Part 4 of the Floquet-Ising series: from two qubits to prethermal oscillations
A quantum state can look simple locally while hiding a great deal of information in correlations. We therefore track two different quantities in the toy model: magnetization as a local order parameter and entanglement entropy as a measure of nonlocal quantum correlation.
Two windows onto the same state
The magnetization
\[M(n)=\langle\psi_n|\frac{Z_0+Z_1}{2}|\psi_n\rangle
\]
asks whether the spins still point, on average, in the original Z direction. The entropy S(n) asks how much information about one qubit is accessible only through the other. Neither quantity determines the full state.
A product state can have zero magnetization without being entangled. A Bell state can likewise have zero magnetization while being maximally entangled. The combination M and S is therefore more informative than either curve alone.
The exact toy trajectory
| Cycle | Magnetization M | Entropy S (bit) |
|---|---|---|
| 0 | 1.000000 | 0.000 |
| 1 | 0.281041 | 0.093 |
| 2 | -0.421446 | 0.793 |
| 4 | 0.039504 | 0.949 |
| 6 | 0.827352 | 0.354 |
| 8 | -0.663285 | 0.508 |
| 10 | 0.510699 | 0.796 |
| 12 | 0.392479 | 0.785 |
The magnetization oscillates strongly, while the entropy does not simply move in the opposite direction. This makes sense: the phases that determine future interference are not fully visible in M, and the same local magnetization can accompany different correlation structures.
Why a small closed system keeps returning
A two-qubit system has only four quasienergy modes. The time series is a finite sum of oscillations at frequencies set by differences between Floquet quasienergies. There is no large internal reservoir in which phase information can appear to disappear irreversibly. Recurrences are therefore normal.
\[U_F|\phi_k\rangle=e^{-i\varepsilon_k T}|\phi_k\rangle,
\qquad
M(n)=\sum_{k,l}c_{kl}
e^{-in(\varepsilon_l-\varepsilon_k)T}.
\]
This explains oscillations in the toy model without invoking prethermalization. In a large system, the quasienergy spectrum becomes extremely dense, and local observables can appear to relax as many frequencies dephase.
Period four in the paper
After an initial transient, the large heavy-hex systems display an oscillatory component with a period of approximately four cycles. The important word is component: the raw magnetization also contains a slowly changing background and decay.
A suitable fit can have the form
\[M(n)=B(n)+A e^{-n/\tau}
\cos\!\left(2\pi n/P+\phi\right),
\]
where B(n) describes the nonoscillating background. The period measurement is credible when the oscillatory model has a clearly better likelihood or information-criterion score than a monotonic fit, and when A is statistically distinguishable from zero.
From oscillation to collective physics
Robust prethermal oscillations require more than P being approximately four at one system size:
- the period remains comparable as systems grow;
- the amplitude can be extracted with uncertainties;
- amplitude scaling distinguishes a finite limit from finite-size decay;
- the dynamics builds substantial state and operator entanglement;
- noise and mitigation do not create the oscillation artificially;
- classical methods agree wherever they are demonstrably converged.
The paper therefore combines exact 21- and 28-qubit simulations, larger hardware measurements, different geometries, and a finite-size fit. The hardware primarily extends the reach of the scaling analysis.
What our 51-qubit measurement adds
In the repository, the public 51-qubit tracker has been run for up to 16 cycles with 512 shots per measurement arm. The raw and locally M3-corrected trajectories contain a strong oscillatory fit around a period of four to five cycles. This is interesting and consistent with the paper’s trend.
It is not yet an independent confirmation of the same many-body conclusion. Our observable is the average of 86 distance-three ZZ pairs, the shot budgets are limited, and there is no exactly converged cycle-16 reference. An independent repetition is therefore part of the validation, not a formality.
The lesson from the toy model
The two curves M and S teach us the language of the large calculation: local order can remain visible while quantum complexity grows. But the small system also teaches restraint. A beautiful oscillation by itself is not evidence of a prethermal phase, time crystal, or quantum advantage.
In part 5, we add noise. We then see how a precisely known curve can be distorted and why extrapolation can look almost perfect in a toy model.
Sources
- Exact toy trajectory.
- E. Leviatan et al., finite-size and oscillation analysis.