Part 3 of the Floquet-Ising series: from two qubits to prethermal oscillations
A toy model is truly educational only when it does not disappear behind a simulator. With two qubits, the state has four amplitudes. The complete Floquet cycle is a four-by-four matrix, and every measurement probability can be checked with ordinary linear algebra.
The basis and initial state
We use the computational basis in the order |00>, |01>, |10>, |11>. The initial state is
\[|\psi_0\rangle=|00\rangle=
\begin{pmatrix}1\\0\\0\\0\end{pmatrix}.
\]
A rotation around a Pauli axis P has the compact form
\[R_P(\theta)=e^{-i\theta P/2}
=\cos\!\left(\frac{\theta}{2}\right)I
-i\sin\!\left(\frac{\theta}{2}\right)P,
\]
because P squared equals I. For two independent qubits, we take the tensor product RX times RX or RZ times RZ. For the interaction, we use P = Z times Z.
The seven gates in one cycle
With A = RX(theta-x) times RX(theta-x), B = RZ(theta-z) times RZ(theta-z), and C = RZZ(theta-zz), the time order is A, B, C, A, B, A, B. The composite Floquet matrix is therefore
\[U_F = B A B A C B A.
\]
The rightmost A acts first on the state. This notation prevents a common mistake: a circuit diagram is often read from left to right, while matrix products act from right to left.
The parameters of the educational reduction are
\[\theta_x=\frac{\pi}{6},
\qquad
\theta_z=\frac{\pi}{27},
\qquad
\theta_{zz}=\frac{\pi}{3}.
\]
From amplitudes to probabilities
After one cycle, we calculate
\[|\psi_1\rangle=U_F|\psi_0\rangle
=a_{00}|00\rangle+a_{01}|01\rangle
+a_{10}|10\rangle+a_{11}|11\rangle.
\]
The phases of the amplitudes matter for later interference. In a direct Z measurement, we see only the Born probabilities Pxy = |axy|². The exact code gives
| Bit string | Probability after cycle 1 |
|---|---|
| 00 | 0.453879 |
| 01 | 0.186641 |
| 10 | 0.186641 |
| 11 | 0.172838 |
The sum is one, up to rounding. The equality P01 = P10 follows from the exchange symmetry of the two identical qubits and the symmetric initial state.
Magnetization without matrix multiplication
The magnetization operator is
\[\hat M=\frac{Z_0+Z_1}{2}.
\]
The states |01> and |10> contribute zero to the average: one spin points up and the other down. |00> contributes +1 and |11> contributes -1. Therefore
\[M(1)=P_{00}-P_{11}
=0.453879-0.172838
=0.281041.
\]
This is a useful check because we can obtain the same number in three ways: from the operator expectation value, directly from the probabilities, and with an independent Qiskit statevector.
Entanglement from a two-by-two matrix
Arrange the four amplitudes as a coefficient matrix
\[A=\begin{pmatrix}a_{00}&a_{01}\\a_{10}&a_{11}\end{pmatrix}.
\]
When we trace out qubit 1, the reduced density matrix for qubit 0 is rho0 = A A-dagger. With eigenvalues lambda1 and lambda2, the entanglement entropy is
\[S=-\sum_k \lambda_k\log_2\lambda_k.
\]
For a product state, S = 0. For a maximally entangled two-qubit state, S = 1 bit. After the first cycle, we find approximately 0.093 bit: the interaction has created entanglement, but not the maximum amount.
Why the exact implementation is small but valuable
The NumPy code constructs the gates from Pauli matrices, multiplies the state, and evaluates the observables. There is no sampling noise: all amplitudes are available. The Qiskit route independently constructs the same circuit. Regression tests require the two calculations to agree numerically.
state = |00>
for cycle in 0..N:
measure exact M and S
state = U_F @ stateThis simplicity exposes errors that are easily hidden in a large experiment: gate order, sign conventions, qubit endianness, the factor of one-half in rotations, and the mapping from bit strings to Z eigenvalues.
The limit of four amplitudes
For N qubits, a dense state contains 2 to the power N amplitudes. Four amplitudes are manageable; at 51 qubits, there are more than two quadrillion. The mathematics of one gate does not change, but the classical storage requirement does.
The toy model is therefore both a physical teaching model and a software oracle. It tells us exactly what the implementation must do in a case that needs no approximation. In part 4, we use the full trajectory to read magnetization and entanglement side by side.
Run it yourself
python -m floquet_qem run --cycles 12 --out .
python -m floquet_qem qiskit-check --cycles 12