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Three Hamiltonians are Sufficient for Unitary $k$-Design in Temporal Ensemble

Yi-Neng Zhou, Tian-Gang Zhou, Julian Sonner

Abstract

Unitary $k$-designs are central to quantum information and quantum many-body physics as efficient proxies for Haar-random dynamics. We study how chaotic Hamiltonian evolution can generate unitary $k$-designs. Standard approaches typically rely on many independent Hamiltonian realizations or fine-tuning evolution times. Here we show that unitary designs can instead arise from a quenched temporal ensemble, where Hamiltonians are sampled once and held fixed, while randomness enters only through the evolution times. We analyze a two-step protocol (2SP), applying $H_1$ for time $t_1$ and $H_2$ for time $t_2$, and a three-step protocol (3SP) with an additional quench, with all times randomly drawn from a prescribed distribution. Time averaging imposes energy-index matching in the frame potential (FP), which quantifies the distance to Haar random. Analytically and numerically, we show that 2SP cannot realize a general unitary $k$-design, whereas 3SP can do so for arbitrary $k$. The advantage of 3SP is that the additional random phases impose stronger constraints, eliminating independent permutation degrees of freedom in the FP. For Gaussian unitary ensemble Hamiltonians, we prove these results rigorously and show that under imperfect time averaging, 3SP achieves the same accuracy as 2SP with a parametrically narrower time window.

Three Hamiltonians are Sufficient for Unitary $k$-Design in Temporal Ensemble

Abstract

Unitary -designs are central to quantum information and quantum many-body physics as efficient proxies for Haar-random dynamics. We study how chaotic Hamiltonian evolution can generate unitary -designs. Standard approaches typically rely on many independent Hamiltonian realizations or fine-tuning evolution times. Here we show that unitary designs can instead arise from a quenched temporal ensemble, where Hamiltonians are sampled once and held fixed, while randomness enters only through the evolution times. We analyze a two-step protocol (2SP), applying for time and for time , and a three-step protocol (3SP) with an additional quench, with all times randomly drawn from a prescribed distribution. Time averaging imposes energy-index matching in the frame potential (FP), which quantifies the distance to Haar random. Analytically and numerically, we show that 2SP cannot realize a general unitary -design, whereas 3SP can do so for arbitrary . The advantage of 3SP is that the additional random phases impose stronger constraints, eliminating independent permutation degrees of freedom in the FP. For Gaussian unitary ensemble Hamiltonians, we prove these results rigorously and show that under imperfect time averaging, 3SP achieves the same accuracy as 2SP with a parametrically narrower time window.

Paper Structure

This paper contains 15 sections, 4 theorems, 115 equations, 4 figures.

Table of Contents

  1. Appendix A: Haar random and Weingarten Calculus
  2. Proof of Theorem 1 (leading large-$D$)
  3. An Example in Theorem 1
  4. Proof of Theorem 2 (leading large-$D$)
  5. Averaging $U^{(1)}$ at leading order.
  6. Averaging $U^{(2)}$ at leading order
  7. An example of Theorem 2
  8. Appendix B: Finite-$T$ corrections from imperfect time filtering
  9. 2SP: structure of the leakage
  10. 3SP: why the O(eps) leakage cancels in expectation
  11. Appendix C: Formulation of the time-filter error for Haar-random unitaries
  12. Proof of Theorem 3
  13. Proof of Theorem 4
  14. Leakage for $\Delta F_{(1)}^{(k)}$
  15. Leakage for $\Delta F_{(2)}^{(k)}$

Key Result

Theorem 1

In the perfect-filter limit $T\to\infty$ so that Eq. eq:2step_FP_perfect applies, the overlap matrix $U^{(\bm{1})}\in U(D)$ of GUE Hamiltonians is Haar-random The Haar randomness of the GUE overlap matrix originates from the unitary invariance of the GUE ensemble. However, this symmetry property doe where $!n$ is the derangement number. $\blacktriangleleft$$\blacktriangleleft$

Figures (4)

  • Figure 1: Random ensembles generated by the two-step (2SP) and three-step (3SP) protocols with fixed Hamiltonians $H_1$, $H_2$, and $H_3$. Each box denotes evolution under the corresponding fixed Hamiltonian for a duration $t_1$, $t_2$, or $t_3$, where the times are drawn independently from the same distribution $P(t)$ illustrated by the curve within the central green circle. Theorems \ref{['thm:2step_perfect']} and \ref{['thm:3step_perfect']} show that 2SP cannot realize a general unitary $k$-design, whereas 3SP can.
  • Figure 2: Numerical FPs for the 2SP and 3SP under GUE and cSYK dynamics. For GUE, we take Hilbert-space dimension $D=100$, for cSYK we take fermion number $N=8$ at half filling, and for the rSpin model with dipolar interactions we take spin number $N=8$ and half filling. Each data point is averaged over $10^{6}$ independent time-sampling realizations. (a),(b) FP versus design order $k$ for 2SP and 3SP. The black dashed lines indicate the infinite-time predictions: $F^{(k)}_{\mathrm{2SP}}(\infty)$ from Theorem \ref{['thm:2step_perfect']} in (a), and $F^{(k)}_{\mathrm{3SP}}(\infty)=k!$ from Theorem \ref{['thm:3step_perfect']} in (b). The Haar value $k!$ is shown as a gray dotted line. Blue (red) markers denote GUE (cSYK) numerics. We use $T\approx 10^6$ to approximate the $T\to\infty$ limit. (c),(d) Imperfect time-filter error for GUE at different design orders $k$, defined as $|\delta F^{(k)}_{\mathrm{2SP}}(T)|=\bigl|F^{(k)}_{\mathrm{2SP}}(T)/F^{(k)}_{\mathrm{2SP}}(\infty)-1\bigr|$ in (c) and $|\delta F^{(k)}_{\mathrm{3SP}}(T)|=\bigl|F^{(k)}_{\mathrm{3SP}}(T)/F^{(k)}_{\mathrm{3SP}}(\infty)-1\bigr|$ in (d), and plotted versus the filter window $T$. The horizontal dashed line indicates the $10^{-1}$ threshold.
  • Figure 3: Pairing lines for the example \ref{['eq:SM_i_j_sequences']}. Top row: $m$-pairs fixed by $\pi$. Bottom row: $n$-pairs fixed by $\sigma$. The only shared pair is $\{1,5\}$ (thick), giving two allowed $\alpha\in\{(),(1\,5)\}$ and thus $\bigl|G_m(\pi)\cap G_n(\sigma)\bigr|=2$.
  • Figure 4: Leading $U^{(1)}$ contraction for $k=3$ and $\pi=(12)$. The full-swap $\alpha_{\rm sw}=(1\,5)(2\,4)(3\,6)$ pairs the second-half $f$'s with the first-half $f$'s and the first-half $p$'s with the second-half $p$'s, forcing $\upsilon=\sigma=\pi$ at leading order.

Theorems & Definitions (12)

  • Theorem 1: 2SP, perfect filter
  • proof : Proof sketch
  • Theorem 2: 3SP, perfect filter
  • proof : Proof sketch
  • Theorem 3: 2SP, imperfect filter
  • proof : Proof sketch
  • Theorem 4: 3SP, imperfect filter
  • proof : Proof sketch
  • proof
  • proof
  • ...and 2 more