Variational approach to the quantum separability problem

We present the variational separability verifier (VSV), which is a variational quantum algorithm that determines the closest separable state (CSS) of an arbitrary quantum state with respect to the Hilbert-Schmidt distance (HSD). We first assess the performance of the VSV by investigating the convergence of the optimization procedure for Greenberger-Horne-Zeilinger states of up to seven qubits, using both state-vector and shot-based simulations. We also numerically determine the CSS of maximally entangled mixed $X$ states, and subsequently use the results of the algorithm to surmise the analytical form of the aforementioned CSS. Our results indicate that current noisy intermediate-scale quantum devices may be useful in addressing the NP-hard full separability problem using the VSV, due to the shallow quantum circuit imposed by employing the destructive swap test to evaluate the HSD. The VSV may also possibly lead to the characterization of multipartite quantum states, once the algorithm is adapted and improved to obtain the closest $k$-separable state of a multipartite entangled state.

Figure 1

Destructive swap test for determining the overlap between two quantum states. The bundled controlled-not (cnot) gate here actually represents cnot gates between each pair of qubits of $\rho$ and $\sigma$, with the bundled Hadamard gate equating to performing a Hadamard gate on each qubit of $\rho$. This effectively reduces to a Bell basis transformation for each pair of qubits of $\rho$ and $\sigma$

Figure 2

Diagrammatic representation of the VSV. The inputs to the VSV are the test state $\rho$, and initial parameters ${\mathit{\rho }}_0,{\mathit{\theta }}_0,{\mathit{\varphi }}_0$. The VSV then computes the orange (bottom-left QPU) circuit once, to compute $\text{Tr}\left\{{\rho }^2\right\}$, with the green (top-right QPU) and purple (bottom-right QPU) circuits evaluated to compute overlaps $\langle {\psi }_i\left|\rho \right|{\psi }_i\rangle$ and $\langle {\psi }_i|{\psi }_j\rangle \forall i,j\in \left[s\right]$, which are used in Eqs. (11) and (12), respectively. The red (bottom CPU) section is the classical feedback loop computed by the CPU, consisting of first minimizing the parameters $\mathit{p}$, followed by proposing new parameters $\mathit{\theta }$ and $\mathit{\varphi }$ to evaluate on the QPU. After a sufficient number of iterations, the VSV outputs optimal parameters ${\mathit{\rho }}^{*},{\mathit{\theta }}^{*},{\mathit{\varphi }}^{*}$, along with the computed CSS and corresponding HSE.

Figure 3

Plot of the convergence of HSD with respect to the number of iterations for the state-vector optimization for GHZ states. The solid lines represent the convergence for two to seven qubits (bottom-up). The dashed lines represent the analytical HSE of the corresponding number of qubits, that is $E_{\text{HS}}{\left(\right|\text{GHZ}\rangle }_n)$. The inset figure represents the shot-based optimization convergence with 8192 shots for two to five qubits (bottom-up). Note that there are less iterations for the shot-based cases (although the same number of function evaluations), since it is harder for the GSA optimizer to find a strictly lower HSE at each function evaluation.

Figure 4

Plot of the results of the state-vector optimization applied to two-qubit $X$-MEMS. The solid lines represent the analytical HSE as a function of $\left|\gamma \right|$. The left blue segment represents the range between $0\le \left|\gamma \right|\le \frac{1}{3}$, while the right orange segment represents the range between $\frac{1}{3}\le \left|\gamma \right|\le \frac{1}{2}$. The inset figure represents the data acquired with shot-based optimization using 8192 shots.

Figure 5

Plot of the results of the state-vector optimization applied to three-qubit $X$-MEMS. The solid lines represent the analytical HSE as a function of $\left|\gamma \right|$. The left blue segment represents the range between $0\le \left|\gamma \right|\le \frac{1}{5}$, while the right orange segment represents the range between $\frac{1}{5}\le \left|\gamma \right|\le \frac{1}{2}$. The inset figure represents the data acquired with shot-based optimization using 8192 shots.

Figure 6

Analytical HSE of two- to nine-qubit $X$-MEMS as a function of $\left|\gamma \right|$ (bottom-up). The infinite-qubit (topmost line) case represents the analytical upper bound of ${2\left|\gamma \right|}^2$ for the HSE of $X$-MEMS.

Figure 7

Visualization of the $n\text{th}$ iteration of the QGA. A random pure product state ${\sigma }_n$ is generated that satisfies the preselection criterion. The next step is to find ${\rho }_n=p_n{\rho }_{n-1}+(1-p_n){\sigma }_n$, $p_n\in (0,1)$, such that $D_{\text{HS}}(\rho ,{\rho }_n)<D_{\text{HS}}(\rho ,{\rho }_{n-1})$. Figure adapted from Ref. [32].