Ultra-Precise Quantum Projective Designs in Constant Depth

• URL: http://arxiv.org/abs/2609.03925v1
• Date: Thu, 03 Sep 2026 14:38:21 GMT
• Title: Ultra-Precise Quantum Projective Designs in Constant Depth
• Abstract summary: We introduce an explicit sparse commuting circuit ensemble on $n$ qubits that reproduces low-order Haar moments in the stringent relative-error sense.<n>Our results show that precise Haar-like statistics can emerge from sparse commuting dynamics with remarkably low quantum resources.
• Score: 8.23817891945353
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: 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.

Related papers

• Low-Depth Random Unitaries without Ancillae [2.1517934062472532]
  We prove that random unitaries can be generated in optimal depth without ancillae.<n>Our results minimize the space-time costs for a wide range of quantum protocols.
  arXiv  Detail & Related papers  (2026-09-06T10:48:57Z)
• Verifiable quantum advantage in extremely low depth [52.51019642214249]
  We give a problem that is solvable by shallow quantum circuits, hard for computation in time under lattice-based assumptions.<n>We prove that shallow quantum circuits have sufficient structure to solve certain classically hard tasks whose solutions can be verified efficiently.
  arXiv  Detail & Related papers  (2026-09-01T15:54:34Z)
• Strong unitary designs in optimal depth and space [0.0]
  Unitary designs provide finite-moment approximations to Haar-random unitaries.<n>We show that strong unitary designs can be generated in logarithmic depth using only the system qubits.
  arXiv  Detail & Related papers  (2026-08-13T17:19:01Z)
• A Scalable Distributed Quantum Optimization Framework via Factor Graph Paradigm [46.08923284345648]
  We introduce a structure-aware framework for distributed quantum optimization.<n>We find that for a search space of size $N$, our framework achieves $O(sqrtN)$ query complexity up to processors and separator dependent factors.<n>We show that structure-aware decomposition delivers a practical path to scalable distributed quantum optimization on quantum networks.
  arXiv  Detail & Related papers  (2026-03-08T15:15:52Z)
• Shallow quantum circuit for generating O(1)-entangled approximate state designs [6.161617062225404]
  We find a new ensemble of quantum states that serve as an $epsilon$-approximate state $t$-design while possessing extremely low entanglement, magic, and coherence.<n>These resources can reach their theoretical lower bounds, $Omega(log (t/epsilon))$, which are also proven in this work.<n>A class of quantum circuits proposed in our work offers reduced cost for classical simulation of random quantum states.
  arXiv  Detail & Related papers  (2025-07-23T18:56:19Z)
• Efficient Quantum Circuit Compilation for Near-Term Quantum Advantage [17.38734393793605]
  We propose an approximate method for compiling target quantum circuits into brick-wall layouts.<n>This new circuit design consists of two-qubit CNOT gates that can be directly implemented on real quantum computers.
  arXiv  Detail & Related papers  (2025-01-13T15:04:39Z)
• Exponentially accurate open quantum simulation via randomized dissipation with minimal ancilla [0.0]
  Some quantum algorithms for simulating Lindblad dynamics achieve logarithmically short circuit depth in terms of accuracy $varepsilon$ by coherently encoding all possible jump processes with a large ancilla consumption.<n>We present a quantum algorithm for simulating general Lindblad dynamics with multiple jump operators aimed at an observable estimation, that achieves both a logarithmically short circuit depth and a minimum ancilla size.
  arXiv  Detail & Related papers  (2024-12-27T04:43:19Z)
• Towards large-scale quantum optimization solvers with few qubits [59.63282173947468]
  We introduce a variational quantum solver for optimizations over $m=mathcalO(nk)$ binary variables using only $n$ qubits, with tunable $k>1$. We analytically prove that the specific qubit-efficient encoding brings in a super-polynomial mitigation of barren plateaus as a built-in feature.
  arXiv  Detail & Related papers  (2024-01-17T18:59:38Z)
• Simulation of IBM's kicked Ising experiment with Projected Entangled Pair Operator [71.10376783074766]
  We perform classical simulations of the 127-qubit kicked Ising model, which was recently emulated using a quantum circuit with error mitigation. Our approach is based on the projected entangled pair operator (PEPO) in the Heisenberg picture. We develop a Clifford expansion theory to compute exact expectation values and use them to evaluate algorithms.
  arXiv  Detail & Related papers  (2023-08-06T10:24:23Z)
• A single $T$-gate makes distribution learning hard [56.045224655472865]
  This work provides an extensive characterization of the learnability of the output distributions of local quantum circuits. We show that for a wide variety of the most practically relevant learning algorithms -- including hybrid-quantum classical algorithms -- even the generative modelling problem associated with depth $d=omega(log(n))$ Clifford circuits is hard.
  arXiv  Detail & Related papers  (2022-07-07T08:04:15Z)
• Quantum State Preparation with Optimal Circuit Depth: Implementations and Applications [10.436969366019015]
  We show that any $Theta(n)$-depth circuit can be prepared with a $Theta(log(nd)) with $O(ndlog d)$ ancillary qubits. We discuss applications of the results in different quantum computing tasks, such as Hamiltonian simulation, solving linear systems of equations, and realizing quantum random access memories.
  arXiv  Detail & Related papers  (2022-01-27T13:16:30Z)
• Realization of arbitrary doubly-controlled quantum phase gates [62.997667081978825]
  We introduce a high-fidelity gate set inspired by a proposal for near-term quantum advantage in optimization problems. By orchestrating coherent, multi-level control over three transmon qutrits, we synthesize a family of deterministic, continuous-angle quantum phase gates acting in the natural three-qubit computational basis.
  arXiv  Detail & Related papers  (2021-08-03T17:49:09Z)

This list is automatically generated from the titles and abstracts of the papers in this site.

This site does not guarantee the quality of this site (including all information) and is not responsible for any consequences.