Unified approach to the resources of tensor network and stabilizer simulations

Abstract

We introduce a general resource indicator, called the bra-ket entanglement, which can be used to bound the resource dependence of classical simulations in the tensor network framework and in the stabilizer formalism. For the tensor network framework, our bounds indicate that bra-ket entanglement governs the interplay between two physical resources, the coherence and the magic. As bra-ket entanglement increases, the dominant resource that governs the complexity of the tensor network framework, quantified by entanglement, shifts from coherence to magic. For the stabilizer formalism approach, we find that magic is always the dominant resource regardless of bra-ket entanglement. This conclusion is obtained by developing an operator stabilizer formalism, which extends the standard stabilizer formalism for pure states and has additional advantages in simulating certain quantum circuits. Therefore, our results indicate that as bra-ket entanglement increases, the resource governing the complexity of the two approaches goes from different to the same.

Figures (4)

• Figure 1: Tensor network representation of vectorization of $O$ and two types of bipartitions: space bipartition and bra-ket bipartition. Taking 2-qubit $O$ as an example. When $O=|00\rangle\langle 00|$, $|O\rangle$ is a product state under both space and bra-ket bipartitions. When $O$ is a Bell state density matrix, $|O\rangle$ is a maximally entangled state under the space bipartition but a product state under the bra-ket bipartition. When $O$ is proportional to a 2-qubit Pauli operator, $|O\rangle$ is a product state in the space bipartition but a maximally entangled state in the bra-ket bipartition.
• Figure 2: (a): Gradient diagram of SE of $UOU^\dag$ in different CBC and BBC resources. $U$ is composed of many layers (deep enough to approach allowable SE upper bounds). Each layer is randomly composed of $\{CX,H,CCX,S\}$ with $p_{CCX}$ and $p_H$ denotes the picking rate. (b): SE of $UOU^\dag$ as functions of BKE. $O \in \{X^{\otimes s} \otimes|0\rangle\!\langle 0|^{\otimes (10- s)}\}_{s = 0}^{10}$ are BKE-matching operators with BKE ranging from 0 to 10. $U$ are deep circuits randomly chosen from $\mathcal{F}_{CBC}$ made up of $\{CCX,S,CX\}$ (SM \ref{['whyccx']}), $\mathcal{F}_{BBC}$ made up of $\{H,S,CX\}$, and $\mathcal{F}_{zero}$ made up of $\{S,CX\}$ respectively. In both sub-figures, we choose $\alpha=1$ for $H_{SE}$.
• Figure 3: Relations between tensor network and stabilizer formalism. As BKE increases, SE, which governs the complexity of the tensor network approach, will change from mainly decided by CBC ($H$) to mainly decided by BBC ($T$). On the other hand, OSF is always governed by magic $T$. Therefore, as BKE increases, the related resources between the two approaches will go from mismatching to matching.
• Figure 4: The same setting as Fig \ref{['fig3']} except for $U$ in $\mathcal{F}_{CBC}$, we construct them randomly from $\{T,S,CX\}$ instead of $\{CCX,T,S,CX\}$.

Theorems & Definitions (38)

• Theorem 1
• Theorem 2
• Theorem 3
• Theorem 4
• Definition 1
• Lemma 1
• Definition 2: Vectorization
• Definition 3: SE-Rényi entropy
• Definition 4: BKE-Rényi entropy
• Definition 5: Fourier-BKE Rényi entropy