paths. The HGLE discussion lists approximate or quantum-accelerated leverage-score estimation, integration with classical shadows or compressed sensing of quantum states, application to other VQAs such as molecular VQE and hardware-efficient circuits, and heuristics that adapt $r$58 and $r$59 on the fly based on spectral decay (Mukherjee et al., 5 Jun 2026). Together with the lower-bound and dequantization results, these directions place quantum leverage score approximation at the intersection of randomized numerical linear algebra, quantum algorithms for spectral problems, and hybrid optimization.