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arXiv:2609.07032

A Single Fixed Shallow Circuit for Classical Shadows of Arbitrary n-Qubit States

Yu Wang, Xiuwu Zhu

quant-phSubmitted Sep 7, 2026• Announced Sep 8, 2026• 0 citations
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Abstract

Classical shadows extract quantum properties from reusable classical records, typically obtained through randomized measurement settings. Although individual settings may be shallow, switching among them introduces control, calibration, and reconfiguration costs beyond conventional metrics. Here we construct, for every $n$, a single fixed shallow quantum analyzer whose Born outcomes replace externally sampled settings as labels for reusable shadow snapshots. The analyzer combines a freshly prepared $n$-qubit fiducial register $A$ with parallel Bell readout of $A$ and the unknown system $S$, producing one $2n$-bit record per copy. The same circuit realizes a rank-one minimal informationally complete measurement: one fixed setting replaces the $3^n$ local-Pauli settings conventionally used for complete reconstruction while retaining the minimum $d^2$ outcomes with $d=2^n$. Its Pauli-diagonal frame admits an analytic inverse. For fixed Hermitian observables, the Haar-averaged conditional variance has a dimension-independent coefficient, whereas the state-uniform coefficient is dimension dependent. For $n\ge3$, worst-state Pauli variances remain bounded in the axial sector and scale as $Θ(d)$ in the mixed sector. Fiducial preparation before system contact uses $n-1$ arbitrary two-qubit gates, no work qubits beyond $A$, and logarithmic depth under all-to-all connectivity. The unknown system undergoes one parallel system-ancilla entangling layer followed by local Hadamards and readout. Measurement-assisted preparation achieves $O(1)$ adaptive quantum depth using $n+1$ extra qubits in the heralded route, or $O(n\log n)$ auxiliaries for deterministic completion. These tradeoffs show that part of the measurement-setting randomness and control complexity normally supplied shot by shot can instead be compiled into a fixed reusable analyzer.

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Classical shadows extract quantum properties from reusable classical records, typically obtained through randomized measurement settings. Although individual settings may be shallow, switching among them introduces control, calibration, and reconfiguration costs beyond conventional metrics. Here we construct, for every nnn, a single fixed shallow quantum analyzer whose Born outcomes replace externally sampled settings as labels for reusable shadow snapshots. The analyzer combines a freshly prepared nnn-qubit fiducial register AAA with parallel Bell readout of AAA and the unknown system SSS, producing one 2n2n2n-bit record per copy. The same circuit realizes a rank-one minimal informationally complete measurement: one fixed setting replaces the 3n3^n3n local-Pauli settings conventionally used for complete reconstruction while retaining the minimum d2d^2d2 outcomes with d=2nd=2^nd=2n. Its Pauli-diagonal frame admits an analytic inverse. For fixed Hermitian observables, the Haar-averaged conditional variance has a dimension-independent coefficient, whereas the state-uniform coefficient is dimension dependent. For n≥3n\ge3n≥3, worst-state Pauli variances remain bounded in the axial sector and scale as Θ(d)Θ(d)Θ(d) in the mixed sector. Fiducial preparation before system contact uses n−1n-1n−1 arbitrary two-qubit gates, no work qubits beyond AAA, and logarithmic depth under all-to-all connectivity. The unknown system undergoes one parallel system-ancilla entangling layer followed by local Hadamards and readout. Measurement-assisted preparation achieves O(1)O(1)O(1) adaptive quantum depth using n+1n+1n+1 extra qubits in the heralded route, or O(nlog⁡n)O(n\log n)O(nlogn) auxiliaries for deterministic completion. These tradeoffs show that part of the measurement-setting randomness and control complexity normally supplied shot by shot can instead be compiled into a fixed reusable analyzer.
We provide practical and powerful schemes for learning many properties of an unknown n-qubit quantum state using a sparing number of copies of the state. Specifically, we present a depth-modulated randomized measurement scheme that interpolates between two known classical shadows schemes based on random Pauli measurements and random Clifford measurements. These can be seen within our scheme as the special cases of zero and infinite depth, respectively. We focus on the regime where depth scales logarithmically in n and provide evidence that this retains the desirable properties of both extremal schemes whilst, in contrast to the random Clifford scheme, also being experimentally feasible. We present methods for two key tasks; estimating expectation values of certain observables from generated classical shadows and, computing upper bounds on the depth-modulated shadow norm, thus providing rigorous guarantees on the accuracy of the output estimates. We consider observables that can be written as a linear combination of poly(n) Paulis and observables that can be written as a low bond dimension matrix product operator. For the former class of observables both tasks are solved efficiently in n. For the latter class, we do not guarantee efficiency but present a method that works in practice; by variationally computing a heralded approximate inverses of a tensor network that can then be used for efficiently executing both these tasks.
Reliable and scalable readout strategies are essential for quantum technologies. As quantum processors grow, extracting useful information must remain feasible without measurement circuits becoming a dominant bottleneck. Randomized measurements and classical shadows provide a powerful route, but global estimation is conventionally associated with highly random ensembles that require increasing circuit depth and hence substantial experimental overhead. In this work, we show that substantially less randomness suffices when the readout is meaningfully adapted to the quantities being estimated. We introduce shallow phase shadows, based on a sparse Clifford-IQP ensemble, and prove efficient global estimation of stabilizer-state fidelities despite the ensemble not forming an approximate relative-error design. On all-to-all architectures, the protocol admits a constant-depth implementation using mid-circuit measurements and classical feedforward, or logarithmic depth without auxiliary systems. The protocol requires only controlled-phase entangling gates and offers a tunable trade-off between circuit resources and estimation accuracy, making it particularly amenable to experimentally relevant architectures with long-range connectivity. Our results show that scalable quantum readout need not reproduce generic randomness: task-adapted randomization can enable substantially shallower global characterization protocols.
Classical shadows are a versatile tool to probe many-body quantum systems, consisting of a combination of randomised measurements and classical post-processing computations. In a recently introduced version of the protocol, the randomization step is performed via unitary circuits of variable depth t , defining the so-called shallow shadows. For sufficiently large t , this approach allows one to get around the use of non-local unitaries to probe global properties such as the fidelity with respect to a target state or the purity. Still, shallow shadows involve the inversion of a many-body map, the measurement channel, which requires non-trivial computations in the post-processing step, thus limiting its applicability when the number of qubits N is large. In this work, we put forward a simple approximate post-processing scheme where the infinite-depth inverse channel is applied to the finite-depth classical shadows and study its performance for fidelity and purity estimation. The scheme allows for different circuit connectivity, as we illustrate for geometrically local circuits in one and two spatial dimensions and geometrically non-local circuits made of two-qubit gates. For the fidelity, we find that the resulting estimator coincides with a known linear cross-entropy, achieving an arbitrary small approximation error δ at depth t=O(log⁡(N/δ)) (independent of the circuit connectivity). For the purity, we show that the estimator becomes accurate at a depth O(N) . In addition, at those depths, the variances of both the fidelity and purity estimators display the same scaling with N as in the case of global random unitaries. We establish these bounds by analytic arguments and extensive numerical computations in several cases of interest. Our work extends the applicability of shallow shadows to large system sizes and general circuit connectivity.
Approximate unitary designs reproduce the statistics of Haar-random unitaries to a given order and accuracy. Recent constructions realize such designs with logarithmic-depth circuits, enabling shallow measurement protocols for various quantum state-learning tasks while preserving performance. These results raise the question of whether approximate designs can replace exact designs more generally. We show that even exponentially small design error need not preserve the learning guarantees of exact designs. This motivates deriving learning guarantees directly from the measurement circuit structure. For observable estimation with classical shadows, we prove that logarithmic-depth two-layer Clifford circuits yield an unbiased estimator matching the global Clifford variance scaling for every state and Hermitian observable. Beyond observable estimation, this bound allows shallow measurements to retain global Clifford guarantees for other tasks, including setting-efficient tomography, mixed-state metrology, and stabilizer structure learning. We complement these statistical guarantees with a compact, exact tensor-network representation of the inverse shadow channel. When measurement bases are reused, approximate unitary designs of arbitrarily high order need not uniformly match the Haar variance in the many-shot limit. For the all-to-all random two-local circuits considered, doing so requires nearly linear depth. Together, our results reveal the capabilities and limitations of shallow quantum state learning, highlighting the distinction between approximating Haar randomness and reproducing its learning guarantees.
Extracting information efficiently from quantum systems is crucial for quantum information processing. Classical shadows enable predicting many properties of arbitrary quantum states using few measurements. While random single-qubit measurements are experimentally friendly and suitable for learning low-weight Pauli observables, they perform poorly for nonlocal observables. Introducing a shallow random quantum circuit before measurements improves sample efficiency for high-weight Pauli observables and low-rank properties. However, in practice, these circuits can be noisy and bias the measurement results. Here, we propose the robust shallow shadows, which employs Bayesian inference to learn and mitigate noise in postprocessing. We analyze noise effects on sample complexity and the optimal circuit depth. We provide theoretical guarantees for the success of error mitigation under a wide class of noise processes. Experimental validation on a superconducting quantum processor confirms the advantage of our method, even in the presence of realistic noise, over single-qubit measurements for predicting diverse state properties, such as fidelity and entanglement entropy. Our protocol thus offers a scalable, robust, and sample-efficient method for quantum state characterization on near-term quantum devices. Fast and reliable characterisation of quantum systems is a key part of quantum technologies development. Here, the authors propose and demonstrate a way to embed noise characterisation in classical shadow estimation of nonlocal properties, enabling an efficient way to extract information from noisy quantum systems.
Quantum state tomography is a fundamental technique for characterizing quantum systems, but its scalability remains a significant challenge. This paper investigates an efficient alternative using classical shadows to reconstruct a Bell state with high fidelity. Classical shadows provide a compressed representation of a quantum state using randomized measurements, reducing the measurement complexity compared to full quantum tomography when full state reconstruction is not required. We employed a quantum circuit to generate a Bell state and utilize 1000 snapshots to construct its classical shadow. The reconstructed density matrix is evaluated using fidelity and norm difference metrics against the ideal Bell state. Our analysis demonstrates that as the number of snapshots increases, the fidelity of reconstruction stabilizes around 0.98-1.0, with the norm difference decreasing accordingly. The results also confirm the convergence of reconstructed states towards the ideal Bell state, highlighting the efficiency and accuracy of classical shadows in quantum state estimation. Shallow shadow tomography uses shallow random circuits to estimate global quantum state properties on noisy hardware. By combining low-depth entangling gates with randomized measurements and noise-aware postprocessing, it reduces sample complexity compared to Pauli shadows while remaining experimentally feasible. Recent experiments on superconducting processors confirmed up to fivefold measurement savings.