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arXiv:2510.08129 · Physical Review Letters

Adaptively Secure Unitary Designs with Constant Non-Clifford Cost

Lennart Bittel, Lorenzo Leone

Published Apr 3, 2026• 0 citations
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Abstract

Randomness is a fundamental resource in quantum information, with crucial applications in cryptography, algorithms, and error correction. A central challenge is to construct unitary k designs that closely approximate Haar-random unitaries while minimizing the costly use of non-Clifford operations. In this Letter, we present a protocol able to generate unitary k designs on n qubits, secure against any adversarial quantum measurement, with a system-size-independent number of non-Clifford gates. Our construction applies a k design only to a subsystem of size Θ(k), independent of n. This "seed" design is then "diluted" across the entire n-qubit system by sandwiching it between two random Clifford operators. The resulting ensemble forms an ϵ-approximate unitary k design on n qubits. We prove that this construction achieves full quantum security against adaptive adversaries using only O[over ˜](k^{2}logϵ^{-1}) non-Clifford gates. If one requires security only against polynomial-time adaptive adversaries, the non-Clifford cost decreases to O[over ˜](k+log^{1+c}ϵ^{-1}). This is optimal, since we show that at least Ω(k) non-Clifford gates are required in this setting. Compared to existing approaches, our method significantly reduces non-Clifford overhead while strengthening security guarantees to adaptive security as well as removing artificial assumptions between n and k. These results make high-order unitary designs practically attainable in near-term fault-tolerant quantum architectures.

DOI
10.1103/9hcw-7fl6

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Randomness is a fundamental resource in quantum information, with crucial applications in cryptography, algorithms, and error correction. A central challenge is to construct unitary k designs that closely approximate Haar-random unitaries while minimizing the costly use of non-Clifford operations. In this Letter, we present a protocol able to generate unitary k designs on n qubits, secure against any adversarial quantum measurement, with a system-size-independent number of non-Clifford gates. Our construction applies a k design only to a subsystem of size Θ(k), independent of n. This "seed" design is then "diluted" across the entire n-qubit system by sandwiching it between two random Clifford operators. The resulting ensemble forms an ϵ-approximate unitary k design on n qubits. We prove that this construction achieves full quantum security against adaptive adversaries using only Oover ˜ non-Clifford gates. If one requires security only against polynomial-time adaptive adversaries, the non-Clifford cost decreases to Oover ˜. This is optimal, since we show that at least Ω(k) non-Clifford gates are required in this setting. Compared to existing approaches, our method significantly reduces non-Clifford overhead while strengthening security guarantees to adaptive security as well as removing artificial assumptions between n and k. These results make high-order unitary designs practically attainable in near-term fault-tolerant quantum architectures.
Abstract Many quantum information protocols require the implementation of random unitaries. Because it takes exponential resources to produce Haar-random unitaries drawn from the full n -qubit group, one often resorts to t -designs. Unitary t -designs mimic the Haar-measure up to t -th moments. It is known that Clifford operations can implement at most 3-designs. In this work, we quantify the non-Clifford resources required to break this barrier. We find that it suffices to inject O(t4log⁡2(t)log⁡(1/ε))O(t^{4}\log ^{2}(t)\log (1/\varepsilon ))O(t4log2(t)log(1/ε)) O ( t 4 log 2 ( t ) log ( 1 / ε ) ) many non-Clifford gates into a polynomial-depth random Clifford circuit to obtain an ε\varepsilon ε ε -approximate t -design. Strikingly, the number of non-Clifford gates required is independent of the system size – asymptotically, the density of non-Clifford gates is allowed to tend to zero. We also derive novel bounds on the convergence time of random Clifford circuits to the t -th moment of the uniform distribution on the Clifford group. Our proofs exploit a recently developed variant of Schur-Weyl duality for the Clifford group, as well as bounds on restricted spectral gaps of averaging operators.
We construct nnn-qubit approximate unitary kkk-designs in 1D systems, achieving circuit depth O(log⁡(n/ε)+klog⁡k)O(\log(n/\varepsilon) + k\log k)O(log(n/ε)+klogk) with relative error ε\varepsilonε and requiring O(nklog⁡k)O(nk\log k)O(nklogk) magic gates. This matches existing lower bounds Ω(log⁡(n/ε)+k)Ω(\log(n/\varepsilon) + k)Ω(log(n/ε)+k) for circuit depth, and Ω~(nk)\widetildeΩ(nk)Ω(nk) for the required number of TTT gates, up to a log⁡k\log klogk factor, achieving simultaneous near-optimality in all parameters. Our construction is based on a combination and refinement of two existing results. We reduce the required magic block size for breaking Clifford symmetries in the magic-augmented circuit construction of Zhang et al. from O(klog⁡k)O(k\log k)O(klogk) to O(log⁡k)O(\log k)O(logk). We also improve the breakthrough construction of Chen et al. to construct a generating set of 1D local constant-depth circuits for the unitary group with a constant spectral gap, making O(log⁡k)O(\log k)O(logk)-local random unitaries realizable in depth O(klog⁡k)O(k\log k)O(klogk). As a by-product, we provide a constant-size 1D-local generating set for the Clifford group, which we expect to be of independent interest. Combining the two results with the gluing lemma, we prove the final result.
Recent years have enjoyed a strong interest in exploring properties and applications of random quantum circuits. In this work, we explore the ensemble of t -doped Clifford circuits on n qubits, consisting of Clifford circuits interspersed with t single-qubit non-Clifford gates. We establish rigorous convergence bounds toward unitary k -designs, revealing the intrinsic cost in terms of non-Clifford resources in various flavors. First, we analyze the k th order frame potential, which quantifies how well the ensemble of doped Clifford circuits is spread within the unitary group. We prove that a quadratic doping level, t = Θ ~ ( k 2 ) , is both necessary and sufficient to approximate the frame potential of the full unitary group. As a consequence, we refine existing upper bounds on the convergence of the ensemble toward state k -designs. Second, we derive tight bounds on the convergence of t -doped Clifford circuits toward relative-error k -designs, showing that t = Θ ~ ( n k ) is both necessary and sufficient for the ensemble to form a relative ε -approximate k -design. Similarly, t = Θ ~ ( n ) is required to generate pseudorandom unitaries. All these results highlight that generating random unitaries is extremely costly in terms of non-Clifford resources, and that such ensembles fundamentally lie beyond the classical simulability barrier. Additionally, we analyze high-order doped-Clifford-Weingarten functions to derive analytic expressions for the twirling operator over the ensemble of random doped Clifford circuits, and we establish their asymptotic behavior in relevant regimes.
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.
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.
Many quantum information protocols require the implementation of random unitaries. Because it takes exponential resources to produce Haar-random unitaries drawn from the full nnn-qubit group, one often resorts to ttt-designs. Unitary ttt-designs mimic the Haar-measure up to ttt-th moments. It is known that Clifford operations can implement at most 333-designs. In this work, we quantify the non-Clifford resources required to break this barrier. We find that it suffices to inject O(t4log⁡2(t)log⁡(1/ε))O(t^{4}\log^{2}(t)\log(1/\varepsilon))O(t4log2(t)log(1/ε)) many non-Clifford gates into a polynomial-depth random Clifford circuit to obtain an ε\varepsilonε-approximate ttt-design. Strikingly, the number of non-Clifford gates required is independent of the system size -- asymptotically, the density of non-Clifford gates is allowed to tend to zero. We also derive novel bounds on the convergence time of random Clifford circuits to the ttt-th moment of the uniform distribution on the Clifford group. Our proofs exploit a recently developed variant of Schur-Weyl duality for the Clifford group, as well as bounds on restricted spectral gaps of averaging operators.