High-Precision Fidelity Estimation with Common Randomized Measurements

Abstract

Efficient fidelity estimation of multiqubit quantum states is crucial to many applications in quantum information processing. However, to estimate the infidelity $ε$ with multiplicative precision, conventional estimation protocols require (order) $1/ε^2$ different circuits in addition to $1/ε^2$ samples, which is quite resource-intensive for high-precision fidelity estimation. Here we introduce an efficient estimation protocol by virtue of common randomized measurements (CRM) integrated with shadow estimation based on the Clifford group, which only requires $1/ε$ circuits. Moreover, in many scenarios of practical interest, in the presence of depolarizing or Pauli noise for example, our protocol only requires a constant number of circuits, irrespective of the infidelity $ε$ and the qubit number. For large and intermediate quantum systems, quite often one circuit is already sufficient. In the course of study, we clarify the performance of CRM shadow estimation based on the Clifford group and 4-designs and highlight its advantages over standard and thrifty shadow estimation.

Figures (16)

• Figure 1: Schematic of CRM shadow estimation. Here $\rho$ is the unknown system state, $\sigma$ is the prior state stored in a classical computer, $R$ is the number of circuit reusing, and $N_U$ is the number of circuits sampled. The procedure involving quantum measurements, shown in the orange box, is the same as in thrifty shadow estimation, while the prior state $\sigma$ is employed to simulate the outcomes of randomized measurements and to construct the CRM estimator, which can usually reduce the variance.
• Figure 2: Circuit sample costs $N_U$ required for HPFE in THR (T) and CRM (C) shadow estimation based on Clifford measurements. Here, $r = 0.25$, $\delta = 0.01$, and $R = \lceil 10/\epsilon^2 \rceil$. The target and prior state has the form $\sigma=|S_{7,k}\rangle \langle S_{7,k}|$ with $k=1,4,7$. Different system states $\rho$ are generated by applying random local rotations described in Sec. \ref{['sec:DescriptRandomRotation']} (left plot) and random Pauli channels described in Sec. \ref{['sec:DescriptRandomPauli']} (right plot) to the target state $\sigma$. The black solid and dashed lines represent $N_U =8000$ and $N_U =8000 /\epsilon^2$, respectively, in the left plot, while they represent $N_U =5000$ and $N_U =5000/\epsilon^2$ in the right plot.
• Figure 3: Circuit sample costs $N_U$ required for HPFE in CRM shadow estimation based on three measurement ensembles, plotted as functions of the qubit number $n$. Here $r=0.25,\delta=0.01$, $\epsilon=0.001$, and $R=2\times10^{10}$. The target state $\sigma=|\mathrm{MC}_n\rangle\langle\mathrm{MC}_n|$ is the $n$-qubit magic cluster state with $n=2,3,\ldots,40$. Different system states $\rho$ are generated by applying depolarizing noise to the target state: $\rho =(1-p)\sigma +p\mathbbm{1}/d$ with $p=\epsilon/(1-d^{-1})$.
• Figure 4: Circuit sample costs $N_U$ required for HPFE in THR (T) and CRM (C) shadow estimation based on Clifford (Cl) and 4-design (4D) measurements. Here, $r = 0.25$, $\delta = 0.01$, and $R = \lceil d/\epsilon^2 \rceil$. The target and prior state has the form $\sigma=|S_{7,k}\rangle \langle S_{7,k}|$ with $k=1,7$, and different system states $\rho$ are generated by applying random local rotations described in SM Sec. \ref{['sec:DescriptRandomRotation']} (left plot) and random Pauli channels described in SM Sec. \ref{['sec:DescriptRandomPauli']} (right plot) to the target state $\sigma$. The results on 4-design measurements are almost independent of $k$, especially in the right plot.
• Figure 5: Circuit sample costs $N_U$ required for HPFE in CRM shadow estimation based on 4-design and Clifford measurements. Here, $r = 0.25$, $\delta = 0.01$, and $R = \lceil d/\epsilon^2 \rceil$. The target and prior states $\sigma$ are random seven-qubit pure states that have the form in Eq. (\ref{['eq:SampleCliffordAdv4design']}), and the system states have the form $\rho=\frac{1}{2}\sigma+\frac{1}{2} Z_1\sigma Z_1$, where $Z_1$ denotes the Pauli $Z$ operator acting on the first qubit.

Theorems & Definitions (54)

• Lemma 1
• Lemma 2
• Lemma 3
• Theorem 1
• Theorem 2
• Theorem 3
• Theorem 4
• Theorem 5
• Proposition 1
• Proposition 2