Entanglement Barriers from Computational Complexity: Matrix-Product-State Approach to Satisfiability

TL;DR

This work argues based on careful analysis of the structure imprinted onto the MPS by the 3-SAT instances that this barrier arises from classical computational complexity, and elucidates with stochastic models the specific relationship between the classical hardness of the NP-complete counting problem and the entanglement properties of the quantum state.

Abstract

We approach the 3-SAT satisfiability problem with the quantum-inspired method of imaginary time propagation (ITP) applied to matrix product states (MPS) on a classical computer. This ansatz is fundamentally limited by a quantum entanglement barrier that emerges in imaginary time, reflecting the exponential hardness expected for this NP-complete problem. Strikingly, we argue based on careful analysis of the structure imprinted onto the MPS by the 3-SAT instances that this barrier arises from classical computational complexity. To reveal this connection, we elucidate with stochastic models the specific relationship between the classical hardness of the $\sharp$P $\supseteq$ NP-complete counting problem $\sharp$3-SAT and the entanglement properties of the quantum state. Our findings illuminate the limitations of this quantum-inspired approach and demonstrate how purely classical computational complexity can manifest in quantum entanglement. Furthermore, we present estimates of the non-stabilizerness required by the protocol, finding a similar resource barrier. Specifically, the necessary amount of non-Clifford operations scales superlinearly in system size, thus implying extensive resource requirements of ITP on different architectures such as Clifford circuits or gate-based quantum computers.

Figures (12)

• Figure 1: An initial state $\vert \psi_0 \rangle$ is transformed into $\vert \psi_s \rangle$ containing the solution to a hard combinatorial problem by evolution of the imaginary time $\tau$. Both states are unentangled and thus trivially representable in the MPS ansatz space $\mathcal{M}$. The protocol is obstructed by a bump of the half-chain entanglement entropy $S$ at intermediate imaginary times. The plot shows data for a uniquely solvable 3-SAT instance with $n=18$ variables and $m=76$ clauses ($\alpha=4.22$). Solving the combinatorial problem as measured by the total weight of the solution (dashed) is only possible after facing the entanglement bump of height $\hat{S}$ at time $\hat{\tau}$.
• Figure 2: Maximal height $\hat{S}$ of the half-chain entanglement bump as illustrated in \ref{['fig:illu-entanglement-bump']} averaged over $1000$ instances. In (a) the scaling of the mean $\mu = \langle \hat{S} \rangle$ with the number of variables is shown to be linear at $\alpha=\alpha_c \approx 4.27$ while the standard deviation $\sigma$ (error bars) remains constant. Compared to the maximally possible entropy $S_{\mathrm{max}} = n \ln 2 / 2$ (hatched region) and that of a typical random states given by the Page law $S_{\mathrm{typ}} \approx S_{\mathrm{max}} - 1/2$PhysRevLett.71.1291 (dash-dot line), the entanglement bump height is smaller as discussed in \ref{['sec:stat_entanglement_model']}. The inset confirms that $\mu$ and $\sigma$ are sufficient to characterize the distribution of $\hat{S}$ by comparing the numerically obtained probability density function to a Gaussian $f_{\mu,\sigma}$ ($\mu=2.181, \sigma=0.207$). In (b) the mean of $\hat{S} / n$ is shown for a range of interesting $\alpha$ values stressing the universal properties independent of system size. Note that $S_{\mathrm{max}} / n = \ln 2 / 2 \approx 0.15$. The temporal position of the entanglement bump $\hat{\tau}$ is investigated in \ref{['app:entanglement_scaling']} .
• Figure 3: In (a) the entanglement entropy $S$ and the $\alpha=1$ ($\alpha=2$) stabilizer Rényi entropy $M_1$ ($M_2$) for a uniquely solvable 3-SAT instance with $n=15$ variables at $\alpha_c$; for the stabilizer Rényi entropies $1000.0$ ($200000.0$) samples were drawn to estimate them. Both measures of non-stabilizerness feature the same characteristics as the entanglement entropy. In (b) we show the average non-stabilizerness $\langle \hat{M}_1\rangle$---normalized to its maximal value $M_{\mathrm{max}} = n \ln 2$---and its uncertainty at different clause to variable ratios $\alpha$ sampled from $100$ instances for different number of variables $n$. Lines are provided for visual guidance only.
• Figure 4: The three complexity regimes reflected in the structure of a typical 3-SAT quantum state $\vert \psi \rangle$ and its Schmidt decomposition for $n=8$ variables at different $\alpha$ ratios. The weights of $\vert \psi \rangle$ are shown with the computational basis spatially separated into left $\vert l_i \rangle$ (vertical) and right $\vert r_i \rangle$ (horizontal) states, which is the decomposition the encodes at the central bond. For small clause-to-variable ratios $\alpha \ll 1$ (top row), the Schmidt states merely subtract weight for forbidden bitstrings from to the initial state, thus being similar to listing all states excluded from satisfying an instance. Similarly, for ratios $\alpha \approx \alpha_c$ (bottom row) in which only few disjoint bitstrings remain viable, the Schmidt weights are formed by superpositions of their computational basis states. In the hard regime $\alpha \approx \alpha^\star$ (middle row), the Schmidt weights are formed by grouping residual correlations between the variables, leading to the formation of logically correlated states. In the example, the first, third and fourth Schmidt vectors group to represent the allowed computational basis states in the bottom half of the full state.
• Figure 5: Entanglement entropy of the combinatorial random states (purple) and those states originating from constraints of 3-SAT (green). The data (dots) is shown along the respective statistical models (lines, bounds as area). States corresponding to 3-SAT instances are differently entangled than random combinatorial ones. The random combinatorial state approaches the dominant diagonal model as described above in the thermodynamic limit; compare, e.g., $n=10$ and $n=20$. Similarly, the model for the 3-SAT entanglement in \ref{['eq:markov_chain_model']} strongly agrees with the numerical data.