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arXiv:2609.03925

Ultra-Precise Quantum Projective Designs in Constant Depth

Qingyue Zhang, Junjie Chen, Zhou You, You Zhou

quant-phSubmitted Sep 3, 2026• Announced Sep 4, 2026• 0 citations
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Abstract

Random quantum objects are powerful resources for quantum information processing, yet exact Haar randomness is costly and typically unnecessary. We introduce an explicit sparse commuting circuit ensemble on $n$ qubits that reproduces low-order Haar moments in the stringent relative-error sense. The circuit consists of a sparse Clifford phase layer followed by independent single-qubit Clifford gates. Acting on a simple product state, the resulting ensemble forms $ε$-approximate projective $2$- and $3$-designs in relative error, with the required logarithmic interaction degree being asymptotically optimal within this circuit family. It admits an ancilla-free implementation of quantum depth $O(\log(n/ε))$ on an all-to-all architecture, as well as an adaptive constant-depth implementation---in fact, depth seven---using $O(n\log(n/ε))$ ancilla qubits. Departing from existing shallow-design paradigms, our analysis exploits the intrinsic moment structure of commuting phase circuits; at third order, this requires a new block decomposition and combinatorial analysis that also suggests a route toward higher-order shallow designs. Our results show that precise Haar-like statistics can emerge from sparse commuting dynamics with remarkably low quantum resources, with applications to randomized characterization, quantum metrology, quantum algorithms, and many-body physics.

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9+35 pages, 3 figures

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Random quantum objects are powerful resources for quantum information processing, yet exact Haar randomness is costly and typically unnecessary. We introduce an explicit sparse commuting circuit ensemble on nnn qubits that reproduces low-order Haar moments in the stringent relative-error sense. The circuit consists of a sparse Clifford phase layer followed by independent single-qubit Clifford gates. Acting on a simple product state, the resulting ensemble forms εεε-approximate projective 222- and 333-designs in relative error, with the required logarithmic interaction degree being asymptotically optimal within this circuit family. It admits an ancilla-free implementation of quantum depth O(log⁡(n/ε))O(\log(n/ε))O(log(n/ε)) on an all-to-all architecture, as well as an adaptive constant-depth implementation---in fact, depth seven---using O(nlog⁡(n/ε))O(n\log(n/ε))O(nlog(n/ε)) ancilla qubits. Departing from existing shallow-design paradigms, our analysis exploits the intrinsic moment structure of commuting phase circuits; at third order, this requires a new block decomposition and combinatorial analysis that also suggests a route toward higher-order shallow designs. Our results show that precise Haar-like statistics can emerge from sparse commuting dynamics with remarkably low quantum resources, with applications to randomized characterization, quantum metrology, quantum algorithms, and many-body physics.
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.
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.
We introduce magic-augmented Clifford circuits—architectures in which Clifford circuits are preceded and/or followed by constant-depth circuits of non-Clifford (“magic”) gates—as a resource-efficient way to realize approximate k -designs, with reduced circuit depth and usage of magic. We prove that shallow-Clifford circuits, when augmented with constant-depth circuits of magic gates, can generate approximate unitary and state k -designs with ε relative error. The total circuit depth for these constructions on N qubits is O ( log  ( N / ε ) ) + 2 O ( k log  k ) in one dimension and O ( log  log  ( N / ε ) ) + 2 O ( k log  k ) in all-to-all circuits using ancillas, which improves upon previous results for small k ≥ 4 . Furthermore, our construction of relative-error state k -designs only involves states with strictly local magic. The required number of magic gates is parametrically reduced when considering k -designs with bounded additive error. As an example, we show that shallow-Clifford circuits followed by O ( k 2 ) single-qubit magic gates, independent of system size, can generate an additive-error state k -design. We develop a classical statistical mechanics description of our random circuit architectures, which provides a quantitative understanding of the required depth and number of magic gates for additive-error state k -designs. We also prove no-go theorems for various architectures to generate designs with bounded relative error.
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.
Unitary designs provide finite-moment approximations to Haar-random unitaries, with wide-ranging applications across physics and quantum information, from scrambling and black-hole dynamics to foundational primitives in quantum algorithms. Strong unitary designs capture a more demanding operational notion of approximation, requiring indistinguishability from Haar randomness even for quantum algorithms that may access a unitary not only in the forward direction, but also through its inverse, transpose, and complex conjugate. Motivated by the physical requirement that scrambling arise within the system itself, Schuster, Ma, Lombardi, Brandão, and Huang (arXiv:2509.26310) left open whether strong unitary designs can be generated in logarithmic depth using only the system qubits. For every fixed design order kkk and measurable-error tolerance, we construct strong approximate unitary kkk-designs in optimal Θ(log⁡n)Θ(\log n)Θ(logn) all-to-all circuit depth using only the nnn original system qubits. Our new ingredient is a logarithmic-depth Pauli-mixing bound for the perfect-matching ensemble, whose layers pair the qubits uniformly at random and apply independent random two-qubit gates. This bound controls the mixed forward-reverse two-query case, which we combine with existing design and gluing results to obtain strong unitary designs of arbitrary fixed order.
Constructing ensembles of circuits which efficiently approximate the Haar measure over various groups is a long-standing and fundamental problem in quantum information theory. Recently it was shown that one can obtain approximate designs over the unitary group with depths scaling logarithmically in the number of qubits, but that no sublinear-depth approximate designs exist over the orthogonal group. Here we derive, for any group GGG possessing an invariant state G⊗k∣Ψ⟩=∣Ψ⟩G^{\otimes k} \lvertΨ\rangle= \lvertΨ\rangleG⊗k∣Ψ⟩=∣Ψ⟩, a lower bound on the diamond distance between the kkk\textsuperscript{th} moment operator of any ensemble of elements of GGG, and that of the Haar measure over GGG. We then use this bound to prove that for many groups of interest, no subset of GGG consisting of sublinear-depth one-dimensional circuits with local gates can form an approximate kkk-design over GGG. More generally, on a DDD-dimensional lattice, our results imply that such group designs require depths scaling at least as n1/Dn^{1/D}n1/D. Moreover, for most of the groups we consider we find that such ensembles can, with high probability, be distinguished from kkk-designs by a single shot of a constant-depth measurement. Among other examples, we show that there is a constant separation between (a) the maximum depth and gate count for which no circuit can approximate even the second moment of random matchgate circuits, and (b) the depth and gate count required to implement the matchgate Haar distribution exactly. We furthermore rule out the existence of sublinear-depth 888-designs over the Clifford group. Finally, we relax the assumption of working with local gates, and prove the impossibility of obtaining approximate designs over GGG using any circuit comprised of a sublinear number of gates generated by Pauli strings.