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arXiv:2507.19018

Approximate k-uniform states: definition, construction and applications

Kaiyi Guo, Fei Shi, You Zhou, Qi Zhao

Published Jul 25, 2025• 0 citations
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Abstract

$k$-Uniform states are fundamental to quantum information and computing, with applications in multipartite entanglement and quantum error-correcting codes (QECCs). Prior work has primarily focused on constructing exact $k$-uniform states or proving their nonexistence. However, due to inevitable theoretical approximations and experimental imperfections, generating exact $k$-uniform states is neither feasible nor necessary in practice. In this work, we initiate the study of approximate $k$-uniform states, demonstrating that they are locally indistinguishable from their exact counterparts unless massive measurements are performed. We prove that such states can be constructed with high probability from the Haar-random ensemble and, more efficiently, via shallow random quantum circuits. Furthermore, we establish a connection between approximate $k$-uniform states and approximate QECCs, showing that Haar random constructions yield high-performance codes with linear rates, vanishing proximity, and exponentially small failure probability while random circuits can't construct codes with linear code rate in shallow depth. Finally, we investigate the relationship between approximate QECCs and approximate quantum information masking. Our work lays the foundation for the practical application of $k$-uniform states.

DOI
10.48550/arxiv.2507.19018

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Approximate Unitary k-Designs from Shallow, Low-Communication Circuits (Extended Abstract)

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It is impossible to mask an arbitrary quantum state into the correlations between two subsystems such that the original information is completely unknown to each local system. This is the no-masking theorem proposed by Modi et al. [K. Modi, A. K. Pati, A. Sen(De), and U. Sen, Phys. Rev. Lett. 120, 230501 (2018)]. In this work, we propose the concept of $k$-uniform quantum information masking in multipartite systems and indicate the relation between quantum error-correcting codes (QECCs) in heterogeneous systems and quantum information masking. As a consequence, we show that the no-masking theorem is a special case of the quantum Singleton bound for QECCs in heterogeneous systems essentially, and we give a more general no-masking theorem in multipartite systems based on the quantum Singleton bound. We also solve two open questions proposed by Li and Wang [M.-S. Li and Y.-L. Wang, Phys. Rev. A 98, 062306 (2018)]. That is, an arbitrary state of level $d$ cannot be masked in a tripartite system of level $d$ such that the marginal states are not proportional to identity, and an arbitrary state of level $d$ cannot be masked in a tripartite system of level $n$ with $n<d$. Further, we give some methods for constructing new QECCs from old QECCs in heterogeneous systems.

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Approximate unitary k-designs are ensembles of unitary matrices whose first k statistical moments approximate the Haar (uniform) measure. Shallow, random quantum circuits with temporally and spatially structured architectures are known to yield approximate k-designs in depth that is logarithmic in the number of qubits. It was open whether this structure is physically necessary. Shown here is shallow relative entropy convergence for standard notions of less structured, time-periodic circuits, implying additive-error approximate k-designs in nearly logarithmic depth (up to sub-logarithmic factors). These architectures include 1-dimensional brickwork and gates placed randomly according to a connected, bounded-degree graph. Relative entropy recombination methods derived herein are potentially of independent interest.

arXiv:2510.08537•Oct 9, 2025•0 citations
kkk-Uniform states are fundamental to quantum information and computing, with applications in multipartite entanglement and quantum error-correcting codes (QECCs). Prior work has primarily focused on constructing exact kkk-uniform states or proving their nonexistence. However, due to inevitable theoretical approximations and experimental imperfections, generating exact kkk-uniform states is neither feasible nor necessary in practice. In this work, we initiate the study of approximate kkk-uniform states, demonstrating that they are locally indistinguishable from their exact counterparts unless massive measurements are performed. We prove that such states can be constructed with high probability from the Haar-random ensemble and, more efficiently, via shallow random quantum circuits. Furthermore, we establish a connection between approximate kkk-uniform states and approximate QECCs, showing that Haar random constructions yield high-performance codes with linear rates, vanishing proximity, and exponentially small failure probability while random circuits can't construct codes with linear code rate in shallow depth. Finally, we investigate the relationship between approximate QECCs and approximate quantum information masking. Our work lays the foundation for the practical application of kkk-uniform states.
A pure state of NNN parties with local dimension ddd is called a kkk-uniform state if all the reductions to kkk parties are maximally mixed. Based on the connections among kkk-uniform states, orthogonal arrays and linear codes, we give general constructions for kkk-uniform states. We show that when d≥4k−2d\geq 4k-2d≥4k−2 (resp. d≥2k−1d\geq 2k-1d≥2k−1) is a prime power, there exists a kkk-uniform state for any N≥2kN\geq 2kN≥2k (resp. 2k≤N≤d+12k\leq N\leq d+12k≤N≤d+1). Specially, we give the existence of 4,54,54,5-uniform states for almost every NNN-qudits. Further, we generalize the concept of quantum information masking in bipartite systems given by [Modi \emph{et al.} {Phys. Rev. Lett. \textbf{120}, 230501 (2018)}] to kkk-uniform quantum information masking in multipartite systems, and we show that kkk-uniform states and quantum error-correcting codes can be used for kkk-uniform quantum information masking.
Random unitaries are useful in quantum information and related fields but hard to generate with limited resources. An approximate unitary k-design is a measure over an ensemble of unitaries such that the average is close to a Haar (uniformly) random ensemble up to the first k moments. A strong notion of approximation bounds the distance from Haar randomness in relative error: the weighted twirl induced by an approximate design can be written as a convex combination involving that of an exact design and vice versa. The main focus of our work is on efficient constructions of approximate designs, in particular whether relative-error designs in sublinear depth are possible. We give a positive answer to this question as part of our main results: 1. Twirl-Swap-Twirl: Let A and B be systems of the same size. Consider a protocol that locally applies k-design unitaries to A^k and B^k respectively, then exchanges l qudits between each copy of A and B respectively, then again applies local k-design unitaries. This protocol yields an ε-approximate relative k-design when l = O(k log k + log(1/ε)). In particular, this bound is independent of the size of A and B as long as it is sufficiently large compared to k and 1/ε. 2. Twirl-Crosstwirl: Let A_1, … , A_P be subsystems of a multipartite system A. Consider the following protocol for k copies of A: (1) locally apply a k-design unitary to each A_p for p = 1, … , P; (2) apply a "crosstwirl" k-design unitary across a joint system combining l qudits from each A_p. Assuming each A_p’s dimension is sufficiently large compared to other parameters, one can choose l to be of the form 2 (Pk + 1) log_q k + log_q P + log_q(1/ε) + O(1) to achieve an ε-approximate relative k-design. As an intermediate step, we show that this protocol achieves a k-tensor-product-expander, in which the approximation error is in 2 → 2 norm, using communication logarithmic in k. 3. Recursive Crosstwirl: Consider an m-qudit system with connectivity given by a lattice in spatial dimension D. For every D = 1, 2, …, we give a construction of an ε-approximate relative k-design using unitaries of spatially local circuit depth O ((log m + log(1/ε) + k log k ) k polylog(k)). Moreover, across the boundaries of spatially contiguous sub-regions, unitaries used in the design ensemble require only area law communication up to corrections logarithmic in m. Hence they generate only that much entanglement on any product state input. These constructions use the alternating projection method to analyze overlapping Haar twirls, giving a bound on the convergence speed to the full twirl with respect to the 2-norm. Using von Neumann subalgebra indices to replace system dimension, the 2-norm distance converts to relative error without introducing system size. The Recursive Crosstwirl construction answers one variant of [Harrow and Mehraban, 2023, Open Problem 1], showing that with a specific, layered architecture, random circuits produce relative error k-designs in sublinear depth. Moreover, it addresses [Harrow and Mehraban, 2023, Open Problem 7], showing that structured circuits in spatial dimension D of depth << m^{1/D} may achieve approximate k-designs.
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.
Abstract Random unitaries are useful in quantum information and related fields, but hard to generate with limited resources. An approximate unitary k -design is an ensemble of unitaries with an underlying measure over which the average is close to a Haar random ensemble up to the first k moments. A particularly strong notion of approximation bounds the distance from Haar randomness in relative error. Such relative-error approximate designs are secure against queries by an adaptive adversary trying to distinguish it from a Haar ensemble. We construct relative-error approximate unitary k -design ensembles for which communication between subsystems is O (1) in the system size. These constructions use the alternating projection method to analyze overlapping Haar averages, giving a bound on the convergence speed to the full averaging with respect to the 2-norm. Using von Neumann subalgebra indices to replace system dimension, the 2-norm distance converts to relative error without introducing any additional dimension dependence. We use these constructions as the building blocks of a two-step protocol that achieves a relative-error design in O((log⁡m+log⁡(1/ϵ)+klog⁡k)k polylog(k))O \big ( (\log m + \log (1/\epsilon ) + k \log k ) k\, \text {polylog}(k) \big )O((logm+log(1/ϵ)+klogk)kpolylog(k)) O ( ( log m + log ( 1 / ϵ ) + k log k ) k polylog ( k ) ) depth, where m is the number of qudits in the complete system and ϵ\epsilon ϵ ϵ the approximation error. This sublinear depth construction answers a variant of [21, Harrow and Mehraban 2023, Section 1.5, Open Question 1] and [21, Harrow and Mehraban 2023, Section 1.5, Open Question 7]. Moreover, entanglement generated by the sublinear depth scheme follows area laws on spatial lattices up to corrections logarithmic in the full system size.
It is impossible to mask an arbitrary quantum state into the correlations between two subsystems such that the original information is completely unknown to each local system. This is the no-masking theorem proposed by Modi et al. [K. Modi, A. K. Pati, A. Sen(De), and U. Sen, Phys. Rev. Lett. 120, 230501 (2018)]. In this work, we propose the concept of kkk-uniform quantum information masking in multipartite systems and indicate the relation between quantum error-correcting codes (QECCs) in heterogeneous systems and quantum information masking. As a consequence, we show that the no-masking theorem is a special case of the quantum Singleton bound for QECCs in heterogeneous systems essentially, and we give a more general no-masking theorem in multipartite systems based on the quantum Singleton bound. We also solve two open questions proposed by Li and Wang [M.-S. Li and Y.-L. Wang, Phys. Rev. A 98, 062306 (2018)]. That is, an arbitrary state of level ddd cannot be masked in a tripartite system of level ddd such that the marginal states are not proportional to identity, and an arbitrary state of level ddd cannot be masked in a tripartite system of level nnn with n<dn<dn<d. Further, we give some methods for constructing new QECCs from old QECCs in heterogeneous systems.
Approximate unitary k-designs are ensembles of unitary matrices whose first k statistical moments approximate the Haar (uniform) measure. Shallow, random quantum circuits with temporally and spatially structured architectures are known to yield approximate k-designs in depth that is logarithmic in the number of qubits. It was open whether this structure is physically necessary. Shown here is shallow relative entropy convergence for standard notions of less structured, time-periodic circuits, implying additive-error approximate k-designs in nearly logarithmic depth (up to sub-logarithmic factors). These architectures include 1-dimensional brickwork and gates placed randomly according to a connected, bounded-degree graph. Relative entropy recombination methods derived herein are potentially of independent interest.