(Almost) quadruply optimal unitary designs in 1D

Abstract: We construct $n$-qubit approximate unitary $k$-designs in 1D systems, achieving circuit depth $O(\log(n/\varepsilon) + k\log k)$ with relative error $\varepsilon$ and requiring $O(nk\log k)$ magic gates. This matches existing lower bounds $Ω(\log(n/\varepsilon) + k)$ for circuit depth, and $\widetildeΩ(nk)$ for the required number of $T$ gates, up to a $\log k$ 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 $O(k\log k)$ to $O(\log k)$. 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 $O(\log k)$-local random unitaries realizable in depth $O(k\log k)$. 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.

• The paper constructs explicit ancilla-free nearest-neighbor circuits that achieve ε-approximate unitary k-designs for k=O(n) in depth O(log(n/ε)+k log k), matching known lower bounds up to a log k factor.
• The paper reduces the required magic-block locality from O(k log k) to O(log k), lowering the non-Clifford resource cost to O(nk log k) T gates independently of the target precision ε.
• The paper develops constant-depth, constant-spectral-gap 1D generators for Clifford and permutation groups, enabling efficient local circuit implementations and leaving open whether the remaining log k overhead can be removed.‌

The paper constructs one-dimensional (1D) nearest-neighbor circuits that generate approximate unitary $k$-designs with relative error $\varepsilon$ in depth $O(\log(n/\varepsilon) + k\log k)$ without ancillas, for design order $k = O(n)$ (2608.18650). This matches the known lower bound of $\Omega(\log(n/\varepsilon) + k)$ (Cui et al., 8 Jul 2025) simultaneously in all three parameters — system size, error, and design order — up to a single $\log k$ factor in the design direction. The construction additionally requires only $O(nk\log k)$ non-Clifford ($T$) gates, independent of $\varepsilon$, improving on prior bounds for $t$-doped Clifford circuits and approaching the lower bound $\varepsilon$0 on non-Clifford resources. The result is obtained by combining two technical advances with the gluing lemma: a reduction of the required magic block locality from $\varepsilon$1 to $\varepsilon$2, and a constant-spectral-gap, constant-depth, 1D-local generating set for the unitary group.

Main theorem and circuit architecture

The central result states that for any $\varepsilon$3-qubit system and $\varepsilon$4, an explicit 1D-local, ancilla-free circuit of depth

$\varepsilon$5

forms an $\varepsilon$6-relative-error approximate unitary $\varepsilon$7-design, using $\varepsilon$8 $\varepsilon$9 gates. The architecture is a two-layer block-brickwork over patches of size $O(\log(n/\varepsilon) + k\log k)$0, where each $O(\log(n/\varepsilon) + k\log k)$1-qubit block is a magic-augmented circuit: a random Clifford unitary sandwiched between two layers of tensor-product $O(\log(n/\varepsilon) + k\log k)$2-local approximate unitary $O(\log(n/\varepsilon) + k\log k)$3-designs with $O(\log(n/\varepsilon) + k\log k)$4, $O(\log(n/\varepsilon) + k\log k)$5. The gluing lemma of Schuster et al. then promotes these local blocks to a global design provided each block is an $O(\log(n/\varepsilon) + k\log k)$6-approximate design and $O(\log(n/\varepsilon) + k\log k)$7.

Because the patch size satisfies $O(\log(n/\varepsilon) + k\log k)$8, the construction inherently operates in the regime $O(\log(n/\varepsilon) + k\log k)$9. The authors note that combining their gluing approach with the constant spectral gap of 1D brickwork circuits yields depth $k = O(n)$0 beyond this regime, but decoupling $k = O(n)$1 from $k = O(n)$2 at high design order remains open.

Magic-augmented circuits with logarithmic locality

The first core theorem shows that a magic-augmented circuit on $k = O(n)$3 qubits forms an approximate $k = O(n)$4-design with error $k = O(n)$5, provided $k = O(n)$6 and the local blocks have relative error $k = O(n)$7. The proof expands the $k = O(n)$8-fold twirling channel in the Clifford commutant basis $k = O(n)$9 of stochastic Lagrangian subspaces and splits it into three contributions: permutation elements, diagonal non-permutation elements, and off-diagonal terms.

Three estimates control the error. First, the discrepancy between the Clifford and Haar Weingarten functions restricted to the permutation block is bounded by $\Omega(\log(n/\varepsilon) + k)$0, established via a Schur complement analysis of the normalized Clifford Gram matrix, using the fact that its inverse has bounded operator norm when $\Omega(\log(n/\varepsilon) + k)$1. Second, sums of absolute values of the Clifford Weingarten function are controlled by $\Omega(\log(n/\varepsilon) + k)$2 via Neumann series expansion. Third — the key novelty — the sum over non-permutation commutant elements of $\Omega(\log(n/\varepsilon) + k)$3, where $\Omega(\log(n/\varepsilon) + k)$4 measures distance to the unitary commutant, is bounded by $\Omega(\log(n/\varepsilon) + k)$5 whenever $\Omega(\log(n/\varepsilon) + k)$6. This bound exploits permutation invariance of $\Omega(\log(n/\varepsilon) + k)$7 to restrict to canonical representatives minimizing the distance $\Omega(\log(n/\varepsilon) + k)$8, combined with Stirling-number counting and the identity $\Omega(\log(n/\varepsilon) + k)$9 for representatives at distance $\log k$0.

The consequence is that locality $\log k$1 suffices to break all non-permutation Clifford symmetries, whereas prior work required $\log k$2. Since each magic block costs depth $\log k$3 and $\log k$4 $\log k$5 gates, this directly compresses both the depth contribution to $\log k$6 per patch and the total $\log k$7 count to $\log k$8. Replacing exact Haar blocks by approximate designs incurs only a factor-of-two loss in the exponent through a Russo–Dye argument showing that any relative-error design perturbs $\log k$9 by at most $O(nk\log k)$0.

Constant-depth 1D generators with constant spectral gap

The second core ingredient replaces Haar-random $O(nk\log k)$1-qubit blocks by efficient circuits. Building on the CPZPC ensemble of Chen et al. (Chen et al., 2024), which comprises two random Cliffords, two random permutations, and a Pauli $O(nk\log k)$2, the paper constructs a distribution $O(nk\log k)$3 in which each sample is a 1D nearest-neighbor circuit of depth $O(nk\log k)$4 and whose $O(nk\log k)$5-th moment operator has spectral gap $O(nk\log k)$6, valid for $O(nk\log k)$7. Gap amplification then gives an $O(nk\log k)$8-qubit $O(nk\log k)$9-relative-error design in depth $T$0; setting $T$1 completes the main theorem.

Two supporting results make this work:

Kassabov generators in open-boundary 1D. Kassabov's expander generators for the alternating group were previously realizable in constant depth only on all-to-all architectures or periodic 1D chains. The paper introduces a folded cell layout: the $T$2 qubits are grouped into cells $T$3 of 18 bits placed in the order $T$4, so that every cyclic-neighbor pair $T$5 used by the cyclic shift matrix sits within constant physical distance. All generator types — intracell CNOT layers, single CNOTs, and Toffoli layers controlled by register bits — satisfy a constant-size routing observation guaranteeing $T$6 depth. For $T$7 not divisible by 18, an overlap decomposition into two overlapping intervals of size $T$8, combined with the permutation overlap theorem, restores the global spectral gap with exponentially small overhead under $T$9.

A 451-generator Clifford expander. The paper proves that the full Clifford group $\varepsilon$0 admits a symmetric generating set of at most 451 elements, each implementable in constant 1D depth, with Kazhdan constant $\varepsilon$1. The construction proceeds through the chain $\varepsilon$2. For the special linear group, an explicit 14-generator set (28 for generic $\varepsilon$3 via a three-copy decomposition proved by a subspace-pairing argument requiring $\varepsilon$4) achieves Kazhdan constant exceeding $\varepsilon$5, following Kassabov's universal lattice construction. Nikolov's product decomposition expresses every symplectic element as a product of at most 130 conjugates of the embedded special linear group, yielding at most 336 symplectic generators whose quantum lifts are parallel Hadamard, phase, and CZ layers — all depth 1. Extension to the projective Clifford group uses Hadad's abelian-extension theorem with the Pauli kernel, whose average Kazhdan constant exceeds 2 with respect to the orbit of a single-qubit $\varepsilon$6 gate. Symmetrization adds 112 generators, and lifting to the full Clifford group adds the two phase generators $\varepsilon$7, which alone provide constant Kazhdan constant in representations nontrivial on the center.

This Clifford expander is of independent interest: it provides a system-size-independent, 1D-local generating set for random Clifford circuits, applicable to randomized benchmarking and device verification. A concurrent work obtains a smaller (28-generator) set but only for $\varepsilon$8 on a circle; the folded layout and overlap lemmas here remove both restrictions.

Limitations and open questions

Several qualifications attach to the main result. The residual $\varepsilon$9 gap to the lower bound would require reducing the magic-block locality from $t$0 to $t$1, which demands analyzing the Weingarten function in the singular regime $t$2; the paper does not resolve this. On the magic side, the construction leaves a gap of scaling $t$3 relative to the conjectured true lower bound $t$4, with the authors suggesting the existing lower bound of Leone et al. is lossy by a factor $t$5 — closing this in either direction remains open. The regime constraint $t$6 stems from reversibility of the Clifford Weingarten function, which fails for $t$7; high-design-order constructions retain a $t$8 dependence that is not decoupled. Finally, the assumption that $t$9 is a multiple of $\varepsilon$00 (and $\varepsilon$01 of $\varepsilon$02) is handled by a partitioning argument costing only constant-factor error overhead, so it does not affect asymptotics.

Conclusion

The paper nearly closes the problem of depth-optimal approximate unitary designs in 1D, achieving simultaneous optimality in system size, error, and design order up to one $\varepsilon$03 factor, with near-optimal non-Clifford gate count independent of precision. Its two technical contributions — the $\varepsilon$04-locality magic-augmented design and the constant-size, constant-depth, constant-gap 1D generating sets for the Clifford and permutation groups — are likely to be useful beyond design synthesis, particularly wherever shallow random Clifford or permutation circuits are required.