Quantum contextuality with mixed states of 1D symmetry-protected topological order

Abstract

Bell theorems of many-body nonlocality and contextuality serve as a benchmark for proving quantum advantage in that a quantum computer outperforms a classical computer for a certain problem. In practice, however, near-term quantum devices do not prepare perfectly pure states but rather mixed states produced from noisy channels. We investigate noisy quantum advantage by considering thermal mixed states of one-dimensional many-body systems with a symmetry-protected topological (SPT) order. In the pure-state (or zero-temperature) case, these states are known to be useful for measurement-based quantum computation, and to outperform classical computers in a many-body contextuality game, provided string order parameters (SOPs) of SPT are sufficiently large. Here, we show that quantum advantage in mixed states is measured by a combination of twisted SOP and symmetry representation expectation values. Using the minimally entangled typical thermal states algorithm, it is demonstrated that quantum advantage persists to a nonzero critical temperature for finite-sized instances of the many-body contextuality game. While this critical temperature goes to zero in the thermodynamic limit, it is relatively robust to system size, suggesting that these states remain useful for demonstrating genuine "quantumness" of noisy hardware in a scalable fashion. Finally, we show that the quantum winning probability is lower bounded by the global fidelity with the 1D cluster state, so that our contextuality game can provide an operational meaning to benchmark the capacity to create long-range order like SPT states in near-term experimental devices.

Figures (9)

• Figure 1: Symmetry operators associated with $\mathds{Z}_2\times\mathds{Z}_2$ SPTO on a spin-$1/2$ chain with $n=12$ qubits. The four $\mathds{Z}_2\times\mathds{Z}_2$ group elements are denoted compactly as $(a,b)$, where $a,b\in\{0,1\}$. The top qubit chain shows explicitly the $\mathds{Z}_2\times\mathds{Z}_2$ symmetry representations and cluster state string order parameters (SOPs). The left and right SOP boundary blocks of SOPs are labeled by $p$ and $q$; the figure illustrates the case of $p=2,q=5$. The bottom qubit chain shows that the twisted string order parameter is constructed by "twisting" a symmetry operator and an SOP, and is equivalent to the product of the two up to the twist phase $\Omega(g,h)$ which measures the commutation properties of projective symmetry representations $V^{L}(g)$ or $V^{R}(g)$.
• Figure 2: Nontrivial SPT phase and quantum advantage of the ground states of our Hamiltonian Eq. \ref{['eq:Hamiltonian']}. The boundary of the SPT phase is shown in white, witnessed when the expectation value of any one of the three ($g=x,y,$ or $z$) string order parameters, shown in Fig. \ref{['fig:SOP']}, is nonzero. The heatmap shows the minimum string order parameter expectation value $\ev{\mathcal{S}}$ given by Eq. \ref{['eq:classicalbound']}. If this quantity is larger than $1/3$, shown by the red contour, the state can be used to show quantum advantage in the multiplayer contextual triangle game.
• Figure 3: Multiplayer contextual triangle game and definition of quantum advantage. (a) Setup of the game. Three players $\alpha,\beta,\gamma$, equally spaced within $N$ total players, are each given inputs $\{0,1\}$ uniformly at random, while all other players are given input $0$ and thus only ever output $(a_p,b_p,c_p)$. We can imagine players $\alpha,\beta,\gamma$ at the corners of a triangle. For each triangle edge, denoted $\mu\nu=\{\mu+1,\hdots,\nu-1\}$ for $\mu\nu\in\{\alpha\beta,\beta\gamma,\gamma\alpha\}$ and each variable $\sigma\in\{a,b,c\}$, an "edge string" $\sigma_{\mu\nu}$ is defined as the sum of all variables of type $\sigma$ on the edge $\mu\nu$. Winning conditions of the game are shown in the brackets and are analogous to the three-player game conditions Eqs. \ref{['eq:win1']}-\ref{['eq:win4']} up to addition of edge strings from the remaining $N-3$ players. (b) Classical strategies. We consider strategies that consist of $R$ rounds of nearest-neighbor communication where players can share their input information. When $R$ reaches $N/3$ the players $\alpha,\beta,\gamma$ are able to share input information and win with unit probability. However, when $R<N/3$ all classical strategies are bounded by $P^{(C)}\le7/8$. Our notion of quantum advantage is defined by a multiplayer triangle game strategy which uses quantum resources to achieve $P^{(Q)}>7/8$. (c) Quantum strategy. Each player $p$ holds 2 adjacent qubits $2p-1$, $2p$ of an $n=2N$-qubit state $\ket{\phi}$ (which we take to be translationally invariant) and outputs measurement outcomes of the cluster state symmetry operators shown in the table. We refer to this measurement strategy as the cluster state measurement strategy. The global measurements corresponding to win conditions $(ii)$ and $(iii)$ form global symmetries and those corresponding to $(iv)$-$(vi)$ form twisted string order parameters. An SPTO ground state is symmetric so that the winning probability only depends on its expectation value of the cluster state twisted SOP. Correspondingly, the cluster state $\ket{C_n}$ wins with unit probability.
• Figure 4: Behavior of the minimum winning probability $P_{n}^{(Q,min)}$ in a thermal cluster state as a function of temperature and system size. The quantum advantageous contour $P_{n}^{(Q,min)}=7/8$ is shown in red. Although the critical temperature $T_c$ (where $P_{n}^{(Q,min)}=7/8$ for fixed $n$) decays to zero in the thermodynamic limit, it is robust for finite-sized systems.
• Figure 5: Analysis of accuracy and precision of the METTS algorithm by plotting METTS estimates of triangle game winning probability against exact solutions along (a) the $J_X$ axis and (b) the $J_{ZZ}$ axis. Exact solutions are shown as solid lines and METTS estimates are plotted as points with error bars corresponding to relative error. The algorithm performs well across these axes.

Theorems & Definitions (6)

• Theorem 1
• proof
• Corollary 1
• Theorem 2
• proof
• Corollary 2