Dorian Rudolph

The theory of Total Function NP (TFNP) and its subclasses says that, even if one is promised an efficiently verifiable proof exists for a problem, finding this proof can be intractable. Despite being a classical complexity class, however, TFNP has made a surprise appearance in the study of Quantum Satisfiability (QSAT): If a QSAT instance has a System of Distinct Representatives (SDR), then it has a product-state solution [Laumann, Läuchli, Moessner, Scardicchio, and Sondhi 2010]. Efficiently finding this product-state solution, however, has remained elusive. In this work, we introduce a new framework based on Weighted SDRs (WSDR), which among other results, allows us to: (1) significantly simplify and extend the results of [LLMSS 2010] to qudit systems, (2) establish a connection to the Bézout number for multihomogeneous polynomial systems, and (3) apply the parameterized algorithm of [Aldi, de Beaudrap, Gharibian, Saeedi 2021] to solve new instances of QSAT efficiently on qudits. The second of these, in particular, allows us to define the first ""quantum-inspired"" subclass of TFNP, for which we show QSAT with SDR is complete. Thus, we obtain the first evidence that QSAT with SDR is, in fact, intractable.

Despite the fundamental role the Quantum Satisfiability (QSAT) problem has played in quantum complexity theory, a central question remains open: At which local dimension does the complexity of QSAT transition from ""easy"" to ""hard""? Here, we study QSAT with each constraint acting on a k-dimensional and l-dimensional qudit pair, denoted (k,l)-QSAT. Our first main result shows that, surprisingly, QSAT on qubits can remain QMA_1-hard, in that (2,5)-QSAT is QMA_1-complete. (QMA_1 is a quantum analogue of MA with perfect completeness.) In contrast, (2,2)-QSAT (i.e. Quantum 2-SAT on qubits) is well-known to be poly-time solvable [Bravyi, 2006]. Our second main result proves that (3,d)-QSAT on the 1D line with d = O(1) is also QMA_1-hard. Finally, we initiate the study of (2,d)-QSAT on the 1D line by giving a frustration-free 1D Hamiltonian with a unique, entangled ground state. As implied by our title, our first result uses a direct embedding: We combine a novel clock construction with the 2D circuit-to-Hamiltonian construction of [Gosset and Nagaj, 2013]. Of note is a new simplified and analytic proof for the latter (as opposed to a partially numeric proof in [GN13]). This exploits Unitary Labelled Graphs [Bausch, Cubitt, Ozols, 2017] together with a new ""Nullspace Connection Lemma"", allowing us to break low energy analyses into small patches of projectors, and to improve the soundness analysis of [GN13] from Omega(1/T^6) to Omega(1/T^2), for T the number of gates. Our second result goes via black-box reduction: Given an arbitrary 1D Hamiltonian H on d'-dimensional qudits, we show how to embed it into an effective 1D (3,d)-QSAT instance, for d = O(1). Our approach may be viewed as a weaker notion of ""simulation"" (à la [Bravyi, Hastings 2017], [Cubitt, Montanaro, Piddock 2018]). As far as we are aware, this gives the first ""black-box simulation""-based QMA_1-hardness result.