Classical Shadows Algorithm Overview

• The Classical Shadows Algorithm is a protocol that uses randomized joint, entangling measurements to produce a concise classical representation of a quantum state for efficient observable estimation.
• The PECS extension targets the principal eigenstate of a mixed quantum state, achieving near-optimal sample complexity and unifying pure, mixed, and ground-state tomography regimes.
• The method leverages symmetry and averaging techniques to reduce variance and cost, overcoming state preparation challenges while accommodating various spectral purity scenarios.

A classical shadows algorithm is a randomized measurement protocol that produces a succinct classical representation of a quantum state, enabling simultaneous estimation of a large collection of expectation values with rigorous sample-complexity guarantees. The principal eigenstate classical shadows (PECS) protocol—also called the principal eigenstate shadow—extends this methodology to the task of learning a classical surrogate for the top eigenstate of a mixed quantum state, allowing efficient estimation of expectation values on the principal eigenvector even when the underlying state is only partially pure. PECS achieves near-optimal sample complexity over a full range of principal eigenvalue parameters and unifies the regimes of pure-state tomography, mixed-state shadow tomography, and top-eigenvector learning with joint measurements.

1. Problem Definition and Principal Eigenstate Setting

Given an unknown density matrix $\rho$ acting on a $d$-dimensional Hilbert space, suppose $\rho$ possesses a unique largest eigenvalue $\lambda>1/2$, associated with a rank-one projector $\phi=|\phi\rangle\langle\phi|$, and spectral gap $2\lambda-1>0$ to the rest. Denoting the principal deviation by $\eta=1-\lambda<1/2$, the goal is to efficiently learn a classical description $\hat\phi$ of $|\phi\rangle$ such that, for any observable $O$ with $\|O\|\le 1$ or squared Hilbert–Schmidt norm $\|O\|_2^2\le B$, one can accurately estimate $\langle\phi|O|\phi\rangle$ to additive accuracy $\epsilon$ with failure probability at most $\delta$, using as few copies of $\rho$ as possible.

This setting arises naturally in applications such as principal component analysis of quantum states, learning ground states of mixed-state ensembles, and quantum algorithms for dominant eigenvector estimation. A key constraint modeled in PECS is that state preparation is expensive, but collective (joint) measurements on small batches of copies are allowed (Grier et al., 22 May 2024).

2. Joint Symmetric Measurement Protocol

The PECS methodology is based on a generalized classical shadows protocol utilizing joint, entangling measurements across $n$ copies of $\rho$. One performs the standard symmetric POVM—a continuous measurement with elements

$$\left\{F_\psi = \binom{n+d-1}{n} \, |\psi\rangle\langle\psi|^{\otimes n}\,d\psi,\ \psi\in\mathbb{CP}^{d-1}\right\}$$

plus a fail element $F_{\rm fail}=I-\Pi_{\text{sym}}$, where $\Pi_{\text{sym}}$ projects onto the $n$-fold symmetric subspace.

The experiment samples $n$ copies of $\rho$ and performs this symmetric POVM:

• If the outcome is $\psi\neq$ "fail", the protocol outputs a classical description of the observed pure state $\psi$ (the Haar outcome).
• If the outcome is "fail", the experiment is repeated.

This protocol reduces to single-copy classical shadows for $n=1$, but crucially, for $n>1$, the symmetric joint measurement amplifies overlap with the unknown principal component $\phi$, enabling efficient variance reduction for $\lambda\gg1/2$ (Grier et al., 22 May 2024).

3. Classical Estimation and Averaging Procedure

Upon each successful $n$-copy measurement, the algorithm forms the raw estimator

$$M = \frac{(d+n)\,|\psi\rangle\langle\psi| - I_d}{n}\;.$$

Averaging theory (Kitaev–Massar–Popescu moments) shows that $E[M]=M_1$, where $M_1$ is an unbiased proxy for $\phi$ constructed from $n$-copy moments of $\rho$ and the symmetric subspace projector.

To reduce variance, the procedure is repeated $b$ times (each on fresh blocks of $n$ copies); the final estimator is

$\hat\phi = \frac{1}{b}\sum_{j=1}^b M^{(j)}.$

To estimate $\langle\phi|O|\phi\rangle$ for a target $O$, output $\operatorname{Tr}(O\hat\phi)$. For simultaneous estimation of $M$ observables, a median-of-means protocol is applied with $O(\log(M/\delta))$ independent shadow estimators (Grier et al., 22 May 2024).

4. Sample Complexity and Three-Regime Performance

The PECS protocol’s sample complexity for target additive error $\epsilon$ exhibits three parametric regimes as a function of the principal deviation $\eta=1-\lambda$:

• Regime I: Nearly pure $\phi$ ($\eta \le 1/s^*$)

$$N = \Theta(s^*) = \Theta(\sqrt{B}/\epsilon + 1/\epsilon^2)$$

where $s^* = \Theta(\sqrt{B}/\epsilon + 1/\epsilon^2)$. This matches the optimal pure-state shadows complexity and the lower bound for “state compression” (Grier et al., 2022).

• Regime II: Moderately pure ($1/s^* \lesssim \eta \le \sqrt{\epsilon}$)

$N = \Theta\big((B\eta + 1)/\epsilon^2\big)$

• Regime III: Fairly mixed ($\sqrt{\epsilon} \lesssim \eta < 1/2$)

$N = \Theta(B\eta/\epsilon^2 + \eta/\epsilon^{5/2})$

To guarantee accuracy $\epsilon$ with probability $\ge 1-\delta$ for $M$ observables, one multiplies $N$ by $O(\log(M/\delta))$ due to the median-of-means bound. As $\lambda \to 1$ ($\eta\to0$), the sample complexity recovers the pure-state bound $\Theta(\sqrt{B}/\epsilon + 1/\epsilon^2)$; for highly mixed states, PECS remains optimal among protocols using joint measurements (Grier et al., 22 May 2024).

5. Comparative Analysis and Optimality

PECS improves over and sometimes strictly outperforms all other natural strategies in the principal-eigenstate learning regime:

• Single-copy classical shadows require $N=\Theta(B/\epsilon^2)$ for $\eta\ll\epsilon$ and degrade to $N=\Theta(B\eta/\epsilon^3)$ for $\eta\gg\epsilon$.
• Purification followed by shadows (first apply $k$-copy purification to reduce $\eta\to O(\eta/k)$, then standard shadows) requires $N=\Omega(B\eta/\epsilon^3)$ in typical regimes.
• Purification plus single joint measurement (no averaging) gives $N=\Theta(s^*)$ for $\eta\le1/s^*$ and $N=\Theta(\eta(s^*)^2)$ for $\eta\ge1/s^*$, suboptimal compared to the three-regime PECS strategy.

A key theorem asserts that PECS is sample-optimal for $\eta\lesssim1/s^*$—including the pure limit—and always at least as good as hybrid alternatives even as the spectral gap closes (Grier et al., 22 May 2024).

6. Pseudocode Summary and Robustness Analysis

Algorithm PECS (single observable version)

1. (Optional) Estimate $\eta$ using 2-copy symmetric measurements (fail rate $\approx\eta$).
2. Select regime (I/II/III) given $\eta$, set purification parameter $k$, joint measurement block size $n$, and repetition count $b$.
3. (If $k>1$) Apply purification using $k$ copies to obtain a purer $\rho'$ with $\eta'=O(\eta/k)$.
4. For $j=1,\ldots, b$:
   • Measure $n$ fresh copies of $\rho'$ via the symmetric POVM to obtain $\psi_j$ or "fail".
   • If "fail", discard the block and repeat.
   • Compute $M^{(j)} = ((d+n)\psi_{j}\psi_{j} - I)/n$.
5. Output $\hat\phi = (1/b)\sum_{j=1}^b M^{(j)}$.
6. Estimate $\langle\phi|O|\phi\rangle$ via $\operatorname{Tr}(O\hat\phi)$.
7. (If estimating $M$ observables) Use median-of-means across $O(\log(M/\delta))$ runs.

Key theorem (joint measurement robustness): For $\rho = (1-\eta)\phi + \eta\sigma$ on $n$ copies, the symmetric POVM succeeds with probability at least $(1-\eta)^{n-1}$. On success, the raw estimator $M$ satisfies bias $O(\eta/n)$ and variance $O(1/n^2)$—so bias/variance are efficiently controlled by block size $n$ and spectral purity (Grier et al., 22 May 2024).

Proof techniques utilize Schur-Weyl duality, analysis of moments in symmetric subspaces, explicit computation of conditional distributions on eigenvalue counts, and the derivation of closed-form bias and variance for both the success conditioned random estimator and its implications for subsequent observable estimation.

7. Extensions, Limitations, and Outlook

PECS provides a sample-optimal protocol for principal eigenvector learning in the joint measurement setting, smoothly interpolating between previously distinct shadow-tomography regimes (pure-state, mixed-state, ground-state learning). While the focus is on the unique top eigenstate scenario ($\lambda>1/2$ and spectral gap), extensions to degenerate or near-degenerate principal eigenspaces may require further developments, as does adaptation to settings with hardware-induced noise or constraints on feasible entangling measurements.

The algorithm’s optimality and efficiency rely on the availability of collective symmetric measurements, which are natural in many photonic, atomic, and trapped-ion architectures supporting permutation-symmetric POVMs. As the field advances, further generalizations to higher-rank eigenprojectors, dynamical learning of time-evolving dominant components, and error-mitigated or symmetry-adapted PECS protocols are plausible research directions (Grier et al., 22 May 2024).