Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:2507.17871 · Physical Review Letters

Shallow Quantum Circuit for Generating Extremely Low-Entangled Approximate State Designs

Wonjun Lee, Minki Hhan, Gil Young Cho, Hyukjoon Kwon

Published Mar 24, 2026• 1 citation
arXivHTMLPDF

Abstract

Random quantum states have various applications in quantum information science. We discover a new ensemble of quantum states that serve as an ε-approximate state t-design while possessing extremely low entanglement, magic, and coherence. These resources can reach their theoretical lower bounds, Ω(log(t/ε)), which are also proven in this Letter. This implies that, for fixed t and ε, entanglement, magic, and coherence do not scale with the system size, i.e., O(1) with respect to the total number of qubits n. Moreover, we explicitly construct an ancilla-free shallow quantum circuit for generating such states by transforming k-qubit approximate state designs into n-qubit ones without increasing the support size. The depth of such a quantum circuit, O(t[logt]^{3}log n log[1/ε]), is the most efficient among existing algorithms without ancilla qubits. A class of quantum circuits proposed in our Letter offers reduced cost for classical simulation of random quantum states, leading to potential applications in quantum information processing. As a concrete example, we propose classical shadow tomography using an estimator with superpositions between only two states, from which almost all quantum states can be efficiently certified by requiring only O(1) measurements and classical postprocessing time.

DOI
10.1103/p966-lcgs

Related papers

Designs from Magic-Augmented Clifford Circuits

Yuzhen Zhang, Sagar Vijay, Yingfei Gu +1 more

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.

arXiv:2507.02828•PRX Quantum•Jan 20, 2026•8 citations

Low-Depth Random Unitaries without Ancillae

Zhenyu Du, Siyuan Cheng, Xiongfeng Ma

Random unitaries are fundamental to quantum information and many-body physics, with widespread applications ranging from quantum learning and metrology to device benchmarking. A central pursuit is to minimize the space and circuit depth required to generate them. However, existing methods for generating low-depth random unitaries rely heavily on an extensive number of ancillary qubits, imposing severe spatial overhead. In this work, we prove that random unitaries can be generated in optimal depth without ancillae. For multiplicative-error approximate $k$-designs on $n$ qubits, our circuits achieve a depth of $\widetilde{O} (k) (\log n)^{1/δ}$ on $δ$-dimensional architectures and $\widetilde{O}(k) \log \log n$ with all-to-all connectivity. Furthermore, by introducing a general exactification lemma, we lift our construction to optimal-depth exact $k$-designs, yielding an exponential resource reduction over state-of-the-art exact constructions. Our results minimize the space-time costs for a wide range of quantum protocols.

quant-pharXiv:2609.06528•Sep 8, 2026•0 citations

A computationally efficient approach to quantum state reconstruction using robust classical shadows

Sanjay Sharma, Shyam Akashe, Govind Murari Upadhyay +1 more

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.

Scientific Reports•Feb 2, 2026•0 citations

Ultra-Precise Quantum Projective Designs in Constant Depth

Qingyue Zhang, Junjie Chen, Zhou You +1 more

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.

quant-pharXiv:2609.03925•Sep 4, 2026•0 citations

Preparation of Hamming-Weight-Preserving Quantum States with Log-Depth Quantum Circuits

Yu Li, Guojing Tian, He, Xiaoyu +3 more

Hamming-Weight-Preserving (HWP) states, $\vertψ_{\mathrm{H}}\rangle=\sum_{\mathrm{HW}(x)=k}α_x\vert x\rangle$, represent a fundamental class of quantum states on $n$ qubits with fixed excitation number $k$, playing a pivotal role in quantum simulation, machine learning, and optimization, with prominent examples including the Bethe eigenstates of integrable models and Configuration Interaction states. However, existing preparation methodologies typically rely on sequential architectures whose depth scales linearly with the subspace dimension, whereas practical deployment on realistic quantum hardware demands shallow circuits to mitigate decoherence. In this work, we present quantum circuits that achieve the information theoretic limit of depth logarithmic in the number of populated computational basis states while maintaining asymptotically optimal size. Specifically, for general HWP states, we provide a construction that matches the lower bounds of $Θ(\log \binom{n}{k})$ depth and $Θ(\binom{n}{k})$ size. Furthermore, we propose a resource efficient scheme utilizing only $O(\log k)$ ancillary qubits. For the $k=2$ case, where HWP states naturally admit a graph representation, we show that states whose support graphs are trees or grids admit ancilla free preparation with $Θ(\log n)$ depth and $Θ(m)$ size. Overall, our work provides logarithmic depth state preparation algorithms across various regimes while establishing a connection between quantum state synthesis and Krawtchouk polynomials.

arXiv:2508.14470•Aug 20, 2025•0 citations

Shallow shadows: Expectation estimation using low-depth random Clifford circuits

Christian Bertoni, Jonas Haferkamp, Marcel Hinsche +3 more

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.

arXiv:2209.12924•Sep 26, 2022•8 citations
Random quantum states have various applications in quantum information science. We discover a new ensemble of quantum states that serve as an ε-approximate state t-design while possessing extremely low entanglement, magic, and coherence. These resources can reach their theoretical lower bounds, Ω(log(t/ε)), which are also proven in this Letter. This implies that, for fixed t and ε, entanglement, magic, and coherence do not scale with the system size, i.e., O(1) with respect to the total number of qubits n. Moreover, we explicitly construct an ancilla-free shallow quantum circuit for generating such states by transforming k-qubit approximate state designs into n-qubit ones without increasing the support size. The depth of such a quantum circuit, O(t[logt]^{3}log n log[1/ε]), is the most efficient among existing algorithms without ancilla qubits. A class of quantum circuits proposed in our Letter offers reduced cost for classical simulation of random quantum states, leading to potential applications in quantum information processing. As a concrete example, we propose classical shadow tomography using an estimator with superpositions between only two states, from which almost all quantum states can be efficiently certified by requiring only O(1) measurements and classical postprocessing time.
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.
Random unitaries are fundamental to quantum information and many-body physics, with widespread applications ranging from quantum learning and metrology to device benchmarking. A central pursuit is to minimize the space and circuit depth required to generate them. However, existing methods for generating low-depth random unitaries rely heavily on an extensive number of ancillary qubits, imposing severe spatial overhead. In this work, we prove that random unitaries can be generated in optimal depth without ancillae. For multiplicative-error approximate kkk-designs on nnn qubits, our circuits achieve a depth of O~(k)(log⁡n)1/δ\widetilde{O} (k) (\log n)^{1/δ}O(k)(logn)1/δ on δδδ-dimensional architectures and O~(k)log⁡log⁡n\widetilde{O}(k) \log \log nO(k)loglogn with all-to-all connectivity. Furthermore, by introducing a general exactification lemma, we lift our construction to optimal-depth exact kkk-designs, yielding an exponential resource reduction over state-of-the-art exact constructions. Our results minimize the space-time costs for a wide range of quantum protocols.
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.
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.
Hamming-Weight-Preserving (HWP) states, ∣ψH⟩=∑HW(x)=kαx∣x⟩\vertψ_{\mathrm{H}}\rangle=\sum_{\mathrm{HW}(x)=k}α_x\vert x\rangle∣ψH​⟩=∑HW(x)=k​αx​∣x⟩, represent a fundamental class of quantum states on nnn qubits with fixed excitation number kkk, playing a pivotal role in quantum simulation, machine learning, and optimization, with prominent examples including the Bethe eigenstates of integrable models and Configuration Interaction states. However, existing preparation methodologies typically rely on sequential architectures whose depth scales linearly with the subspace dimension, whereas practical deployment on realistic quantum hardware demands shallow circuits to mitigate decoherence. In this work, we present quantum circuits that achieve the information theoretic limit of depth logarithmic in the number of populated computational basis states while maintaining asymptotically optimal size. Specifically, for general HWP states, we provide a construction that matches the lower bounds of Θ(log⁡(nk))Θ(\log \binom{n}{k})Θ(log(kn​)) depth and Θ((nk))Θ(\binom{n}{k})Θ((kn​)) size. Furthermore, we propose a resource efficient scheme utilizing only O(log⁡k)O(\log k)O(logk) ancillary qubits. For the k=2k=2k=2 case, where HWP states naturally admit a graph representation, we show that states whose support graphs are trees or grids admit ancilla free preparation with Θ(log⁡n)Θ(\log n)Θ(logn) depth and Θ(m)Θ(m)Θ(m) size. Overall, our work provides logarithmic depth state preparation algorithms across various regimes while establishing a connection between quantum state synthesis and Krawtchouk polynomials.
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.