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arXiv:2610.04621

Exact Clifford Optimality for Two-Sided Locally Randomized Classical Shadows

Xiaotian Nie, Tao Zhang, Yadong Wu, Linghui Chen

quant-phSubmitted Oct 3, 2026• Announced Oct 6, 2026• 0 citations• Revised Oct 29, 2026 (v2)
v1 · Oct 3, 2026v2 · Oct 29, 2026
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Abstract

Entangling measurements can reduce the cost of learning many-body correlations. We consider classical shadows in which a fixed unitary on a $k$-qubit region is sandwiched between two layers of random single-qubit Clifford gates, and ask which middle unitary best estimates Pauli strings with full support on the region. Combining symplectic support counting with a constraint from operator-size parity, we prove that within this architecture the minimum squared shadow norm over all Clifford middle unitaries is $C_k^{\rm opt}=4\cdot 9^k/(3\cdot 5^k+3^k-6)$ for odd $k$ and $C_k^{\rm opt}=4\cdot 9^k/(3\cdot 5^k-3^k+2)$ for even $k$. A dual-axis rotation circuit $U_*$ attains this minimum for every $k$. Asymptotically $C_k^{\rm opt}\simeq(4/3)\cdot(9/5)^k$: the exponential base of Wu et al.'s contractive unitary is optimal, and the prefactor improves from $2$ to $4/3$. For $1\leq k\leq7$, exact certificates show that $U_*$ is optimal over the entire unitary group $\mathrm U(2^k)$, and we conjecture that this holds for every $k$.

Comments
15 pages, 5 figures

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Entangling measurements can reduce the cost of learning many-body correlations. We consider classical shadows in which a fixed unitary on a kkk-qubit region is sandwiched between two layers of random single-qubit Clifford gates, and ask which middle unitary best estimates Pauli strings with full support on the region. Combining symplectic support counting with a constraint from operator-size parity, we prove that within this architecture the minimum squared shadow norm over all Clifford middle unitaries is Ckopt=4⋅9k/(3⋅5k+3k−6)C_k^{\rm opt}=4\cdot 9^k/(3\cdot 5^k+3^k-6)Ckopt​=4⋅9k/(3⋅5k+3k−6) for odd kkk and Ckopt=4⋅9k/(3⋅5k−3k+2)C_k^{\rm opt}=4\cdot 9^k/(3\cdot 5^k-3^k+2)Ckopt​=4⋅9k/(3⋅5k−3k+2) for even kkk. A dual-axis rotation circuit U∗U_*U∗​ attains this minimum for every kkk. Asymptotically Ckopt≃(4/3)⋅(9/5)kC_k^{\rm opt}\simeq(4/3)\cdot(9/5)^kCkopt​≃(4/3)⋅(9/5)k: the exponential base of Wu et al.'s contractive unitary is optimal, and the prefactor improves from 222 to 4/34/34/3. For 1≤k≤71\leq k\leq71≤k≤7, exact certificates show that U∗U_*U∗​ is optimal over the entire unitary group U(2k)\mathrm U(2^k)U(2k), and we conjecture that this holds for every kkk.
Efficiently learning expectation values of a quantum state using classical shadow tomography has become a fundamental task in quantum information theory. In a classical shadows protocol, one measures a state in a chosen basis W after it has evolved under a unitary transformation randomly sampled from a chosen distribution U. In this work we study the case where U corresponds to either local or global orthogonal Clifford gates, and W consists of real-valued vectors. Our results show that for various situations of interest, this ``real'' classical shadow protocol improves the sample complexity over the standard scheme based on general Clifford unitaries. For example, when one is interested in estimating the expectation values of arbitrary real-valued observables, global orthogonal Cliffords decrease the required number of samples by a factor of two. More dramatically, for k-local observables composed only of real-valued Pauli operators, sampling local orthogonal Cliffords leads to a reduction by an exponential-in-k factor in the sample complexity over local unitary Cliffords. Finally, we show that by measuring in a basis containing complex-valued vectors, orthogonal shadows can, in the limit of large system size, exactly reproduce the original unitary shadows protocol.
The real classical shadows protocol of West et al. replaces the unitary (Clifford) ensemble of the Huang--Kueng--Preskill scheme by the orthogonal (real Clifford) ensemble, and for symmetric observables achieves strictly smaller estimator variances: a factor approaching two for global evolution and an exponential factor (3/2)k(3/2)^k(3/2)k for kkk-local real Pauli observables. Real hardware, however, never implements the ideal evolution. Building on the noisy classical shadows framework of Koh and Grewal, we give a complete theory of the real classical shadows protocol in the presence of a known completely positive trace-preserving noise channel acting after the orthogonal evolution. We derive the noisy global and local orthogonal shadow channels from first principles using the Weingarten calculus of the orthogonal group, prove that each is a depolarizing channel acting on the symmetric (respectively locally symmetric) component of its input, and derive from it the exact single-shot variance in closed form, together with the associated shadow seminorm, two-sided bounds on it, and the resulting sample-complexity guarantees. We prove that the noiseless sample-complexity advantages survive intact under noise. Because the variances are exact rather than bounded, the ratio is controlled by a single dimensionless parameter, which gives a closed-form criterion for when the factor of two is attainable: both the second-moment and the variance ratio reach it exactly when the observable's norm profile grows, and the noise enters that limit only through a factor lying within 2/(d+2)2/(d+2)2/(d+2) of two, so the advantage is uniform in the noise. For rank-one targets it is provably unattainable, saturating strictly below two. The local real-Pauli advantage remains (3/2)k(3/2)^k(3/2)k. We treat complex measurement bases through a reality parameter and a transposed-noise scalar, recovering unitary shadows in the appropriate limit.
Abstract Efficiently learning expectation values of a quantum state using classical shadow tomography has become a fundamental task in quantum information theory. In a classical shadows protocol, one measures a state in a chosen basis W after it has evolved under a unitary transformation randomly sampled from a chosen distribution U . In this work we study the case where U corresponds to either local or global orthogonal Clifford gates, and W consists of real-valued vectors. Our results show that for various situations of interest, this ‘real’ classical shadow protocol improves the sample complexity over the standard scheme based on general Clifford unitaries. For example, when one is interested in estimating the expectation values of arbitrary real-valued observables, global orthogonal Cliffords typically decrease the required number of samples by a factor of two. More dramatically, for k -local observables composed only of real-valued Pauli operators, sampling local orthogonal Cliffords leads to a reduction by an exponential-in- k factor in the sample complexity over local unitary Cliffords. Finally, we show that by measuring in a basis containing complex-valued vectors, orthogonal shadows can, in the limit of large system size, exactly reproduce the original unitary shadows protocol.
Encoding quantum information with low circuit overhead is a fundamental challenge in fault-tolerant quantum computation. Random circuits provide a natural mechanism for rapidly spreading logical information through simple gates applied in parallel. Brown and Fawzi showed that random Clifford circuits on two-qubit Clifford gates provide such encoders that achieve the quantum Gilbert-Varshamov rate-distance tradeoff with depth O(log⁡3n)O(\log^3 n)O(log3n). We show that the same asymptotic tradeoff is attained in optimal O(log⁡n)O(\log n)O(logn) depth under a gate distribution with a more restricted support. For every fixed δ>0δ>0δ>0 and sufficiently large nnn, if kn<1−H(dn)−dnlog⁡23−δ\frac kn < 1 - H(\frac{d}{n}) - \frac{d}{n}\log_2 3 - δnk​<1−H(nd​)−nd​log2​3−δ, we can construct random circuits of depth O(log⁡n)O(\log n)O(logn) which define, with high probability, an [n,k][n,k][n,k] stabilizer code of distance at least d+1d+1d+1, which matches the Ω(log⁡n)Ω(\log n)Ω(logn) light-cone lower bound for linear distance encoders. Our ensemble employs a random matching circuit architecture consisting of TTT independent permutation-invariant layers. In each layer, the qubits are paired up by a uniformly random perfect matching, and a random independent two-qubit Clifford gate is applied to each pair. The gate distribution need not be uniform over, or even have full support on, the two-qubit Clifford group; rather, we allow for very general distributions on Clifford gates satisfying three regularity conditions. In particular, the construction can be implemented using n/2n/2n/2 CNOT gates on randomly matched pairs in each layer, with parallel one-qubit Clifford twirls. These regularity conditions allow us to reduce the second-moment dynamics of our random circuits to a reversible Markov chain on binary support strings. We establish logarithmic hitting-time bounds for this Markov chain and comparisons of its stationary distribution to prove the coding properties of the circuits.
Properties of quantum systems can be estimated using classical shadows, which implement measurements based on random ensembles of unitaries. Originally derived for global Clifford unitaries and products of single-qubit Clifford gates, practical implementations are limited to the latter scheme for moderate numbers of qubits. Beyond local gates, the accurate implementation of very short random circuits with two-local gates is still experimentally feasible and, therefore, interesting for implementing measurements in near-term applications. In this work, we derive closed-form analytical expressions for shadow estimation using brickwork circuits with two layers of parallel two-local Haar-random (or Clifford) unitaries. Besides the construction of the classical shadow, our results give rise to sample-complexity guarantees for estimating Pauli observables.We then compare the performance of shadow estimation with brickwork circuits to the established approach using local Clifford unitaries and find improved sample complexity in the estimation of observables supported on sufficiently many qubits.
Predicting properties of large-scale quantum systems is crucial for the development of quantum science and technology. Shadow estimation is an efficient method for this task based on randomized measurements, where many-qubit random Clifford circuits are used for estimating global properties like quantum fidelity. Here we introduce the minimal Clifford measurement (MCM) to reduce the number of possible random circuits to the minimum, while keeping the effective postprocessing channel in shadow estimation. In particular, we show that MCM requires 2n+1{2}^{n}+12n+1 distinct Clifford circuits, and it can be realized by mutually unbiased bases, with nnn as the total qubit number. By applying the ZZZ-tableau formalism, this ensemble of circuits can be synthesized to the $\ensuremath{-}\mathrm{S}\ensuremath{-}\mathrm{CZ}\ensuremath{-}\mathrm{H}\ensuremath{-}$ structure, which can be decomposed to $2n\ensuremath{-}1$ fixed circuit modules, and the total circuit depth is at most n+1n+1n+1. Compared to the original Clifford measurements, our MCM reduces the circuit complexity and the compilation costs. In addition, we find the sampling advantage of MCM on estimating off-diagonal operators, and extend this observation to the biased-MCM scheme to enhance the sampling improvement further.