Classical Shadow Protocols for Quantum Estimation

• Classical shadow protocols are a family of randomized measurement schemes that convert quantum state data into succinct classical representations for efficient observable estimation.
• They leverage random unitaries and inverse channel operations to provide provable sample efficiency, often reducing exponential resource demands to polynomial regimes.
• Variants such as product, global dual pairs, local dual pairs, and dual product protocols offer trade-offs between circuit depth and sample complexity, especially in symmetry-adapted settings.

Classical shadow protocols are a family of randomized measurement schemes and postprocessing techniques devised for efficient estimation of many expectation values (properties) of high-dimensional quantum states. They provide an alternative to full quantum state tomography, leveraging random measurements and succinct classical representations to achieve provable sample efficiency for a broad class of observables. The framework is particularly notable for enabling exponential improvements—sometimes interpolating between exponential and polynomial resources—when prior knowledge about symmetry or subsystem structure is available.

1. Classical Shadow Framework: Principles and Unbiased Estimation

The classical-shadow paradigm, initiated by Huang, Kueng, and Preskill (Bringewatt et al., 4 Nov 2025), is defined by three core steps:

1. Randomized Measurement Channel: On each quantum shot, one samples a unitary $U$ from a predetermined distribution (e.g., single-qubit Cliffords) and measures in the computational basis. The quantum channel

$$\mathcal{M}(\rho) = \mathbb{E}_{U}\sum_{\mathbf{b}} U^\dagger |\mathbf{b}\rangle\langle\mathbf{b}| U\, \rho\, U^\dagger|\mathbf{b}\rangle\langle\mathbf{b}| U$$

transforms the state into an ensemble of classical outcomes indexed by $(U, \mathbf{b})$.

1. Classical-Shadow Estimator: Each outcome is mapped to a single-shot estimator via

$$\hat\rho = \mathcal{M}^{-1}(U^\dagger|\mathbf{b}\rangle\langle\mathbf{b}|U)$$

where $\mathcal{M}^{-1}$ is the explicit (channel-dependent) inverse.

1. Expectation Value Prediction: For $N$ independent shadows $\{\hat\rho^{(s)}\}_{s=1}^N$, the empirical mean

$$\frac{1}{N} \sum_{s=1}^N \operatorname{Tr}(O\, \hat\rho^{(s)})$$

is an unbiased estimator of $\operatorname{Tr}(O \rho)$ for any observable $O$.

The sample complexity for estimating $M$ observables $\{ O_j \}$ (up to error $\epsilon$ and failure probability $\delta$) scales as

$$N = \mathcal{O}\left( \frac{ \log M }{ \epsilon^2 } \max_j \| O_j \|_{\rm shadow}^2 \right)$$

where $\| \cdot \|_{\rm shadow}$ is a norm set by the shadow protocol (e.g., $4^k$ for $k$-local observables with the product protocol).

2. Gauge Symmetry and Symmetry-Adapted Shadow Protocols

In lattice gauge theories, physical quantum states reside in a symmetry sector defined by local Gauss-law constraints, forming a subspace $\mathcal{H}_{\rm phys}$ of the full Hilbert space—typically of exponentially smaller dimension. Standard, symmetry-agnostic shadow protocols (e.g., the product protocol) ignore this structure, resulting in wasteful over-sampling and exponential scaling with the locality of gauge-invariant observables.

By invoking dualities—mapping the lattice gauge theory to, for example, a dual Ising model on plaquette variables—one can construct symmetry-adapted shadow protocols that only probe physically relevant degrees of freedom. The dual space has dimension $2^{V-1}$ versus $2^{2V}$ on an $N_x \times N_y$ lattice with $V = N_x N_y$, enabling polynomial resource scaling.

The paper (Bringewatt et al., 4 Nov 2025) introduces three symmetry-adapted protocols:

• Global Dual Pairs: Entangling pairs of dual qubits (plaquettes) across the lattice with Haar-random two-qubit gates preserving $\mathbb{Z}_2$ parity, mapped back to link variables via the inverse duality. Circuit depth scales as $\mathcal{O}(V^2)$.
• Local Dual Pairs: Tiling the lattice into $L \times L$ patches and applying local pairings inside each, with resource savings for observables local to patches. Circuit depth $\mathcal{O}(L^4)$, sampling cost polynomial in $L$.
• Dual Product Protocol: Employing single-qubit Clifford randomization in the dual space (with an ancilla qubit to resolve parity sectors), realizing circuit depth $\mathcal{O}(V^2)$. For observables of fixed dual-locality, the sampling cost is independent of system size.

3. Sample Complexity and Protocol-Specific Scalings

The sample complexity for estimating $M$ gauge-invariant observables $\{O_i\}$ up to error $\epsilon$ and failure probability $\delta$ is lower bounded by

$$N \ge \frac{C}{\epsilon^2} \log\left( \frac{M}{\delta} \right) \max_i \mathrm{Var}_{\rm shadow}(O_i)$$

with $\mathrm{Var}_{\rm shadow}(O)$ the channel-specific variance.

Protocol	Depth	Classical Cost/shot	Sample Complexity
Product	$1$	$O(k)$	$\tilde O(4^k/\epsilon^2)$
Global Dual Pairs	$O(V^2)$	poly$(V)$	$\tilde O(\mathrm{poly}(V)/\epsilon^2)$
Local Dual Pairs	$O(L^4)$	poly$(V)$	$\tilde O(\mathrm{poly}(L)/\epsilon^2)$
Dual Product	$O(V^2)$	$O(V)$	$\tilde O(1/\epsilon^2)$

The variance for the product protocol is upper-bounded by $4^k \|O\|_\infty^2$ for $k$-local observables, implying an exponential scaling in $k$. The dual product protocol achieves variance $4^{k'}$ where $k'$ is the dual-locality; if $k' = \mathcal{O}(1)$, the scaling in samples becomes constant in system size for fixed-observable complexity.

Exponential savings are realized precisely when the dual mapping compresses the support of gauge-invariant observables.

4. Worked Example: $\mathbb{Z}_2$ Lattice Gauge Theory

In the concrete setting of a $2+1$D $\mathbb{Z}_2$ lattice gauge theory, link qubits correspond to gauge field degrees of freedom with Hamiltonian

$$H = -\sum_{\square} \prod_{\ell \in \square} \sigma^z_\ell - g \sum_{\ell} \sigma^x_\ell$$

and Gauss-law constraints $\prod_{\ell\ni s} \sigma^x_\ell = +1$ at each site $s$. The duality to a plaquette Ising model defines dual qubits with mapping:

$$\left[ \prod_{\ell\in\square} \sigma^z_\ell \right] \mapsto Z_p,\quad \sigma^x_\ell \mapsto X_p\, X_{p'}$$

with global parity constraints set by boundary conditions.

In this setting:

• Product protocol is trivial to implement (circuit depth 1), but sample complexity is exponential in the Pauli weight $k$ of the observable—a poor match for gauge-invariant operators whose support grows with $V=N_xN_y$.
• Global dual pairs achieve polynomial scaling in $V$ by mapping the measurement problem to the dual space, but at the price of deep circuits (ribbon operators of length $\mathcal{O}(V)$ and depth $\mathcal{O}(V^2)$).
• Local dual pairs localize both circuit depth and sample complexity to observables' spatial support ($O(L^4)$ and $\operatorname{poly}(L)$), optimal for local probes.
• Dual product protocol enables constant sample complexity for dual-local observables, saturating the lower bound at the cost of long-range, albeit structured, circuit operations.

5. Protocol Selection: Trade-offs and Implementation Criteria

The following trade-offs are essential for protocol selection:

• Product protocol: Optimal for devices limited in two-qubit gate depth, or for observables of small Pauli weight.
• Global dual pairs: Useful for arbitrary gauge-invariant observables or for small system sizes; balance between sampling and circuit depth may be favorable in intermediate regimes.
• Local dual pairs: Provide the best balance for highly local observables in extended systems, with both circuit and sampling resources scaling polynomially with the observable's spatial size $L$.
• Dual product protocol: Asymptotically optimal sample complexity when computational overhead (circuit depth $O(V^2)$) is not limiting, e.g., in fault-tolerant or measurement-limited regimes.

Practical considerations dictate that in NISQ architectures, protocols requiring deep or highly nonlocal circuits (e.g., global dual pairs, dual product) may be infeasible; then one reverts to product or localized protocols, accepting higher sample complexity as a trade-off for circuit simplicity.

6. Significance, Limitations, and Outlook

The development of gauge-symmetry-adapted classical shadow protocols demonstrates that detailed exploitation of prior knowledge—specifically, symmetry structure—can lead to exponential or polynomial reductions in experimental resource requirements for quantum property estimation.

However, the improvement in sample complexity often comes at the cost of increased quantum circuit depth and classical postprocessing complexity. Implementations on near-term devices are limited by circuit errors and the ability to realize long-range or non-product entangling gates. The dual product and global dual pairs protocols, while asymptotically optimal in sampling, require engineered circuits of depth $\mathcal{O}(V^2)$ built from Pauli strings acting over the lattice.

Further advances are expected in:

• The design of symmetry-adapted protocols for other non-Abelian gauge theories and higher symmetry groups.
• Hybrid schemes, interpolating between circuit depth and sample complexity, to match the capabilities of evolving quantum hardware.
• Automated protocol selection based on problem instance size, desired observables, and architecture constraints.

Classical shadow protocols in the presence of symmetry—especially as applied to gauge theories—set a new quantitative standard for sample-efficient quantum property estimation, and their trade-offs will guide both future theoretical development and the architecture of large-scale quantum simulators (Bringewatt et al., 4 Nov 2025).