Krylov Shadow Tomography for QFI Estimation

• KST is a hybrid framework that combines Krylov subspace methods with shadow tomography to efficiently estimate the quantum Fisher information.
• It constructs a strict hierarchy of non-polynomial lower bounds that converge exponentially fast to the exact QFI, outperforming traditional polynomial approximations.
• KST leverages classical shadows from local randomized measurements, making QFI estimation experimentally feasible for large or low-rank quantum systems.

Searching arXiv for papers on Krylov shadow tomography and QFI estimation. Krylov shadow tomography (KST) is a hybrid framework for estimating the quantum Fisher information (QFI) of an unknown quantum state by combining the Krylov subspace method with shadow tomography. It was introduced as a way to make QFI estimation experimentally feasible for large quantum systems, where direct evaluation is difficult because the QFI depends on the density operator in a highly nonlinear manner (Zhang et al., 3 Mar 2025). Subsequent work established that KST yields a strict hierarchy of non-polynomial lower bounds on the QFI, that these bounds converge exponentially fast with Krylov order, that they are tighter than state-of-the-art polynomial lower bounds at comparable resource level, and that for certain low-rank states the hierarchy terminates at the exact QFI already at low order (Wang et al., 19 Feb 2026).

1. QFI estimation problem and the motivation for KST

For a quantum state $\rho=\sum_k p_k |k\rangle\langle k|$ and Hermitian generator $H$, the QFI is given by

$$F_Q = 2\sum_{p_k+p_l>0}\frac{(p_k-p_l)^2}{p_k+p_l}\, |\langle k|H|l\rangle|^2.$$

This quantity is central in quantum metrology because it determines the ultimate precision limit in parameter estimation, and it is also used in entanglement detection and related tasks (Zhang et al., 3 Mar 2025). The difficulty is that $F_Q$ is highly nonlinear in $\rho$, so it is not directly accessible through the low-order polynomial observables naturally handled by modern randomized-measurement methods.

Earlier work therefore concentrated on experimentally accessible lower bounds on the QFI, including Legendre-transform bounds, sub-QFI bounds, and Taylor-series-based bounds (Wang et al., 19 Feb 2026). These polynomial lower bounds are measurable, but they exhibit an unavoidable systematic gap: no polynomial lower bound can coincide with the QFI for all states (Wang et al., 19 Feb 2026). Previous asymptotic polynomial hierarchies can approach $F_Q$, but only at high order, with rapidly increasing measurement and postprocessing cost (Zhang et al., 3 Mar 2025).

KST addresses this limitation by replacing the polynomial-approximation viewpoint with a hierarchy of non-polynomial Krylov bounds. This shifts the task from approximating the QFI by polynomial surrogates to approximating an operator associated with the QFI inside a structured Krylov subspace. A plausible implication is that the central advantage of KST is not merely tighter lower bounds, but a qualitatively different approximation principle.

2. Operator-theoretic construction

The mathematical setup begins with the weighted inner product

$$\langle X,Y\rangle_\rho = \operatorname{tr}\!\left[\rho(XY+YX)/2\right],$$

and the associated norm $\|X\|_\rho=\sqrt{\langle X,X\rangle_\rho}$. The relevant superoperator is the symmetrization map

$\mathcal{R}_\rho(X)=\frac{1}{2}(\rho X+X\rho).$

In the eigenbasis of $\rho$, this acts as

$H$0

with inverse on the relevant operator subspace

$H$1

Defining

$H$2

one obtains the compact representation

$H$3

(Zhang et al., 3 Mar 2025).

KST then constructs the nested Krylov subspaces

$H$4

Within each $H$5, one chooses the best approximation $H$6 to $H$7 in the $H$8-weighted norm, and defines the corresponding Krylov bound

$H$9

Equivalently,

$$F_Q = 2\sum_{p_k+p_l>0}\frac{(p_k-p_l)^2}{p_k+p_l}\, |\langle k|H|l\rangle|^2.$$0

Because $$F_Q = 2\sum_{p_k+p_l>0}\frac{(p_k-p_l)^2}{p_k+p_l}\, |\langle k|H|l\rangle|^2.$$1 is the orthogonal projection of $$F_Q = 2\sum_{p_k+p_l>0}\frac{(p_k-p_l)^2}{p_k+p_l}\, |\langle k|H|l\rangle|^2.$$2 onto $$F_Q = 2\sum_{p_k+p_l>0}\frac{(p_k-p_l)^2}{p_k+p_l}\, |\langle k|H|l\rangle|^2.$$3, the exact decomposition

$$F_Q = 2\sum_{p_k+p_l>0}\frac{(p_k-p_l)^2}{p_k+p_l}\, |\langle k|H|l\rangle|^2.$$4

holds, and therefore

$$F_Q = 2\sum_{p_k+p_l>0}\frac{(p_k-p_l)^2}{p_k+p_l}\, |\langle k|H|l\rangle|^2.$$5

(Zhang et al., 3 Mar 2025).

The operator space is restricted to

$$F_Q = 2\sum_{p_k+p_l>0}\frac{(p_k-p_l)^2}{p_k+p_l}\, |\langle k|H|l\rangle|^2.$$6

so that the weighted inner product is positive definite even when $$F_Q = 2\sum_{p_k+p_l>0}\frac{(p_k-p_l)^2}{p_k+p_l}\, |\langle k|H|l\rangle|^2.$$7 is singular. The nontrivial case assumes $$F_Q = 2\sum_{p_k+p_l>0}\frac{(p_k-p_l)^2}{p_k+p_l}\, |\langle k|H|l\rangle|^2.$$8; otherwise $$F_Q = 2\sum_{p_k+p_l>0}\frac{(p_k-p_l)^2}{p_k+p_l}\, |\langle k|H|l\rangle|^2.$$9 (Zhang et al., 3 Mar 2025).

3. Hierarchy, strict improvement, and finite termination

A defining feature of KST is that it yields not a single bound but a strict hierarchy of increasingly tight lower bounds. The Krylov subspaces satisfy

$F_Q$0

for some finite stopping index $F_Q$1, and the exact operator $F_Q$2 belongs to $F_Q$3 but not to any smaller $F_Q$4 with $F_Q$5 (Zhang et al., 3 Mar 2025). Consequently,

$F_Q$6

This strict hierarchy is one of the principal structural results of the framework (Wang et al., 19 Feb 2026).

The highest Krylov bound is exact because once the Krylov subspace contains $F_Q$7, the projection error vanishes. In this sense, KST is not limited to asymptotic approximation: at finite Krylov dimension it can reproduce the exact QFI (Zhang et al., 3 Mar 2025). This directly differentiates it from previously studied polynomial lower-bound schemes, which cannot coincide with the QFI for all states (Wang et al., 19 Feb 2026).

A common misconception is to regard KST as simply another polynomial lower-bound construction. That characterization is inaccurate. The bound $F_Q$8 is non-polynomial in $F_Q$9, even though it is assembled from polynomial moments. The distinction is essential because the exactness and finite termination properties of KST lie beyond the reach of polynomial lower bounds in general (Zhang et al., 3 Mar 2025).

4. Convergence properties and comparison with polynomial bounds

The 2026 analysis established that low-order Krylov bounds are already practically effective. Its central quantitative result is an exponential convergence theorem for the relative error

$\rho$0

namely

$\rho$1

Here $\rho$2 is a condition-number-like quantity determined by the spectrum of $\rho$3; for full rank it is essentially the ratio of largest to smallest eigenvalue, with an appropriate modification for non-full-rank states (Wang et al., 19 Feb 2026). The theorem implies exponential convergence of the Krylov hierarchy to the exact QFI as the Krylov order increases, with faster convergence for more mixed states.

The same work proved a direct superiority statement against the Taylor hierarchy $\rho$4:

$\rho$5

This comparison is described as resource-fair because $\rho$6 and $\rho$7 require estimating polynomial functions of $\rho$8 of comparable degree (Wang et al., 19 Feb 2026). The result shows that, at essentially the same estimation cost, the Krylov bound is generally tighter than the corresponding Taylor bound.

The significance of this comparison is methodological as well as numerical. Polynomial lower bounds approximate the nonlinear QFI through polynomial surrogates and therefore retain a systematic bias. KST instead constructs a hierarchy of non-polynomial approximants through projection in Krylov space. This suggests that the observed superiority is structurally rooted in the approximation scheme rather than in a favorable choice of constants.

5. Exactness for low-rank states

KST has a particularly sharp exactness theory for low-rank states. Let

$\rho$9

be the set of sums of distinct eigenvalue pairs. The stopping order satisfies

$F_Q$0

where $F_Q$1 denotes the number of distinct values in $F_Q$2 (Wang et al., 19 Feb 2026). Since $F_Q$3 for a rank-$F_Q$4 state,

$F_Q$5

Thus low-rank structure can force exact recovery of the QFI at very small Krylov order.

The Methods section further refines this by giving the exact formula

$F_Q$6

where $F_Q$7 consists of those eigenvalue sums $F_Q$8 for which all relevant matrix elements of $F_Q$9 vanish between the corresponding eigenspaces (Wang et al., 19 Feb 2026). This characterizes when the Krylov sequence terminates earlier than the generic rank-based upper bound would suggest.

The original KST paper presented explicit examples with $$\langle X,Y\rangle_\rho = \operatorname{tr}\!\left[\rho(XY+YX)/2\right],$$0. For the pseudo-pure state

$$\langle X,Y\rangle_\rho = \operatorname{tr}\!\left[\rho(XY+YX)/2\right],$$1

it showed that $$\langle X,Y\rangle_\rho = \operatorname{tr}\!\left[\rho(XY+YX)/2\right],$$2 for any $$\langle X,Y\rangle_\rho = \operatorname{tr}\!\left[\rho(XY+YX)/2\right],$$3, any pure state $$\langle X,Y\rangle_\rho = \operatorname{tr}\!\left[\rho(XY+YX)/2\right],$$4, and any $$\langle X,Y\rangle_\rho = \operatorname{tr}\!\left[\rho(XY+YX)/2\right],$$5, so that

$$\langle X,Y\rangle_\rho = \operatorname{tr}\!\left[\rho(XY+YX)/2\right],$$6

For the bound entangled state

$$\langle X,Y\rangle_\rho = \operatorname{tr}\!\left[\rho(XY+YX)/2\right],$$7

with $$\langle X,Y\rangle_\rho = \operatorname{tr}\!\left[\rho(XY+YX)/2\right],$$8, it likewise proved $$\langle X,Y\rangle_\rho = \operatorname{tr}\!\left[\rho(XY+YX)/2\right],$$9, hence again $\|X\|_\rho=\sqrt{\langle X,X\rangle_\rho}$0 (Zhang et al., 3 Mar 2025).

The broader relevance of these results follows from the observation that low-rank and nearly low-rank states are common in quantum information processing, for example when ideal pure states are weakly corrupted by noise (Wang et al., 19 Feb 2026). In such settings, low-order Krylov bounds can eliminate the systematic error that affects polynomial approximations.

6. Estimation by classical shadows, practical behavior, and limitations

KST becomes experimentally accessible because each bound can be expressed through quantities

$\|X\|_\rho=\sqrt{\langle X,X\rangle_\rho}$1

which are polynomial functionals of $\|X\|_\rho=\sqrt{\langle X,X\rangle_\rho}$2 (Wang et al., 19 Feb 2026). Writing

$\|X\|_\rho=\sqrt{\langle X,X\rangle_\rho}$3

the bound is computed as

$\|X\|_\rho=\sqrt{\langle X,X\rangle_\rho}$4

Equivalently, the original derivation formulates $\|X\|_\rho=\sqrt{\langle X,X\rangle_\rho}$5 as a positive-definite Hankel matrix for $\|X\|_\rho=\sqrt{\langle X,X\rangle_\rho}$6, so the matrix inverse is well defined (Zhang et al., 3 Mar 2025). The non-polynomial character of $\|X\|_\rho=\sqrt{\langle X,X\rangle_\rho}$7 thus enters through the inverse of a moment matrix built from polynomial data.

The measurement protocol is local. One samples

$\|X\|_\rho=\sqrt{\langle X,X\rangle_\rho}$8

where each $\|X\|_\rho=\sqrt{\langle X,X\rangle_\rho}$9 is drawn independently from a suitable single-qubit ensemble such as the single-qubit Clifford group or $\mathcal{R}_\rho(X)=\frac{1}{2}(\rho X+X\rho).$0, applies $\mathcal{R}_\rho(X)=\frac{1}{2}(\rho X+X\rho).$1 to $\mathcal{R}_\rho(X)=\frac{1}{2}(\rho X+X\rho).$2, measures in the computational basis, and records a bitstring $\mathcal{R}_\rho(X)=\frac{1}{2}(\rho X+X\rho).$3 (Zhang et al., 3 Mar 2025). Repeating this procedure yields classical shadows

$\mathcal{R}_\rho(X)=\frac{1}{2}(\rho X+X\rho).$4

with $\mathcal{R}_\rho(X)=\frac{1}{2}(\rho X+X\rho).$5. The moments $\mathcal{R}_\rho(X)=\frac{1}{2}(\rho X+X\rho).$6 can be estimated simultaneously from the same shadow data using $\mathcal{R}_\rho(X)=\frac{1}{2}(\rho X+X\rho).$7-statistics, and the batch-shadow method can accelerate post-processing (Wang et al., 19 Feb 2026). The estimator in the original proposal uses a median-of-means construction and finally computes

$\mathcal{R}_\rho(X)=\frac{1}{2}(\rho X+X\rho).$8

(Zhang et al., 3 Mar 2025).

The formal guarantee states that for

$\mathcal{R}_\rho(X)=\frac{1}{2}(\rho X+X\rho).$9

and $\rho$0 chosen according to an explicit bound depending on the operator norms or variances of the relevant $\rho$1, the final estimate satisfies

$\rho$2

with probability at least $\rho$3 (Zhang et al., 3 Mar 2025). The papers emphasize that the protocol uses only single-qubit gates and local measurements, and that its measurement cost is comparable to the best existing randomized-measurement approaches for QFI lower bounds (Zhang et al., 3 Mar 2025).

At the same time, the resource cost grows exponentially with Krylov order, which is why the practical emphasis falls on low-order bounds (Wang et al., 19 Feb 2026). This is the main limitation of current KST implementations. The method is therefore most attractive precisely in the regimes highlighted by the theory: moderate Krylov order, rapid convergence, or low-rank structure yielding early exactness. Numerical simulations support this picture: low-order Krylov bounds are close to $\rho$4 already for moderate mixed states, outperform Taylor bounds in direct comparisons, and for rank-2 examples the highest relevant Krylov bound matches the QFI exactly (Wang et al., 19 Feb 2026). The original benchmarks further reported that $\rho$5 for pseudo-pure states obtained from $\rho$6 classical shadows tracks $\rho$7 well, with sample complexity for relative error $\rho$8 scaling like $\rho$9, while for the bound entangled example numerical results using $H$00 classical shadows showed sample cost scaling like $H$01 (Zhang et al., 3 Mar 2025).

KST is therefore best understood as a framework for QFI estimation that combines three features: a Krylov-subspace approximation to the SLD-related operator $H$02, a strict hierarchy of non-polynomial lower bounds $H$03, and shadow-based estimation of the polynomial moments required to evaluate those bounds (Zhang et al., 3 Mar 2025). In quantum metrology, entanglement detection, sensing, variational algorithms, and resource quantification, its importance lies in reducing or removing the systematic error inherent to polynomial lower-bound methods while preserving experimental accessibility (Wang et al., 19 Feb 2026).