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Measurement-driven quantum advantages in shallow circuits

Chenfeng Cao, Jens Eisert

TL;DR

The paper tackles the challenge of achieving quantum advantage with shallow circuits under hardware constraints by leveraging mid-circuit measurements and feed-forward. It introduces measurement-driven fan-out staircases that compress dense IQP circuits into constant depth, yielding $k$-local diagonal interactions and strong anti-concentration on bounded-degree 2D lattices. Beyond sampling, the approach enables constant-depth, globally entangling quantum reservoirs that can classify phases in an extended SSH model, demonstrating robust performance across Pauli sectors and resilience to noise. Under plausible complexity-theoretic assumptions, the work argues that approximate sampling from this family is classically hard, highlighting a practical route to NISQ+-level advantages on near-term hardware.

Abstract

Quantum advantage schemes probe the boundary between classically simulatable and classically intractable quantum dynamics. We explore the impact of mid-circuit measurements on the computational power of quantum circuits. To this effect, we focus on quantum sampling and introduce a constant-depth measurement-driven approach for efficiently sampling from a broad class of commuting diagonal quantum circuits and associated structured phase states, previously requiring polynomial-depth unitary circuits. By interleaving mid-circuit measurements with feed-forward in randomized "fan-out staircases", our dynamical circuits bypass Lieb-Robinson light-cone constraints, enabling global entanglement with flexible auxiliary qubit usage on bounded-degree lattices (e.g., two-dimensional grids). The generated phase states exhibit random-matrix statistics and anti-concentration comparable to fully random architectures. We further demonstrate measurement-driven feature maps that distinguish phases of an extended SSH model from random eigenstates in a quantum machine-learning benchmark (reservoir computing). Technologically, our results harness mid-circuit measurements to realize quantum advantages on bounded-degree hardware with a favorable topology. Conceptually, they provide complexity-theoretic support for quantum speedups by mid-circuit measurements.

Measurement-driven quantum advantages in shallow circuits

TL;DR

The paper tackles the challenge of achieving quantum advantage with shallow circuits under hardware constraints by leveraging mid-circuit measurements and feed-forward. It introduces measurement-driven fan-out staircases that compress dense IQP circuits into constant depth, yielding -local diagonal interactions and strong anti-concentration on bounded-degree 2D lattices. Beyond sampling, the approach enables constant-depth, globally entangling quantum reservoirs that can classify phases in an extended SSH model, demonstrating robust performance across Pauli sectors and resilience to noise. Under plausible complexity-theoretic assumptions, the work argues that approximate sampling from this family is classically hard, highlighting a practical route to NISQ+-level advantages on near-term hardware.

Abstract

Quantum advantage schemes probe the boundary between classically simulatable and classically intractable quantum dynamics. We explore the impact of mid-circuit measurements on the computational power of quantum circuits. To this effect, we focus on quantum sampling and introduce a constant-depth measurement-driven approach for efficiently sampling from a broad class of commuting diagonal quantum circuits and associated structured phase states, previously requiring polynomial-depth unitary circuits. By interleaving mid-circuit measurements with feed-forward in randomized "fan-out staircases", our dynamical circuits bypass Lieb-Robinson light-cone constraints, enabling global entanglement with flexible auxiliary qubit usage on bounded-degree lattices (e.g., two-dimensional grids). The generated phase states exhibit random-matrix statistics and anti-concentration comparable to fully random architectures. We further demonstrate measurement-driven feature maps that distinguish phases of an extended SSH model from random eigenstates in a quantum machine-learning benchmark (reservoir computing). Technologically, our results harness mid-circuit measurements to realize quantum advantages on bounded-degree hardware with a favorable topology. Conceptually, they provide complexity-theoretic support for quantum speedups by mid-circuit measurements.
Paper Structure (16 sections, 5 theorems, 84 equations, 12 figures, 2 algorithms)

This paper contains 16 sections, 5 theorems, 84 equations, 12 figures, 2 algorithms.

Table of Contents

  1. Conclusion.
  2. Acknowledgments.
  3. Proof:
  4. Appendix A: Fan-out ladders and staircases
  5. One-dimensional topology
  6. Single repetition of the $\mathrm{CX}$ layer.
  7. Two repetitions of the $\mathrm{CX}$ layer.
  8. Generalization and optimization
  9. Appendix B: Measurement-driven IQP sampling
  10. Proof of Proposition 1.
  11. Proof of Lemma 1.
  12. Proof of Theorem 1.
  13. Spectral (eigenvalue) requirement.
  14. Appendix C: Geometrically local quantum circuit cost in two-dimensional lattices
  15. Appendix D: Measurement‑based multibody XY quantum reservoir
  16. ...and 1 more sections

Key Result

Proposition 1

Fully random architectures $\mathbf{A} \in \mathbb{Z}_2^{s \times n}$ achieve anti-concentration with $s\in\mathcal{O}(n)$.

Figures (12)

  • Figure 1: Schematic of measurement-based fan-out staircases for dense IQP sampling. (a) Multi-loop fan-out staircases (alternating-direction ladders) generated by random dynamic circuits with measurement and feed-forward. (b) Conceptual bipartite (system–auxiliary) coupling graph. Our geometrically local, bounded-degree 2D layouts are subgraphs of this schematic. (c) Constant-depth implementation of dense IQP sampling via interleaved phase rotations and randomized fan-out staircases, where measurement outcomes inform subsequent corrections. Staircase construction and transfer-matrix feed-forward are detailed in the End Matter and SM.
  • Figure 2: (a) Checkerboard layout of alternating system and auxiliary qubits on a 2D square lattice, with a sampled Hamiltonian path covering all qubits. (b) Comparison of eigenvalue distributions from randomized fan-out staircases with the theoretical Marchenko–Pastur law for a representative $41 \times 41$-qubit system. (c, d) Minimal circuit depths required to satisfy Criterion 1, contrasting measurement-driven fan-out staircases (blue squares) with circuits without auxiliary qubits (red circles), for (c) 2D grid and (d) all-to-all connectivity.
  • Figure 3: Comparison of measurement-driven IQP sampling (blue) with standard IQP sampling (red/gray) at matched constant two-qubit depth on a 2D nearest-neighbor grid. (a) Relative entanglement-based circuit cost $\xi/\xi_{\text{lin}}$ for fixed-depth circuits $\mathfrak{L}=2, \mathfrak{D}=2$ versus system size. (b) Collision probability ratio $\chi/\chi_{\text{Haar}}$ versus system size. (c) Total variation distance between noisy and ideal output distributions under gate depolarizing errors. (d) Total variation distance under pure dephasing noise assuming fixed gate duration.
  • Figure 4: Phase-classification accuracy versus Floquet cycle for different reservoirs. Each violin distribution is obtained from 50 random realizations of architectures and coupling strengths. The measurement-driven multibody XY reservoir achieves high accuracy at short times, outperforming local reservoirs (i)-(iii), while removing feed-forward (FF) adaptivity significantly reduces performance.
  • Figure 5: An example of the split-and-mend procedure for constructing a random directed Hamiltonian path on a 2D grid.
  • ...and 7 more figures

Theorems & Definitions (8)

  • Proposition 1: Anti-concentration
  • Theorem 1: Measurement-driven short IQP circuits
  • Conjecture 1: Average-case hardness
  • Theorem 2: Expressivity separation for local Hamiltonian reservoirs
  • Proposition 2: Measurement-driven fan-out staircases
  • Example 1: Transfer matrix for a CX ladder Baumer2024Measurement
  • Lemma 1: Circuit synthesis
  • Remark 1: Alternative circuit synthesis strategies