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Design boosters: from constant-time quantum chaos to $\infty$-designs and beyond

Soumik Ghosh, Arjun Mirani, Yihui Quek, Michelle Xu

Abstract

We study a counterintuitive property of 'conditioning' on the result of measuring a subsystem of a quantum state: such conditioning can boost design quality, at the cost of increased system size. We work in the setting of deep thermalization from many-body physics: starting from a bipartite state on a global system $(A,B)$ drawn from a $k$-design, we measure system $B$ in the computational basis, keep the outcome and examine the state that remains in system $A$, approximating the overall ensemble (the 'projected ensemble') by a $k'$-design. We ask: how does the design quality change due to this procedure, or how does $k'$ compare to $k$? We give the first rigorous example of unitary dynamics generating a state such that, projection at very early (constant) times can boost design randomness. These dynamics are those of quantum chaos, modeled by the evolution of a Hamiltonian drawn from the Gaussian Unitary Ensemble (GUE). We show that, even though a state generated by such dynamics at constant time only forms a $k=\mathcal{O}(1)$ design, the projected ensemble is Haar-random (or a $k'=\infty$ design) in the thermodynamic limit (i.e. when $N_B=\infty$). This phenomenon persists even with weaker and more physically realistic assumptions; our results can be appropriately applied to non-GUE Hamiltonians that nevertheless show likely chaotic signatures in their eigenbases. Moreover, we show that with no assumption on how the global state was generated, a $k$-design experiences a degradation in design quality to $k' = \lfloor k/2 \rfloor$. This improves upon best prior results on the deep thermalization of designs. Together, our contributions argue for design boosting as a result of chaos and showcase a novel mechanism to generate good designs.

Design boosters: from constant-time quantum chaos to $\infty$-designs and beyond

Abstract

We study a counterintuitive property of 'conditioning' on the result of measuring a subsystem of a quantum state: such conditioning can boost design quality, at the cost of increased system size. We work in the setting of deep thermalization from many-body physics: starting from a bipartite state on a global system drawn from a -design, we measure system in the computational basis, keep the outcome and examine the state that remains in system , approximating the overall ensemble (the 'projected ensemble') by a -design. We ask: how does the design quality change due to this procedure, or how does compare to ? We give the first rigorous example of unitary dynamics generating a state such that, projection at very early (constant) times can boost design randomness. These dynamics are those of quantum chaos, modeled by the evolution of a Hamiltonian drawn from the Gaussian Unitary Ensemble (GUE). We show that, even though a state generated by such dynamics at constant time only forms a design, the projected ensemble is Haar-random (or a design) in the thermodynamic limit (i.e. when ). This phenomenon persists even with weaker and more physically realistic assumptions; our results can be appropriately applied to non-GUE Hamiltonians that nevertheless show likely chaotic signatures in their eigenbases. Moreover, we show that with no assumption on how the global state was generated, a -design experiences a degradation in design quality to . This improves upon best prior results on the deep thermalization of designs. Together, our contributions argue for design boosting as a result of chaos and showcase a novel mechanism to generate good designs.

Paper Structure

This paper contains 24 sections, 24 theorems, 137 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Our contributions
  3. Our setup
  4. Our work in context
  5. Our results
  6. Projected designs using GUEs
  7. Projected designs using more physical Hamiltonians
  8. Modifications on the spectrum
  9. Modifications on the eigenbasis
  10. Projected designs without the thermodynamic limit
  11. Projected designs without exact times
  12. Projected designs on subsystems of global designs
  13. Discussion
  14. The frame potential
  15. Facts about GUE
  16. ...and 9 more sections

Key Result

Theorem 2.1

Let $G$ be an $N_{AB}$-dimensional random matrix drawn from the GUE that acts on $\mathcal{H}_{A}\otimes\mathcal{H}_B$. Then, the projected ensemble $\mathcal{E}$ obtained by evolving with $G$ as a Hamiltonian and then measuring out subsystem $B$ approaches an exact quantum state design in the therm

Figures (3)

  • Figure 1: We evolve an initial state under a Hamiltonian $H$, measure the bath comprising of $n_B$ qubits, and get a projected ensemble on $n_A$ qubits.
  • Figure 2: A plot of the first expected trace moment for the Gaussian Hamiltonian plotted against time $t$. At the zeros, marked in red, the projected ensemble of the Gaussian Hamiltonian is an exact design in the thermodynamic limit of the bath. This begins at early (i.e. constant) times and extends for infinite points. In the finite system size limit though, those points are corrected by small subleading permutations and hence are designs. Moreover, at later times, we expect the amplitude of the trace moment decreases enough such that subleading permutations will be small even when the time is not exactly at the roots. Then, we conjecture the projected ensemble for the Gaussian Hamiltonian will always be an approximate design past a certain time.
  • Figure 3: A tensor contraction or Weingarten diagram for $(UDU^\dagger\ketbra{\phi} UD^\dagger U^\dagger)^{\otimes R}$. Notably the bottom factors of $D$ and $D^\dagger$ are all contracted together via $\sigma\in S_{2R}$, and the top has empty legs where the $\mathcal{Q}_B$ and overall trace will come in.

Theorems & Definitions (34)

  • Theorem 2.1: Informal
  • Lemma 1: Informal
  • Lemma 2
  • Lemma 3
  • Lemma 4
  • Theorem 2.2
  • Lemma 5
  • Lemma 6
  • Lemma 7: Bounding the trace distance from any $k$-th moment operator to the $k$-th Haar moment
  • proof
  • ...and 24 more