Computational complexity of injective projected entangled pair states

TL;DR

The paper analyzes the computational complexity of contracting two-dimensional PEPS as a function of injectivity. It proves a dichotomy: strongly injective PEPS with $\delta>\delta_{\mathrm{easy}}$ admit polynomial-time classical contraction (via a gapped parent Hamiltonian and patch contraction) and can be efficiently prepared on a quantum computer, while weakly injective PEPS with $\delta<\delta_{\mathrm{hard}}$ render the local observable problem postBQP-hard by embedding fault-tolerant, noisy postselected quantum circuits into PEPS. The key technical advance is a fault-tolerant postselection scheme that mitigates noise in such embeddings, enabling a rigorous hardness proof, and a parallel construction showing the easy regime via exponential clustering and uniform gaps. The results connect PEPS contraction complexity to Hamiltonian complexity and fault-tolerant quantum computation, with implications for normalization, translation-invariant cases, and potential learning tasks.

Abstract

Projected entangled pair states (PEPS) constitute a variational family of quantum states with area-law entanglement. PEPS are particularly relevant and successful for studying ground states of spatially local Hamiltonians. However, computing local expectation values in these states is known to be \postBQP-hard. Injective PEPS, where all constituent tensors fulfil an injectivity constraint, are generally believed to be better behaved, because they are unique ground states of spatially local Hamiltonians. In this work, we therefore examine how the computational hardness of contraction depends on the injectivity. We establish that below a constant positive injectivity threshold, evaluating local observables remains \postBQP-complete, while above a different constant nontrivial threshold there exists an efficient classical algorithm for the task, resolving an open question from (Anshu et al., STOC `24). We do this by proving that noisy postselected quantum computation can be made fault-tolerant.

Figures (3)

• Figure 1: Embedding circuits into PEPS Malz2025. (a) A (postselected) circuit is built out of gates (and projections) $U$ acting on a Hilbert space of dimension $q$ (here, $q=2\times2$). (b) In the noisy circuit considered here, each gate/projection is followed by depolarizing noise at rate $\eta$, yielding the map $\Phi_\eta$. (c) The PEPS tensors $T$ are obtained by considering the Stinespring representation of the map using an ancilla of dimension $q^2$ (red). To obtain $\delta$-injective tensors, we need $\eta=\delta^2/(4(1+3\delta^2))$.
• Figure 2: (a) Ideal postselected quantum computation. A circuit consisting of $\mathop{\mathrm{poly}}\nolimits(n)$ gates acts on $n$ qubits. The first output qubit is postselected on the $|0\rangle$ state, and the second output qubit is measured in the computational basis. (b) Fault-tolerant circuit for noisy postselected computation. $n$ logical qubits are encoded via $k$-fold concatenation of a quantum error-correcting code, and given as input to a fault-tolerant version of the unitary $U$. Moreover, $m-1$ additional qubits are also encoded and the postselection register is "copied" onto them using logical $\mathop{\mathrm{CNOT}}\nolimits$ gates. The resulting $m$ postselection qubits are the noisily decoded and projected onto the $|0\rangle$ state. Meanwhile, the output qubit is decoded and measured.
• Figure 3: (a) Circuit for a postselection gate onto the $|0\rangle$ state using a closed timelike curve (CTC). (b) A Pauli $X$ error on the CTC corresponds to incorrectly postselecting onto $|1\rangle$; $Y$ and $Z$ errors can be ignored, as these result in zero amplitude over the CTC.

Theorems & Definitions (16)

• Theorem : Informal
• Definition 1: PEPS
• Definition 2: $\delta$-injective PEPS
• Definition 3: Norm
• Definition 4: Nev, Normalized expectation value of local observables
• Lemma 5: Uniform gap for strongly injective PEPS Schuch2011
• Corollary 6: Nev is efficient in strongly injective PEPS
• proof
• Theorem 7: Threshold theorem Aharonov1999
• Lemma 8: Concatenated decoding of errors