Weak distillation of quantum resources

Abstract

Importance sampling based on quasi-probability decomposition is the backbone of many widely used techniques, such as error mitigation, circuit knitting, and, more generally, virtual quantum resource distillation, as it allows one to simulate operations that are not accessible in a given setting. However, this class of protocols faces a fundamental problem -- it only allows to estimate expectation values. Here, we provide a general framework that lifts any quasi-probability-based protocol from expectation value estimation to a weak simulator, realizing sampling from the desired distribution only using a restricted class of quantum resources. Our method runs with the sampling cost proportional to the negativity of the quasi-probability, in stark contrast to the naive estimation-based approach that requires a large number of samples even in the case of small negativity. We show that our method requires significantly fewer samples in a number of relevant scenarios, such as error mitigation, entanglement distillation and magic state distillation. Our framework realizes the weak simulation of quantum resources without actually distilling the state, introducing a new notion of quantum resource distillation.

Figures (1)

• Figure 1: Comparison between the total variation distance of two frameworks with respect to sampling cost. For all plots, we take the average of 20 outcomes. Blue line named "Rejection TVD" represents the numerical simulation's outcome of the actual total variation distance we obtain with our framework. The orange line named "Estimation TVD" represents the outcome with the probability distribution estimation method. To make a valid probability distribution, we normalized the distribution and calculated the total variation distance for the estimation method. Dotted lines represent the tightest bounds shown in the theorems. Since the total variation distance is at most 1, we take 1 as the maximum value for both bounds. We use $\delta = 0.1$. (a) The simulation of error mitigation for local depolarizing noise. For each qubit, we use the decomposition $\rho = \frac{1}{1-p}\mathcal{E}(\rho) - \frac{p}{1-p}\frac{I}{2}$ where $\mathcal{E}(\cdot)$ is the local depolarizing noise channel. We take $p = 0.005$, and prepared a random $4$ qubit system. (b) The simulation of entanglement distillation. The goal is to obtain a noiseless $n$-pair of Bell states. Consider the decomposition $\Phi^{\otimes n} = \frac{1}{1-p}\rho^{n}_p- \frac{p}{1-p}\frac{I-\Phi^{\otimes n}}{2^{2n}-1}$ where $\rho^n_\alpha = (1-\alpha)\Phi^{\otimes n} + \alpha \frac{I - \Phi^{\otimes n}}{2^{2n}-1}$ is the isotropic state, we simulate the setting for 10 qubit case with $p = 0.01$. (c) The simulation of the 5 qubit $T$-doped IQP circuit, which has 5 $T$ gates in the circuit. With the dephased $T$ state defined as $\rho^T_p = (1-p)T + p \frac{I}{2}$, and dephased flipped $T$ state $\rho^{\bar{T}}_p$, we know the decomposition $T = \frac{2-p}{2(1-p)}\rho^T_p - \frac{p}{2(1-p)}\rho^{\bar{T}}_p$. We implement the $T$ gate with $T$-state injection, with the decomposition parameter set to $p = 0.1$.

Theorems & Definitions (13)

• Proposition 1
• Theorem 1
• Proposition 2
• proof
• Theorem 2: Theorem \ref{['thm: Sampling cost for rejection']} in the main text
• proof
• Definition 1: Negative association
• Lemma 1
• proof
• Theorem 3