Passive verification protocol for thermal graph states

Graph states are entangled resource states for universal measurement-based quantum computation. Although matter qubits such as superconducting circuits and trapped ions are promising candidates to generate graph states, it is technologically hard to entangle a large number of them due to several types of noise. Since they must be sufficiently cooled to maintain their quantum properties, thermal noise is one of the major ones. In this paper, we show that, for any temperature $T$, the fidelity $\langle G\left|{\rho }_T\right|G\rangle$ between an ideal graph state $|G\rangle$ at zero temperature and a thermal graph state ${\rho }_T$, which is a graph state at temperature $T$, can be efficiently estimated by using only one measurement setting. A remarkable property of our protocol is that it is passive, while existing protocols are active, namely, they switch between at least two measurement settings. Since thermal noise is equivalent to an independent phase-flip error, our estimation protocol also works for that error. By generalizing our protocol to hypergraph states, we apply our protocol to the quantum-computational-supremacy demonstration with instantaneous quantum polynomial time circuits. Our results should make the characterization of entangled matter qubits extremely feasible under thermal noise.

Figure 1

Schematic diagram of our passive verification protocol. A quantum computer affected by thermal noise generates thermal graph states ${\rho }_T^{\otimes N}$ and sends each qubit one by one. The verifier just measures $S_{1^k{0}^k}$ on each received state ${\rho }_T$ by using only single-qubit Pauli measurements. No quantum memory is required for the verifier.

Figure 2

Concrete examples of our protocol. Each circle and line represent $|+\rangle$ and the $\mathrm{CZ}$ gate, respectively. $X$, $Y$, and $Z$ represent the Pauli $X$, $Y$, and $Z$ measurements, respectively. The graph states depicted on the left-hand side are the ideal states. (a) $S_{10}=X\otimes Z$. (b) $S_{1100}=Y\otimes Y\otimes Z\otimes Z$. (c) $S_{111000}=-Y\otimes X\otimes Y\otimes Z\otimes Z\otimes Z$. The effect of the minus sign can be incorporated by a classical postprocessing.

Figure 3

Temperature dependence of our estimated value $F_{\mathrm{est}}$ and the value $F_{\mathrm{ub}}$ obtained with the union bound for $n=50$ and 100. The solid black, dotted yellow, and dashed red lines represent the fidelity $F\equiv \langle G\left|{\rho }_T\right|G\rangle$, $F_{\mathrm{est}}$, and $F_{\mathrm{ub}}$, respectively. In (a,b), the label of the horizontal axis is $K_{B}T$. In (c,d), it is converted to $p_{\beta }$. In (a,c) [(b,d)], fidelity is plotted in the case of $n=50\left(100\right)$.

Figure 4

A ten-qubit hypergraph state with ${\tilde{E}}_2=\varnothing$. For $1\le i\le 10$, the $i\mathrm{th}$ circle represents the $i\mathrm{th}$ qubit. The solid-line blue, dotted-line orange, dashed-line green, and thin-solid-line red triangles represent $\mathrm{CCZ}$ gates applied on ${\left\{(4j-3,4j-2,4j-1)\right\}}_{j=1}^2$, ${\left\{(4j-3,4j-1,4j)\right\}}_{j=1}^2$, ${\left\{(4j-1,4j,4j+1)\right\}}_{j=1}^2$, and ${\left\{(4j-1,4j+1,4j+2)\right\}}_{j=1}^2$, respectively.

Figure 5

A 20-qubit hypergraph state with four column qubits. For $1\le i\le 20$, the $i\mathrm{th}$ circle represents the $i\mathrm{th}$ qubit. The solid-line blue triangles represent $\mathrm{CCZ}$ gates.