Characterising the efficiency with which designs over various subsets of the unitary group may be constructed is an important goal of quantum information theory. While it is now known that approximate unitary designs can be realised in depth logarithmic in the system size, it has recently been shown that ensembles of local nearest-neighbour sublinear-depth one-dimensional circuits over the matchgate, orthogonal, and symplectic groups cannot form approximate 2-designs over their parent groups; similarly, sublinear-depth ensembles of Cliffords cannot form a Clifford 4-design. In this note we show that this remarkable exponential separation is not merely an artefact of restricting to ensembles consisting of unitaries from the subgroups themselves, but rather that no ensemble of local nearest-neighbour sublinear-depth unitaries can realise approximate designs in the aforementioned cases, even when employing "ambient" unitaries from beyond the subgroup itself (possibly acting on ancilla qubits). This implies that various natural tomography and benchmarking schemes which involves sampling from these groups suffer from a dramatic circuit depth overhead compared to similar protocols which involve sampling from the full unitary group. We additionally conclude that, in all of the above cases, the known linear-depth design constructions are up to constant factors optimal.
Characterising the efficiency with which designs over various subsets of the unitary group may be constructed is an important goal of quantum information theory. While it is now known that approximate unitary designs can be realised in depth logarithmic in the system size, it has recently been shown that ensembles of local nearest-neighbour sublinear-depth one-dimensional circuits over the matchgate, orthogonal, and symplectic groups cannot form approximate 2-designs over their parent groups; similarly, sublinear-depth ensembles of Cliffords cannot form a Clifford 4-design. In this note we show that this remarkable exponential separation is not merely an artefact of restricting to ensembles consisting of unitaries from the subgroups themselves, but rather that no ensemble of local nearest-neighbour sublinear-depth unitaries can realise approximate designs in the aforementioned cases, even when employing "ambient" unitaries from beyond the subgroup itself (possibly acting on ancilla qubits). This implies that various natural tomography and benchmarking schemes which involves sampling from these groups suffer from a dramatic circuit depth overhead compared to similar protocols which involve sampling from the full unitary group. We additionally conclude that, in all of the above cases, the known linear-depth design constructions are up to constant factors optimal.
Understanding the resources required to generate approximately random unitaries over various groups is a natural goal of quantum information theory. With respect to one notion of approximation, that of a design, it is known that the full unitary group can be approximated in logarithmic depth by one-dimensional circuits of nearest-neighbour 2-local gates. On the other hand, remarkably, circuits with this connectivity cannot form designs over the matchgate group in sublinear depth. Here we show that this dramatic slowdown can disappear when using a general qubit connectivity graph G of routing number rt(G). Indeed, in this setting one can obtain (strong) ε-approximate relative error matchgate k-designs in depth O(k2rt(G)lognlog(n/ε)). For all-to-all connectivity, rt(G)=2. Our construction is conceptually simple, involving a random walk on the matchgate group, and no ancillae. As a technical byproduct, we improve upon the state of the art for fermionic routing, obtaining an O(rt(G)logn) depth router. Additionally, for k=3, we obtain an exact strong matchgate design in O(rt(G)logn) depth, again without ancillae. Under all-to-all connectivity, our fermionic router and 3-designs are optimal. Notably, our results imply that quantum algorithms for fermionic tomography which require drawing from a matchgate 3-design may be exponentially sped up on quantum computers with all-to-all connectivity, relative to their strictly one-dimensional counterparts.
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 G possessing an invariant state G⊗k∣Ψ⟩=∣Ψ⟩, a lower bound on the diamond distance between the k\textsuperscript{th} moment operator of any ensemble of elements of G, and that of the Haar measure over G. We then use this bound to prove that for many groups of interest, no subset of G consisting of sublinear-depth one-dimensional circuits with local gates can form an approximate k-design over G. More generally, on a D-dimensional lattice, our results imply that such group designs require depths scaling at least as n1/D. Moreover, for most of the groups we consider we find that such ensembles can, with high probability, be distinguished from k-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 8-designs over the Clifford group. Finally, we relax the assumption of working with local gates, and prove the impossibility of obtaining approximate designs over G using any circuit comprised of a sublinear number of gates generated by Pauli strings.
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 construct ε-approximate unitary k-designs on n qubits in circuit depth O(logkloglognk/ε). The depth is exponentially improved over all known results in all three parameters n, k, ε. We further show that each dependence is optimal up to exponentially smaller factors. Our construction uses O~(nk) ancilla qubits and O(nk) bits of randomness, which are also optimal up to log(nk) factors. An alternative construction achieves a smaller ancilla count O~(n) with circuit depth O(kloglognk/ε). To achieve these efficient unitary designs, we introduce a highly-structured random unitary ensemble that leverages long-range two-qubit gates and low-depth implementations of random classical hash functions. We also develop a new analytical framework for bounding errors in quantum experiments involving many queries to random unitaries. As an illustration of this framework's versatility, we provide a succinct alternative proof of the existence of pseudorandom unitaries.
We construct n-qubit approximate unitary k-designs in 1D systems, achieving circuit depth O(log(n/ε)+klogk) with relative error ε and requiring O(nklogk) magic gates. This matches existing lower bounds Ω(log(n/ε)+k) for circuit depth, and Ω(nk) for the required number of T gates, up to a logk 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(klogk) to 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(logk)-local random unitaries realizable in depth 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.
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.