Measuring the Rényi entropy of a two-site Fermi-Hubbard model on a trapped ion quantum computer

The efficient simulation of correlated quantum systems is a promising near-term application of quantum computers. Here, we present a measurement of the second Rényi entropy of the ground state of the two-site Fermi-Hubbard model on a five-qubit programmable quantum computer based on trapped ions. Our work illustrates the extraction of a nonlinear characteristic of a quantum state using a controlled-swap gate acting on two copies of the state. This scalable measurement of entanglement on a universal quantum computer will, with more qubits, provide insights into many-body quantum systems that are impossible to simulate on classical computers.

Figure 1

The quantum circuit for the adiabatic evolution of two copies (labeled 1 and 2) of the two-site Fermi Hubbard model (each realized in two qubits $A$ and $B$) and the measurement of the second Rényi entropy. The noninteracting ground state is prepared via the application of Hadamard gates, followed by the digitized adiabatic evolution to a finite value of $U$ by repeated application of the central code block. $m$ is an integer referring to the $m\mathrm{th}$ step of the adiabatic evolution; $\delta$ and $\tau$ are given in the text. The angles in the rotation gates are in radians. The Rényi entropy $R_2$ is measured by applying a cswap gate controlled by an ancilla qubit (anc.) on subsystems $A$ and subsequent detection of the state of the ancilla. By applying additional Hadamard gates and measuring the system qubits, experimental errors can be detected and discarded.

Figure 2

Expectation value of the Hamiltonian $\langle \widehat{H}\rangle$ of the Fermi-Hubbard model after adiabatic evolution, to the values of $U$ given by the abscissa. Method I (M. I, open diamonds) sees an increase in experimental error with $U$ (shaded area) since $N_{\mathrm{steps}}$ increases with $U$. Subtracting a straight line of slope 0.063 best matches the data points to the theory curve (filled diamonds). For method II (M. II, open circles), $N_{\mathrm{steps}}=5$ is constant and the experimental error in $\langle \widehat{H}\rangle$ is seen as an offset. Even for large systems, this can be measured for the integrable case $U=0$ and then subtracted (filled circles). The corrected results (measured offset 0.58) follow the theory curve well. Method II is a better match to the exact solution.

Figure 3

Implementation of the cswap or Fredkin gate. The plot shows results for all eight input states, reproducing the evolution operator [Eq. (9)] with an average success probability of $86.8\left(3\right)\%$, while the state of the control qubit is correct with $94.0\left(2\right)\%$ probability. The results have been corrected for $\sim 1\%$ state preparation and measurement error.

Figure 4

Results of the measurement of $R_2$ after the digitized adiabatic evolution to different interaction strengths $U$ according to method I (a) and II (b) compared to the expected curve (solid line) and the exact solution (dashed line). The open symbols show the original data while the filled symbols show the same data after postselection based on the symmetry of the cswap gate.

Figure 5

The value of $R_2$ output from the quantum algorithm compared to the exact value at different values of the adiabatic evolution time $\tau$ and Trotter step size $\delta$ for the first-order Trotter approximation.

Figure 6

(a) The error in $R_2,{\epsilon }_{R_2}$ (defined in the text) as a function of evolution time $\tau$ at fixed $\delta =0.05$ fits well to a line of slope $-1.9$. (b) ${\epsilon }_{R_2}$ as a function of Trotter step size $\delta$ at fixed $\tau =10$ fits well to lines of slope 3.3. (c) The error in the wave function ${\epsilon }_{\mathrm{\Psi }}$ (defined in the text) at fixed $\tau =10$ as a function of Trotter step size $\delta$ fits well to a line of slope 2.2.

Figure 7

In first Trotter approximation, (a) the parallel circuit depth $D$ as a function of evolution time $\tau$ at fixed $\delta =0.05$ fits well to lines of slope 1, and (b) $D$ as a function of Trotter step size $\delta$ at fixed $\tau =10$ fits well to lines of slope $-1$. Here $D$ is defined as the circuit depth for adiabatic evolution to $U=10$.

Figure 8

The value of $R_2$ resulting from the quantum algorithm compared to the exact value for different values of the adiabatic evolution time $\tau$ and Trotter step size $\delta$ for (a) method I with fixed $\tau /\delta =1$ and (b) method II with fixed $N_{\text{steps}}=5$.

Figure 9

Controlled-swap (or Fredkin) gate implementation using entangling xx$\left(\chi \right)$ gates, and single-qubit rotations $R_x\left(\theta \right),R_y\left(\theta \right)$, and $R(\theta ,\varphi )$. The parameters are $\alpha =\text{sgn}\left({\chi }_{12}\right),\beta =\text{sgn}\left({\chi }_{23}\right),\gamma =\text{sgn}\left({\chi }_{13}\right)$, and $P=arcsin\sqrt{\frac{2}{3}}$.

Figure 10

The original output of the circuit shown in Fig. 1 averaged over 2000–2500 runs, corrected for SPAM errors. Method I is shown in panel (a) and method II in panel (b).

Figure 11

The output of the circuit shown in Fig. 1 averaged over 2000–2500 runs, corrected for SPAM errors and after discarding in postselection any runs that resulted in the states given in the text. Method I is shown in panel (a) and method II in panel (b). The legend gives the yields, i.e., the fractions of experimental runs that were kept.