The strong exponential-time hypothesis (SETH) is a commonly used conjecture in the field of complexity theory. It essentially states that determining whether a CNF formula is satisfiable can not be done faster than exhaustive search over all possible assignments. This hypothesis and its variants gave rise to a fruitful field of research, fine-grained complexity, obtaining (mostly tight) lower bounds for many problems in P whose unconditional lower bounds are very likely beyond current techniques. In this work, we introduce an extensive framework of Quantum Strong Exponential-Time Hypotheses, as quantum analogues to what SETH is for classical computation. Using the QSETH framework, we are able to translate quantum query lower bounds on black-box problems to conditional quantum time lower bounds for many problems in P. As an example, we provide a conditional quantum time lower bound of Ω(n^1.5) for the Longest Common Subsequence and Edit Distance problems. We also show that the n^2 SETH-based lower bound for a recent scheme for Proofs of Useful Work carries over to the quantum setting using our framework, maintaining a quadratic gap between verifier and prover. Lastly, we show that the assumptions in our framework can not be simplified further with relativizing proof techniques, as they are false in relativized worlds.
The strong exponential-time hypothesis (SETH) is a commonly used conjecture in the field of complexity theory. It essentially states that determining whether a CNF formula is satisfiable can not be done faster than exhaustive search over all possible assignments. This hypothesis and its variants gave rise to a fruitful field of research, fine-grained complexity, obtaining (mostly tight) lower bounds for many problems in P whose unconditional lower bounds are very likely beyond current techniques. In this work, we introduce an extensive framework of Quantum Strong Exponential-Time Hypotheses, as quantum analogues to what SETH is for classical computation. Using the QSETH framework, we are able to translate quantum query lower bounds on black-box problems to conditional quantum time lower bounds for many problems in P. As an example, we provide a conditional quantum time lower bound of Ω(n^1.5) for the Longest Common Subsequence and Edit Distance problems. We also show that the n^2 SETH-based lower bound for a recent scheme for Proofs of Useful Work carries over to the quantum setting using our framework, maintaining a quadratic gap between verifier and prover. Lastly, we show that the assumptions in our framework can not be simplified further with relativizing proof techniques, as they are false in relativized worlds.
The strong exponential-time hypothesis (SETH) is a commonly used conjecture in the field of complexity theory. It states that CNF formulas cannot be analyzed for satisfiability with a speedup over exhaustive search. This hypothesis and its variants gave rise to a fruitful field of research, fine-grained complexity, obtaining (mostly tight) lower bounds for many problems in P whose unconditional lower bounds are hard to find. In this work, we introduce a framework of Quantum Strong Exponential-Time Hypotheses, as quantum analogues to SETH. Using the QSETH framework, we are able to translate quantum query lower bounds on black-box problems to conditional quantum time lower bounds for many problems in BQP. As an example, we illustrate the use of the QSETH by providing a conditional quantum time lower bound of Ω(n1.5) for the Edit Distance problem. We also show that the n2 SETH-based lower bound for a recent scheme for Proofs of Useful Work, based on the Orthogonal Vectors problem holds for quantum computation assuming QSETH, maintaining a quadratic gap between verifier and prover.
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.
Despite the wide range of problems for which quantum computers offer a computational advantage over their classical counterparts, there are also many problems for which the best known quantum algorithm provides a speedup that is only quadratic, or even subquadratic. Such a situation could also be desirable if we don't want quantum computers to solve certain problems fast - say problems relevant to post-quantum cryptography. When searching for algorithms and when analyzing the security of cryptographic schemes, we would like to have evidence that these problems are difficult to solve on quantum computers; but how do we assess the exact complexity of these problems? For most problems, there are no known ways to directly prove time lower bounds, however it can still be possible to relate the hardness of disparate problems to show conditional lower bounds. This approach has been popular in the classical community, and is being actively developed for the quantum case [Aaronson et al., 2020; Buhrman et al., 2021; Harry Buhrman et al., 2022; Andris Ambainis et al., 2022]. In this paper, by the use of the QSETH framework [Buhrman et al., 2021] we are able to understand the quantum complexity of a few natural variants of CNFSAT, such as parity-CNFSAT or counting-CNFSAT, and also are able to comment on the non-trivial complexity of approximate versions of counting-CNFSAT. Without considering such variants, the best quantum lower bounds will always be quadratically lower than the equivalent classical bounds, because of Grover’s algorithm; however, we are able to show that quantum algorithms will likely not attain even a quadratic speedup for many problems. These results have implications for the complexity of (variations of) lattice problems, the strong simulation and hitting set problems, and more. In the process, we explore the QSETH framework in greater detail and present a useful guide on how to effectively use the QSETH framework.
In the upcoming era of quantum computation, those successful classical, number-theoretical public-key cryptosystems that get widely deployed nowadays, such as RSA or ElGamal, all face severe insecurity if the adversary gets access to quantum computers. More urgently, time-insensitive secrets on today’s systems are facing immediate threats of the same level due to the “harvest now, decrypt later” attack. The field of post-quantum cryptography aims at building quantum-safe systems to prepare for the future. Among all candidate post-quantum proposals, lattice-based cryptography, which builds cryptosystems based on lattice problems, so far provides the best results in terms of performance, simplicity, and versatility. It is thus important to better understand the complexity of lattice problems. In this thesis, we study a couple of new lower and upper bounds for various lattice problems under various complexity notions. These new results include: 1. (Fine-grained hardness/lower bounds.) The fine-grained (exponential-time) complexity studies the precise constant c in the 2^{n^c} or 2^{c·n} running time for solving a computational problem, usually based on (variants of) the exponential time hypothesis (ETH) or the strong exponential time hypothesis (SETH). Under this complexity notion, for the bounded distance decoding problem (BDD), we give quantitatively improved bounds on the approximation factor α for which BDD_α cannot be solved in 2^{o(n)} or 2^{c·n} time, assuming variants of ETH or SETH, respectively, and for part of our results also assuming a geometric conjecture known as exponential lattice kissing number. For the shortest vector problem (SVP), we show a qualitatively improved result that for some constant approximation factor γ>1, SVP_γ cannot be solved in 2^{c·n} time, assuming a variant of SETH; previously, only analogous result for non-approximate, exact SVP was known. 2. (“Standard” NP-hardness.) The NP-hardness is a classical flag for identifying a computational problem as being hard. For SVP, previously only NP-hardness results under randomized reductions are known (except in corner cases), and whether SVP is “standardly” NP-hard under deterministic reductions is a notoriously longstanding open problem. We propose conjectures about ways to derandomize existing reductions, which would lead to the “standard” NP-hardness of approximate SVP. We also present empirical evidences that support the conjectures. 3. (Parallel algorithms/upper bounds.) The parallel complexity/sequentiality of a computational problem is crucial for its applications in time-release/timed cryptography. We show logarithmic-depth parallel algorithms for a sequential version of the short integer solution problem (SIS), which is used in a certain proof of sequential work (PoSW) timed-cryptographic protocol. This means that sequential SIS has low parallel complexity, as opposed to its previous conjecture. Moreover, we show a polylogarithmic-depth parallel attack that (almost) breaks that PoSW protocol.
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.
Buhrman, Patro, and Speelman presented a framework of conjectures that together form a quantum analogue of the strong exponential-time hypothesis and its variants. They called it the QSETH framework. In this paper, using a notion of quantum natural proofs (built from natural proofs introduced by Razborov and Rudich), we show how part of the QSETH conjecture that requires properties to be `compression oblivious' can in many cases be replaced by assuming the existence of quantum-secure pseudorandom functions, a standard hardness assumption. Combined with techniques from Fourier analysis of Boolean functions, we show that properties such as PARITY and MAJORITY are compression oblivious for certain circuit class Λ if subexponentially secure quantum pseudorandom functions exist in Λ, answering an open question in [Buhrman-Patro-Speelman 2021].