We show a linear-size reduction from gap Max-2-Lin(2) (a generalization of the approximate Maximum Cut, or gap Max-Cut, problem) to γ-CVP_p for γ = O(1) and finite p ≥ 1, as well as a no-go theorem against poly-sized non-adaptive quantum reductions from k-SAT to CVP₂. This implies three headline results: (i) Faster algorithms for γ-CVP_p are also faster algorithms for Max-2-Lin(2) and Max-Cut. Depending on the approximation regime, even a 2^{0.78n}-time or 2^{0.3n}-time algorithm would improve upon state-of-the-art algorithms such as Williams' 2004 algorithm [TCS 2005] or Arora, Barak, and Steurer’s 2010 algorithm [JACM 2015]. This provides evidence that γ-CVP_p for γ = O(1) requires exponential time, improving upon the previous exponential lower-bound for γ-CVP₂ with γ < 3 by Bennett, Golovnev, and Stephens-Davidowitz [FOCS 2017]. (ii) A new almost 2^{(1/2 + ε/4ς + o(1)) n}-time classical algorithm and a new almost 2^{(1/3 + ε/6ς + o(1)) n}-time quantum algorithm for (1-ε, 1-ς)-gap Max-Cut. This algorithm is faster than the algorithm of Arora, Barak and Steurer [JACM 2015], as well as the algorithm of Williams [TCS 2005], and the algorithm of Manurangsi and Trevisan [APPROX 2018] when c₀ ε < ς < c₁ ε for constants c₀, c₁. (iii) If the Quantum Strong Exponential Time Hypothesis (QSETH) can be used to show a 2^{δ n}-time lower-bound for Max-Cut, Max-2-Lin(2), or CVP₂ for any constant δ > 0, it must be via an adaptive quantum reduction unless NP ⊆ pr-QSZK. This illuminates some difficulties in characterizing the hardness of approximate constraint satisfaction problems and shows that the post-quantum security of lattice-based cryptography likely cannot be supported by QSETH. This result complements the no-go results of Aggarwal and Kumar [FOCS 2023], who showed that the classical security of lattice-based cryptography likely cannot be supported by the classical Strong Exponential Time Hypothesis (SETH).
We show a linear-size reduction from gap Max-2-Lin(2) (a generalization of the approximate Maximum Cut, or gap Max-Cut, problem) to γ-CVP_p for γ = O(1) and finite p ≥ 1, as well as a no-go theorem against poly-sized non-adaptive quantum reductions from k-SAT to CVP₂. This implies three headline results: (i) Faster algorithms for γ-CVP_p are also faster algorithms for Max-2-Lin(2) and Max-Cut. Depending on the approximation regime, even a 2^{0.78n}-time or 2^{0.3n}-time algorithm would improve upon state-of-the-art algorithms such as Williams' 2004 algorithm [TCS 2005] or Arora, Barak, and Steurer’s 2010 algorithm [JACM 2015]. This provides evidence that γ-CVP_p for γ = O(1) requires exponential time, improving upon the previous exponential lower-bound for γ-CVP₂ with γ < 3 by Bennett, Golovnev, and Stephens-Davidowitz [FOCS 2017]. (ii) A new almost 2^{(1/2 + ε/4ς + o(1)) n}-time classical algorithm and a new almost 2^{(1/3 + ε/6ς + o(1)) n}-time quantum algorithm for (1-ε, 1-ς)-gap Max-Cut. This algorithm is faster than the algorithm of Arora, Barak and Steurer [JACM 2015], as well as the algorithm of Williams [TCS 2005], and the algorithm of Manurangsi and Trevisan [APPROX 2018] when c₀ ε < ς < c₁ ε for constants c₀, c₁. (iii) If the Quantum Strong Exponential Time Hypothesis (QSETH) can be used to show a 2^{δ n}-time lower-bound for Max-Cut, Max-2-Lin(2), or CVP₂ for any constant δ > 0, it must be via an adaptive quantum reduction unless NP ⊆ pr-QSZK. This illuminates some difficulties in characterizing the hardness of approximate constraint satisfaction problems and shows that the post-quantum security of lattice-based cryptography likely cannot be supported by QSETH. This result complements the no-go results of Aggarwal and Kumar [FOCS 2023], who showed that the classical security of lattice-based cryptography likely cannot be supported by the classical Strong Exponential Time Hypothesis (SETH).
The local Hamiltonian (LH) problem is the canonical QMA-complete problem introduced by Kitaev. In this paper, we show its hardness in a very strong sense: we show that the 3-local Hamiltonian problem on n qubits cannot be solved classically in time O(2(1−ε)n) for any ε>0 under the Strong Exponential-Time Hypothesis (SETH), and cannot be solved quantumly in time O(2(1−ε)n/2) for any ε>0 under the Quantum Strong Exponential-Time Hypothesis (QSETH). These lower bounds give evidence that the currently known classical and quantum algorithms for LH cannot be significantly improved.
Furthermore, we are able to demonstrate fine-grained complexity lower bounds for approximating the quantum partition function (QPF) with an arbitrary constant relative error. Approximating QPF with relative error is known to be equivalent to approximately counting the dimension of the solution subspace of QMA problems. We show the SETH and QSETH hardness to estimate QPF with constant relative error. We then provide a quantum algorithm that runs in O(2n) time for an arbitrary 1/poly(n) relative error, matching our lower bounds and improving the state-of-the-art algorithm by Bravyi, Chowdhury, Gosset, and Wocjan (Nature Physics 2022) in the low-temperature regime.
To prove our fine-grained lower bounds, we introduce the first size-preserving circuit-to-Hamiltonian construction that encodes the computation of a T-time quantum circuit acting on N qubits into a (d+1)-local Hamiltonian acting on N+O(T1/d) qubits. This improves the standard construction based on the unary clock, which uses N+O(T) qubits.
The local Hamiltonian problem is the canonical QMA-complete problem, and O(2n) time classical algorithms and O(2n/2) time quantum algorithms are known to solve the problem in the worst case. It is not clear how to improve these brute force strategies for a broad class of the problem because ground states are highly entangled in general, and we cannot directly apply known strategies for classical CSPs.
In this work, we present exponentially faster classical and quantum algorithms under two mild assumptions: (1) the Hamiltonian is frustration-free on YES instances, and (2) it is approximately regular, meaning that every qubit is acted upon by approximately the same number of constraints.
We complement these upper bounds by showing that, assuming (Q)SETH, quantum 5-SAT admits no non-trivial worst-case speedup. Our lower bound further demonstrates that the dependence of our algorithms on regularity is in some sense nearly optimal. Specifically, quantum 5-SAT remains (Q)SETH-hard even for Hamiltonians in which all but O(n) qubits participate in only constantly many constraints, while the remaining O(n) qubits each participate in O(n) constraints. By contrast, if either the size of this high-degree subset or the degrees of its qubits is reduced by a factor of nδ, for any δ>0, our algorithm solves the problem in time O(2(1−ε)n) for some ε>0.
Together, our upper and lower bounds establish a fine-grained complexity dichotomy for quantum satisfiability.
In this work, we analyze the average-case hardness of approximation for the Quantum Hypergraph Max-Cut problem using the theoretical framework of the Quantum Overlap Gap Property (QOGP). We establish two main results. Our first result applies to a wide class of stable quantum algorithms, satisfying a Lipschitz property with respect to the quantum Wasserstein distance of order 2. We show a weak hardness result, demonstrating that for any Lipschitz constant L, there is some k such that L-stable algorithms cannot approximate the optimal solution to Quantum Hypergraph Max-Cut on k-uniform hypergraphs in the average case. Additionally, we establish a strong hardness result where k is independent of L, but only for a more restricted class of local quantum algorithms defined using the quantum Wasserstein distance of order ∞. We apply these results to establish concrete depth lower bounds for popular quantum algorithms for preparing near-optimal states for this problem.
We investigate the space complexity of two graph streaming problems: MAX-CUT and its quantum analogue, QUANTUM MAX-CUT. Previous work by Kapralov and Krachun [STOC 19] resolved the classical complexity of the classical problem, showing that any (2 – ε)-approximation requires Ω(n) space (a 2-approximation is trivial with O(log n) space). We generalize both of these qualifiers, demonstrating Ω(n) space lower bounds for (2 – ε)-approximating MAX-CUT and QUANTUM MAX-CUT, even if the algorithm is allowed to maintain a quantum state. As the trivial approximation algorithm for QUANTUM MAX-CUT only gives a 4-approximation, we show tightness with an algorithm that returns a (2 + ε)-approximation to the QUANTUM MAX-CUT value of a graph in O(log n) space. Our work resolves the quantum and classical approximability of quantum and classical Max-Cut using o(n) space.We prove our lower bounds through the techniques of Boolean Fourier analysis. We give the first application of these methods to sequential one-way quantum communication, in which each player receives a quantum message from the previous player, and can then perform arbitrary quantum operations on it before sending it to the next. To this end, we show how Fourier-analytic techniques may be used to understand the application of a quantum channel.
What is the power of polynomial-time quantum computation with access to an NP oracle? In this work, we focus on two fundamental tasks from the study of Boolean satisfiability (SAT) problems: search-to-decision reductions, and approximate counting. We first show that, in strong contrast to the classical setting where a poly-time Turing machine requires Θ(n) queries to an NP oracle to compute a witness to a given SAT formula, quantumly Θ(logn) queries suffice. We then show this is tight in the black-box model - any quantum algorithm with "NP-like" query access to a formula requires Ω(logn) queries to extract a solution with constant probability. Moving to approximate counting of SAT solutions, by exploiting a quantum link between search-to-decision reductions and approximate counting, we show that existing classical approximate counting algorithms are likely optimal. First, we give a lower bound in the "NP-like" black-box query setting: Approximate counting requires Ω(logn) queries, even on a quantum computer. We then give a "white-box" lower bound (i.e. where the input formula is not hidden in the oracle) - if there exists a randomized poly-time classical or quantum algorithm for approximate counting making o(logn) NP queries, then BPPNP[o(n)] contains a PNP-complete problem if the algorithm is classical and FBQPNP[o(n)] contains an FPNP-complete problem if the algorithm is quantum.
For general max-k-XORSAT with k≥3, no polynomial-time algorithm can do substantially better than random guessing on worst-case instances unless P=NP: approximating beyond the random-assignment value of 1/2 is NP-hard. The picture changes when each variable appears in at most D constraints. In that bounded-degree setting, polynomial-time algorithms can provably beat the random baseline by an additive amount of order 1/D. For Boolean instances, this scaling is known to be optimal: the matching hardness result is due to Trevisan, while the corresponding algorithmic guarantee was established by Barak et al. Whether the same holds over general finite fields, and what it implies for quantum algorithms, has not been established. We make this connection explicit and extend the hardness to max-Ek-LINSAT(q,r) with bounded degree D and over arbitrary finite fields Fq, proving that it is NP-hard to exceed r/q+Oq,r(1/D). These results provide the complexity-theoretic benchmark for the bounded-degree instances targeted by decoded quantum interferometry (DQI), QAOA, and classical heuristics. Any quantum advantage on bounded-degree instances is therefore confined to the constant prefactor. We further show that in the context of DQI and on (k,D)-regular instances, this prefactor is sensitive to the nature of the decoder: DQI with classical decoders faces an information-theoretic 1/DlogD barrier that prevents it from matching the hardness scaling, while DQI with quantum decoders is compatible with the 1/D scaling -- identifying quantum decoding as the key ingredient for matching the complexity-theoretic scaling with DQI.