Variational quantum algorithms to estimate rank, quantum entropies, fidelity, and Fisher information via purity minimization

Variational quantum algorithms (VQAs) that estimate values of widely used physical quantities such as the rank, the quantum entropies, the Bures fidelity, and the quantum Fisher information of mixed quantum states are developed. In addition, variations of these VQAs are also adapted to perform other useful functions such as quantum state learning and approximate fractional inverses. The common theme shared by the proposed algorithms is that their cost functions are all based on minimizing the quantum purity of a quantum state. Strategies to mitigate or avoid the problem of exponentially vanishing cost function gradients are also discussed.

Figure 1

Example circuits to compute (left) the numerator and (right) the denominator of the cost function $C\left(\theta \right)$ with $k=1$ [see Eq. (6)]. In both cases, the output is obtained by measuring a single qubit in the computational basis

Figure 2

Exact and estimated values of the (left) rank, (middle) Rényi entropy for $\alpha =1/2$, and (right) fidelity using the proposed VQAs. The input state is parametrized by $\varphi$ and is given by $\rho ={cos}^2(\varphi /2)|0\rangle \langle 0|+{sin}^2(\varphi /2)|1\rangle \langle 1|$. For the fidelity calculation $F(\rho ,\sigma )$, the second input state was chosen to be the maximally mixed state $\sigma ={\mathbb{1}}_2/2$.

Figure 3

Exact (dashed lines) and estimated (triangle points) values of the (left) rank, (middle) Rényi entropy for $\alpha =1/2$, and (right) fidelity for $n=3,6,9$ qubit states. The input state is a product state given by $\rho =\left[{cos}^2(\varphi /2)\right|0\rangle \langle 0|+{sin}^2(\varphi /2)|1\rangle {\langle 1\left|\right]}^{\otimes n}$. For the fidelity calculation $F(\rho ,\sigma )$, the second input state was chosen to be the maximally mixed state $\sigma ={\mathbb{1}}_d/d$, where $d=2^n$. Here, the cost function is directly computed rather than obtained via simulated quantum circuits as in Fig. 2.

Figure 4

Left: Average magnitude of the gradient components of cost function (6) for $k=4$ and for $R=2,3,...,20$, with $\rho$ being the completely mixed state of rank $R$. Error bars are smaller than the dot radius. The red curve is a least-squares fit of the function $f\left(x\right)=a{x}^c+b{x}^{c-1}$ to the data; optimal parameters are $a\approx 1.41, b\approx -1.35, c\approx -1.25$. Right: Grayscale plot of $C\left(\theta \right)$ defined by (26) and (27) with $\lambda =1/2, \mu ={\lambda }^{1/2k}/[{\lambda }^{1/2k}+{(1-\lambda )}^{1/2k}]$, and $n=2$. The gradient vanishes with $n$ in the center region near the solution.